diff --git "a/data/ContemporaryMathematics-WEB.txt" "b/data/ContemporaryMathematics-WEB.txt" new file mode 100644--- /dev/null +++ "b/data/ContemporaryMathematics-WEB.txt" @@ -0,0 +1,71988 @@ + + +! +! + + + + + +Contemporary +Mathematics + + + + + + + + + +SENIOR CONTRIBUTING AUTHORS +DONNA KIRK, UNIVERSITY OF WISCONSIN AT SUPERIOR + + + + + + + + + +! +! +OpenStax +Rice University +6100 Main Street MS-375 +Houston, Texas 77005 + +To learn more about OpenStax, visit https://openstax.org. +Individual print copies and bulk orders can be purchased through our website. + +©2023 Rice University. Textbook content produced by OpenStax is licensed under a Creative Commons +Attribution 4.0 International License (CC BY 4.0). Under this license, any user of this textbook or the textbook +contents herein must provide proper attribution as follows: + +- +If you redistribute this textbook in a digital format (including but not limited to PDF and HTML), then you +must retain on every page the following attribution: +“Access for free at openstax.org.” +- +If you redistribute this textbook in a print format, then you must include on every physical page the +following attribution: +“Access for free at openstax.org.” +- +If you redistribute part of this textbook, then you must retain in every digital format page view (including +but not limited to PDF and HTML) and on every physical printed page the following attribution: +“Access for free at openstax.org.” +- +If you use this textbook as a bibliographic reference, please include +https://openstax.org/details/books/contemporary-mathematics in your citation. + +For questions regarding this licensing, please contact support@openstax.org. + +Trademarks +The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, OpenStax CNX logo, +OpenStax Tutor name, Openstax Tutor logo, Connexions name, Connexions logo, Rice University name, and +Rice University logo are not subject to the license and may not be reproduced without the prior and express +written consent of Rice University. + +HARDCOVER BOOK ISBN-13 +978-1-711470-55-9 +B&W PAPERBACK BOOK ISBN-13 +978-1-711470-54-2 +DIGITAL VERSION ISBN-13 +978-1-951693-68-8 +ORIGINAL PUBLICATION YEAR +2023 +1 2 3 4 5 6 7 8 9 10 TAM 23 + + + + + + + + + + + + + + + + +! +! +OPENSTAX + +OpenStax provides free, peer-reviewed, openly licensed textbooks for introductory college and Advanced +Placement® courses and low-cost, personalized courseware that helps students learn. A nonprofit ed tech +initiative based at Rice University, we’re committed to helping students access the tools they need to complete +their courses and meet their educational goals. + +RICE UNIVERSITY + +OpenStax, OpenStax CNX, and OpenStax Tutor are initiatives of Rice University. As a leading research university +with a distinctive commitment to undergraduate education, Rice University aspires to path-breaking research, +unsurpassed teaching, and contributions to the betterment of our world. It seeks to fulfill this mission by +cultivating a diverse community of learning and discovery that produces leaders across the spectrum of human +endeavor. + + + +PHILANTHROPIC SUPPORT + +OpenStax is grateful for the generous philanthropic partners who advance our mission to improve educational +access and learning for everyone. To see the impact of our supporter community and our most updated list of +partners, please visit openstax.org/impact. + +Arnold Ventures +Chan Zuckerberg Initiative +Chegg, Inc. +Arthur and Carlyse Ciocca Charitable Foundation +Digital Promise +Ann and John Doerr +Bill & Melinda Gates Foundation +Girard Foundation +Google Inc. +The William and Flora Hewlett Foundation +The Hewlett-Packard Company +Intel Inc. +Rusty and John Jaggers +The Calvin K. Kazanjian Economics Foundation +Charles Koch Foundation +Leon Lowenstein Foundation, Inc. +The Maxfield Foundation +Burt and Deedee McMurtry +Michelson 20MM Foundation +National Science Foundation +The Open Society Foundations +Jumee Yhu and David E. Park III +Brian D. Patterson USA-International Foundation +The Bill and Stephanie Sick Fund +Steven L. Smith & Diana T. Go +Stand Together +Robin and Sandy Stuart Foundation +The Stuart Family Foundation +Tammy and Guillermo Treviño +Valhalla Charitable Foundation +White Star Education Foundation +Schmidt Futures +William Marsh Rice University + +Study where you want, what +you want, when you want. +Access. The future of education. +openstax.org +When you access your book in our web view, you can use our new online +highlighting and note-taking features to create your own study guides. +Our books are free and flexible, forever. +Get started at openstax.org/details/books/contemporary-mathematics + +Contents +Preface +1 +Sets +5 +1 +Introduction +5 +1.1 Basic Set Concepts +6 +1.2 Subsets +14 +1.3 Understanding Venn Diagrams +20 +1.4 Set Operations with Two Sets +28 +1.5 Set Operations with Three Sets +40 +Chapter Summary +51 +Logic +59 +2 +Introduction +59 +2.1 Statements and Quantifiers +60 +2.2 Compound Statements +70 +2.3 Constructing Truth Tables +78 +2.4 Truth Tables for the Conditional and Biconditional +87 +2.5 Equivalent Statements +97 +2.6 De Morgan’s Laws +102 +2.7 Logical Arguments +108 +Chapter Summary +116 +Real Number Systems and Number Theory +127 +3 +Introduction +127 +3.1 Prime and Composite Numbers +128 +3.2 The Integers +149 +3.3 Order of Operations +158 +3.4 Rational Numbers +165 +3.5 Irrational Numbers +191 +3.6 Real Numbers +203 +3.7 Clock Arithmetic +211 +3.8 Exponents +218 +3.9 Scientific Notation +227 +3.10 Arithmetic Sequences +239 +3.11 Geometric Sequences +246 +Chapter Summary +254 +Number Representation and Calculation +267 +4 +Introduction +267 +4.1 Hindu-Arabic Positional System +268 +4.2 Early Numeration Systems +273 +4.3 Converting with Base Systems +287 +4.4 Addition and Subtraction in Base Systems +300 +4.5 Multiplication and Division in Base Systems +314 + +Chapter Summary +327 +Algebra +333 +5 +Introduction +333 +5.1 Algebraic Expressions +334 +5.2 Linear Equations in One Variable with Applications +346 +5.3 Linear Inequalities in One Variable with Applications +356 +5.4 Ratios and Proportions +366 +5.5 Graphing Linear Equations and Inequalities +376 +5.6 Quadratic Equations with Two Variables with Applications +404 +5.7 Functions +428 +5.8 Graphing Functions +447 +5.9 Systems of Linear Equations in Two Variables +474 +5.10 Systems of Linear Inequalities in Two Variables +491 +5.11 Linear Programming +518 +Chapter Summary +530 +Money Management +543 +6 +Introduction +543 +6.1 Understanding Percent +544 +6.2 Discounts, Markups, and Sales Tax +552 +6.3 Simple Interest +566 +6.4 Compound Interest +579 +6.5 Making a Personal Budget +589 +6.6 Methods of Savings +601 +6.7 Investments +612 +6.8 The Basics of Loans +630 +6.9 Understanding Student Loans +645 +6.10 Credit Cards +656 +6.11 Buying or Leasing a Car +675 +6.12 Renting and Homeownership +687 +6.13 Income Tax +697 +Chapter Summary +706 +Probability +723 +7 +Introduction +723 +7.1 The Multiplication Rule for Counting +724 +7.2 Permutations +728 +7.3 Combinations +732 +7.4 Tree Diagrams, Tables, and Outcomes +738 +7.5 Basic Concepts of Probability +751 +7.6 Probability with Permutations and Combinations +763 +7.7 What Are the Odds? +766 +7.8 The Addition Rule for Probability +771 +7.9 Conditional Probability and the Multiplication Rule +777 +7.10 The Binomial Distribution +790 +7.11 Expected Value +797 +Access for free at openstax.org + +Chapter Summary +808 +Statistics +815 +8 +Introduction +815 +8.1 Gathering and Organizing Data +816 +8.2 Visualizing Data +827 +8.3 Mean, Median and Mode +857 +8.4 Range and Standard Deviation +873 +8.5 Percentiles +879 +8.6 The Normal Distribution +883 +8.7 Applications of the Normal Distribution +903 +8.8 Scatter Plots, Correlation, and Regression Lines +907 +Chapter Summary +934 +Metric Measurement +943 +9 +Introduction +943 +9.1 The Metric System +944 +9.2 Measuring Area +952 +9.3 Measuring Volume +960 +9.4 Measuring Weight +970 +9.5 Measuring Temperature +977 +Chapter Summary +987 +Geometry +993 +10 +Introduction +993 +10.1 Points, Lines, and Planes +994 +10.2 Angles +1007 +10.3 Triangles +1019 +10.4 Polygons, Perimeter, and Circumference +1035 +10.5 Tessellations +1052 +10.6 Area +1068 +10.7 Volume and Surface Area +1087 +10.8 Right Triangle Trigonometry +1098 +Chapter Summary +1117 +Voting and Apportionment +1129 +11 +Introduction +1129 +11.1 Voting Methods +1130 +11.2 Fairness in Voting Methods +1162 +11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem +1179 +11.4 Apportionment Methods +1191 +11.5 Fairness in Apportionment Methods +1212 +Chapter Summary +1226 + +Graph Theory +1237 +12 +Introduction +1237 +12.1 Graph Basics +1238 +12.2 Graph Structures +1249 +12.3 Comparing Graphs +1265 +12.4 Navigating Graphs +1283 +12.5 Euler Circuits +1309 +12.6 Euler Trails +1326 +12.7 Hamilton Cycles +1340 +12.8 Hamilton Paths +1351 +12.9 Traveling Salesperson Problem +1364 +12.10 Trees +1380 +Chapter Summary +1401 +Math and... +1417 +13 +Introduction +1417 +13.1 Math and Art +1418 +13.2 Math and the Environment +1425 +13.3 Math and Medicine +1431 +13.4 Math and Music +1438 +13.5 Math and Sports +1444 +Chapter Summary +1448 +Co-Req Appendix: Integer Powers of 10 +1459 +A +Answer Key +1463 +Index +1563 +Access for free at openstax.org + +Preface +About OpenStax +OpenStax is part of Rice University, which is a 501(c)(3) nonprofit charitable corporation. As an educational initiative, it's +our mission to transform learning so that education works for every student. Through our partnerships with +philanthropic organizations and our alliance with other educational resource companies, we're breaking down the most +common barriers to learning. Because we believe that everyone should and can have access to knowledge. +About OpenStax Resources +Customization +Contemporary Mathematics is licensed under a Creative Commons Attribution 4.0 International (CC BY) license, which +means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its +content contributors. +Because our books are openly licensed, you are free to use the entire book or select only the sections that are most +relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and +sections in your syllabus, in the order that you prefer. You can even provide a direct link in your syllabus to the sections in +the web view of your book. +Instructors also have the option of creating a customized version of their OpenStax book. Visit the Instructor Resources +section of your book page on OpenStax.org for more information. +Art attribution +In Contemporary Mathematics, art contains attribution to its title, creator or rights holder, host platform, and license +within the caption. Because the art is openly licensed, anyone may reuse the art as long as they provide the same +attribution to its original source. For illustrations (e.g., graphs, charts, etc.) that are not credited, use the following +attribution: Copyright Rice University, OpenStax, under CC BY 4.0 license. +Errata +All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors +sometimes occur. Since our books are web-based, we can make updates periodically when deemed pedagogically +necessary. If you have a correction to suggest, submit it through the link on your book page on OpenStax.org. Subject +matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you +will also find a list of past and pending errata changes on your book page on OpenStax.org. +Format +You can access this textbook for free in web view or PDF through OpenStax.org, and for a low cost in print. +About Contemporary Mathematics +Contemporary Mathematics is designed to meet the requirements for a liberal arts mathematics course. The textbook +covers a range of topics that are typically found in a liberal arts course as well as some topics to connect mathematics to +the world around us. The text provides stand-alone sections with a focus on showing relevance in the features as well as +the examples, exercises, and exposition. +Pedagogical Foundation +Learning Objectives +Every section begins with a set of clear and concise learning objectives, which have been thoroughly revised to be both +measurable and more closely aligned with current teaching practice. These objectives are designed to help the instructor +decide what content to include or assign and to guide student expectations of learning. After completing the section and +end-of-section exercises, students should be able to demonstrate mastery of the learning objectives. +Key Features +Check Your Understanding: Concept checks to confirm students understand content at the end of every section +immediately before the exercise sets are provided to help bolster confidence before embarking on homework. +People in Mathematics: A mix of historic and contemporary profiles aimed to incorporate extensive diversity in gender +and ethnicity. The profiles incorporate how the person’s contribution has benefitted students or is relevant to their lives +in some way. +Who Knew?: A high-interest feature designed to showcase something interesting related to the section contents. These +features are crafted to offer something students might be surprised to find is so relevant to them. +Preface +1 + +Work It Out: Offers some activity ideas in line with the sections to support the learning objectives. +Tech Check: Highlights technologies that support content in the section. +Projects: A feature designed to put students in the driver’s seat researching a topic using various online resources. It is +intended to be primarily or wholly non-computational. Projects utilize online research and writing to summarize their +findings. +Section Summaries +Section summaries distill the information in each section for both students and instructors down to key, concise points +addressed in the section. +Key Terms +Key terms are bold and are followed by a definition in context. +Answers and Solutions to Questions in the Book +Answers for Your Turn and Check Your Understanding exercises are provided in the Answer Key at the end of the book. +The Section Exercises, Chapter Reviews, and Chapter Tests are intended for homework assignments or assessment; thus, +student-facing solutions are provided in the Student Solution Manual for only a subset of the exercises. Solutions for all +exercises are provided in the Instructor Solution Manual for instructors to share with students at their discretion, as is +standard for such resources. +About the Authors +Senior Contributing Author +Donna Kirk, University of Wisconsin at Superior +Donna Kirk received her B.S. in Mathematics from the State University of New York at Oneonta and her master’s degree +from City University – Seattle in Educational Technology and Curriculum Design. After teaching math in higher education +for more than twenty years, she joined University of Wisconsin’s Education Department in 2021, teaching math +education for teacher preparation. She is also the director of a STEM institute focused on connecting underrepresented +students with access to engaging and innovative experiences to empower themselves to pursue STEM related careers. +Contributing Authors +Barbara Boschmans Beaudrie, Northern Arizona University +Brian Beaudrie, Northern Arizona University +Matthew Cathey, Wofford College +Valeree Falduto, Palm Beach State College +Maureen Gerlofs, Texas State University +Quin Hearn, Broward College +Ian Walters, D’Youville College +Reviewers +Anna Pat Alpert, Navarro College +Mario Barrientos, Angelo State University +Keisha Brown, Perimeter College at Georgia State University +Hugh Cornell, University of North Florida +David Crombecque, University of Southern California +Shari Davis, Old Dominion University +Angela Everett, Chattanooga State Community College +David French, Tidewater Community College +Michele Gribben, McDaniel College +Celeste Hernandez, Dallas College-Richland +Trevor Jack, Illinois Wesleyan University +2 +Preface +Access for free at openstax.org + +Kristin Kang, Grand View University +Karla Karstens, University of Vermont +Sergio Loch, Grand View University +Andrew Misseldine, Southern Utah University +Carla Monticelli, Camden County College +Cindy Moss, Skyline College +Jill Rafael, Sierra College +Gary Rosen, University of Southern California +Faith Willman, Harrisburg Area Community College +Additional Resources +Student and Instructor Resources +We’ve compiled additional resources for both students and instructors, including student solution manuals, instructor +solution manuals, and PowerPoint lecture slides. Instructor resources require a verified instructor account, which you +can apply for when you log in or create your account on OpenStax.org. Take advantage of these resources to +supplement your OpenStax book. +Academic Integrity +Academic integrity builds trust, understanding, equity, and genuine learning. While students may encounter significant +challenges in their courses and their lives, doing their own work and maintaining a high degree of authenticity will result +in meaningful outcomes that will extend far beyond their college career. Faculty, administrators, resource providers, and +students should work together to maintain a fair and positive experience. +We realize that students benefit when academic integrity ground rules are established early in the course. To that end, +OpenStax has created an interactive to aid with academic integrity discussions in your course. +attribution: Copyright Rice University, OpenStax, under CC BY 4.0 license +Visit our academic integrity slider (https://openstax.org/r/academic-integrity-slider) . Click and drag icons along the +continuum to align these practices with your institution and course policies. You may then include the graphic on your +syllabus, present it in your first course meeting, or create a handout for students. +At OpenStax we are also developing resources supporting authentic learning experiences and assessment. Please visit +this book’s page for updates. For an in-depth review of academic integrity strategies, we highly recommend visiting the +International Center of Academic Integrity (ICAI) website (https://openstax.org/r/academicintegrity) . +Community Hubs +OpenStax partners with the Institute for the Study of Knowledge Management in Education (ISKME) to offer +Community Hubs on OER Commons—a platform for instructors to share community-created resources that support +OpenStax books, free of charge. Through our Community Hubs, instructors can upload their own materials or download +resources to use in their own courses, including additional ancillaries, teaching material, multimedia, and relevant +course content. We encourage instructors to join the hubs for the subjects most relevant to your teaching and research +as an opportunity both to enrich your courses and to engage with other faculty. To reach the Community Hubs, visit +Preface +3 + +www.oercommons.org/hubs/openstax (https://openstax.org/r/community-hub). +Technology partners +As allies in making high-quality learning materials accessible, our technology partners offer optional low-cost tools that +are integrated with OpenStax books. To access the technology options for your text, visit your book page on +OpenStax.org. +4 +Preface +Access for free at openstax.org + +Figure 1.1 A flatware drawer is like a set in that it contains distinct objects. (credit: modification of work “silverware” by Jo +Naylor/Flickr, CC BY 2.0) +Chapter Outline +1.1 Basic Set Concepts +1.2 Subsets +1.3 Understanding Venn Diagrams +1.4 Set Operations with Two Sets +1.5 Set Operations with Three Sets +Introduction +Think of a drawer in your kitchen used to store flatware. This drawer likely holds forks, spoons, and knives, and possibly +other items such as a meat thermometer and a can opener. The drawer in this case represents a tool used to group a +collection of objects. The members of the group are the individual items in the drawer, such as a fork or a spoon. +The members of a set can be anything, such as people, numbers, or letters of the alphabet. In statistical studies, a set is +a well-defined collection of objects used to identify an entire population of interest. For example, in a research study +examining the effects of a new medication, there can be two sets of people: one set that is given the medication and a +different set that is given a placebo (control group). In this chapter, we will discuss sets and Venn diagrams, which are +graphical ways to show relationships between different groups. +1 +SETS +1 • Introduction +5 + +1.1 Basic Set Concepts +Figure 1.2 A spoon, fork, and knife are elements of the set of flatware. (credit: modification of work “Cupofjoy” Wikimedia +CC0 1.0 Public Domain Dedication) +Learning Objectives +After completing this section, you should be able to: +1. +Represent sets in a variety of ways. +2. +Represent well-defined sets and the empty set with proper set notation. +3. +Compute the cardinal value of a set. +4. +Differentiate between finite and infinite sets. +5. +Differentiate between equal and equivalent sets. +Sets and Ways to Represent Them +Think back to your kitchen organization. If the drawer is the set, then the forks and knives are elements in the set. Sets +can be described in a number of different ways: by roster, by set-builder notation, by interval notation, by graphing on a +number line, and by Venn diagrams. Sets are typically designated with capital letters. The simplest way to represent a set +with only a few members is the roster (or listing) method, in which the elements in a set are listed, enclosed by curly +braces and separated by commas. For example, if +represents our set of flatware, we can represent +by using the +following set notation with the roster method: +EXAMPLE 1.1 +Writing a Set Using the Roster or Listing Method +Write a set consisting of your three favorite sports and label it with a capital +. +Solution +There are multiple possible answers depending on what your three favorite sports are, but any answer must list three +different sports separated by commas, such as the following: +YOUR TURN 1.1 +1. Write a set consisting of four small hand tools that might be in a toolbox and label it with a capital +. +All the sets we have considered so far have been well-defined sets. A well-defined set clearly communicates whether an +element is a member of the set or not. The members of a well-defined set are fixed and do not change over time. +Consider the following question. What are your top 10 songs of 2021? You could create a list of your top 10 favorite +songs from 2021, but the list your friend creates will not necessarily contain the same 10 songs. So, the set of your top +10 songs of 2021 is not a well-defined set. On the other hand, the set of the letters in your name is a well-defined set +because it does not vary (unless of course you change your name). The NFL wide receiver, Chad Johnson, famously +6 +1 • Sets +Access for free at openstax.org + +changed his name to Chad Ochocinco to match his jersey number of 85. +EXAMPLE 1.2 +Identifying Well-Defined Sets +For each of the following collections, determine if it represents a well-defined set. +1. +The group of all past vice presidents of the United States. +2. +A group of old cats. +Solution +1. +The group of all past vice presidents of the United States is a well-defined set, because you can clearly identify if any +individual was or was not a member of that group. For example, Britney Spears is not a member of this set, but Joe +Biden is a member of this set. +2. +A group of old cats is not a well-defined set because the word old is ambiguous. Some people might consider a +seven-year-old cat to be old, while others might think a cat is not old until it is 13 years old. Because people can +disagree on what is and what is not a member of this group, the set is not well-defined. +YOUR TURN 1.2 +For each of the following collections, determine if it represents a well-defined set. +1. A collection of medium-sized potatoes. +2. The original members of the Black Eyed Peas musical group. +On January 20, 2021, Kamala Harris was sworn in as the first woman vice president of the United States of America. If we +were to consider the set of all women vice presidents of the United States of America prior to January 20, 2021, this set +would be known as an empty set; the number of people in this set is 0, since there were no women vice presidents +before Harris. The empty set, also called the null set, is written symbolically using a pair of braces, +, or a zero with a +slash through it, +. +The set containing the number +, is a set with one element in it. It is not the same as the empty set, +, which +does not have any elements in it. Symbolically: +. +EXAMPLE 1.3 +Representing the Empty Set Symbolically +Represent each of the following sets symbolically. +1. +The set of prime numbers less than 2. +2. +The set of birds that are also mammals. +Solution +1. +A prime number is a natural number greater than 1 that is only divisible by one and itself. Since there are no prime +numbers less than 2, this set is empty, and we can represent it symbolically as follows: +These two different +symbols for the empty set can be used interchangeably. +2. +The set of birds and the set of mammals do not intersect, so the set of birds that are also mammals is empty, and +we can represent it symbolically as +YOUR TURN 1.3 +1. Represent the set of all numbers divisible by 0 symbolically. +1.1 • Basic Set Concepts +7 + +WHO KNEW? +The Number Zero +We use the number zero to represent the concept of nothing every day. The machine language of computers is +binary, consisting only of zeros and ones, and even way before that, the number zero was a powerful invention that +allowed our understanding of mathematics and science to develop. The historical record shows the Babylonians first +used zeros around 300 B.C., while the Mayans developed and began using zero separately around 350 A.D. What is +considered the first formal use of zero in arithmetic operations was developed by the Indian mathematician +Brahmagupta around 650 A.D. +Brahmagupta, Mathematician and Astronomer (https://openstax.org/r/brahmagupta.html) +Another interesting feature of the number zero is that although it is an even number, it is the only number that is +neither negative nor positive. +For larger sets that have a natural ordering, sometimes an ellipsis is used to indicate that the pattern continues. It is +common practice to list the first three elements of a set to establish a pattern, write the ellipsis, and then provide the last +element. Consider the set of all lowercase letters of the English alphabet, +. This set can be written symbolically as +. +The sets we have been discussing so far are finite sets. They all have a limited or fixed number of elements. We also use +an ellipsis for infinite sets, which have an unlimited number of elements, to indicate that the pattern continues. For +example, in set theory, the set of natural numbers, which is the set of all positive counting numbers, is represented as +ℕ +. +Notice that for this set, there is no element following the ellipsis. This is because there is no largest natural number; you +can always add one more to get to the next natural number. Because the set of natural numbers grows without bound, it +is an infinite set. +EXAMPLE 1.4 +Writing a Finite Set Using the Roster Method and an Ellipsis +Write the set of even natural numbers including and between 2 and 100, and label it with a capital +. Include an ellipsis. +Solution +Write the label, +, followed by an equal sign and then a bracket. Write the first three even numbers separated by +commas, beginning with the number two to establish a pattern. Next, write the ellipsis followed by a comma and the last +number in the list, 100. Finally, write the closing bracket to complete the set. +Write the label, +, followed by an equal sign and then a bracket. +Write the first three even numbers separated by commas, beginning with the number 2 to establish a pattern. +Next, write the ellipsis followed by a comma and the last number in the list, 100. +Finally, write the closing bracket to complete the set. +8 +1 • Sets +Access for free at openstax.org + +YOUR TURN 1.4 +1. Use an ellipsis to write the set of single digit numbers greater than or equal to zero and label it with a capital +. +Our number system is made up of several different infinite sets of numbers. The set of integers, ℤ is another infinite +set of numbers. It includes all the positive and negative counting numbers and the number zero. There is no largest or +smallest integer. +EXAMPLE 1.5 +Writing an Infinite Set Using the Roster Method and Ellipses +Write the set of integers using the roster method, and label it with a ℤ . +Solution +Step 1: As always, we write the label and then the opening bracket. Because the negative counting numbers are infinite, +to represent that the pattern continues without bound to the left, we must use an ellipsis as the first element in our list. +Step 2: We place a comma and follow it with at least three consecutive integers separated by commas to establish a +pattern. +Step 3: Add an ellipsis to the end of the list to show that the set of integers continues without bound to the right. +Complete the list with a closing bracket. The set of integers may be represented as follows: +ℤ +YOUR TURN 1.5 +1. Write the set of odd numbers greater than 0 and label it with a capital +. +A shorthand way to write sets is with the use of set builder notation, which is a verbal description or formula for the +set. For example, the set of all lowercase letters of the English alphabet, +, written in set builder notation is: +This is read as, “Set +is the set of all elements +such that +is a lowercase letter of the English alphabet.” +EXAMPLE 1.6 +Writing a Set Using Set Builder Notation +Using set builder notation, write the set +of all types of balls. Explain what the notation means. +Solution +The verbal description of the set is, “Set +is the set of all elements +such that +is a ball.” This set can be written in set +builder notation as follows: +YOUR TURN 1.6 +1. Using set builder notation, write the set +of all types of cars. +EXAMPLE 1.7 +Writing Sets Using Various Methods +Consider the set of letters in the word “happy.” Determine the best way to represent this set, and then write the set using +1.1 • Basic Set Concepts +9 + +either the roster method or set builder notation, whichever is more appropriate. +Solution +Because the letters in the word “happy” consist of a small finite set, the best way to represent this set is with the roster +method. Choose a label to represent the set, such as +. +. +Notice that the letter “p” is only represented one time. This occurs because when representing members of a set, each +unique element is only listed once no matter how many times it occurs. Duplicate elements are never repeated when +representing members of a set. +YOUR TURN 1.7 +1. Use the roster method or set builder notation to represent the collection of all musical instruments. +Computing the Cardinal Value of a Set +Almost all the sets most people work with outside of pure mathematics are finite sets. For these sets, the cardinal value +or cardinality of the set is the number of elements in the set. For finite set +, the cardinality is denoted symbolically as +. For example, a set that contains four elements has a cardinality of 4. +How do we measure the cardinality of infinite sets? The ‘smallest’ infinite set is the set of natural numbers, or counting +numbers, ℕ +. This set has a cardinality of +(pronounced "aleph-null"). All sets that have the same +cardinality as the set of natural numbers are countably infinite. This concept, as well as notation using aleph, was +introduced by mathematician Georg Cantor who once said, “A set is a Many that allows itself to be thought of as a One.” +EXAMPLE 1.8 +Computing the Cardinal Value of a Set +Write the cardinal value of each of the following sets in symbolic form. +1. +2. +The empty set. +Solution +1. +There are 5 distinct elements in set +: a fork, a spoon, a knife, a meat thermometer, and a can opener. Therefore, +the cardinal value of set +is 5 and written symbolically as +2. +Because the empty set does not have any elements in it, the cardinality of the empty set is zero. Symbolically we +write this as: +YOUR TURN 1.8 +Write the cardinal value of each of the following sets in symbolic form. +1. Set +is the set of prime numbers less than 2. +2. Set +is the set of lowercase letters of the English alphabet, +. +Now that we have learned to represent finite and infinite sets using both the roster method and set builder notation, we +should also be able to determine if a set is finite or infinite based on its verbal or symbolic description. One way to +determine if a set is finite or not is to determine the cardinality of the set. If the cardinality of a set is a natural number, +then the set is finite. +10 +1 • Sets +Access for free at openstax.org + +EXAMPLE 1.9 +Differentiating Between Finite and Infinite Sets +Classify each of the following sets as infinite or finite. +1. +2. +is the set of lowercase letters of the English Alphabet, +. +3. ℚ +Solution +1. +. Since 5 is a natural number, the set is finite. +2. +. Since 26 is a natural number, the set is finite. +3. +Set ℚ is the set of rational numbers or fractions. Because the set of integers is a subset of the set of rational +numbers, and the set of integers is infinite, the set of rational numbers is also infinite. There is no smallest or +largest rational number. +YOUR TURN 1.9 +Classify each of the following sets as infinite or finite. +1. +2. +Equal versus Equivalent Sets +When speaking or writing we tend to use equal and equivalent interchangeably, but there is an important distinction +between their meanings. Consider a new Ford Escape Hybrid and a new Toyota Rav4 Hybrid. Both cars are hybrid electric +sport utility vehicles; in that sense, they are equivalent. They will both get you from place to place in a relatively fuel- +efficient way. In this example we are comparing the single member set {Toyota Rav4 Hybrid} to the single member set +{Ford Escape Hybrid}. Since these two sets have the same number of elements, they are also equivalent mathematically, +meaning they have the same cardinality. But they are not equal, because the two cars have different looks and features, +and probably even handle differently. Each manufacturer will emphasize the features unique to their vehicle to persuade +you to buy it; if the SUVs were truly equal, there would be no reason to choose one over the other. +Now consider two Honda CR-Vs that are made with exactly the same parts, on the same assembly line within a few +minutes of each other—these SUVs are equal. They are identical to each other, containing the same elements without +regard to order, and the only differentiator when making a purchasing decision would be varied pricing at different +dealerships. The set {Honda CR-V} is equal to the set {Honda CR-V}. Symbolically, we represent equal sets as +and +equivalent sets as +. +Now, let us consider a Toyota dealership that has 10 RAV4s on the lot, 8 Prii, 7 Highlanders, and 12 Camrys. There is a +one-to-one relationship between the set of vehicles on the lot and the set consisting of the number of each type of +vehicle on the lot. Therefore, these two sets are equivalent, but not equal. The set {RAV4, Prius, Highlander, Camry} is +equivalent to the set {10, 8, 7, 12} because they have the same number of elements. +VIDEO +Equal and Equivalent Sets (https://openstax.org/r/Equal_and_Equivalent_Sets) +If two sets are equal, they are also equivalent, because equal sets also have the same cardinality. +EXAMPLE 1.10 +Differentiating Between Equivalent and Equal Sets +Determine if the following pairs of sets are equal, equivalent, or neither. +1.1 • Basic Set Concepts +11 + +1. +and +2. +The empty set and the set of prime numbers less than 2. +3. +The set of vowels in the word happiness and the set of consonants in the word happiness. +Solution +1. +Sets E and F both have a cardinal value of 5, but the elements in these sets are different. So, the two sets are +equivalent, but they are not equal: +. +2. +The set of prime numbers consists of the set of counting numbers greater than one that can only be divided evenly +by one and itself. The set of prime numbers less than 2 is an empty set, since there are no prime numbers less than +2. Therefore, these two sets are equal (and equivalent). +3. +The set of vowels in the word happiness is +and the set of consonants in the word happiness is +The cardinal value of these sets two sets is +and +respectively. +Because the cardinality of the two sets differs, they are not equivalent. Further, their elements are not identical, so +they are also not equal. +YOUR TURN 1.10 +Determine if the following pairs of sets are equal, equivalent, or neither. +1. Set +and set +2. Set +and set +3. Set +and set +PEOPLE IN MATHEMATICS +Georg Cantor +Figure 1.3 Georg Cantor (credit: Wikimedia, public domain) +Georg Cantor, the father of modern set theory, was born during the year 1845 in Saint Petersburg, Russa and later +moved to Germany as a youth. Besides being an accomplished mathematician, he also played the violin. Cantor +received his doctoral degree in Mathematics at the age of 22. +In 1870, at the age of 25 he established the uniqueness theorem for trigonometric series. His most significant work +happened between 1874 and 1884, when he established the existence of transcendental numbers (also called +irrational numbers) and proved that the set of real numbers are uncountably infinite—despite the objections of his +former professor Leopold Kronecker. +Cantor published his final treatise on set theory in 1897 at the age of 52, and was awarded the Sylvester Medial from +the Royal Society of London in 1904 for his contributions to the field. At the heart of Cantor’s work was his goal to +solve the continuum problem, which later influenced the works of David Hilbert and Ernst Zermelo. +References: +12 +1 • Sets +Access for free at openstax.org + +Wikipedia contributors. “Cantor.” Wikipedia, The Free Encyclopedia, 23 Mar. 2021. Web. 20 Jul. 2021. +Akihiro Kanamori, “Set Theory from Cantor to Cohen,” Editor(s): Dov M. Gabbay, Akihiro Kanamori, John Woods, +Handbook of the History of Logic, North-Holland, Volume 6, 2012. +Check Your Understanding +1. A _____________ is a well-defined collection of objects. +2. The _________________ of a finite set +, denoted +, is the number of elements in set +. +3. Determine if the following description describes a well-defined set: “The top 5 pizza restaurants in Chicago.” +4. The United States is the only country to have landed people on the moon as of March 21, 2021. What is the +cardinality of the set of all people who have walked on the moon prior to this date? +5. Set +is a set of a dozen distinct donuts, and set +is a set of a dozen different types of apples. Is set +equal to set +, equivalent to set +, or neither? +6. Is the set of all butterflies in the world a finite set or an infinite set? +7. Represent the set of all upper-case letters of the English alphabet using both the roster method and set builder +notation. +SECTION 1.1 EXERCISES +For the following exercises, represent each set using the roster method. +1. The set of primary colors: red, yellow, and blue. +2. A set of the following flowers: rose, tulip, marigold, iris, and lily. +3. The set of natural numbers between 50 and 100. +4. The set of natural numbers greater than 17. +5. The set of different pieces in a game of chess. +6. The set of natural numbers less than 21. +For the following exercises, represent each set using set builder notation. +7. The set of all types of lizards. +8. The set of all stars in the universe. +9. The set of all integer multiples of 3 that are greater than zero. +10. The set of all integer multiples of 4 that are greater than zero. +11. The set of all plants that are edible. +12. The set of all even numbers. +For the following exercises, represent each set using the method of your choice. +13. The set of all squares that are also circles. +14. The set of natural numbers divisible by zero. +15. The set of Mike and Carol’s children on the TV show, The Brady Bunch. +16. The set of all real numbers. +17. The set of polar bears that live in Antarctica. +18. The set of songs written by Prince. +19. The set of children’s books written and illustrated by Mo Willems. +20. The set of seven colors commonly listed in a rainbow. +For the following exercises, determine if the collection of objects represents a well-defined set or not. +21. The names of all the characters in the book, The Fault in Our Stars by John Green. +22. The five greatest soccer players of all time. +23. A group of old dogs that are able to learn new tricks. +24. A list of all the movies directed by Spike Lee as of 2021. +25. The group of all zebras that can fly an airplane. +26. The group of National Baseball League Hall of Fame members who have hit over 700 career home runs. +For the following exercises, compute the cardinal value of each set. +27. +1.1 • Basic Set Concepts +13 + +28. +29. +30. +31. +32. +33. +34. +35. +36. The set of numbers on a standard 6-sided die. +For the following exercises, determine whether set +and set +are equal, equivalent or neither. +37. +; +. +38. +; +. +39. +; +. +40. +; +. +41. +; +. +42. +; +. +43. +44. +; +For the following exercises, determine if the set described is finite or infinite. +45. The set of natural numbers. +46. The empty set. +47. The set consisting of all jazz venues in New Orleans, Louisiana. +48. The set of all real numbers. +49. The set of all different types of cheeses. +50. The set of all words in Merriam-Webster's Collegiate Dictionary, Eleventh Edition, published in 2020. +1.2 Subsets +Figure 1.4 The players on a soccer team who are actively participating in a game are a subset of the greater set of team +members. (Credit: “PAFC-Mezokovesd-108” by Puskás Akadémia/Flickr, Public Domain Mark 1.0) +Learning Objectives +After completing this section, you should be able to: +1. +Represent subsets and proper subsets symbolically. +2. +Compute the number of subsets of a set. +3. +Apply concepts of subsets and equivalent sets to finite and infinite sets. +The rules of Major League Soccer (MLS) allow each team to have up to 30 players on their team. However, only 18 of +these players can be listed on the game day roster, and of the 18 listed, 11 players must be selected to start the game. +14 +1 • Sets +Access for free at openstax.org + +How the coaches and general managers form the team and choose the starters for each game will determine the +success of the team in any given year. +The entire group of 30 players is each team’s set. The group of game day players is a subset of the team set, and the +group of 11 starters is a subset of both the team set and the set of players on the game day roster. +Set +is a subset of set +if every member of set +is also a member of set +. Symbolically, this relationship is written as +. +Sets can be related to each other in several different ways: they may not share any members in common, they may share +some members in common, or they may share all members in common. In this section, we will explore the way we can +select a group of members from the whole set. +Every set is also a subset of itself, +Recall the set of flatware in our kitchen drawer from Section 1.1, +. Suppose you are preparing to eat dinner, so you pull a fork and +a knife from the drawer to set the table. The set +is a subset of set +, because every member or +element of set +is also a member of set +. More specifically, set +is a proper subset of set +, because there are other +members of set +not in set +. This is written as +. The only subset of a set that is not a proper subset of the set +would be the set itself. +The empty set or null set, +, is a proper subset of every set, except itself. +Graphically, sets are often represented as circles. In the following graphic, set +is represented as a circle completely +enclosed inside the circle representing set +, showing that set +is a proper subset of set +. The element +represents +an element that is in both set +and set +. +Figure 1.5 +While we can list all the subsets of a finite set, it is not possible to list all the possible subsets of an infinite set, as it +would take an infinitely long time. +EXAMPLE 1.11 +Listing All the Proper Subsets of a Finite Set +Set +is a set of reading materials available in a shop at the airport, +. List all the +subsets of set +. +Solution +Step 1: It is best to begin with the set itself, as every set is a subset of itself. In our example, the cardinality of set +is +. There is only one subset of set +that has the same number of elements of set +. +Step 2: Next, list all the proper subsets of the set containing +elements. In this case, +. There are three +subsets that each contain two elements: +, +, and +. +Step 3: Continue this process by listing all the proper subsets of the set containing +elements. In this case, +. There are three subsets that contain one element: +, +, and +. +Step 4: Finally, list the subset containing 0 elements, or the empty set: +. +1.2 • Subsets +15 + +YOUR TURN 1.11 +1. Consider the set of possible outcomes when you flip a coin, +. List all the possible subsets of set +EXAMPLE 1.12 +Determining Whether a Set Is a Proper Subset +Consider the set of common political parties in the United States, +. +Determine if the following sets are proper subsets of +. +1. +2. +3. +Solution +1. +is a proper subset of +, written symbolically as +because every member of +is a member of set +, but +also contains at least one element that is not in +. +2. +is a single member proper subset of +, written symbolically as +because Green is a member of set +, but +also contains other members (such as Democratic) that are not in +. +3. +is subset of +because every member of +is also a member of +, but it is not a proper subset of +because there +are no members of +that are not also in set +. We can represent the relationship symbolically as +or more +precisely, set +is equal to set +, +YOUR TURN 1.12 +Consider the set of generation I legendary Pokémon, +. Give an example +of a proper subset containing: +1. one member. +2. three members. +3. no members. +EXAMPLE 1.13 +Expressing the Relationship between Sets Symbolically +Consider the subsets of a standard deck of cards: +; +; +; and +. +Express the relationship between the following sets symbolically. +1. +Set +and set +. +2. +Set +and set +. +3. +Set +and +. +Solution +1. +. +is a proper subset of set +. +2. +. +is a proper subset of set +. +3. +. +is subset of itself, but not a proper subset of itself because +is equal to itself. +YOUR TURN 1.13 +1. Express the relationship between the set of natural numbers, +and the set of even numbers, +. +16 +1 • Sets +Access for free at openstax.org + +Exponential Notation +So far, we have figured out how many subsets exist in a finite set by listing them. Recall that in Example 1.11, when we +listed all the subsets of the three-element set +we saw that there are eight subsets. In +Your Turn 1.11, we discovered that there are four subsets of the two-element subset, +. A one-element +set has two subsets, the empty set and itself. The only subset of the empty set is the empty set itself. But how can we +easily figure out the number of subsets in a very large finite set? It turns out that the number of subsets can be found by +raising 2 to the number of elements in the set, using exponential notation to represent repeated multiplication. For +example, the number of subsets of the set +is equal to +. Exponential +notation is used to represent repeated multiplication, +, where +appears as a factor +times. +FORMULA +The number of subsets of a finite set +is equal to 2 raised to the power of +, where +is the number of +elements in set +: +. +Note that +, so this formula works for the empty set, also. +EXAMPLE 1.14 +Computing the Number of Subsets of a Set +Find the number of subsets of each of the following sets. +1. +The set of top five scorers of all time in the NBA: +2. +The set of the top four bestselling albums of all time: +. +3. +. +Solution +1. +. So, the total number of subsets of +. +2. +. Therefore, the total number of subsets of +. +3. +. So, the total number of subsets of +. +YOUR TURN 1.14 +1. Compute the total number of subsets in the set of the top nine tennis grand slam singles winners, +. +Equivalent Subsets +In the early 17th century, the famous astronomer Galileo Galilei found that the set of natural numbers and the subset of +the natural numbers consisting of the set of square numbers, +, are equivalent. Upon making this discovery, he +conjectured that the concepts of less than, greater than, and equal to did not apply to infinite sets. +Sequences and series are defined as infinite subsets of the set of natural numbers by forming a relationship between +the sequence or series in terms of a natural number, +. For example, the set of even numbers can be defined using set +builder notation as +. The formula in this case replaces every natural number with +two times the number, resulting in the set of even numbers, +. The set of even numbers is also equivalent to +the set of natural numbers. +1.2 • Subsets +17 + +WHO KNEW? +Employment Opportunities +You can make a career out of working with sets. Applications of equivalent sets include relational database design +and analysis. +Relational databases that store data are tables of related information. Each row of a table has the same number of +columns as every other row in the table; in this way, relational databases are examples of set equivalences for finite +sets. In a relational database, a primary key is set up to identify all related information. There is a one-to-one +relationship between the primary key and any other information associated with it. +Database design and analysis is a high demand career with a median entry-level salary of about $85,000 per year, +according to salary.com. +EXAMPLE 1.15 +Writing Equivalent Subsets of an Infinite Set +Using natural numbers, multiples of 3 are given by the sequence +. Write this set using set builder notation by +expressing each multiple of 3 using a formula in terms of a natural number, +. +Solution +or +. In this example, +is a multiple of 3 and +is a +natural number. The symbol +is read as “is a member or element of.” Because there is a one-to-one correspondence +between the set of multiples of 3 and the natural numbers, the set of multiples of 3 is an equivalent subset of the natural +numbers. +YOUR TURN 1.15 +1. Using natural numbers, multiples of 5 are given by the sequence +. Write this set using set builder +notation by associating each multiple of 5 in terms of a natural number, +. +EXAMPLE 1.16 +Creating Equivalent Subsets of a Finite Set That Are Not Equal +A fast-food restaurant offers a deal where you can select two options from the following set of four menu items for $6: a +chicken sandwich, a fish sandwich, a cheeseburger, or 10 chicken nuggets. Javier and his friend Michael are each +purchasing lunch using this deal. Create two equivalent, but not equal, subsets that Javier and Michael could choose to +have for lunch. +Solution +The possible two-element subsets are: {chicken sandwich, fish sandwich}, {chicken sandwich, cheeseburger}, {chicken +sandwich, chicken nuggets}, {fish sandwich, cheeseburger}, {fish sandwich, chicken nuggets}, and {cheeseburger, +chicken nuggets}. One possible solution is that Javier picked the set {chicken sandwich, chicken nuggets}, while Michael +chose the {cheeseburger, chicken nuggets}. Because Javier and Michael both picked two items, but not exactly the same +two items, these sets are equivalent, but not equal. +YOUR TURN 1.16 +1. Serena and Venus Williams walk into the same restaurant as Javier and Michael, but they order the same pair of +items, resulting in equal sets of choices. If Venus ordered a fish sandwich and chicken nuggets, what did Serena +order? +18 +1 • Sets +Access for free at openstax.org + +EXAMPLE 1.17 +Creating Equivalent Subsets of a Finite Set +A high school volleyball team at a small school consists of the following players: {Angie, Brenda, Colleen, Estella, Maya, +Maria, Penny, Shantelle}. Create two possible equivalent starting line-ups of six players that the coach could select for +the next game. +Solution +There are actually 28 possible ways that the coach could choose his starting line-up. Two such equivalent subsets are +{Angie, Brenda, Maya, Maria, Penny, Shantelle} and {Angie, Brenda, Colleen, Estella, Maria, Shantelle}. Each subset has +six members, but they are not identical, so the two sets are equivalent but not equal. +YOUR TURN 1.17 +1. Consider the same group of volleyball players from above: {Angie, Brenda, Colleen, Estella, Maya, Maria, Penny, +Shantelle}. The team needs to select a captain and an assistant captain from their members. List two possible +equivalent subsets that they could select. +Check Your Understanding +8. Every member of a __________ of a set is also a member of the set. +9. Explain what distinguishes a proper subset of a set from a subset of a set. +10. The __________ set is a proper subset of every set except itself. +11. Is the following statement true or false? +12. If the cardinality of set +is +, then set +has a total of ___________ subsets. +13. Set +is ______________ to set +if +14. If every member of set +is a member of set +and every member of set +is also a member set +, then set +is +____________ to set +. +SECTION 1.2 EXERCISES +For the following exercises, list all the proper subsets of each set. +1. +2. +3. +4. +For the following exercises, determine the relationship between the two sets and write the relationship symbolically. +and +5. +and +6. +and +7. +and +8. +and +9. +and +10. +and +11. +and +12. +and +13. +and +14. +and +For the following exercises, calculate the total number of subsets of each set. +15. +16. +1.2 • Subsets +19 + +17. +18. +19. +20. +21. +22. +23. Set +if +24. Set +if +For the following exercises, use the set of letters in the word largest as the set, +25. Find a subset of +that is equivalent, but not equal, to the set: +26. Find a subset of +that is equal to the set: +27. Find a subset of +that is equal to the set: +28. Find a subset of +that is equivalent, but not equal, to the set +29. Find a subset of +that is equivalent, but not equal, to the set: +30. Find a subset of +that is equal to the set: +31. Find two three-character subsets of set +that are equivalent, but not equal, to each other. +32. Find two three-character subsets of set +that are equal to each other. +33. Find two five-character subsets of set +that are equal to each other. +34. Find two five-character subsets of set +that are equivalent, but not equal, to each other. +For the following exercises, use the set of integers as the set +35. Find two equivalent subset of +with a cardinality of 7. +36. Find two equal subsets of +with a cardinality of 4. +37. Find a subset of +that is equivalent, but not equal to, +38. Find a subset of +that is equivalent, but not equal to, +39. True or False. The set of natural numbers, +, is equivalent to set +40. True or False. Set +is an equivalent subset of the set of rational numbers, +1.3 Understanding Venn Diagrams +Figure 1.6 When assembling furniture, instructions with images are easier to follow, just like how set relationships are +easier to understand when depicted graphically. (credit: "Time to assemble more Ikea furniture!" by Rod Herrea/Flickr, CC +BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Utilize a universal set with two sets to interpret a Venn diagram. +2. +Utilize a universal set with two sets to create a Venn diagram. +3. +Determine the complement of a set. +20 +1 • Sets +Access for free at openstax.org + +Have you ever ordered a new dresser or bookcase that required assembly? When your package arrives you excitedly +open it and spread out the pieces. Then you check the assembly guide and verify that you have all the parts required to +assemble your new dresser. Now, the work begins. Luckily for you, the assembly guide includes step-by-step instructions +with images that show you how to put together your product. If you are really lucky, the manufacturer may even provide +a URL or QR code connecting you to an online video that demonstrates the complete assembly process. We can likely all +agree that assembly instructions are much easier to follow when they include images or videos, rather than just written +directions. The same goes for the relationships between sets. +Interpreting Venn Diagrams +Venn diagrams are the graphical tools or pictures that we use to visualize and understand relationships between sets. +Venn diagrams are named after the mathematician John Venn, who first popularized their use in the 1880s. When we use +a Venn diagram to visualize the relationships between sets, the entire set of data under consideration is drawn as a +rectangle, and subsets of this set are drawn as circles completely contained within the rectangle. The entire set of data +under consideration is known as the universal set. +Consider the statement: All trees are plants. This statement expresses the relationship between the set of all plants and +the set of all trees. Because every tree is a plant, the set of trees is a subset of the set of plants. To represent this +relationship using a Venn diagram, the set of plants will be our universal set and the set of trees will be the subset. Recall +that this relationship is expressed symbolically as: +To create a Venn diagram, first we draw a rectangle +and label the universal set “ +” Then we draw a circle within the universal set and label it with the word “Trees.” +Figure 1.7 +This section will introduce how to interpret and construct Venn diagrams. In future sections, as we expand our +knowledge of relationships between sets, we will also develop our knowledge and use of Venn diagrams to explore how +multiple sets can be combined to form new sets. +EXAMPLE 1.18 +Interpreting the Relationship between Sets in a Venn Diagram +Write the relationship between the sets in the following Venn diagram, in words and symbolically. +Figure 1.8 +Solution +The set of terriers is a subset of the universal set of dogs. In other words, the Venn diagram depicts the relationship that +all terriers are dogs. This is expressed symbolically as +YOUR TURN 1.18 +1. Write the relationship between the sets in the following Venn diagram, in words and symbolically. +1.3 • Understanding Venn Diagrams +21 + +So far, the only relationship we have been considering between two sets is the subset relationship, but sets can be +related in other ways. Lions and tigers are both different types of cats, but no lions are tigers, and no tigers are lions. +Because the set of all lions and the set of all tigers do not have any members in common, we call these two sets disjoint +sets, or non-overlapping sets. +Two sets +and +are disjoint sets if they do not share any elements in common. That is, if +is a member of set +, then +is not a member of set +. If +is a member of set +, then +is not a member of set +. To represent the relationship +between the set of all cats and the sets of lions and tigers using a Venn diagram, we draw the universal set of cats as a +rectangle and then draw a circle for the set of lions and a separate circle for the set of tigers within the rectangle, +ensuring that the two circles representing the set of lions and the set of tigers do not touch or overlap in any way. +Figure 1.9 +EXAMPLE 1.19 +Describing the Relationship between Sets +Describe the relationship between the sets in the following Venn diagram. +Figure 1.10 +Solution +The set of triangles and the set of squares are two disjoint subsets of the universal set of two-dimensional figures. The +set of triangles does not share any elements in common with the set of squares. No triangles are squares and no +squares are triangles, but both squares and triangles are 2D figures. +YOUR TURN 1.19 +1. Describe the relationship between the sets in the following Venn diagram. +22 +1 • Sets +Access for free at openstax.org + +Creating Venn Diagrams +The main purpose of a Venn diagram is to help you visualize the relationship between sets. As such, it is necessary to be +able to draw Venn diagrams from a written or symbolic description of the relationship between sets. +Procedure +To create a Venn diagram: +1. +Draw a rectangle to represent the universal set, and label it +. +2. +Draw a circle within the rectangle to represent a subset of the universal set and label it with the set name. +If there are multiple disjoint subsets of the universal set, their separate circles should not touch or overlap. +EXAMPLE 1.20 +Drawing a Venn Diagram to Represent the Relationship Between Two Sets +Draw a Venn diagram to represent the relationship between each of the sets. +1. +All rectangles are parallelograms. +2. +All women are people. +Solution +1. +The set of rectangles is a subset of the set of parallelograms. +First, draw a rectangle to represent the universal set and label it with +, then draw a circle +completely within the rectangle, and label it with the name of the set it represents, +. +Figure 1.11 +In this example, both letters and names are used to represent the sets involved, but this is not necessary. You may use +either letters or names alone, as long as the relationship is clearly depicted in the diagram, as shown below. +Figure 1.12 +or +1.3 • Understanding Venn Diagrams +23 + +Figure 1.13 +2. +The universal set is the set of people, and the set of all women is a subset of the set of people. +Figure 1.14 +YOUR TURN 1.20 +1. Draw a Venn diagram to represent the relationship between each of the sets. All natural numbers are +integers. +2. +. Draw a Venn diagram to represent this relationship. +EXAMPLE 1.21 +Drawing a Venn Diagram to Represent the Relationship Between Three Sets +All bicycles and all cars have wheels, but no bicycle is a car. Draw a Venn diagram to represent this relationship. +Solution +Step 1: The set of bicycles and the set of cars are both subsets of the set of things with wheels. The universal set is the +set of things with wheels, so we first draw a rectangle and label it with +. +Step 2: Because the set of bicycles and the set of cars do not share any elements in common, these two sets are disjoint +and must be drawn as two circles that do not touch or overlap with the universal set. +Figure 1.15 +YOUR TURN 1.21 +1. Airplanes and birds can fly, but no birds are airplanes. Draw a Venn diagram to represent this relationship. +The Complement of a Set +Recall that if set +is a proper subset of set +, the universal set (written symbolically as +), then there is at least one +element in set +that is not in set +. The set of all the elements in the universal set +that are not in the subset +is +called the complement of set +, +. In set builder notation this is written symbolically as: +The +24 +1 • Sets +Access for free at openstax.org + +symbol +is used to represent the phrase, “is a member of,” and the symbol +is used to represent the phrase, “is not a +member of.” In the Venn diagram below, the complement of set +is the region that lies outside the circle and inside the +rectangle. The universal set +includes all of the elements in set +and all of the elements in the complement of set +, +and nothing else. +Figure 1.16 +Consider the set of digit numbers. Let this be our universal set, +Now, let set +be the +subset of +consisting of all the prime numbers in set +, +The complement of set +is +The following Venn diagram represents this relationship graphically. +Figure 1.17 +EXAMPLE 1.22 +Finding the Complement of a Set +For both of the questions below, +is a proper subset of +. +1. +Given the universal set +and set +, find +2. +Given the universal set +and +, find +Solution +1. +The complement of set +is the set of all elements in the universal set +that are not in set +. +2. +The complement of set +is the set of all dogs that are not beagles. All members of set +are in the universal set +because they are dogs, but they are not in set +because they are not beagles. This relationship can be expressed in +set build notation as follows: +, +, or +YOUR TURN 1.22 +For both of the questions below, +is a proper subset of +. +1. Given the universal set +and set +, +find +. +2. Given the universal set +and set +find +. +Check Your Understanding +15. A Venn diagram is a graphical representation of the _____________ between sets. +16. In a Venn diagram, the set of all data under consideration, the _____________ set, is drawn as a rectangle. +17. Two sets that do not share any elements in common are _____________ sets. +1.3 • Understanding Venn Diagrams +25 + +18. The _____________ of a subset +or the universal set, +, is the set of all members of +that are not in +. +19. The sets +and +are _____________ subsets of the universal set. +SECTION 1.3 EXERCISES +For the following exercises, interpret each Venn diagram and describe the relationship between the sets, symbolically +and in words. +1. +2. +3. +4. +5. +6. +7. +26 +1 • Sets +Access for free at openstax.org + +8. +For the following exercises, create a Venn diagram to represent the relationships between the sets. +9. All birds have wings. +10. All cats are animals. +11. All almonds are nuts, and all pecans are nuts, but no almonds are pecans. +12. All rectangles are quadrilaterals, and all trapezoids are quadrilaterals, but no rectangles are trapezoids. +13. Lizards +Reptiles. +14. Ladybugs +Insects. +15. Ladybugs +Insects and Ants +Insects, but no Ants are Ladybugs. +16. Lizards +Reptiles and Snakes +Reptiles, but no Lizards are Snakes. +17. +and +are disjoint subsets of +18. +and +are disjoint subsets of +. +19. +is a subset of +. +20. +is a subset of +. +21. +Jazz, +Music, and +. +22. +Reggae, +Music, and +. +23. +Jazz, +Reggae, and +Music are sets with the following relationships: +and +is +disjoint from +. +24. +Jazz, +Bebop, and +Music are sets with the following relationships: +and +. +For the following exercises, the universal set is the set of single digit numbers, +. Find the +complement of each subset of +. +25. +26. +27. +28. +29. +30. +31. +32. +For the following exercises, the universal set is +{Bashful, Doc, Dopey, Grumpy, Happy, Sleepy, Sneezy}. Find the +complement of each subset of +. +33. +{Happy, Bashful, Grumpy} +34. +{Sleepy, Sneezy} +35. +{Doc} +36. +{Doc, Dopey} +37. +38. +{Doc, Grumpy, Happy, Sleepy, Bashful, Sneezy, Dopey} +For the following exercises, the universal set is +. Find the complement of each subset of +. +39. +40. +41. +42. +For the following exercises, use the Venn diagram to determine the members of the complement of set +. +43. +1.3 • Understanding Venn Diagrams +27 + +44. +45. +46. +1.4 Set Operations with Two Sets +Figure 1.18 A large, multigenerational family contains an intersection and a union of sets. (credit: “Family Photo Shoot +Bani Syakur” by Mainur Risyada/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Determine the intersection of two sets. +2. +Determine the union of two sets. +3. +Determine the cardinality of the union of two sets. +4. +Apply the concepts of AND and OR to set operations. +5. +Draw conclusions from Venn diagrams with two sets. +The movie Yours, Mine, and Ours was originally released in 1968 and starred Lucille Ball and Henry Fonda. This movie, +which is loosely based on a true story, is about the marriage of Helen, a widow with eight children, and Frank, a widower +with ten children, who then have an additional child together. The movie is a comedy that plays on the interpersonal and +organizational struggles of feeding, bathing, and clothing twenty people in one household. +If we consider the set of Helen's children and the set of Frank's children, then the child they had together is the +intersection of these two sets, and the collection of all their children combined is the union of these two sets. In this +section, we will explore the operations of union and intersection as it relates to two sets. +28 +1 • Sets +Access for free at openstax.org + +The Intersection of Two Sets +The members that the two sets share in common are included in the intersection of two sets. To be in the intersection +of two sets, an element must be in both the first set and the second set. In this way, the intersection of two sets is a +logical AND statement. Symbolically, +intersection +is written as: +. +intersection +is written in set builder +notation as: +. +Let us look at Helen's and Frank's children from the movie Yours, Mine, and Ours. Helen's children consist of the set +and Frank's children are included in the set +. +intersection +is the set +of children they had together. +, because Joseph is in both set +and set +. +EXAMPLE 1.23 +Finding the Intersection of Set +and Set +Set +and +Find +intersection +Solution +The intersection of sets +and +include the elements that set +and +have in common: 3, 5, and 7. +YOUR TURN 1.23 +1. Set +and +. Find +intersection +. +Notice that if sets +and +are disjoint sets, then they do not share any elements in common, and +intersection +is the +empty set, as shown in the Venn diagram below. +Figure 1.19 +EXAMPLE 1.24 +Determining the Intersection of Disjoint Sets +Set +and set +Find +Solution +Because sets +and +are disjoint, they do not share any elements in common. So, the intersection of set +and set +is +the empty set. +YOUR TURN 1.24 +1. Set +and set +. Find +. +Notice that if set +is a subset of set +, then +intersection +is equal to set +, as shown in the Venn diagram below. +1.4 • Set Operations with Two Sets +29 + +Figure 1.20 +EXAMPLE 1.25 +Finding the Intersection of a Set and a Subset +Set +and set +ℕ +Find +Solution +Because set +is a subset of set +, +intersection +is equal to set +. +the set of odd natural +numbers. +YOUR TURN 1.25 +1. Set +and set +. Find +. +The Union of Two Sets +Like the union of two families in marriage, the union of two sets includes all the members of the first set and all the +members of the second set. To be in the union of two sets, an element must be in the first set, the second set, or both. In +this way, the union of two sets is a logical inclusive OR statement. Symbolically, +union +is written as: +union +is written in set builder notation as: +Let us consider the sets of Helen's and Frank's children from the movie Yours, Mine, and Ours again. Helen's children is +set +and Frank's children is set +. The union of these two +sets is the collection of all nineteen of their children, +Notice, Joseph is in both set +and set +, but he is only one child, so, he is only listed once in the union. +EXAMPLE 1.26 +Finding the Union of Sets +and +When +and +Overlap +Set +and set +. Find +union +. +Solution +union +is the set formed by including all the unique elements in set +, set +, or both sets +and +: +The first five elements of the union are the five unique elements in set +. Even though 3, 5, and +7 are also members of set +, these elements are only listed one time. Lastly, set +includes the unique element 2, so 2 is +also included as part of the union of sets +and +. +YOUR TURN 1.26 +1. Set +and set +. Find +union +. +When observing the union of sets +and +, notice that both set +and set +are subsets of +union +. Graphically, +union +can be represented in several different ways depending on the members that they have in common. If +and +are disjoint sets, then +union +would be represented with two disjoint circles within the universal set, as shown in the +30 +1 • Sets +Access for free at openstax.org + +Venn diagram below. +Figure 1.21 +If sets +and +share some, but not all, members in common, then the Venn diagram is drawn as two separate circles +that overlap. +Figure 1.22 +If every member of set +is also a member of set +, then +is a subset of set +, and +union +would be equal to set +. +To draw the Venn diagram, the circle representing set +should be completely enclosed in the circle containing set +. +Figure 1.23 +EXAMPLE 1.27 +Finding the Union of Sets +and +When +and +Are Disjoint +Set +and set +Find +Solution +Because sets +and +are disjoint, the union is simply the set containing all the elements in both set +and set +. +YOUR TURN 1.27 +1. Set +and set +. Find +. +EXAMPLE 1.28 +Finding the Union of Sets +and +When One Set is a Subset of the Other +Set +and set +ℕ +Find +Solution +Because set +is a subset of set +, +union +is equal to set +. +ℕ +1.4 • Set Operations with Two Sets +31 + +YOUR TURN 1.28 +1. Set +and set +. Find +. +VIDEO +The Basics of Intersection of Sets, Union of Sets and Venn Diagrams (https://openstax.org/r/operation-on-Sets) +TECH CHECK +Set Operation Practice +Sets Challenge is an application available on both Android and iPhone smartphones that allows you to practice and +gain familiarity with the operations of set union, intersection, complement, and difference. +Figure 1.24 Google Play Store image of Sets Challenge game. (credit: screenshot from Google Play) +The Sets Challenge application/game uses some notation that differs from the notation covered in the text. +• +The complement of set +in this text is written symbolically as +but the Sets Challenge game uses +to +represent the complement operation. +• +In the text we do not cover set difference between two sets +and +, represented in the game as +In the +game this operation removes from set +all the elements in +For example, if set +and set +are subsets of the universal set +then +and +There is a project at the end of the chapter to research the set +difference operation. +Determining the Cardinality of Two Sets +The cardinality of the union of two sets is the total number of elements in the set. Symbolically the cardinality of +union +is written, +. If two sets +and +are disjoint, the cardinality of +union +is the sum of the cardinality of +32 +1 • Sets +Access for free at openstax.org + +set +and the cardinality of set +. If the two sets intersect, then +intersection +is a subset of both set +and set +. This +means that if we add the cardinality of set +and set +, we will have added the number of elements in +intersection +twice, so we must then subtract it once as shown in the formula that follows. +FORMULA +The cardinality of +union +is found by adding the number of elements in set +to the number of elements in set +, then subtracting the number of elements in the intersection of set +and set +. +or +If sets +and +are disjoint, then +and the formula is still valid, but simplifies to +EXAMPLE 1.29 +Determining the Cardinality of the Union of Two Sets +The number of elements in set +is 10, the number of elements in set +is 20, and the number of elements in +intersection +is 4. Find the number of elements in +union +. +Solution +Using the formula for determining the cardinality of the union of two sets, we can say +YOUR TURN 1.29 +1. If +and +then find +. +EXAMPLE 1.30 +Determining the Cardinality of the Union of Two Disjoint Sets +If +and +are disjoint sets and the cardinality of set +is 37 and the cardinality of set +is 43, find the cardinality of +union +. +Solution +To find the cardinality of +union +, apply the formula, +Because sets +and +are +disjoint, +is the empty set, therefore +and +YOUR TURN 1.30 +1. If +and +then find +. +Applying Concepts of “AND” and “OR” to Set Operations +To become a licensed driver, you must pass some form of written test and a road test, along with several other +requirements depending on your age. To keep this example simple, let us focus on the road test and the written test. If +you pass the written test but fail the road test, you will not receive your license. If you fail the written test, you will not be +allowed to take the road test and you will not receive a license to drive. To receive a driver's license, you must pass the +written test AND the road test. For an “AND” statement to be true, both conditions that make up the statement must be +true. Similarly, the intersection of two sets +and +is the set of elements that are in both set +and set +. To be a +member of +intersection +, an element must be in set +and also must be in set +. The intersection of two sets +corresponds to a logical "AND" statement. +The union of two sets is a logical inclusive "OR" statement. Say you are at a birthday party and the host offers Leah, +1.4 • Set Operations with Two Sets +33 + +Lenny, Maya, and you some cake or ice cream for dessert. Leah asks for cake, Lenny accepts both cake and ice cream, +Maya turns down both, and you choose only ice cream. Leah, Lenny, and you are all having dessert. The “OR” statement +is true if at least one of the components is true. Maya is the only one who did not have cake or ice cream; therefore, she +did not have dessert and the “OR” statement is false. To be in the union of two sets +and +, an element must be in set +or set +or both set +and set +. +EXAMPLE 1.31 +Applying the "AND" or "OR" Operation +and +Find the set consisting of elements in: +1. +2. +3. +4. +Solution +1. +because only the elements 0 and 12 are members of both set +and set +. +2. +because the set +or +is the collection of all elements in set +or set +, +or both. +3. +because the set +or +is the collection of all elements in set +or set +, or both. +4. +Parentheses are evaluated first: +because the only +member that both set +and set +share in common is 8. So, now we need to find +Because +the word translates to the union operation, the problem becomes +which is equal to +YOUR TURN 1.31 +and +and +Find the set consisting of elements in: +1. +. +2. +. +3. +. +4. +. +EXAMPLE 1.32 +Determine and Apply the Appropriate Set Operations to Solve the Problem +Don Woods is serving cake and ice cream at his Juneteenth celebration. The party has a total of 54 guests in attendance. +Suppose 30 guests requested cake, 20 guests asked for ice cream, and 12 guests did not have either cake or ice cream. +1. +How many guests had cake or ice cream? +2. +How many guests had cake and ice cream? +Solution +1. +The total number of people at the party is 54, and 12 people did not have cake or ice cream. Recall that the total +number of elements in the universal set is always equal to the number of elements in a subset plus the number of +elements in the complement of the set, +That means +, or equivalently, +A total of 42 people at the party had cake +or ice cream. +2. +To determine the number of people who had both cake and ice cream, we need to find the intersection of the set of +people who had cake and the set of people who had ice cream. From Question 1, the number of people who had +34 +1 • Sets +Access for free at openstax.org + +cake or ice cream is 42. This is the union of the two sets. The formula for the union of two sets is +Use the information given in the problem and substitute the known values into +the formula to solve for the number of people in the intersection: +Adding 30 and 20, the +equation simplifies to +Which means +YOUR TURN 1.32 +Ravi and Priya are serving soup and salad along with the main course at their wedding reception. The reception will +have a total of 150 guests in attendance. A total of 92 soups and 85 salads were ordered, while 23 guests did not +order any soup or salad. +1. How many guests had soup or salad or both? +2. How many guests had both soup and a salad? +WHO KNEW? +The Real Inventor of the Venn Diagram +John Venn, in his writings, references works by both John Boole and Augustus De Morgan, who referred to the circle +diagrams commonly used to present logical relationships as Euler's circles. Leonhard Euler's works were published +over 100 years prior to Venn's, and Euler may have been influenced by the works of Gottfried Leibniz. +So, why does John Venn get all the credit for these graphical depictions? Venn was the first to formalize the use of +these diagrams in his book Symbolic Logic, published in 1881. Further, he made significant improvements in their +design, including shading to highlight the region of interest. The mathematician C.L. Dodgson, also known as Lewis +Carroll, built upon Venn’s work by adding an enclosing universal set. +Invention is not necessarily coming up with an initial idea. It is about seeing the potential of an idea and applying it to +a new situation. +References: +Margaret E. Baron. "A Note on the Historical Development of Logic Diagrams: Leibniz, Euler and Venn." The +Mathematical Gazette, vol. 53, no. 384, 1969, pp. 113-125. JSTOR, www.jstor.org/stable/3614533. Accessed 15 July +2021. +Deborah Bennett. "Drawing Logical Conclusions." Math Horizons, vol. 22, no. 3, 2015, pp. 12-15. JSTOR, www.jstor.org/ +stable/10.4169/mathhorizons.22.3.12. Accessed 15 July 2021. +Drawing Conclusions from a Venn Diagram with Two Sets +All Venn diagrams will display the relationships between the sets, such as subset, intersecting, and/or disjoint. In +addition to displaying the relationship between the two sets, there are two main additional details that Venn diagrams +can include: the individual members of the sets or the cardinality of each disjoint subset of the universal set. +A Venn diagram with two subsets will partition the universal set into 3 or 4 sections depending on whether they are +disjoint or intersecting sets. Recall that the complement of set +, written +is the set of all elements in the universal set +that are not in set +Figure 1.25 Side-by-side Venn diagrams with disjoint and intersecting sets, respectively. +1.4 • Set Operations with Two Sets +35 + +EXAMPLE 1.33 +Using a Venn Diagram to Draw Conclusions about Set Membership +Figure 1.26 +1. +Find +2. +Find +3. +Find +. +4. +Find +Solution +1. +because +union +is the collection of all elements in set +or set +or both. +2. +Because +and +are disjoint sets, there are no elements that are in both +and +. Therefore, +intersection +is +the empty set, +3. +The complement of set +is the set of all elements in the universal set that are not in set +: +4. +The cardinality, or number of elements in set +YOUR TURN 1.33 +Venn diagram with two intersecting sets and members. +1. Find +. +2. Find +. +3. Find +. +4. Find +. +EXAMPLE 1.34 +Using a Venn Diagram to Draw Conclusions about Set Cardinality +Figure 1.27 Venn diagram with two intersecting sets and number of elements in each section indicated. +1. +Find +2. +Find +3. +Find +36 +1 • Sets +Access for free at openstax.org + +Solution +1. +The number of elements in +or +is the number of elements in +union +: +2. +The number of elements in +and +is the number of elements in +intersection +: +3. +The number of elements in set +is the sum of all the numbers enclosed in the circle representing set +: +YOUR TURN 1.34 +Venn diagram with two disjoint sets and number of elements in each section. +1. Find +. +2. Find +. +3. Find +. +Check Your Understanding +20. The ___________ of two sets +and +is the set of all elements that they share in common. +21. The ___________ of two sets +and +is the collection of all elements that are in set +or set +, or both set +and set +. +22. The union of two sets +and +is represented symbolically as __________. +23. The intersection of two sets +and +is represented symbolically as ___________. +24. If set +is a subset of set +, then +intersection +is equal to set ___________. +25. If set +is a subset of set +, then +union +is equal to set ___________. +26. If set +and set +are disjoint sets, then +intersection +is the ___________ set. +27. The cardinality of +union +, +, is found using the formula: ___________. +SECTION 1.4 EXERCISES +For the following exercises, determine the union or intersection of the sets as indicated. +, +, +, and +. +1. +2. +3. +4. +5. +6. +7. +8. +9. +10. +11. +12. +For the following exercises, use the sets provided to apply the “AND” or “OR” operation as indicated to find the resulting +set. +1.4 • Set Operations with Two Sets +37 + +, +, +, +, +, and +. +13. Find the set consisting of elements in +and +. +14. Find the set consisting of elements in +or +. +15. Find the set consisting of elements in +or +. +16. Find the set consisting of elements in +and +. +17. Find the set consisting of elements in +and +. +18. Find the set consisting of elements in +or +. +19. Find the set consisting of the elements in +or +or +. +20. Find the set consisting of the elements in +or +or +. +21. Find the set consisting of the elements in ( +or +) and +. +22. Find the set consisting of the elements in +or ( +and +). +23. Find the set consisting of elements in +or ( +and +). +24. Find the set consisting of elements in ( +or +) and +. +For the following exercises, use the Venn diagram provided to answer the following questions about the sets. +25. Find +. +26. Find +. +27. Find +. +28. Find +. +29. Find +. +30. Find +. +For the following exercises, use the Venn diagram provided to answer the following questions about the sets. +31. Find +. +32. Find +. +33. Find +. +34. Find +. +35. Find +. +36. Find +. +For the following exercises, use the Venn diagram provided to answer the following questions about the sets. +37. Find +. +38. Find +. +39. Find +. +40. Find +. +41. Find +. +42. Find +. +38 +1 • Sets +Access for free at openstax.org + +For the following exercises, determine the cardinality of the union of set +and set +. +43. If set +and set +, find +. +44. If set +and set +, find the number of elements in +or +. +45. If set +and set +, find the number of elements in +or +. +46. If set +and Set +, find +. +For the following exercises, use the Venn diagram to determine the cardinality of +union +. +47. +48. +49. +50. +1.4 • Set Operations with Two Sets +39 + +1.5 Set Operations with Three Sets +Figure 1.28 Companies like Google collect data on how you use their services, but the data requires analysis to really +mean something. (credit: “Man holding smartphone and searches through google” by Nenad Stojkovic/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Interpret Venn diagrams with three sets. +2. +Create Venn diagrams with three sets. +3. +Apply set operations to three sets. +4. +Prove equality of sets using Venn diagrams. +Have you ever searched for something on the Internet and then soon after started seeing multiple advertisements for +that item while browsing other web pages? Large corporations have built their business on data collection and analysis. +As we start working with larger data sets, the analysis becomes more complex. In this section, we will extend our +knowledge of set relationships by including a third set. +A Venn diagram with two intersecting sets breaks up the universal set into four regions; simply adding one additional set +will increase the number of regions to eight, doubling the complexity of the problem. +Venn Diagrams with Three Sets +Below is a Venn diagram with two intersecting sets, which breaks the universal set up into four distinct regions. +Figure 1.29 +Next, we see a Venn diagram with three intersecting sets, which breaks up the universal set into eight distinct +regions. +40 +1 • Sets +Access for free at openstax.org + +Figure 1.30 +TECH CHECK +Shading Venn Diagrams +Venn Diagram is an Android application that allows you to visualize how the sets are related in a Venn diagram by +entering expressions and displaying the resulting Venn diagram of the set shaded in gray. +Figure 1.31 Google Play Store image of Venn Diagram app. (credit: screenshot from Google Play) +The Venn Diagram application uses some notation that differs from the notation covered in this text. +a. +The complement of set +in this text is written symbolically as +, but the Venn Diagram app uses +to +1.5 • Set Operations with Three Sets +41 + +represent the complement operation. +b. +The set difference operation, +, is available in the Venn Diagram app, although this operation is not covered in +the text. +It is recommended that you explore this application to expand your knowledge of Venn diagrams prior to continuing +with the next example. +In the next example, we will explore the three main blood factors, A, B and Rh. The following background information +about blood types will help explain the relationships between the sets of blood factors. If an individual has blood factor A +or B, those will be included in their blood type. The Rh factor is indicated with a +or a +. For example, if a person has all +three blood factors, then their blood type would be +. In the Venn diagram, they would be in the intersection of all +three sets, +If a person did not have any of these three blood factors, then their blood type would be +and they would be in the set +which is the region outside all three circles. +EXAMPLE 1.35 +Interpreting a Venn Diagram with Three Sets +Use the Venn diagram below, which shows the blood types of 100 people who donated blood at a local clinic, to answer +the following questions. +Figure 1.32 +1. +How many people with a type A blood factor donated blood? +2. +Julio has blood type +If he needs to have surgery that requires a blood transfusion, he can accept blood from +anyone who does not have a type A blood factor. How many people donated blood that Julio can accept? +3. +How many people who donated blood do not have the +blood factor? +4. +How many people had type A and type B blood? +Solution +1. +The number of people who donated blood with a type A blood factor will include the sum of all the values included +in the A circle. It will be the union of sets +2. +In part 1, it was determined that the number of donors with a type A blood factor is 46. To determine the number of +people who did not have a type A blood factor, use the following property, +union is equal to +, which means +and +Thus, 54 people donated blood that Julio can +accept. +3. +This would be everyone outside the +circle, or everyone with a negative Rh factor, +4. +To have both blood type A and blood type B, a person would need to be in the intersection of sets +and +. The two +circles overlap in the regions labeled +and +Add up the number of people in these two regions to get the +total: +This can be written symbolically as +42 +1 • Sets +Access for free at openstax.org + +YOUR TURN 1.35 +Use the same Venn diagram in the example above to answer the following questions. +1. How many people donated blood with a type B blood factor? +2. How many people who donated blood did not have a type B blood factor? +3. How many people who donated blood had a type B blood factor or were Rh+? +WHO KNEW? +Blood Types +Most people know their main blood type of A, B, AB, or O and whether they are +or +, but did you know that +the International Society of Blood Transfusion recognizes twenty-eight additional blood types that have important +implications for organ transplants and successful pregnancy? For more information, check out this article: +Blood mystery solved: Two new blood types identified (https://openstax.org/r/Two-new-blood-types-identified) +Creating Venn Diagrams with Three Sets +In general, when creating Venn diagrams from data involving three subsets of a universal set, the strategy is to work +from the inside out. Start with the intersection of the three sets, then address the regions that involve the intersection of +two sets. Next, complete the regions that involve a single set, and finally address the region in the universal set that does +not intersect with any of the three sets. This method can be extended to any number of sets. The key is to start with the +region involving the most overlap, working your way from the center out. +EXAMPLE 1.36 +Creating a Venn Diagram with Three Sets +A teacher surveyed her class of 43 students to find out how they prepared for their last test. She found that 24 students +made flash cards, 14 studied their notes, and 27 completed the review assignment. Of the entire class of 43 students, 12 +completed the review and made flash cards, nine completed the review and studied their notes, and seven made flash +cards and studied their notes, while only five students completed all three of these tasks. The remaining students did not +do any of these tasks. Create a Venn diagram with subsets labeled: “Notes,” “Flash Cards,” and “Review” to represent how +the students prepared for the test. +Solution +Step 1: First, draw a Venn diagram with three intersecting circles to represent the three intersecting sets: Notes, Flash +Cards, and Review. Label the universal set with the cardinality of the class. +Figure 1.33 +Step 2: Next, in the region where all three sets intersect, enter the number of students who completed all three tasks. +1.5 • Set Operations with Three Sets +43 + +Figure 1.34 +Step 3: Next, calculate the value and label the three sections where just two sets overlap. +a. +Review and flash card overlap. A total of 12 students completed the review and made flash cards, but five of these +twelve students did all three tasks, so we need to subtract: +. This is the value for the region where the +flash card set intersects with the review set. +b. +Review and notes overlap. A total of 9 students completed the review and studied their notes, but again, five of +these nine students completed all three tasks. So, we subtract: +. This is the value for the region where the +review set intersects with the notes set. +c. +Flash card and notes overlap. A total of 7 students made flash cards and studied their notes; subtracting the five +students that did all three tasks from this number leaves 2 students who only studied their notes and made flash +cards. Add these values to the Venn diagram. +Figure 1.35 +Step 4: Now, repeat this process to find the number of students who only completed one of these three tasks. +a. +A total of 24 students completed flash cards, but we have already accounted for +of these. Thus, +students who just made flash cards. +b. +A total of 14 students studied their notes, but we have already accounted for +of these. Thus, +students only studied their notes. +c. +A total of 27 students completed the review assignment, but we have already accounted for +of these, +which means +students only completed the review assignment. +d. +Add these values to the Venn diagram. +44 +1 • Sets +Access for free at openstax.org + +Figure 1.36 +Step 5: Finally, compute how many students did not do any of these three tasks. To do this, we add together each value +that we have already calculated for the separate and intersecting sections of our three sets: +. Because there 43 students in the class, and +, this means only one student +did not complete any of these tasks to prepare for the test. Record this value somewhere in the rectangle, but outside of +all the circles, to complete the Venn diagram. +Figure 1.37 +YOUR TURN 1.36 +1. A group of 50 people attending a conference who preordered their lunch were able to select their choice of soup, +salad, or sandwich. A total of 17 people selected soup, 29 people selected salad and 35 people selected a +sandwich. Of these orders, 11 attendees selected soup and salad, 10 attendees selected soup and a sandwich, +and 18 selected a salad and a sandwich, while eight people selected a soup, a salad, and a sandwich. Create a +Venn diagram with subsets labeled “Soup,” “Salad,” and “Sandwich,” and label the cardinality of each section of +the Venn diagram as indicated by the data. +Applying Set Operations to Three Sets +Set operations are applied between two sets at a time. Parentheses indicate which operation should be performed first. +As with numbers, the inner most parentheses are applied first. Next, find the complement of any sets, then perform any +union or intersections that remain. +1.5 • Set Operations with Three Sets +45 + +EXAMPLE 1.37 +Applying Set Operations to Three Sets +Perform the set operations as indicated on the following sets: +, +and +1. +Find +2. +Find +3. +Find +Solution +1. +Parentheses first, +intersection +equals +the elements common to both +and +. +because the only elements that are in both sets are 0 and 6. +2. +Parentheses first, +union +equals +the collection of all elements in set +or set +or both. +because the intersection of +these two sets is the set of elements that are common to both sets. +3. +Parentheses first, +intersection +equals +Next, find +The complement of set +is the set of +elements in the universal set +that are not in set +Finally, find +YOUR TURN 1.37 +Using the same sets from Example 1.37, perform the set operations indicated. +1. Find +. +2. Find +. +3. Find +. +Notice that the answers to the Your Turn are the same as those in the Example. This is not a coincidence. The following +equivalences hold true for sets: +• +and +These are the associative property for set intersection +and set union. +• +and +These are the commutative property for set intersection and set union. +• +and +These are the distributive property for sets +over union and intersection, respectively. +Proving Equality of Sets Using Venn Diagrams +To prove set equality using Venn diagrams, the strategy is to draw a Venn diagram to represent each side of the equality, +then look at the resulting diagrams to see if the regions under consideration are identical. +Augustus De Morgan was an English mathematician known for his contributions to set theory and logic. De Morgan’s law +for set complement over union states that +. In the next example, we will use Venn diagrams to prove +De Morgan’s law for set complement over union is true. But before we begin, let us confirm De Morgan’s law works for a +specific example. While showing something is true for one specific example is not a proof, it will provide us with some +reason to believe that it may be true for all cases. +Let +and +We will use these sets in the equation +To begin, find the value of the set defined by each side of the equation. +Step 1: +is the collection of all unique elements in set +or set +or both. +The complement +of A union B, +, is the set of all elements in the universal set that are not in +. So, the left side the equation +is equal to the set +Step 2: The right side of the equation is +is the set of all members of the universal set +that are not in set +. +Similarly, +Step 3: Finally, +is the set of all elements that are in both +and +The numbers 1 and 7 are common to both +sets, therefore, +Because, +we have demonstrated that De Morgan’s law for set +46 +1 • Sets +Access for free at openstax.org + +complement over union works for this particular example. The Venn diagram below depicts this relationship. +Figure 1.38 +EXAMPLE 1.38 +Proving De Morgan’s Law for Set Complement over Union Using a Venn Diagram +De Morgan’s Law for the complement of the union of two sets +and +states that: +Use a Venn +diagram to prove that De Morgan’s Law is true. +Solution +Step 1: First, draw a Venn diagram representing the left side of the equality. The regions of interest are shaded to +highlight the sets of interest. +is shaded on the left, and +is shaded on the right. +Figure 1.39 +Step 2: Next, draw a Venn diagram to represent the right side of the equation. +is shaded and +is shaded. Because +and +mix to form +is also shaded. +Figure 1.40 Venn diagram of intersection of the complement of two sets. +Step 3: Verify the conclusion. Because the shaded region in the Venn diagram for +matches the shaded region in +the Venn diagram for +, the two sides of the equation are equal, and the statement is true. This completes the +proof that De Morgan’s law is valid. +YOUR TURN 1.38 +1. De Morgan’s Law for the complement of the intersection of two sets +and +states that +. +Use a Venn diagram to prove that De Morgan’s Law is true. +Check Your Understanding +28. When creating a Venn diagram with two or more subsets, you should begin with the region involving the most +_____________, then work your way from the center outward. +29. To construct a Venn diagram with three subsets, draw and label three circles that overlap in a common +1.5 • Set Operations with Three Sets +47 + +_____________ region inside the rectangle of the universal set to represent each of the three subsets. +30. In a Venn diagram with three sets, the area where all three sets, +, +, and +overlap is equal to the set +_____________. +31. When performing set operations with three or more sets, the order of operations is inner most _____________ first, +then find the ___________ of any sets, and finally perform any union or intersection operations that remain. +32. To prove set equality using Venn diagrams, draw a Venn diagram to represent each side of the ______________ and +then compare the diagrams to determine if they match or not. If they match, the statement is ____________, +otherwise it is not. +SECTION 1.5 EXERCISES +A gamers club at Baily Middle School consisting of 25 members was surveyed to find out who played board games, +card games, or video games. Use the results depicted in the Venn diagram below to answer the following exercises. +1. How many gamers club members play all three types of games: board games, card games, and video games? +2. How many gamers are in the set Board +Video? +3. If Javier is in the region with a total of three members, what type of games does he play? +4. How many gamers play video games? +5. How many gamers are in the set Board +Card? +6. How many members of the gamers club do not play video games? +7. How many members of this club only play board games? +8. How many members of this club only play video games? +9. How many members of the gamers club play video and card games? +10. How many members of the gamers club are in the set +? +A blood drive at City Honors High School recently collected blood from 140 students, staff, and faculty. Use the results +depicted in the Venn diagram below to answer the following exercises. +11. Blood type +is the universal acceptor. Of the 140 people who donated at City Honors, how many had blood +48 +1 • Sets +Access for free at openstax.org + +type +? +12. Blood type +is the universal donor. Anyone needing a blood transfusion can receive this blood type. How +many people who donated blood during this drive had +blood? +13. How many people donated with a type A blood factor? +14. How many people donated with a type A and type B blood factor (that is, they had type AB blood). +15. How many donors were +? +16. How many donors were not +? +17. Opal has blood type +. If she needs to have surgery that requires a blood transfusion, she can accept blood +from anyone who does not have a type B blood factor. How many people donated blood during this drive at City +Honors that Opal can accept? +18. Find +. +19. Find +. +20. Find +. +For the following exercises, create a three circle Venn diagram to represent the relationship between the described +sets. +21. The number of elements in the universal set, +, is +. Sets +, +, and +are subsets of +: +, and +. Also, +, and +. +22. The number of elements in the universal set, +, is +. Sets +, +, and +are subsets of +: +: +. Also, +, +, and +. +23. The number of elements in the universal set, +, is +. Sets +, +, and +are subsets of +: +, +, and +. Also, +, and +. +24. The number of elements in the universal set, +, is +. Sets +, +, and +are subsets of +: +, +, and +. Also, +, and +. +25. The universal set, +, has a cardinality of 36. +, +, and +26. The universal set, +, has a cardinality of 63. +, +, and +. +27. The universal set, +, has a cardinality of 72. +, and +. +28. The universal set, +, has a cardinality of 81. +, and +. +29. The anime drawing club at Pratt Institute conducted a survey of its 42 members and found that 23 of them +sketched with pastels, 28 used charcoal, and 17 used colored pencils. Of these, 10 club members used all three +mediums, 18 used charcoal and pastels, 11 used colored pencils and charcoal, and 12 used colored pencils and +pastels. The remaining club members did not use any of these three mediums. +30. A new SUV is selling with three optional packages: a sport package, a tow package, and an entertainment +package. A dealership gathered the following data for all 31 of these vehicles sold during the month of July. A +total of 18 SUVs included the entertainment package, 11 included the tow package, and 16 included the sport +package. Of these, five SUVs included all three packages, seven were sold with both the tow package and sport +package, 11 were sold with the entertainment and sport package, and eight were sold with the tow package +and entertainment package. The remaining SUVs sold did not include any of these optional packages. +For the following exercises, perform the set operations as indicated on the following sets: +, and +. +31. Find +. +32. Find +. +33. Find +. +34. Find +. +35. Find +. +36. Find +. +For the following exercises, perform the set operations as indicated on the following sets: +, and +. +1.5 • Set Operations with Three Sets +49 + +37. Find +. +38. Find +. +39. Find +. +40. Find +. +41. Find +. +42. Find +. +For the following exercises, use Venn diagrams to prove the following properties of sets: +43. Commutative property for the union of two sets: +. +44. Commutative property for the intersection of two sets: +. +45. Associative property for the intersection of three sets: +. +46. Associative property for the union of three sets: +. +47. Distributive property for set intersection over set union: +. +48. Distributive property for set union over set intersection: +. +50 +1 • Sets +Access for free at openstax.org + +Chapter Summary +Key Terms +1.1 Basic Set Concepts +• +set +• +elements +• +well-defined set +• +empty set +• +roster method +• +finite set +• +infinite set +• +natural numbers +• +integer +• +set-builder notation +• +cardinality of a set +• +countably infinite +• +equal sets +• +equivalent sets +1.2 Subsets +• +subset +• +proper subset +• +equivalent subsets +• +exponential notation +1.3 Understanding Venn Diagrams +• +Venn diagram +• +universal set +• +disjoint set +• +complement of a set +1.4 Set Operations with Two Sets +• +intersection of two sets +• +union of two sets +Key Concepts +1.1 Basic Set Concepts +• +Identify a set as being a well-defined collection of objects and differentiate between collections that are not well- +defined and collections that are sets. +• +Represent sets using both the roster or listing method and set builder notation which includes a description of the +members of a set. +• +In set theory, the following symbols are universally used: +ℕ - The set of natural numbers, which is the set of all positive counting numbers. +ℕ +ℤ - The set of integers, which is the set of all the positive and negative counting numbers and the number zero. +ℤ +ℚ - The set of rational numbers or fractions. +ℚ +• +Distinguish between finite sets, infinite sets, and the empty set to determine the size or cardinality of a set. +• +Distinguish between equal sets which have exactly the same members and equivalent sets that may have different +members but must have the same cardinality or size. +1 • Chapter Summary +51 + +1.2 Subsets +• +Every member of a subset of a set is also a member of the set containing it. +• +A proper subset of a set does not contain all the members of the set containing it. There is a least one member of +set +that is not a member of set +. +• +The number subsets of a finite set +with +members is equal to 2 raised to the +power. +• +The empty set is a subset of every set and must be included when listing all the subsets of a set. +• +Understand how to create and distinguish between equivalent subsets of finite and infinite sets that are not equal +to the original set. +1.3 Understanding Venn Diagrams +• +A Venn diagram is a graphical representation of the relationship between sets. +• +In a Venn diagram, the universal set, +is the largest set under consideration and is drawn as a rectangle. All +subsets of the universal set are drawn as circles within this rectangle. +• +The complement of set +includes all the members of the universal set that are not in set +. A set and its +complement are disjoint sets, they do not share any elements in common. +• +To find the complement of set +remove all the elements of set +from the universal set +, the set that includes +only the remaining elements is the complement of set +, +. +• +Determine the complement of a set using Venn diagrams, the roster method and set builder notation. +1.4 Set Operations with Two Sets +• +The intersection of two sets, +is the set of all elements that they have in common. Any member of +intersection +must be is both set +and set +. +• +The union of two sets, +, is the collection of all members that are in either in set +, set +or both sets +and +combined. +• +Two sets that share at least one element in common, so that they are not disjoint are represented in a Venn +Diagram using two circles that overlap. +◦ +The region of the overlap is the set +intersection +, +◦ +The regions that include everything in the circle representing set +or the circle representing set +or their +overlap is the set +union +, +• +Apply knowledge of set union and intersection to determine cardinality and membership using Venn Diagrams, the +roster method and set builder notation. +1.5 Set Operations with Three Sets +• +A Venn diagram with two overlapping sets breaks the universal set up into four distinct regions. When a third +overlapping set is added the Venn diagram is broken up into eight distinct regions. +• +Analyze, interpret, and create Venn diagrams involving three overlapping sets. +◦ +Including the blood factors: A, B and Rh +◦ +To find unions and intersections. +◦ +To find cardinality of both unions and intersections. +• +When performing set operations with three or more sets, the order of operations is inner most parentheses first, +then fine the complement of any sets, then perform any union or intersection operations that remain. +• +To prove set equality using Venn diagrams the strategy is to draw a Venn diagram to represent each side of the +equality or equation, then look at the resulting diagrams to see if the regions under consideration are identical. If +they regions are identical the equation represents a true statement, otherwise it is not true. +Videos +1.1 Basic Set Concepts +• +Equal and Equivalent Sets (https://openstax.org/r/Equal_and_Equivalent_Sets) +1.4 Set Operations with Two Sets +• +The Basics of Intersection of Sets, Union of Sets and Venn Diagrams (https://openstax.org/r/operation-on-Sets) +Formula Review +1.2 Subsets +The number of subsets of a finite set +is equal to 2 raised to the power of +, where +is the number of elements +52 +1 • Chapter Summary +Access for free at openstax.org + +in set +: Number of Subsets of Set +. +1.4 Set Operations with Two Sets +The cardinality of +union +Projects +Cardinality of Infinite Sets +In set theory, it has been shown that the set of irrational numbers has a cardinality greater than the set of natural +numbers. That is, the set of irrational numbers is so large that it is uncountably infinite. +1. +Perform a search with the phrase, “Who first proved that the real numbers are uncountable?” +a. +Who first proved that the real numbers are uncountable? +b. +What was the significance of this proof to the development of set theory and by extension other fields of +mathematics? +2. +Recent discoveries in the field of set theory include the solution to a 70-year-old problem previously thought to be +unprovable. To learn more read this article (https://openstax.org/r/measure-infinities): +a. +What does it mean for two infinite sets to have the same size? +b. +The real numbers are sometimes referred to as what? +c. +Summarize your understanding of the problem known as the “Continuum Hypothesis.” +d. +Malliaris and Shelah’s proof of this 70-year-old problem is opening up investigation in what two fields of +mathematics? +3. +Summarize your understanding of infinity. +a. +Define what it means to be infinite. +b. +Explain the difference between countable and uncountable sets. +c. +Research the difference between a discrete set and a continuous set, then summarize your findings. +Set Notation +In arithmetic, the operation of addition is represented by the plus sign, +, but multiplication is represented in multiple +ways, including +and parentheses, such as 5(3). Several set operations also are written in different forms based on +the preferences of the mathematician and often their publisher. +1. +Search for “Set Complement” on the internet and list at least three ways to represent the complement of a set. +2. +Both the Set Challenge and Venn Diagram smartphone apps highlighted in the Tech Check sections have an +operation for set difference. List at least two ways to represent set difference and provide a verbal description of +how to calculate the difference between two sets +and +. +3. +When researching possible Venn diagram applications, the Greek letter delta, +appeared as a symbol for a set +operator. List at least one other symbol used for this same operation. +4. +Search for “List of possible set operations and their symbols.” Find and select two symbols that were not presented +in this chapter. +The Real Number System +The set of real numbers and their properties are studied in elementary school today, but how did the number system +evolve? The idea of natural numbers or counting numbers surfaced prior to written words, as evidenced by tally marks in +cave writing. Create a timeline for significant contributions to the real number system. +1. +Use the following phrase to search online for information on the origins of the number zero: “History of the number +zero.” Then, record significant dates for the invention and common use of the number zero on your timeline. +2. +Find out who is credited for discovering that the +is irrational and add this information to your timeline. Hint: +Search for, “Who was the first to discover irrational numbers?” +3. +Research Georg Cantor’s contribution to the representation of real numbers as a continuum and add this to your +timeline. +4. +Research Ernst Zermelo’s contribution to the real number system and add this to your timeline. +1 • Chapter Summary +53 + +Chapter Review +Basic Set Concepts +1. A ______________ is a well-defined collection of distinct objects. +2. A collection of well-defined objects without any members in it is called the ________ _______. +3. Write the set consisting of the last five letters of the English alphabet using the roster method. +4. Write the set consisting of the numbers 1 through 20 inclusive using the roster method and an ellipsis. +5. Write the set of all zebras that do not have stripes in symbolic form. +6. Write the set of negative integers using the roster method and an ellipsis. +7. Use set builder notation to write the set of all even integers. +8. Write the set of all letters in the word Mississippi and label it with a capital +. +9. Determine whether the following collection describes a well-defined set: "A group of these five types of apples: +Granny Smith, Red Delicious, McIntosh, Fuji, and Jazz." +10. Determine whether the following collection describes a well-defined set: "A group of five large dogs." +11. Determine the cardinality of the set +. +12. Determine whether the following set is a finite set or an infinite set: +. +13. Determine whether sets +and +are equal, equivalent, or neither: +and +. +14. Determine if sets +and +are equal, equivalent, or neither: +and +. +15. Determine if sets +and +are equal, equivalent, or neither: +and +. +Subsets +16. If every member of set +is also a member of set +, then set +is a _________ of set +. +17. Determine whether set +is a subset, proper subset, or neither a subset nor proper subset of set +: +and +. +18. Determine whether set +is a subset, proper subset, or neither a subset nor proper subset of set +: +and +. +19. Determine whether set +is a subset, proper subset, or neither a subset nor proper subset of set +: +and +. +20. List all the subsets of the set +. +21. List all the subsets of the set +. +22. Calculate the total number of subsets of the set {Scooby, Velma, Daphne, Shaggy, Fred}. +23. Calculate the total number of subsets of the set {top hat, thimble, iron, shoe, battleship, cannon}. +24. Find a subset of the set +that is equivalent, but not equal, to +. +25. Find a subset of the set +that is equal to +. +26. Find two equivalent finite subsets of the set of natural numbers, +, with a cardinality of 4. +27. Find two equal finite subsets of the set of natural numbers, +, with a cardinality of 3. +Understanding Venn Diagrams +28. Use the Venn diagram below to describe the relationship between the sets, symbolically and in words: +29. Use the Venn diagram below to describe the relationship between the sets, symbolically and in words: +54 +1 • Chapter Summary +Access for free at openstax.org + +30. Draw a Venn diagram to represent the relationship between the described sets: Falcons +Raptors. +31. Draw a Venn diagram to represent the relationship between the described sets: Natural numbers +Integers +Real numbers. +32. The universal set is the set +. Find the complement of the set +. +33. The universal set is the set +. Find the complement of the set +. +34. Use the Venn diagram below to determine the members of the set +. +35. Use the Venn diagram below to determine the members of the set +. +Set Operations with Two Sets +Determine the union and intersection of the sets indicated: +, +, +, +, +, and +. +36. What is +? +37. What is +? +38. Write the set containing the elements in sets +. +39. Write the set containing all the elements is both sets +. +40. Find +. +41. Find +. +42. Find the cardinality of +. +43. Find +. +44. Use the Venn diagram below to find +. +45. Use the Venn diagram below to find +. +1 • Chapter Summary +55 + +Set Operations with Three Sets +Use the Venn diagram below to answer the following questions. +46. Find +. +47. Find +. +48. A food truck owner surveyed a group of 50 customers about their preferences for hamburger condiments. After +tallying the responses, the owner found that 24 customers preferred ketchup, 11 preferred mayonnaise, and 31 +preferred mustard. Of these customers, eight preferred ketchup and mayonnaise, one preferred mayonnaise +and mustard, and 13 preferred ketchup and mustard. No customer preferred all three. The remaining +customers did not select any of these three condiments. Draw a Venn diagram to represent this data. +49. Given +, +, +, and +, find +. +50. Use Venn diagrams to prove that if +, then +. +Chapter Test +1. Determine whether the following collection describes a well-defined set: “A group of small tomatoes.” +Classify each of the following sets as either finite or infinite. +2. +3. +4. +5. +6. +Use the sets provided to answer the following questions: +, +, +, and +. +7. Find +. +8. Find +. +9. Determine if set +is equivalent to, equal to, or neither equal nor equivalent to set +. Justify your answer. +10. Find +. +11. Find +. +12. Find +. +13. Find +. +Use the Venn diagram below to answer the following questions. +14. Find +. +15. Find +. +16. Find +. +17. Draw a Venn diagram to represent the relationship between the two sets: “All flowers are plants.” +For the following questions, use the Venn diagram showing the blood types of all donors at a recent mobile blood +56 +1 • Chapter Summary +Access for free at openstax.org + +drive. +18. Find the number of donors who were +; that is, find +. +19. Find the number of donors who were +. +20. Use Venn diagrams to prove that if +, then +. +1 • Chapter Summary +57 + +58 +1 • Chapter Summary +Access for free at openstax.org + +Figure 2.1 Logic is key to a well-reasoned argument, in both math and law. (credit: modification of work "Lady Justicia +holding sword and scale bronze figurine with judge hammer on wooden table" by Jernej Furman/Flickr, CC BY 2.0) +Chapter Outline +2.1 Statements and Quantifiers +2.2 Compound Statements +2.3 Constructing Truth Tables +2.4 Truth Tables for the Conditional and Biconditional +2.5 Equivalent Statements +2.6 De Morgan’s Laws +2.7 Logical Arguments +Introduction +What is logic? Logic is the study of reasoning, and it has applications in many fields, including philosophy, law, +psychology, digital electronics, and computer science. +In law, constructing a well-reasoned, logical argument is extremely important. The main goal of arguments made by +lawyers is to convince a judge and jury that their arguments are valid and well-supported by the facts of the case, so the +case should be ruled in their favor. Think about Thurgood Marshall arguing for desegregation in front of the U.S. +Supreme Court during the Brown v. Board of Education of Topeka lawsuit in 1954, or Ruth Bader Ginsburg arguing for +equality in social security benefits for both men and women under the law during the mid-1970s. Both these great minds +were known for the preparation and thoroughness of their logical legal arguments, which resulted in victories that +advanced the causes they fought for. Thurgood Marshall and Ruth Bader Ginsburg would later become well respected +justices on the U.S. Supreme Court themselves. +In this chapter, we will explore how to construct well-reasoned logical arguments using varying structures. Your ability to +form and comprehend logical arguments is a valuable tool in many areas of life, whether you're planning a dinner date, +negotiating the purchase of a new car, or persuading your boss that you deserve a raise. +2 +LOGIC +2 • Introduction +59 + +2.1 Statements and Quantifiers +Figure 2.2 Construction of a logical argument, like that of a house, requires you to begin with the right parts. (credit: +modification of work “Barn Raising” by Robert Stinnett/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify logical statements. +2. +Represent statements in symbolic form. +3. +Negate statements in words. +4. +Negate statements symbolically. +5. +Translate negations between words and symbols. +6. +Express statements with quantifiers of all, some, and none. +7. +Negate statements containing quantifiers of all, some, and none. +Have you ever built a club house, tree house, or fort with your friends? If so, you and your friends likely started by +gathering some tools and supplies to work with, such as hammers, saws, screwdrivers, wood, nails, and screws. +Hopefully, at least one member of your group had some knowledge of how to use the tools correctly and helped to +direct the construction project. After all, if your house isn't built on a strong foundation, it will be weak and could +possibly fall apart during the next big storm. This same foundation is important in logic. +In this section, we will begin with the parts that make up all logical arguments. The building block of any logical +argument is a logical statement, or simply a statement. A logical statement has the form of a complete sentence, and it +must make a claim that can be identified as being true or false. +When making arguments, sometimes people make false claims. When evaluating the strength or validity of a logical +argument, you must also consider the truth values, or the identification of true or false, of all the statements used to +support the argument. While a false statement is still considered a logical statement, a strong logical argument starts +with true statements. +60 +2 • Logic +Access for free at openstax.org + +Identifying Logical Statements +Figure 2.3 Not all roses are red. (credit: “assorted pink yellow white red roses macro” by ProFlowers/Flickr, CC BY 2.0) +An example of logical statement with a false truth value is, “All roses are red.” It is a logical statement because it has the +form of a complete sentence and makes a claim that can be determined to be either true or false. It is a false statement +because not all roses are red: some roses are red, but there are also roses that are pink, yellow, and white. Requests, +questions, or directives may be complete sentences, but they are not logical statements because they cannot be +determined to be true or false. For example, suppose someone said to you, “Please, sit down over there.” This request +does not make a claim and it cannot be identified as true or false; therefore, it is not a logical statement. +EXAMPLE 2.1 +Identifying Logical Statements +Determine whether each of the following sentences represents a logical statement. If it is a logical statement, determine +whether it is true or false. +1. +Tiger Woods won the Master’s golf championship at least five times. +2. +Please, sit down over there. +3. +All cats dislike dogs. +Solution +1. +This is a logical statement because it is a complete sentence that makes a claim that can be identified as being true +or false. As of 2021, this statement is true: Tiger Woods won the Master’s in 1997, 2001, 2002, 2005 and 2019. +2. +This is not a logical statement because, although it is a complete sentence, this request does not make a claim that +can be identified as being either true or false. +3. +This is a logical statement because it is a complete sentence that makes a claim that can be identified as being true +or false. This statement is false because some cats do like some dogs. +YOUR TURN 2.1 +Determine whether each of the following sentences represents a logical statement. If it is a logical statement, +determine whether it is true or false. +1. The Buffalo Bills defeated the New York Giants in Super Bowl XXV. +2. Michael Jackson’s album Thriller was released in 1982. +3. Would you like some coffee or tea? +Representing Statements in Symbolic Form +When analyzing logical arguments that are made of multiple logical statements, symbolic form is used to reduce the +amount of writing involved. Symbolic form also helps visualize the relationship between the statements in a more +2.1 • Statements and Quantifiers +61 + +concise way in order to determine the strength or validity of an argument. Each logical statement is represented +symbolically as a single lowercase letter, usually starting with the letter +. +To begin, you will practice how to write a single logical statement in symbolic form. This skill will become more useful as +you work with compound statements in later sections. +EXAMPLE 2.2 +Representing Statements Using Symbolic Form +Write each of the following logical statements in symbolic form. +1. +Barry Bonds holds the Major League Baseball record for total career home runs. +2. +Some mammals live in the ocean. +3. +Ruth Bader Ginsburg served on the U.S. Supreme Court from 1993 to 2020. +Solution +1. +: Barry Bonds holds the Major League Baseball record for total career home runs. The statement is labeled with a +Once the statement is labeled, use +as a replacement for the full written statement to write and analyze the +argument symbolically. +2. +: Some mammals live in the ocean. The letter +is used to distinguish this statement from the statement in question +1, but any lower-case letter may be used. +3. +: Ruth Bader Ginsburg served on the U.S. Supreme Court from 1993 to 2020. When multiple statements are present +in later sections, you will want to be sure to use a different letter for each separate logical statement. +YOUR TURN 2.2 +Write each of the following logical statements in symbolic form. +1. The movie Gandhi won the Academy Award for Best Picture in 1982. +2. Soccer is the most popular sport in the world. +3. All oranges are citrus fruits. +WHO KNEW? +Mathematics is not the only language to use symbols to represent thoughts or ideas. The Chinese and Japanese +languages use symbols known as Hanzi and Kanji, respectively, to represent words and phrases. At one point, the +American musician Prince famously changed his name to a symbol representing love. +BBC Prince Symbol Article (https://openstax.org/r/magazine-36107590) +Negating Statements +Consider the false statement introduced earlier, “All roses are red.” If someone said to you, “All roses are red,” you might +respond with, “Some roses are not red.” You could then strengthen your argument by providing additional statements, +such as, “There are also white roses, yellow roses, and pink roses, to name a few.” +The negation of a logical statement has the opposite truth value of the original statement. If the original statement is +false, its negation is true, and if the original statement is true, its negation is false. Most logical statements can be +negated by simply adding or removing the word not. For example, consider the statement, “Emma Stone has green +eyes.” The negation of this statement would be, “Emma Stone does not have green eyes.” The table below gives some +other examples. +62 +2 • Logic +Access for free at openstax.org + +Logical Statement +Negation +Gordon Ramsey is a chef. +Gordon Ramsey is not a chef. +Tony the Tiger does not have spots. +Tony the Tiger has spots. +Table 2.1 +The way you phrase your argument can impact its success. If someone presents you with a false statement, the ability to +rebut that statement with its negation will provide you with the tools necessary to emphasize the correctness of your +position. +EXAMPLE 2.3 +Negating Logical Statements +Write the negation of each logical statement in words. +1. +Michael Phelps was an Olympic swimmer. +2. +Tom is a cat. +3. +Jerry is not a mouse. +Solution +1. +Add the word not to negate the affirmative statement: “Michael Phelps was not an Olympic swimmer.” +2. +Add the word not to negate the affirmative statement: “Tom is not a cat.” +3. +Remove the word not to negate the negative statement: “Jerry is a mouse.” +YOUR TURN 2.3 +Write the negation of each logical statement in words. +1. Ted Cruz was not born in Texas. +2. Adele has a beautiful singing voice. +3. Leaves convert carbon dioxide to oxygen during the process of photosynthesis. +Negating Logical Statements Symbolically +The symbol for negation, or not, in logic is the tilde, ~. So, not +is represented as +. To negate a statement symbolically, +remove or add a tilde. The negation of not (not +) is +. Symbolically, this equation is +EXAMPLE 2.4 +Negating Logical Statements Symbolically +Write the negation of each logical statement symbolically. +1. +: Michael Phelps was an Olympic swimmer. +2. +: Tom is not a cat. +3. +: Jerry is not a mouse. +Solution +1. +To negate an affirmative logical statement symbolically, add a tilde: +. +2. +Because the symbol for this statement is , its negation is +. +3. +The symbol for this statement is +, so to negate it we simply remove the ~, because +The answer is . +YOUR TURN 2.4 +Write the negation of each logical statement symbolically. +2.1 • Statements and Quantifiers +63 + +1. +: Ted Cruz was not born in Texas. +2. +: Adele has a beautiful voice. +3. +: Leaves convert carbon dioxide to oxygen during the process of photosynthesis. +Translating Negations Between Words and Symbols +In order to analyze logical arguments, it is important to be able to translate between the symbolic and written forms of +logical statements. +EXAMPLE 2.5 +Translating Negations Between Words and Symbols +Given the statements: +: Elmo is a red Muppet. +: Ketchup is not a vegetable. +1. +Write the symbolic form of the statement, “Elmo is not a red Muppet.” +2. +Translate the statement +into words. +Solution +1. +“Elmo is not a red muppet” is the negation of “Elmo is a red muppet,” which is represented as . Thus, we would +write “Elmo is not a red muppet” symbolically as +. +2. +Because +is the symbol representing the statement, “Ketchup is not a vegetable,” +is equivalent to the statement, +“Ketchup is a vegetable.” +YOUR TURN 2.5 +Given the statements: +: Woody and Buzz Lightyear are best friends. +: Wonder Woman is not stronger than Captain Marvel. +1. Write the symbolic form of the statement, "Wonder Woman is stronger than Captain Marvel." +2. Translate the statement +into words. +Expressing Statements with Quantifiers of All, Some, or None +A quantifier is a term that expresses a numerical relationship between two sets or categories. For example, all squares +are also rectangles, but only some rectangles are squares, and no squares are circles. In this example, all, some, and +none are quantifiers. In a logical argument, the logical statements made to support the argument are called premises, +and the judgment made based on the premises is called the conclusion. Logical arguments that begin with specific +premises and attempt to draw more general conclusions are called inductive arguments. +Consider, for example, a parent walking with their three-year-old child. The child sees a cardinal fly by and points it out. +As they continue on their walk, the child notices a robin land on top of a tree and a duck flying across to land on a pond. +The child recognizes that cardinals, robins, and ducks are all birds, then excitedly declares, "All birds fly!" The child has +just made an inductive argument. They noticed that three different specific types of birds all fly, then synthesized this +information to draw the more general conclusion that all birds can fly. In this case, the child's conclusion is false. +The specific premises of the child's argument can be paraphrased by the following statements: +• +Premise: Cardinals are birds that fly. +• +Premise: Robins are birds that fly. +• +Premise: Ducks are birds that fly. +The general conclusion is: “All birds fly!” +All inductive arguments should include at least three specific premises to establish a pattern that supports the general +conclusion. To counter the conclusion of an inductive argument, it is necessary to provide a counter example. The parent +can tell the child about penguins or emus to explain why that conclusion is false. +64 +2 • Logic +Access for free at openstax.org + +On the other hand, it is usually impossible to prove that an inductive argument is true. So, inductive arguments are +considered either strong or weak. Deciding whether an inductive argument is strong or weak is highly subjective and +often determined by the background knowledge of the person making the judgment. Most hypotheses put forth by +scientists using what is called the “scientific method” to conduct experiments are based on inductive reasoning. +In the following example, we will use quantifiers to express the conclusion of a few inductive arguments. +EXAMPLE 2.6 +Drawing General Conclusions to Inductive Arguments Using Quantifiers +For each series of premises, draw a logical conclusion to the argument that includes one of the following quantifiers: all, +some, or none. +1. +Squares and rectangles have four sides. A square is a parallelogram, and a rectangle is a parallelogram. What +conclusion can be drawn from these premises? +2. +and +Of these, 1 and 2, 6 and 7, and 23 and 24 are consecutive integers; 3, 13, +and 47 are odd numbers. What conclusion can be drawn from these premises? +3. +Sea urchins live in the ocean, and they do not breathe air. Sharks live in the ocean, and they do not breathe air. Eels +live in the ocean, and they do not breath air. What conclusion can be drawn from these premises? +Solution +1. +The conclusion you would likely come to here is “Some four-sided figures are parallelograms.” However, it would be +incorrect to say that all four-sided figures are parallelograms because there are some four-sided figures, such as +trapezoids, that are not parallelograms. This is a false conclusion. +2. +From these premises, you may draw the conclusion “All sums of two consecutive counting numbers result in an odd +number.” Most inductive arguments cannot be proven true, but several mathematical properties can be. If we let +represent our first counting number, then +would be the next counting number and +. +Because +is divisible by two, it is an even number, and if you add one to any even number the result is always an +odd number. Thus, the conclusion is true! +3. +Based on the premises provided, with no additional knowledge about whales or dolphins, you might conclude “No +creatures that live in the ocean breathe air.” Even though this conclusion is false, it still follows from the known +premises and thus is a logical conclusion based on the evidence presented. Alternatively, you could conclude “Some +creatures that live in the ocean do not breathe air.” The quantifier you choose to write your conclusion with may vary +from another person’s based on how persuasive the argument is. There may be multiple acceptable answers. +YOUR TURN 2.6 +For each series of premises, draw a logical conclusion to the argument that includes one of the following quantifiers: +all, some, or none. +1. +, +, and +. Of these, 1 and 2 are consecutive integers, 5 and 6 are consecutive +integers, and 14 and 15 are consecutive integers. Also, their sums, 3, 11, and 29 are all prime numbers. Prime +numbers are positive integers greater than one that are only divisible by one and the number itself. What +conclusion can you draw from these premises? +2. A robin is a bird that lays blue eggs. A chicken is a bird that typically lays white and brown eggs. An ostrich is a +bird that lays exceptionally large eggs. If a bird lays eggs, then they do not give live birth to their young. What +conclusion can you draw from these premises? +3. All parallelograms have four sides. All rectangles are parallelograms. All squares are rectangles. What +additional conclusion can you make about squares from these premises? +It is not possible to prove definitively that an inductive argument is true or false in most cases. +Negating Statements Containing Quantifiers +Recall that the negation of a statement will have the opposite truth value of the original statement. There are four basic +forms that logical statements with quantifiers take on. +• +All +are +. +2.1 • Statements and Quantifiers +65 + +• +Some +are +. +• +No +are +. +• +Some +are not +. +The negation of logical statements that use the quantifiers all, some, or none is a little more complicated than just +adding or removing the word not. +For example, consider the logical statement, “All oranges are citrus fruits.” This statement expresses as a subset +relationship. The set of oranges is a subset of the set of citrus fruit. This means that there are no oranges that are +outside the set of citrus fruit. The negation of this statement would have to break the subset relationship. To do this, you +could say, “At least one orange is not a citrus fruit.” Or, more concisely, “Some oranges are not citrus fruit.” It is tempting +to say "No oranges are citrus fruit," but that would be incorrect. Such a statement would go beyond breaking the subset +relationship, to stating that the two sets have nothing in common. The negation of " +is a subset of +" would be to state +that " +is not a subset of +," as depicted by the Venn diagram in Figure 2.4. +Figure 2.4 +The statement, “All oranges are citrus fruit,” is true, so its negation, “Some oranges are not citrus fruit,” is false. +Now, consider the statement, “No apples are oranges.” This statement indicates that the set of apples is disjointed from +the set of oranges. The negation must state that the two are not disjoint sets, or that the two sets have a least one +member in common. Their intersection is not empty. The negation of the statement, “ +intersection +is the empty set,” +is the statement that " +intersection +is not empty," as depicted in the Venn diagram in Figure 2.5. +Figure 2.5 +The negation of the true statement “No apples are oranges,” is the false statement, “Some apples are oranges.” +Table 2.2 summarizes the four different forms of logical statements involving quantifiers and the forms of their +associated negations, as well as the meanings of the relationships between the two categories or sets +and +. +66 +2 • Logic +Access for free at openstax.org + +Logical Statements with Quantifiers +Negation of Logical Statements w/Quantifiers +Form: All +are +. +Means: +is a subset of +, +All zebras have stripes. (True) +Form: Some +are not +. +Means: +is not a subset of +, +Some zebras do not have stripes. (False) +Form: Some +are +. +Means: +intersection +is not empty, +Some fish are sharks. (True) +Form: No +are +. +Means: +intersection +is empty, +No fish are sharks. (False) +Form: No +are +. +Means: +intersection +is empty, +No trees are evergreens. (False) +Form: Some +are +. +Means: +intersection +is not empty, +Some trees are evergreens. (True) +Form: Some +are not +. +Means: +is not a subset of +, +Some horses are not mustangs. (True) +Form: All +are +. +Means: +is a subset of +, +All horses are mustangs. (False) +Table 2.2 +We covered sets in great detail in Chapter 1. To review, " +is a subset of +" means that every member of set +is also a +member of set +. The intersection of two sets +and +is the set of all elements that they share in common. If +intersection +is the empty set, then sets +and +do not have any elements in common. The two sets do not overlap. +They are disjoint. +VIDEO +Logic Part 1A: Logic Statements and Quantifiers (https://openstax.org/r/Logic_Statements_and_Quantifiers) +EXAMPLE 2.7 +Negating Statements Containing Quantifiers All, Some, or None +Given the statements: +: All leopards have spots. +: Some apples are red. +: No lemons are sweet. +Write each of the following symbolic statements in words. +1. +2. +~ +3. +Solution +1. +The statement “All leopards have spots” is +and has the form “All +are +.” According to Table 2.2, the negation will +have the form “Some +are not +.” The negation of +is the statement, “Some leopards do not have spots.” +2. +The statement “Some apples are red” has the form “Some +are +.” This indicates that the categories +and +overlap or intersect. According to Table 2.2, the negation will have the form, “No +are +,” indicating that +and +do not intersect. This results in the opposite truth value of the original statement, so the negation of “Some apples +are red” is the statement: “No apples are red.” +3. +Because +is the statement: “No lemons are sweet,” +is asserting that the set of lemons does not intersect with the +set of sweet things. The negation of , +, must make the opposite claim. It must indicate that the set of lemons +intersects with the set of sweet things. This means at least one lemon must be sweet. The statement, “Some lemons +are sweet” is +. The negation of the statement, “No +are +,” is the statement, “Some +are +,” as indicated in +Table 2.2. +2.1 • Statements and Quantifiers +67 + +YOUR TURN 2.7 +Given the statements: +: Some apples are not sweet. +: No triangles are squares. +: Some vegetables are green. Write each of the following symbolic statements in words. +1. +2. +3. +Check Your Understanding +1. A _________ __________ is a complete sentence that makes a claim that may be either true or false. +2. The _________________ of a logical statement has the opposite truth value of the original statement. +3. If +represents the logical statement, “Marigolds are yellow flowers,” then ______ represents the statement, +“Marigolds are not yellow flowers.” +4. The statement +has the same truth value as the statement _______. +5. The logical statements used to support the conclusion of an argument are called ____________. +6. _______________________ arguments attempt to draw a general conclusion from specific premises. +7. All, some, and none are examples of ______________________, words that assign a numerical relationship between two +or more groups. +8. The negation of the statement, “All giraffes are tall,” is _______________________________. +SECTION 2.1 EXERCISES +For the following exercises, determine whether the sentence represents a logical statement. If it is a logical statement, +determine whether it is true or false. +1. A loan used to finance a house is called a mortgage. +2. All odd numbers are divisible by 2. +3. Please, bring me that notebook. +4. Robot, what’s your function? +5. In English, a conjunction is a word that connects two phrases or parts of a sentence together. +6. +. +7. +. +8. What is 7 plus 3? +For the following exercises, write each statement in symbolic form. +9. Grammy award winning singer, Lady Gaga, was not born in Houston, Texas. +10. Bruno Mars performed during the Super Bowl halftime show twice. +11. Coco Chanel said, “The most courageous act is still to think for yourself. Aloud.” +12. Bruce Wayne is not Superman. +For the following exercises, write the negation of each statement in words. +13. Bozo is not a clown. +14. Ash is Pikachu’s trainer and friend. +15. Vanilla is the most popular flavor of ice cream. +16. Smaug is a fire breathing dragon. +17. Elephant and Piggy are not best friends. +18. Some dogs like cats. +19. Some donuts are not round. +20. All cars have wheels. +21. No circles are squares. +22. Nature’s first green is not gold. +23. The ancient Greek philosopher Plato said, “The greatest wealth is to live content with little.” +68 +2 • Logic +Access for free at openstax.org + +24. All trees produce nuts. +For the following exercises, write the negation of each statement symbolically and in words. +25. +: Their hair is red. +26. +: My favorite superhero does not wear a cape. +27. +: All wolves howl at the moon. +28. +: Nobody messes with Texas. +29. +: I do not love New York. +30. +: Some cats are not tigers. +31. +: No squares are not parallelograms. +32. +: The President does not like broccoli. +For the following exercises, write each of the following symbolic statements in words. +33. Given: +: Kermit is a green frog; translate +into words. +34. Given: +: Mick Jagger is not the lead singer for The Rolling Stones; translate +into words. +35. Given: : All dogs go to heaven; translate +into words. +36. Given: +: Some pizza does not come with pepperoni on it; translate +into words. +37. Given: +: No pizza comes with pineapple on it; translate +into words. +38. Given: : Not all roses are red; translate +into words. +39. Given: +: Thelonious Monk is not a famous jazz pianist; translate +into words. +40. Given: +: Not all violets are blue; translate +into words. +For the following exercises, draw a logical conclusion from the premises that includes one of the following quantifiers: +all, some, or none. +41. The Ford Motor Company builds cars in Michigan. General Motors builds cars in Michigan. Chrysler builds cars +in Michigan. What conclusion can be drawn from these premises? +42. Michelangelo Buonarroti was a great Renaissance artist from Italy. Raphael Sanzio was a great Renaissance +artist from Italy. Sandro Botticelli was a great Renaissance artist from Italy. What conclusion can you draw from +these premises? +43. Four is an even number and it is divisible by 2. Six is an even number and it is divisible by 2. Eight is an even +number and it is divisible by 2. What conclusion can you draw from these premises? +44. Three is an odd number and it is not divisible by 2. Seven is an odd number and it is not divisible by 2. Twenty- +one is an odd number and it is not divisible by 2. What conclusion can you draw from these premises? +45. The odd number 5 is not divisible by 3. The odd number 7 is not divisible by 3. The odd number 29 is not +divisible by 3. What conclusion can you draw from these premises? +46. Penguins are flightless birds. Emus are flightless birds. Ostriches are flightless birds. What conclusion can you +draw from these premises? +47. Plants need water to survive. Animals need water to survive. Bacteria need water to survive. What conclusion +can you draw from these premises? +48. A chocolate chip cookie is not sour. An oatmeal cookie is not sour. An Oreo cookie is not sour. What conclusion +can you draw from these premises? +2.1 • Statements and Quantifiers +69 + +2.2 Compound Statements +Figure 2.6 A person seeking their driver's license must pass two tests. A compound statement can be used to explain +performance on both tests at once. (credit: modification of work “Drivers License -Teen driver” by State Farm/Flickr, CC BY +2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Translate compound statements into symbolic form. +2. +Translate compound statements in symbolic form with parentheses into words. +3. +Apply the dominance of connectives. +Suppose your friend is trying to get a license to drive. In most places, they will need to pass some form of written test +proving their knowledge of the laws and rules for driving safely. After passing the written test, your friend must also pass +a road test to demonstrate that they can perform the physical task of driving safely within the rules of the law. +Consider the statement: "My friend passed the written test, but they did not pass the road test." This is an example of a +compound statement, a statement formed by using a connective to join two independent clauses or logical +statements. The statement, “My friend passed the written test,” is an independent clause because it is a complete +thought or idea that can stand on its own. The second independent clause in this compound statement is, “My friend did +not pass the road test.” The word "but" is the connective used to join these two statements together, forming a +compound statement. So, did your friend acquire their driving license?. +This section introduces common logical connectives and their symbols, and allows you to practice translating compound +statements between words and symbols. It also explores the order of operations, or dominance of connectives, when a +single compound statement uses multiple connectives. +Common Logical Connectives +Understanding the following logical connectives, along with their properties, symbols, and names, will be key to applying +the topics presented in this chapter. The chapter will discuss each connective introduced here in more detail. +The joining of two logical statements with the word "and" or "but" forms a compound statement called a conjunction. In +logic, for a conjunction to be true, all the independent logical statements that make it up must be true. The symbol for a +conjunction is +. Consider the compound statement, “Derek Jeter played professional baseball for the New York Yankees, +and he was a shortstop.” If +represents the statement, “Derrick Jeter played professional baseball for the New York +Yankees,” and if +represents the statement, “Derrick Jeter was a short stop,” then the conjunction will be written +symbolically as +The joining of two logical statements with the word “or” forms a compound statement called a disjunction. Unless +otherwise specified, a disjunction is an inclusive or statement, which means the compound statement formed by joining +two independent clauses with the word or will be true if a least one of the clauses is true. Consider the compound +statement, "The office manager ordered cake for for an employee’s birthday or they ordered ice cream.” This is a +disjunction because it combines the independent clause, “The office manager ordered cake for an employee’s birthday,” +with the independent clause, “The office manager ordered ice cream,” using the connective, or. This disjunction is true if +70 +2 • Logic +Access for free at openstax.org + +the office manager ordered only cake, only ice cream, or they ordered both cake and ice cream. Inclusive or means you +can have one, or the other, or both! +Joining two logical statements with the word implies, or using the phrase “if first statement, then second statement,” is +called a conditional or implication. The clause associated with the "if" statement is also called the hypothesis or +antecedent, while the clause following the "then" statement or the word implies is called the conclusion or consequent. +The conditional statement is like a one-way contract or promise. The only time the conditional statement is false, is if the +hypothesis is true and the conclusion is false. Consider the following conditional statement, “If Pedro does his +homework, then he can play video games.” The hypothesis/antecedent is the statement following the word if, which is +“Pedro does/did his homework.” The conclusion/consequent is the statement following the word then, which is “Pedro +can play his video games.” +Joining two logical statements with the connective phrase “if and only if” is called a biconditional. The connective phrase +"if and only if" is represented by the symbol, +In the biconditional statement, +is called the hypothesis or +antecedent and +is called the conclusion or consequent. For a biconditional statement to be true, the truth values of +and +must match. They must both be true, or both be false. +The table below lists the most common connectives used in logic, along with their symbolic forms, and the common +names used to describe each connective. +Connective +Symbol +Name +and +but +conjunction +or +disjunction, inclusive or +not +~ +negation +if +, then implies +conditional, implication +if and only if +biconditional +Table 2.3 +These connectives, along with their names, symbols, and properties, will be used throughout this chapter and +should be memorized. +EXAMPLE 2.8 +Associate Connectives with Symbols and Names +For each of the following connectives, write its name and associated symbol. +1. +or +2. +implies +3. +but +Solution +1. +A compound statement formed with the connective word or is called a disjunction, and it is represented by the +symbol. +2. +A compound statement formed with the connective word implies or phrase “if …, then” is called a conditional +statement or implication and is represented by the +symbol. +3. +A compound statement formed with the connective words but or and is called a conjunction, and it is represented +by the +symbol. +2.2 • Compound Statements +71 + +YOUR TURN 2.8 +For each connective write its name and associated symbol. +1. not +2. and +3. if and only if +Translating Compound Statements to Symbolic Form +To translate a compound statement into symbolic form, we take the following steps. +1. +Identify and label all independent affirmative logical statements with a lower case letter, such as +, , or . +2. +Identify and label any negative logical statements with a lowercase letter preceded by the negation symbol, such as +, +, or +. +3. +Replace the connective words with the symbols that represent them, such as +Consider the previous example of your friend trying to get their driver’s license. Your friend passed the written test, but +they did not pass the road test. Let +represent the statement, “My friend passed the written test.” And, let +represent +the statement, “My friend did not pass the road test.” Because the connective but is logically equivalent to the word and, +the symbol for but is the same as the symbol for and; replace but with the symbol +The compound statement is +symbolically written as: +. My friend passed the written test, but they did not pass the road test. +When translating English statements into symbolic form, do not assume that every connective “and” means you are +dealing with a compound statement. The statement, “The stripes on the U.S. flag are red and white,” is a simple +statement. The word “white” doesn’t express a complete thought, so it is not an independent clause and does not +get its own symbol. This statement should be represented by a single letter, such as : The stripes on the U.S. flag +are red and white. +EXAMPLE 2.9 +Translating Compound Statements into Symbolic Form +Let +represent the statement, “It is a warm sunny day,” and let +represent the statement, “the family will go to the +beach.” Write the symbolic form of each of the following compound statements. +1. +If it is a warm sunny day, then the family will go to the beach. +2. +The family will go to the beach, and it is a warm sunny day. +3. +The family will not go to the beach if and only if it is not a warm sunny day. +4. +The family not go to the beach, or it is a warm sunny day. +Solution +1. +Replace “it is a warm sunny day” with +. Replace “the family will go to the beach.” with . Next. Next, because the +connective is if …, then place the conditional symbol, +, between +and . The compound statement is written +symbolically as: +2. +Replace “The family will go to the beach” with . Replace “it is a warm sunny day.” with +. Next, because the +connective is and, place the +symbol between +and +. The compound statement is written symbolically as: +3. +Replace “The family will not go to the beach. with +. Replace “it is not a warm sunny day” with +. Next, because +the connective is or, if and only if, place the biconditional symbol, +between +and +. The compound statement +is written symbolically as: +4. +Replace “The family will not go to the beach” with +. Replace “it is a warm sunny day” with +. Next, because the +connective is or, place the +symbol between +and +. The compound statement is written symbolically as: +YOUR TURN 2.9 +Let +represent the statement, “Last night it snowed,” and let +represent the statement, “Today we will go skiing.” +Write the symbolic form of each of the following compound statements: +1. Today we will go skiing, but last night it did not snow. +72 +2 • Logic +Access for free at openstax.org + +2. Today we will go skiing if and only if it snowed last night. +3. Last night is snowed or today we will not go skiing. +4. If it snowed last night, then today we will go skiing. +Translating Compound Statements in Symbolic Form with Parentheses into Words +When parentheses are written in a logical argument, they group a compound statement together just like when +calculating numerical or algebraic expressions. Any statement in parentheses should be treated as a single component +of the expression. If multiple parentheses are present, work with the inner most parentheses first. +Consider your friend’s struggles to get their license to drive. Let +represent the statement, “My friend passed the written +test,” let +represent the statement, “My friend passed the road test,” and let +represent the statement, “My friend +received a driver’s license.” The statement +can be translated into words as follows: the statement +is +grouped together to form the hypothesis of the conditional statement and +is the conclusion. The conditional statement +has the form “if +then +” Therefore, the written form of this statement is: “If my friend passed the written test and +they passed the road test, then my friend received a driver’s license.” +Sometimes a compound statement within parentheses may need to be negated as a group. To accomplish this, add the +phrase, “it is not the case that” before the translation of the phrase in parentheses. For example, using +, , and +of your +friend obtaining a license, let’s translate the statement +into words. +In this case, the hypothesis of the conditional statement is +and the conclusion is +To negate the hypothesis, +add the phrase “it is not the case” before translating what is in parentheses. The translation of the hypothesis is the +sentence, “It is not the case that my friend passed the written test and they passed the road test,” and the translation of +the conclusion is, “My friend did not receive a driver’s license.” So, a translation of the complete conditional statement, +is: “If it is not the case that my friend passed the written test and the road test, then my friend did not +receive a driver’s license.” +It is acceptable to interchange proper names with pronouns and remove repeated phrases to make the written +statement more readable, as long the meaning of the logical statement is not changed. +VIDEO +Logic Part 1B: Compound Statements, Connectives and Symbols (https://openstax.org/r/Compound_Statements) +EXAMPLE 2.10 +Translating Compound Statements in Symbolic Form with Parentheses into Words +Let +represent the statement, “My child finished their homework,” let +represent the statement, “My child cleaned her +room,” let +represent the statement, “My child played video games,” and let +represent the statement, “My child +streamed a movie.” Translate each of the following symbolic statements into words. +1. +2. +3. +Solution +1. +Replace ~ with “It is not the case,” and +with “and.” One possible translation is: “It is not the case that my child +finished their homework and cleaned their room.” +2. +The hypothesis of the conditional statement is, “My child finished their homework and cleaned their room.” The +conclusion of the conditional statement is, “My child played video games or streamed a movie.” One possible +translation of the entire statement is: “If my child finished their homework and cleaned their room, then they played +video games or streamed a movie.” +3. +The hypothesis of the biconditional statement is +and is written in words as: “It is not the case that my child +played video games or streamed a movie.” The conclusion of the biconditional statement is +, which +translates to: “It is not the case that my child finished their homework and cleaned their room.” Because the +biconditional, +translates to if and only if, one possible translation of the statement is: “It is not the case that my +child played video games or streamed a movie if and only if it is not the case that my child finished their homework +and cleaned their room.” +2.2 • Compound Statements +73 + +YOUR TURN 2.10 +Let +represent the statement, “My roommates ordered pizza,” let +represent the statement, “I ordered wings,” and +let +be the statement, “Our friends came over to watch the game.” Translate the following statements into words. +1. +2. +3. +The Dominance of Connectives +The order of operations for working with algebraic and arithmetic expressions provides a set of rules that allow +consistent results. For example, if you were presented with the problem +, and you were not familiar with the +order of operation, you might assume that you calculate the problem from left to right. If you did so, you would add 1 +and 3 to get 4, and then multiply this answer by 2 to get 8, resulting in an incorrect answer. Try inputting this expression +into a scientific calculator. If you do, the calculator should return a value of 7, not 8. +Scientific Calculator (https://openstax.org/r/Scientific_Calculator) +The order of operations for algebraic and arithmetic operations states that all multiplication must be applied prior to any +addition. Parentheses are used to indicate which operation—addition or multiplication—should be done first. Adding +parentheses can change and/or clarify the order. The parentheses in the expression +indicate that 3 should be +multiplied by 2 to get 6, and then 1 should be added to 6 to get 7: +As with algebraic expressions, there is a set of rules that must be applied to compound logical statements in order to +evaluate them with consistent results. This set of rules is called the dominance of connectives. When evaluating +compound logical statements, connectives are evaluated from least dominant to most dominant as follows: +1. +Parentheses are the least dominant connective. So, any expression inside parentheses must be evaluated first. Add +as many parentheses as needed to any statement to specify the order to evaluate each connective. +2. +Next, we evaluate negations. +3. +Then, we evaluate conjunctions and disjunctions from left to right, because they have equal dominance. +4. +After evaluating all conjunctions and disjunctions, we evaluate conditionals. +5. +Lastly, we evaluate the most dominant connective, the biconditional. If a statement includes multiple connectives of +equal dominance, then we will evaluate them from left to right. +See Figure 2.7 for a visual breakdown of the dominance of connectives. +Figure 2.7 +Let’s revisit your friend’s struggles to get their driver’s license. Let +represent the statement, “My friend passed the +written test,” let +represent the statement, “My friend passed the road test,” and let +represent the statement, “My +friend received a driver’s license.” Let's use the dominance of connectives to determine how the compound statement +should be evaluated. +Step 1: There are no parentheses, which is least dominant of all connectives, so we can skip over that. +Step 2: Because negation is the next least dominant, we should evaluate +first. We could add parentheses to the +statement to make it clearer which connecting needs to be evaluated first: +is equivalent to +74 +2 • Logic +Access for free at openstax.org + +Step 3: Next, we evaluate the conjunction, +. +is equivalent to +Step 4: Finally, we evaluate the conditional, +as this is the most dominant connective in this compound statement. +WHO KNEW? +When using spreadsheet applications, like Microsoft Excel or Google Sheets, have you noticed that core functions, +such as sum, average, or rate, have the same name and syntax for use? This is not a coincidence. Various standards +organizations have developed requirements that software developers must implement to meet the conditions set by +governments and large customers worldwide. The OpenDocument Format OASIS Standard enables transferring data +between different office productivity applications and was approved by the International Standards Organization +(ISO) and IEC on May 6, 2006. +Prior to adopting these standards, a government entity, business, or individual could lose access to their own data +simply because it was saved in a format no longer supported by a proprietary software product—even data they were +required by law to preserve, or data they simply wished to maintain access to. +Just as rules for applying the dominance of connectives help maintain understanding and consistency when writing +and interpreting compound logical statements and arguments, the standards adopted for database software +maintain global integrity and accessibility across platforms and user bases. +EXAMPLE 2.11 +Applying the Dominance of Connectives +For each of the following compound logical statements, add parentheses to indicate the order to evaluate the statement. +Recall that parentheses are evaluated innermost first. +1. +2. +3. +Solution +1. +Because negation is the least dominant connective, we evaluate it first: +Because conjunction and +disjunction have the same dominance, we evaluate them left to right. So, we evaluate the conjunction next, as +indicated by the additional set of parentheses: +The only remaining connective is the disjunction, so it +is evaluated last, as indicated by the third set of parentheses. The complete solution is: +2. +Negation has the lowest dominance, so it is evaluated first: +The remaining connectives are the +conditional and the conjunction. Because conjunction has a lower precedence than the conditional, it is evaluated +next, as indicated by the second set of parentheses: +The last step is to evaluate the conditional, as +indicated by the third set of parentheses: +3. +This statement is known as De Morgan’s Law for the negation of a disjunction. It is always true. Section 2.6 of this +chapter will explore De Morgan’s Laws in more detail. +◦ +First, we evaluate the negations on the right side of the biconditional prior to the conjunction. +◦ +Then, we evaluate the disjunction on the left side of the biconditional, followed by the negation of the disjunction +on the left side. +◦ +Lastly, after completely evaluating each side of the biconditional, we evaluate the biconditional. It does not +matter which side you begin with. +The final solution is: +YOUR TURN 2.11 +For each of the following compound logical statements, add parentheses to indicate the order in which to evaluate +the statement. Recall that parentheses are evaluated innermost first. +1. +2. +2.2 • Compound Statements +75 + +3. +WORK IT OUT +Logic Terms and Properties – Matching Game +Materials: For every group of four students, include 30 cards (game, trading, or playing cards), 30 individual clear +plastic gaming card sleeves, and 30 card-size pieces of paper. Prepare a list of 60 questions and answers ahead of +time related to definitions and problems in Statements and Quantifiers and Compound Statements. Provide each +group of four students with 20 questions and their associated answers. Instruct each group to select 15 of the 20 +questions, and then, for each problem selected, create one card with the question and one card with the answer. +Instruct the groups to then place each problem and answer in a separate card sleeve. Once they create 15 problem +cards and 15 answer cards, have students shuffle the set of cards. +To play the game, groups should exchange their card decks to ensure no team begins playing with the deck that they +created. Split each four-person group into teams of two students. After shuffling the cards, one team lays cards face +down on their desk in a five-by-six grid. The other team will go first. +Each player will have a turn to try matching the question with the correct answer by flipping two cards to the face up +position. If a team makes a match, they get to flip another two cards; if they do not get a match, they flip the cards +face down and it is the other team’s turn to flip over two cards. The game continues in this manner until teams match +all question cards with their corresponding answer cards. The team with the most set of matching cards wins. +In the first module of this chapter, we evaluated the truth value of negations. In the following modules, we will discuss +how to evaluate conjunctions, disjunctions, conditionals, and biconditionals, and then evaluate compound logical +statements using truth tables and our knowledge of the dominance of connectives. +Check Your Understanding +9. A __________ __________ is a logical statement formed by combining two or more statements with connecting words, +such as and, or, but, not, and if …, then. +10. A _____ is a word or symbol used to join two or more logical statements together to form a compound statement. +11. The most dominant connective is the _____. +12. _____ are used to specify which logical connective should be evaluated first when evaluating a compound +statement. +13. Both _____ and _____ have equal dominance and are evaluated from left to right when no parentheses are present +in a compound logical statement. +SECTION 2.2 EXERCISES +For the following exercises, translate each compound statement into symbolic form. +Given +: “Layla has two weeks for vacation,” : “Marcus is Layla’s friend,” : “Layla will travel to Paris, France,” and : +“Layla and Marcus will travel together to Niagara Falls, Ontario.” +1. If Layla has two weeks for vacation, then she will travel to Paris, France. +2. Layla and Marcus will travel together to Niagara Falls, Ontario or Layla will travel to Paris, France. +3. If Marcus is not Layla’s friend, then they will not travel to Niagara Falls, Ontario together.” +4. Layla and Marcus will travel to Niagara Falls, Ontario together if and only if Layla and Marcus are friends. +5. If Layla does not have two weeks for vacation and Marcus is Layla’s friend, then Marcus and Layla will travel +together to Niagara Falls, Ontario. +6. If Layla has two weeks for vacation and Marcus is not her friend, then she will travel to Paris, France. +For the following exercises, translate each compound statement into symbolic form. +Given +: “Tom is a cat,” : “Jerry is a mouse,” : “Spike is a dog,” : “Tom chases Jerry,” and : “Spike catches Tom.” +7. Jerry is a mouse and Tom is a cat. +8. If Tom chases Jerry, then Spike will catch Tom. +9. If Spike does not catch Tom, then Tom did not chase Jerry. +76 +2 • Logic +Access for free at openstax.org + +10. Tom is a cat and Spike is a dog, or Jerry is not a Mouse. +11. It is not the case that Tom is not a cat and Jerry is not a mouse. +12. Spike is not a dog and Jerry is a mouse if and only if Tom chases Jerry, but Spike does not catch Tom. +For the following exercises, translate the symbolic form of each compound statement into words. +Given +: “Tracy Chapman plays guitar,” : “Joan Jett plays guitar,” : “All rock bands include guitarists,” and : “Elton John +plays the piano.” +13. +14. +15. +16. +17. +18. +For the following exercises, translate the symbolic form of each compound statement into words. +Given +: “The median is the middle number,” : “The mode is the most frequent number,” : “The mean is the average +number,” : “The median, mean, and mode are equal,” and : “The data set is symmetric.” +19. +20. +21. +22. +23. +24. +For the following exercises, apply the proper dominance of connectives by adding parentheses to indicate the order in +which the statement must be evaluated. +25. +26. +27. +28. +29. +30. +31. +32. +2.2 • Compound Statements +77 + +2.3 Constructing Truth Tables +Figure 2.8 Just like solving a puzzle, a computer programmer must consider all potential solutions in order to account +for each possible outcome. (credit: modification of work “Deadline Xmas 2010” by Allan Henderson/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Interpret and apply negations, conjunctions, and disjunctions. +2. +Construct a truth table using negations, conjunctions, and disjunctions. +3. +Construct a truth table for a compound statement and interpret its validity. +Are you familiar with the Choose Your Own Adventure book series written by Edward Packard? These gamebooks allow +the reader to become one of the characters and make decisions that affect what happens next, resulting in different +sequences of events in the story and endings based on the choices made. Writing a computer program is a little like +what it must be like to write one of these books. The programmer must consider all the possible inputs that a user can +put into the program and decide what will happen in each case, then write their program to account for each of these +possible outcomes. +A truth table is a graphical tool used to analyze all the possible truth values of the component logical statements to +determine the validity of a statement or argument along with all its possible outcomes. The rows of the table correspond +to each possible outcome for the given logical statement identified at the top of each column. A single logical statement +has two possible truth values, true or false. In truth tables, a capital T will represent true values, and a capital F will +represent false values. +In this section, you will use the knowledge built in Statements and Quantifiers and Compound Statements to analyze +arguments and determine their truth value and validity. A logical argument is valid if its conclusion follows from its +premises, regardless of whether those premises are true or false. You will then explore the truth tables for negation, +conjunction, and disjunction, and use these truth tables to analyze compound logical statements containing these +connectives. +Interpret and Apply Negations, Conjunctions, and Disjunctions +The negation of a statement will have the opposite truth value of the original statement. When +is true, +is false, and +when +is false, +is true. +EXAMPLE 2.12 +Finding the Truth Value of a Negation +For each logical statement, determine the truth value of its negation. +1. +: +2. +: All horses are mustangs. +3. +: New Delhi is not the capital of India. +78 +2 • Logic +Access for free at openstax.org + +Solution +1. +is true because 3 + 5 does equal 8; therefore, the negation of +, +, is false. +2. +is false because there are other types of horses besides mustangs, such as Clydesdales or Arabians; therefore, the +negation of , +, is true. +3. +is false because New Delhi is the capital of India; therefore, the negation of +, , is true. +YOUR TURN 2.12 +For each logical statement, determine the truth value of its negation. +1. +: +. +2. +: Some houses are built with bricks. +3. +: Abuja is the capital of Nigeria. +A conjunction is a logical and statement. For a conjunction to be true, both statements that make up the conjunction +must be true. If at least one of the statements is false, the and statement is false. +EXAMPLE 2.13 +Finding the Truth Value of a Conjunction +Given +and +determine the truth value of each conjunction. +1. +2. +3. +Solution +1. +is true, and +is false. Because one statement is true, and the other statement is false, this makes the complete +conjunction false. +2. +is false, so +is true, and +is true. Therefore, both statements are true, making the complete conjunction true. +3. +is false, and +is false. Because both statements are false, the complete conjunction is false. +YOUR TURN 2.13 +Given +: Yellow is a primary color, : Blue is a primary color, and : Green is a primary color, determine the truth value +of each conjunction. +1. +2. +3. +The only time a conjunction is true is if both statements that make up the conjunction are true. +A disjunction is a logical inclusive or statement, which means that a disjunction is true if one or both statements that +form it are true. The only way a logical inclusive or statement is false is if both statements that form the disjunction are +false. +EXAMPLE 2.14 +Finding the Truth Value of a Disjunction +Given +and +determine the truth value of each disjunction. +1. +2. +3. +2.3 • Constructing Truth Tables +79 + +Solution +1. +is true, and +is false. One statement is true, and one statement is false, which makes the complete disjunction +true. +2. +is false, so +is true, and +is true. Therefore, one statement is true, and the other statement is true, which makes +the complete disjunction true. +3. +is false, and +is false. When all of the component statements are false, the disjunction is false. +YOUR TURN 2.14 +Given +: Yellow is a primary color, : Blue is a primary color, and : Green is a primary color, determine the truth value +of each disjunction. +1. +2. +3. +In the next example, you will apply the dominance of connectives to find the truth values of compound statements +containing negations, conjunctions, and disjunctions and use a table to record the results. When constructing a truth +table to analyze an argument where you can determine the truth value of each component statement, the strategy is to +create a table with two rows. The first row contains the symbols representing the components that make up the +compound statement. The second row contains the truth values of each component below its symbol. Include as many +columns as necessary to represent the value of each compound statement. The last column includes the complete +compound statement with its truth value below it. +VIDEO +Logic Part 2: Truth Values of Conjunctions: Is an "AND" statement true or false? (https://openstax.org/r/ +Truth_Values_of_Conjunctions) +Logic Part 3: Truth Values of Disjunctions: Is an "OR" statement true or false? (https://openstax.org/r/ +Truth_Values_of_Disjunctions) +EXAMPLE 2.15 +Finding the Truth Value of Compound Statements +Given +and +construct a truth table to determine the truth value of each +compound statement +1. +2. +3. +Solution +1. +Step 1: The statement “ +” contains three basic logical statements, +, , and , and three connectives, +When we place parentheses in the statement to indicate the dominance of connectives, the statement becomes +Step 2: After we have applied the dominance of connectives, we create a two row table that includes a column for +each basic statement that makes up the compound statement, and an additional column for the contents of each +parentheses. Because we have three sets of parentheses, we include a column for +the innermost parentheses, a +column for +the next set of parentheses, and +in the last column for the third parentheses. +Step 3: Once the table is created, we determine the truth value of each statement starting from left to right. The +truth values of +, , and +are true, false, and true, respectively, so we place T, F, and T in the second row of the table. +Because +is true, +is false. +Step 4: Next, evaluate +from the table: +is false, and +is also false, so +is false, because a conjunction +is only true if both of the statements that make it are true. Place an F in the table below its heading. +Step 5: Finally, using the table, we understand that +is false and +is true, so the complete statement +is false or true, which is true (because a disjunction is true whenever at least one of the statements that +80 +2 • Logic +Access for free at openstax.org + +make it is true). Place a T in the last column of the table. The complete statement +is true. +T +F +T +F +F +T +2. +Step 1: Applying the dominance of connectives to the original compound statement +, we get +Step 2: The table needs columns for +and +Step 3:The truth values of +, , , and +are the same as in Question 1. +Step 4: Next, +is false or false, which is false, so we place an F below this statement in the table. This is the +only time that a disjunction is false. +Step 5: Finally, +and +are the conjunction of the statements, +and , and so the expressions evaluate to +false and true, which is false. Recall that the only time an "and" statement is true is when both statements that form +it are also true. The complete statement +is false. +T +F +T +F +F +F +3. +Step 1: Applying the dominance of connectives to the original statement, we have: +. +Step 2: So, the table needs the following columns: +and +Step 3: The truth values of +, , and +are the same as in Questions 1 and 2. +Step 4: From the table it can be seen that +is true and true, which is true. So the negation of +and +is false, +because the negation of a statement always has the opposite truth value of the original statement. +Step 5: Finally, +is the disjunction of +with , and so we have false or false, which makes the +complete statement false. +T +F +T +T +F +F +YOUR TURN 2.15 +Given +: Yellow is a primary color, : Blue is a primary color, and : Green is a primary color, determine the truth value +of each compound statement, by constructing a truth table. +1. +2. +3. +Construct Truth Tables to Analyze All Possible Outcomes +Recall from Statements and Questions that the negation of a statement will always have the opposite truth value of the +original statement; if a statement +is false, then its negation +is true, and if a statement +is true, then its negation +is false. To create a truth table for the negation of statement +, add a column with a heading of +, and for each row, +input the truth value with the opposite value of the value listed in the column for +, as depicted in the table below. +2.3 • Constructing Truth Tables +81 + +Negation +T +F +F +T +Conjunctions and disjunctions are compound statements formed when two logical statements combine with the +connectives “and” and “or” respectively. How does that change the number of possible outcomes and thus determine the +number of rows in our truth table? The multiplication principle, also known as the fundamental counting principle, +states that the number of ways you can select an item from a group of +items and another item from a group with +items is equal to the product of +and +. Because each proposition +and +has two possible outcomes, true or false, the +multiplication principle states that there will be +possible outcomes: {TT, TF, FT, FF}. +The tree diagram and table in Figure 2.9 demonstrate the four possible outcomes for two propositions +and . +Figure 2.9 +A conjunction is a logical and statement. For a conjunction to be true, both the statements that make up the conjunction +must be true. If at least one of the statements is false, the and statement is false. +A disjunction is a logical inclusive or statement. Which means that a disjunction is true if one or both statements that +make it are true. The only way a logical inclusive or statement is false is if both statements that make up the disjunction +are false. +Conjunction (AND) +Disjunction (OR) +T +T +T +T +T +T +T +F +F +T +F +T +F +T +F +F +T +T +F +F +F +F +F +F +VIDEO +Logic Part 4: Truth Values of Compound Statements with "and", "or", and "not" (https://openstax.org/r/opL9I4tZCC0) +Logic Part 5: What are truth tables? How do you set them up? (https://openstax.org/r/-tdSRqLGhaw) +82 +2 • Logic +Access for free at openstax.org + +EXAMPLE 2.16 +Constructing Truth Tables to Analyze Compound Statements +Construct a truth table to analyze all possible outcomes for each of the following arguments. +1. +2. +3. +Solution +1. +Step 1: Because there are two basic statements, +and , and each of these has two possible outcomes, we will have +rows in our table to represent all possible outcomes: TT, TF, FT, and FF. The columns will include +, , +, +and +Step 2: Every value in column +will have the opposite truth value of the corresponding value in column : F, T, F, +and T. +Step 3: To complete the last column, evaluate each element in column +with the corresponding element in column +using the connective or. +T +T +F +T +T +F +T +T +F +T +F +F +F +F +T +T +2. +Step 1: The columns will include +, , +and +. Because there are two basic statements, +and , the table +will have four rows to account for all possible outcomes. +Step 2: The +column will be completed by evaluating the corresponding elements in columns +and +respectively with the and connective. +Step 3: The final column, +, will be the negation of the +column. +T +T +T +F +T +F +F +T +F +T +F +T +F +F +F +T +3. +Step 1: This statement has three basic statements, +, , and . Because each basic statement has two possible truth +values, true or false, the multiplication principal indicates there are +possible outcomes. So eight rows of +outcomes are needed in the truth table to account for each possibility. Half of the eight possibilities must be true for +the first statement, and half must be false. +Step 2: So, the first column for statement +, will have four T’s followed by four F’s. In the second column for +statement , when +is true, half the outcomes for +must be true and the other half must be false, and the same +pattern will repeat for when +is false. So, column +will have TT, FF, FF, FF. +Step 3: The column for the third statement, , must alternate between T and F. Once, the three basic propositions +are listed, you will need a column for +, +, and +. +Step 4: The column for the negation of , +, will have the opposite truth value of each value in column . +Step 5: Next, fill in the truth values for the column containing the statement +The or statement is true if at +least one of +or +is true, otherwise it is false. +2.3 • Constructing Truth Tables +83 + +Step 6: Finally, fill in the column containing the conjunction +. To evaluate this statement, combine +column +and column +with the and connective. Recall, that only time "and" is true is when both values are +true, otherwise the statement is false. The complete truth table is: +T +T +T +F +T +T +T +T +F +F +T +F +T +F +T +T +T +T +T +F +F +T +T +F +F +T +T +F +F +F +F +T +F +F +F +F +F +F +T +T +T +T +F +F +F +T +T +F +YOUR TURN 2.16 +Construct a truth table to analyze all possible outcomes for each of the following arguments. +1. +2. +3. +VIDEO +Logic Part 6: More on Truth Tables and Setting Up Rows and Column Headings (https://openstax.org/r/j3kKnUNIt6c) +Determine the Validity of a Truth Table for a Compound Statement +A logical statement is valid if it is always true regardless of the truth values of its component parts. To test the validity of +a compound statement, construct a truth table to analyze all possible outcomes. If the last column, representing the +complete statement, contains only true values, the statement is valid. +EXAMPLE 2.17 +Determining the Validity of Compound Statements +Construct a truth table to determine the validity of each of the following statements. +1. +2. +Solution +1. +Step 1: Because there are two statements, +and , and each of these has two possible outcomes, there will be +rows in our table to represent all possible outcomes: TT, TF, FT, and FF. +Step 2: The columns, will include +, , +and +Every value in column +will have the opposite truth value of +the corresponding value in column +. +Step 3: To complete the last column, evaluate each element in column +with the corresponding element in +column +using the connective and . The last column contains at least one false, therefore the statement +is +84 +2 • Logic +Access for free at openstax.org + +not valid. +T +T +F +F +T +F +F +F +F +T +T +T +F +F +T +F +2. +Step 1: Because the statement +only contains one basic proposition, the truth table will only contain two +rows. Statement +may be either true or false. +Step 2: The columns will include +, +, +and +Evaluate column +with the and connective, +because the symbol +represents a conjunction or logical and statement. True and false is false, and false and true +is also false. +Step 3: The final column is the negation of each entry in the third column, both of which are false, so the negation +of false is true. Because all the truth values in the final column are true, the statement +is valid. +T +F +F +T +F +T +F +T +YOUR TURN 2.17 +Construct a truth table to determine the validity of each of the following statements. +1. +2. +Check Your Understanding +14. A logical argument is _____ if its conclusion follows from its premises. +15. A logical statement is valid if it is always _____. +16. A _____ _____ is a tool used to analyze all the possible outcomes for a logical statement. +17. The truth table for the conjunction, +, has _____ rows of truth values. +18. The truth table for the negation of a logical statement, +, has _____ rows of truth values. +SECTION 2.3 EXERCISES +For the following exercises, find the truth value of each statement. +1. +: +. What is the truth value of +? +2. +: The sun revolves around the Earth. What is the truth value of +? +3. +: The acceleration of gravity is +m/sec2. What is the truth value of ? +4. +: Dan Brown is not the author of the book, The Davinci Code. What is the truth value of +? +5. +: Broccoli is a vegetable. What is the truth value of +? +For the following exercises, given +: +, : Five is an even number, and : Seven is a prime number, find the truth +value of each of the following statements. +2.3 • Constructing Truth Tables +85 + +6. +7. +8. +9. +10. +11. +12. +13. +14. +15. +16. +17. +18. +19. +20. +For the following exercises, complete the truth table to determine the truth value of the proposition in the last column. +21. +T +T +T +22. +F +T +F +23. +F +F +F +24. +F +F +F +For the following exercises, given +All triangles have three sides, +Some rectangles are not square, and +A +pentagon has eight sides, determine the truth value of each compound statement by constructing a truth table. +25. +26. +27. +28. +For the following exercises, construct a truth table to analyze all the possible outcomes for the following arguments. +29. +30. +31. +32. +For the following exercises, construct a truth table to determine the validity of each statement. +33. +34. +86 +2 • Logic +Access for free at openstax.org + +35. +36. +2.4 Truth Tables for the Conditional and Biconditional +Figure 2.10 If-then statements use logic to execute directions. (credit: “Coding” by Carlos Varela/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Use and apply the conditional to construct a truth table. +2. +Use and apply the biconditional to construct a truth table. +3. +Use truth tables to determine the validity of conditional and biconditional statements. +Computer languages use if-then or if-then-else statements as decision statements: +• +If the hypothesis is true, then do something. +• +Or, if the hypothesis is true, then do something; else do something else. +For example, the following representation of computer code creates an if-then-else decision statement: +Check value of variable . +If +, then print "Hello, World!" else print "Goodbye". +In this imaginary program, the if-then statement evaluates and acts on the value of the variable . For instance, if +, +the program would consider the statement +as true and “Hello, World!” would appear on the computer screen. If +instead, +, the program would consider the statement +as false (because 3 is greater than 1), and print +“Goodbye” on the screen. +In this section, we will apply similar reasoning without the use of computer programs. +PEOPLE IN MATHEMATICS +The Countess of Lovelace, Ada Lovelace, is credited with writing the first computer program. She wrote an algorithm +to work with Charles Babbage’s Analytical Engine that could compute the Bernoulli numbers in 1843. In doing so, she +became the first person to write a program for a machine that would produce more than just a simple calculation. +The computer programming language ADA is named after her. +Reference: Posamentier, Alfred and Spreitzer Christian, “Chapter 34 Ada Lovelace: English (1815-1852)” pp. 272-278, +Math Makers: The Lives and Works of 50 Famous Mathematicians, Prometheus Books, 2019. +2.4 • Truth Tables for the Conditional and Biconditional +87 + +Use and Apply the Conditional to Construct a Truth Table +A conditional is a logical statement of the form if +, then . The conditional statement in logic is a promise or contract. +The only time the conditional, +is false is when the contract or promise is broken. +For example, consider the following scenario. A child’s parent says, “If you do your homework, then you can play your +video games.” The child really wants to play their video games, so they get started right away, finish within an hour, and +then show their parent the completed homework. The parent thanks the child for doing a great job on their homework +and allows them to play video games. Both the parent and child are happy. The contract was satisfied; true implies true is +true. +Now, suppose the child does not start their homework right away, and then struggles to complete it. They eventually +finish and show it to their parent. The parent again thanks the child for completing their homework, but then informs +the child that it is too late in the evening to play video games, and that they must begin to get ready for bed. Now, the +child is really upset. They held up their part of the contract, but they did not receive the promised reward. The contract +was broken; true implies false is false. +So, what happens if the child does not do their homework? In this case, the hypothesis is false. No contract has been +entered, therefore, no contract can be broken. If the conclusion is false, the child does not get to play video games and +might not be happy, but this outcome is expected because the child did not complete their end of the bargain. They did +not complete their homework. False implies false is true. The last option is not as intuitive. If the parent lets the child +play video games, even if they did not do their homework, neither parent nor child are going to be upset. False implies +true is true. +The truth table for the conditional statement below summarizes these results. +T +T +T +T +F +F +F +T +T +F +F +T +Notice that the only time the conditional statement, +is false is when the hypothesis, +, is true and the +conclusion, , is false. +VIDEO +Logic Part 8: The Conditional and Tautologies (https://openstax.org/r/Conditional_and_Tautologies) +EXAMPLE 2.18 +Constructing Truth Tables for Conditional Statements +Assume both of the following statements are true: +: My sibling washed the dishes, and : My parents paid them $5.00. +Create a truth table to determine the truth value of each of the following conditional statements. +1. +2. +3. +88 +2 • Logic +Access for free at openstax.org + +Solution +1. +Because +is true and +is true, the statement +is, “If my sibling washed the dishes, then my parents paid them +$5.00.” My sibling did wash the dishes, since +is true, and the parents did pay the sibling $5.00, so the contract was +entered and completed. The conditional statement is true, as indicated by the truth table representing this case: +T → T = T. +T +T +T +2. +translates to the statement, “If my sibling washed the dishes, then my parents did not pay them $5.00.” +is +true, but +is false. The sibling completed their end of the contract, but they did not get paid. The contract was +broken by the parents. The conditional statement is false, as indicated by the truth table representing this case: +T → F = F. +T +T +F +F +3. +translates to the statement, “If my sibling did not wash the dishes, then my parents paid them $5.00.” +is +false, but +is true. The sibling did not do the dishes. No contract was entered, so it could not be broken. The parents +decided to pay them $5.00 anyway. The conditional statement is true, as indicated by the truth table representing +this case: F → T = T. +T +T +F +T +YOUR TURN 2.18 +Assume +is true and +is false. +: Kevin vacuumed the living room, and : Kevin's parents did not let him borrow the +car. Create a truth table to determine the truth value of each of the following conditional statements. +1. +2. +3. +EXAMPLE 2.19 +Determining Validity of Conditional Statements +Construct a truth table to analyze all possible outcomes for each of the following statements then determine whether +they are valid. +1. +2. +Solution +1. +Applying the dominance of connectives, the statement +is equivalent to +So, the columns +of the truth table will include +, , +, and +Because there are only two basic propositions, +and , +the table will have +rows of truth values to account for all the possible outcomes. The statement is not valid +because the last column is not all true. +2.4 • Truth Tables for the Conditional and Biconditional +89 + +T +T +T +F +F +T +F +F +T +T +F +T +F +F +T +F +F +F +T +T +2. +Applying the dominance of connectives, the statement +is equivalent to +So, the +columns of the truth table will include +, , +, +and +Because there are only two basic +propositions, +and , the table will have +rows of truth values to account for all the possible outcomes. The +statement is not valid because the last column is not all true. +T +T +F +T +T +T +F +F +F +F +F +T +T +T +T +F +F +T +T +T +YOUR TURN 2.19 +Construct a truth table to analyze all possible outcomes for each of the following statements, then determine +whether they are valid. +1. +2. +Use and Apply the Biconditional to Construct a Truth Table +The biconditional, +, is a two way contract; it is equivalent to the statement +A biconditional +statement, +is true whenever the truth value of the hypothesis matches the truth value of the conclusion, +otherwise it is false. +The truth table for the biconditional is summarized below. +T +T +T +T +F +F +F +T +F +F +F +T +90 +2 • Logic +Access for free at openstax.org + +VIDEO +Logic Part 11B: Biconditional and Summary of Truth Value Rules in Logic (https://openstax.org/r/omKzui0Fytk) +EXAMPLE 2.20 +Constructing Truth Tables for Biconditional Statements +Assume both of the following statements are true: +: The plumber fixed the leak, and : The homeowner paid the +plumber $150.00. Create a truth table to determine the truth value of each of the following biconditional statements. +1. +2. +3. +Solution +1. +Because +is true and +is true, the statement +is “The plumber fixed the leak if and only if the homeowner paid +them $150.00.” Because both +and +are true, the leak was fixed and the plumber was paid, meaning both parties +satisfied their end of the bargain. The biconditional statement is true, as indicated by the truth table representing +this case: T ↔ T = T. +T +T +T +2. +translates to the statement, “The plumber fixed the leak if and only if the homeowner did not pay them +$150.” If the plumber fixed the leak and the homeowner did not pay them, the homeowner will have broken their +end of the contract. The biconditional statement is false, as indicated by the truth table representing this case: +T ↔ F = F. +T +T +F +F +3. +translates to the statement, “The plumber did not fix the leak if and only if the homeowner did not pay +them $150.” In this case, neither party—the plumber nor the homeowner—entered into the contract. The leak was +not repaired, and the plumber was not paid. No agreement was broken. The biconditional statement is true, as +indicated by the truth table representing this case: F ↔ F = T. +T +T +F +F +T +YOUR TURN 2.20 +Assume +is true and +is false: +The contractor fixed the broken window, and +The homeowner paid the +contractor $200. Create a truth table to determine the truth value of each of the following biconditional statements. +1. +2. +3. +The biconditional, +is true whenever the truth values of +and +match, otherwise it is false. +2.4 • Truth Tables for the Conditional and Biconditional +91 + +VIDEO +Logic Part 13: Truth Tables to Determine if Argument is Valid or Invalid (https://openstax.org/r/AQB3svnxxiw) +EXAMPLE 2.21 +Determining Validity of Biconditional Statements +Construct a truth table to analyze all possible outcomes for each of the following statements, then determine whether +they are valid. +1. +2. +3. +4. +Solution +1. +Applying the dominance of connectives, the statement +is equivalent to +So, the +columns of the truth table will include +, , +, +and +Because there are only two +basic propositions, +and , the table will have +rows of truth values to account for all the possible +outcomes. The statement is not valid because the last column is not all true. +T +T +T +F +F +F +T +F +F +T +T +F +F +T +F +F +F +T +F +F +F +T +F +T +2. +Applying the dominance of connectives, the statement +is equivalent to +So, the +columns of the truth table will include +, , +, +and +Because there are only two +basic propositions, +and , the table will have +rows of truth values to account for all the possible +outcomes. The statement is not valid because the last column is not all true. +T +T +T +F +T +T +T +F +T +F +F +F +F +T +T +T +T +T +F +F +F +T +T +F +3. +Applying the dominance of connectives, the statement +is equivalent to +So, the columns of the truth table will include +, , +, +, +and +Because +there are only two basic propositions, +and +the table will have +rows of truth values to account for all the +possible outcomes. The statement is valid because the last column is all true. +92 +2 • Logic +Access for free at openstax.org + +T +T +T +F +F +T +T +T +F +F +T +F +F +T +F +T +T +F +T +T +T +F +F +T +T +T +T +T +4. +Applying the dominance of connectives, the statement +is equivalent to +So, the columns of the truth table will include +, , , +, +and +Because there are three basic propositions, +, , and , the table +will have +rows of truth values to account for all the possible outcomes. The statement is not valid +because the last column is not all true. +T +T +T +F +T +F +T +T +T +T +F +T +T +T +F +T +T +F +T +F +F +T +F +F +T +F +F +T +F +T +F +F +F +T +T +F +F +T +F +F +F +T +F +T +F +T +F +F +F +F +T +F +F +T +F +F +F +F +F +T +F +T +F +F +YOUR TURN 2.21 +Construct a truth table to analyze all possible outcomes for each of the following statements, then determine +whether they are valid. +1. +2. +3. +4. +Check Your Understanding +19. In logic, a conditional statement can be thought of as a _____________. +20. If the hypothesis, +, of a conditional statement is true, the _____, , must also be true for the conditional statement +to be true. +21. If the ______________ of a conditional statement is false, the conditional statement is true. +22. The symbolic form of the _____________________ statement is +. +2.4 • Truth Tables for the Conditional and Biconditional +93 + +23. The _____________________ statement is equivalent to the statement +24. +if and only if +is ____________ whenever the truth value of +matches the truth value of , otherwise it is false. +SECTION 2.4 EXERCISES +For the following exercises, complete the truth table to determine the truth value of the proposition in the last column. +94 +2 • Logic +Access for free at openstax.org + +1. +T +T +2. +T +T +3. +F +T +4. +F +T +5. +F +T +F +6. +F +F +F +7. +F +F +F +8. +T +F +F +9. +F +F +F +10. +T +T +T +For the following exercises, assume these statements are true: +Faheem is a software engineer, +Ann is a project +2.4 • Truth Tables for the Conditional and Biconditional +95 + +manager, +Giacomo works with Faheem, and +The software application was completed on time. Translate each of the +following statements to symbols, then construct a truth table to determine its truth value. +11. If Giacomo works with Faheem, then Faheem is not a software engineer. +12. If the software application was not completed on time, then Ann is not a project manager. +13. The software application was completed on time if and only if Giacomo worked with Faheem. +14. Ann is not a project manager if and only if Faheem is a software engineer. +15. If the software application was completed on time, then Ann is a project manager, but Faheem is not a software +engineer. +16. If Giacomo works with Faheem and Ann is a project manager, then the software application was completed on +time. +17. The software application was not completed on time if and only if Faheem is a software engineer or Giacomo +did not work with Faheem. +18. Faheem is a software engineer or Ann is not a project manager if and only if Giacomo did not work with +Faheem and the software application was completed on time. +19. Ann is a project manager implies Faheem is a software engineer if and only if the software application was +completed on time implies Giacomo worked with Faheem. +20. If Giacomo did not work with Faheem implies that the software application was not completed on time, then +Ann was not the project manager. +For the following exercises, construct a truth table to analyze all the possible outcomes and determine the validity of +each argument. +21. +22. +23. +24. +25. +26. +27. +28. +29. +30. +96 +2 • Logic +Access for free at openstax.org + +2.5 Equivalent Statements +Figure 2.11 How your logical argument is stated affects the response, just like how you speak when holding a +conversation can affect how your words are received. (credit: modification of work by Goelshivi/Flickr, Public Domain +Mark 1.0) +Learning Objectives +After completing this section, you should be able to: +1. +Determine whether two statements are logically equivalent using a truth table. +2. +Compose the converse, inverse, and contrapositive of a conditional statement +Have you ever had a conversation with or sent a note to someone, only to have them misunderstand what you intended +to convey? The way you choose to express your ideas can be as, or even more, important than what you are saying. If +your goal is to convince someone that what you are saying is correct, you will not want to alienate them by choosing +your words poorly. +Logical arguments can be stated in many different ways that still ultimately result in the same valid conclusion. Part of +the art of constructing a persuasive argument is knowing how to arrange the facts and conclusion to elicit the desired +response from the intended audience. +In this section, you will learn how to determine whether two statements are logically equivalent using truth tables, and +then you will apply this knowledge to compose logically equivalent forms of the conditional statement. Developing this +skill will provide the additional skills and knowledge needed to construct well-reasoned, persuasive arguments that can +be customized to address specific audiences. +An alternate way to think about logical equivalence is that the truth values have to match. That is, whenever +is +true, +is also true, and whenever +is false, +is also false. +Determine Logical Equivalence +Two statements, +and , are logically equivalent when +is a valid argument, or when the last column of the truth +table consists of only true values. When a logical statement is always true, it is known as a tautology. To determine +whether two statements +and +are logically equivalent, construct a truth table for +and determine whether it +valid. If the last column is all true, the argument is a tautology, it is valid, and +is logically equivalent to ; otherwise, +is +not logically equivalent to . +EXAMPLE 2.22 +Determining Logical Equivalence with a Truth Table +Create a truth table to determine whether the following compound statements are logically equivalent. +2.5 • Equivalent Statements +97 + +1. +2. +Solution +1. +Construct a truth table for the biconditional formed by using the first statement as the hypothesis and the second +statement as the conclusion, +T +T +T +F +F +T +T +T +F +F +F +T +T +F +F +T +T +T +F +F +F +F +F +T +T +T +T +T +Because the last column it not all true, the biconditional is not valid and the statement +is not logically +equivalent to the statement +. +2. +Construct a truth table for the biconditional formed by using the first statement as the hypothesis and the second +statement as the conclusion, +T +T +T +F +T +T +T +F +F +F +F +T +F +T +T +T +T +T +F +F +T +T +T +T +Because the last column is true for every entry, the biconditional is valid and the statement +is logically +equivalent to the statement +. Symbolically, +YOUR TURN 2.22 +Create a truth table to determine whether the following compound statements are logically equivalent. +1. +2. +Compose the Converse, Inverse, and Contrapositive of a Conditional Statement +The converse, inverse, and contrapositive are variations of the conditional statement, +• +The converse is if +then +, and it is formed by interchanging the hypothesis and the conclusion. The converse is +logically equivalent to the inverse. +• +The inverse is if +then +, and it is formed by negating both the hypothesis and the conclusion. The inverse is +logically equivalent to the converse. +• +The contrapositive is if +then +, and it is formed by interchanging and negating both the hypothesis and the +conclusion. The contrapositive is logically equivalent to the conditional. +The table below shows how these variations are presented symbolically. +98 +2 • Logic +Access for free at openstax.org + +Conditional +Contrapositive +Converse +Inverse +T +T +F +F +T +T +T +T +T +F +F +T +F +F +T +T +F +T +T +F +T +T +F +F +F +F +T +T +T +T +T +T +EXAMPLE 2.23 +Writing the Converse, Inverse, and Contrapositive of a Conditional Statement +Use the statements, +: Harry is a wizard and : Hermione is a witch, to write the following statements: +1. +Write the conditional statement, +, in words. +2. +Write the converse statement, +, in words. +3. +Write the inverse statement, +, in words. +4. +Write the contrapositive statement, +, in words. +Solution +1. +The conditional statement takes the form, “if +, then ,” so the conditional statement is: “If Harry is a wizard, then +Hermione is a witch.” Remember the if … then … words are the connectives that form the conditional statement. +2. +The converse swaps or interchanges the hypothesis, +, with the conclusion, . It has the form, “if , then +.” So, the +converse is: "If Hermione is a witch, then Harry is a wizard." +3. +To construct the inverse of a statement, negate both the hypothesis and the conclusion. The inverse has the form, +“if +, then +,” so the inverse is: "If Harry is not a wizard, then Hermione is not a witch." +4. +The contrapositive is formed by negating and interchanging both the hypothesis and conclusion. It has the form, “if +, then +," so the contrapositive statement is: "If Hermione is not a witch, then Harry is not a wizard." +YOUR TURN 2.23 +Use the statements, +: Elvis Presley wore capes and : Some superheroes wear capes, to write the following +statements: +1. Write the conditional statement, +, in words. +2. Write the converse statement, +, in words. +3. Write the inverse statement, +, in words. +4. Write the contrapositive statement, +, in words. +EXAMPLE 2.24 +Identifying the Converse, Inverse, and Contrapositive +Use the conditional statement, “If all dogs bark, then Lassie likes to bark,” to identify the following. +1. +Write the hypothesis of the conditional statement and label it with a +. +2. +Write the conclusion of the conditional statement and label it with a . +3. +Identify the following statement as the converse, inverse, or contrapositive: “If Lassie likes to bark, then all dogs +bark.” +4. +Identify the following statement as the converse, inverse, or contrapositive: “If Lassie does not like to bark, then +some dogs do not bark.” +5. +Which statement is logically equivalent to the conditional statement? +2.5 • Equivalent Statements +99 + +Solution +1. +The hypothesis is the phrase following the if. The answer is +: All dogs bark. Notice, the word if is not included as +part of the hypothesis. +2. +The conclusion of a conditional statement is the phrase following the then. The word then is not included when +stating the conclusion. The answer is: : Lassie likes to bark. +3. +“Lassie likes to bark” is +and “All dogs bark” is +. So, “If Lassie likes to bark, then all dogs bark,” has the form “if , +then +,” which is the form of the converse. +4. +“Lassie does not like to bark” is +and “Some dogs do not bark” is +. The statement, “If Lassie does not like to bark, +then some dogs do not bark,” has the form “if +, then +,” which is the form of the contrapositive. +5. +The contrapositive +is logically equivalent to the conditional statement +YOUR TURN 2.24 +Use the conditional statement, “If Dora is an explorer, then Boots is a monkey,” to identify the following: +1. Write the hypothesis of the conditional statement and label it with a +. +2. Write the conclusion of the conditional statement and label it with a . +3. Identify the following statement as the converse, inverse, or contrapositive: “If Dora is not an explorer, then +Boots is not a monkey.” +4. Identify the following statement as the converse, inverse, or contrapositive: “If Boots is a monkey, then Dora +is an explorer.” +5. Which statement is logically equivalent to the inverse? +EXAMPLE 2.25 +Determining the Truth Value of the Converse, Inverse, and Contrapositive +Assume the conditional statement, +“If Chadwick Boseman was an actor, then Chadwick Boseman did not star in +the movie Black Panther” is false, and use it to answer the following questions. +1. +Write the converse of the statement in words and determine its truth value. +2. +Write the inverse of the statement in words and determine its truth value. +3. +Write the contrapositive of the statement in words and determine its truth value. +Solution +1. +The only way the conditional statement can be false is if the hypothesis, +: Chadwick Boseman was an actor, is true +and the conclusion, : Chadwick Boseman did not star in the movie Black Panther, is false. The converse is +and it is written in words as: “If Chadwick Boseman did not star in the movie Black Panther, then Chadwick Boseman +was an actor.” This statement is true, because false +true is true. +2. +The inverse has the form “ +” The written form is: “If Chadwick Boseman was not an actor, then Chadwick +Boseman starred in the movie Black Panther.” Because +is true, and +is false, +is false, and +is true. This means +the inverse is false +true, which is true. Alternatively, from Question 1, the converse is true, and because the +inverse is logically equivalent to the converse it must also be true. +3. +The contrapositive is logically equivalent to the conditional. Because the conditional is false, the contrapositive is +also false. This can be confirmed by looking at the truth values of the contrapositive statement. The contrapositive +has the form “ +.” Because +is false and +is true, +is true and +is false. Therefore, +is true +false, which is false. The written form of the contrapositive is “If Chadwick Boseman starred in the movie Black +Panther, then Chadwick Boseman was not an actor.” +YOUR TURN 2.25 +Assume the conditional statement +“If my friend lives in San Francisco, then my friend does not live in +California” is false, and use it to answer the following questions. +1. Write the converse of the statement in words and determine its truth value. +2. Write the inverse of the statement in words and determine its truth value. +100 +2 • Logic +Access for free at openstax.org + +3. Write the contrapositive of the statement in words and determine its truth value. +Check Your Understanding +25. Two statements +and +are logically equivalent to each other if the biconditional statement, +is +________________. +26. The _____ statement has the form, “ +then .” +27. The contrapositive is _____________ ___________ to the conditional statement, and has the form, "if +, then +." +28. The _________________ of the conditional statement has the form, "if +, then +." +29. The _________________ of the conditional statement is logically equivalent to the _______________ and has the form, "if +then +." +SECTION 2.5 EXERCISES +For the following exercises, determine whether the pair of compound statements are logically equivalent by +constructing a truth table. +1. Converse: +and inverse: +2. Conditional: +and contrapositive: +3. Inverse: +and contrapositive: +4. Conditional: +and converse: +5. +and +6. +and +7. +and +8. +and +9. +and +10. +and +For the following exercises, answer the following: +a. +Write the conditional statement +in words. +b. +Write the converse statement +in words. +c. +Write the inverse statement +in words. +d. +Write the contrapositive statement +in words. +11. +: Six is afraid of Seven and : Seven ate Nine. +12. +: Hope is eternal and : Despair is temporary. +13. +: Tom Brady is a quarterback and : Tom Brady does not play soccer. +14. +: Shakira does not sing opera and : Shakira sings popular music. +15. +:The shape does not have three sides and : The shape is not a triangle. +16. +: All birds can fly and : Emus can fly. +17. +: Penguins cannot fly and : Some birds can fly. +18. +: Some superheroes do not wear capes and : Spiderman is a superhero. +19. +: No Pokémon are little ponies and : Bulbasaur is a Pokémon. +20. +: Roses are red, and violets are blue and : Sugar is sweet, and you are sweet too. +For the following exercises,use the conditional statement: “If Clark Kent is Superman, then Lois Lane is not a reporter,” +to answer the following questions. +21. Write the hypothesis of the conditional statement, label it with a +, and determine its truth value. +22. Write the conclusion of the conditional statement, label it with a , and determine its truth value. +23. Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: “If +Clark Kent is not Superman, then Lois Lane is a reporter.” +24. Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: “If +Lois Lane is a reporter, then Clark Kent is not Superman.” +25. Which form of the conditional is logically equivalent to the converse? +For the following exercises, use the conditional statement: “If The Masked Singer is not a music competition, then +Donnie Wahlberg was a member of New Kids on the Block,” to answer the following questions. +26. Write the hypothesis of the conditional statement, label it with a +, and determine its truth value. +2.5 • Equivalent Statements +101 + +27. Write the conclusion of the conditional statement, label it with a , and determine its truth value. +28. Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: “If +Donnie Wahlberg was a member of New Kids on the Block, then The Masked Singer is not a music competition.” +29. Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: “If +The Masked Singer is a music competition, then Donnie Wahlberg was not a member of New Kids on the Block.” +30. Which form of the conditional is logically equivalent to the contrapositive, +? +For the following exercises, use the conditional statement: “If all whales are mammals, then no fish are whales,” to +answer the following questions. +31. Write the hypothesis of the conditional statement, label it with a +, and determine its truth value. +32. Write the conclusion of the conditional statement, label it with a , and determine its truth value. +33. Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: “If +some fish are whales, then some whales are not mammals.” +34. Write the inverse in words and determine its truth value. +35. Write the converse in words and determine its truth value. +For the following exercises, use the conditional statement: “If some parallelograms are rectangles, then some circles +are not symmetrical,” to answer the following questions. +36. Write the hypothesis of the conditional statement, label it with a +, and determine its truth value. +37. Write the conclusion of the conditional statement, label it with a , and determine its truth value. +38. Write the converse in words and determine its truth value. +39. Write the contrapositive in words and determine its truth value. +40. Write the inverse in words and determine its truth value. +2.6 De Morgan’s Laws +Figure 2.12 De Morgan’s Laws were key to the rise of logical mathematical expression and helped serve as a bridge for +the invention of the computer. (credit: modification of work “Golden Gate Bridge (San Francisco Bay, California, USA)” by +James St. John/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Use De Morgan’s Laws to negate conjunctions and disjunctions. +2. +Construct the negation of a conditional statement. +3. +Use truth tables to evaluate De Morgan’s Laws. +The contributions to logic made by Augustus De Morgan and George Boole during the 19th century acted as a bridge to +the development of computers, which may be the greatest invention of the 20th century. Boolean logic is the basis for +computer science and digital electronics, and without it the technological revolution of the late 20th and early 21st +centuries—including the creation of computer chips, microprocessors, and the Internet—would not have been possible. +Every modern computer language uses Boolean logic statements, which are translated into commands understood by +the underlying electronic circuits enabling computers to operate. But how did this logic get its name? +102 +2 • Logic +Access for free at openstax.org + +PEOPLE IN MATHEMATICS +George Boole +Figure 2.13 Boole’s algebra of logic was foundational in the design of digital computer circuits. (credit: “Circuit Board” +by Squeezyboy/Flickr, CC BY 2.0) +George Boole was born in Lincolnshire, England in 1815. He was the son of a cobbler who provided him some initial +education, but Boole was mostly self-taught. He began teaching at 16 years of age, and opened his own school at the +age of 20. In 1849, at the age of 34, he was appointed Professor of Mathematics at Queens College in Cork, Ireland. In +1853, he published the paper, An Investigation of the Laws of Thought, on Which Are Founded the Mathematical +Theories of Logic and Probabilities, which is the treatise that the field of Boolean algebra and digital circuitry was built +on. +Reference: Posamentier, Alfred and Spreitzer Christian, “Chapter 35 George Boole: English (1815-1864)” pp. 279-283, +Math Makers: The Lives and Works of 50 Famous Mathematicians, Prometheus Books, 2019. +Negation of Conjunctions and Disjunctions +In Chapter 1, Example 1.37 used a Venn diagram to prove De Morgan’s Law for set complement over union. Because the +complement of a set is analogous to negation and union is analogous to an or statement, there are equivalent versions +of De Morgan’s Laws for logic. +FORMULA +De Morgan’s Law for negation of a conjunction: +De Morgan’s Law for the negation of a disjunction: +Negation of a conditional: +Writing conditional as a disjunction: +Recall that the symbol for logical equivalence is: +De Morgan’s Laws allow us to write the negation of conjunctions and disjunctions without using the phrase, “It is not the +case that …” to indicate the parentheses. Avoiding this phrase often results in a written or verbal statement that is +clearer or easier to understand. +EXAMPLE 2.26 +Applying De Morgan’s Law for Negation of Conjunctions and Disjunctions +Write the negation of each statement in words without using the phrase, “It is not the case that.” +2.6 • De Morgan’s Laws +103 + +1. +Kristin is a biomedical engineer and Thomas is a chemical engineer. +2. +A person had cake or they had ice cream. +Solution +1. +Kristin is a biomedical engineer and Thomas is a chemical engineer has the form “ +,” where +is the statement, +“Kristin is a biomedical engineer,” and +is the statement, “Thomas is a chemical engineer.” By De Morgan’s Law, the +negation of a conjunction, +, is logically equivalent to +is “Kristen is not a biomedical engineer,” +and +is “Thomas is not a chemical engineer.” By De Morgan’s Law, the solution has the form “ +,” so the +answer is: “Kristin is not a biomedical engineer or Tom is not a chemical engineer.” +2. +A person had cake or they had ice cream has the form “ +” where +is the statement, “A person had cake,” and +is the statement, “A person had ice cream.” By De Morgan’s Law for the negations of a disjunction, +The solution is the statement: “A person did not have cake and they did not have ice cream.” +YOUR TURN 2.26 +Write the negation of each statement in words without using the phrase, it is not the case that. +1. Jackie played softball or she ran track. +2. Brandon studied for his certification exam, and he passed his exam. +Negation of a Conditional Statement +The negation of any statement has the opposite truth values of the original statement. The negation of a conditional, +, is the conjunction of +and not , +Consider the truth table below for the negation of the conditional. +T +T +T +F +T +F +F +T +F +T +T +F +F +F +T +F +The only time the negation of the conditional statement is true is when +is true, and +is false. This means that +is logically equivalent to +as the following truth table demonstrates. +T +T +T +F +F +F +T +T +F +F +T +T +T +T +F +T +T +F +F +F +T +F +F +T +F +T +F +T +EXAMPLE 2.27 +Constructing the Negation of a Conditional Statement +Write the negation of each conditional statement. +104 +2 • Logic +Access for free at openstax.org + +1. +If Adele won a Grammy, then she is a singer. +2. +If Henrik Lundqvist played professional hockey, then he did not win the Stanley Cup. +Solution +1. +The negation of the conditional statement, +is the statement, +The hypothesis of the conditional +statement is +: “Adele won a Grammy,” and conclusion of the conditional statement is : “Adele is a singer.” The +negation of the conclusion, +, is the statement: “She is not a singer.” Therefore, the answer is +“Adele won a +Grammy, and she is not a singer.” +2. +The hypothesis is +: “Henrik Lundqvist played professional hockey,” and the conclusion of the conditional statement +is : “He did not win the Stanley Cup.” The negation of +is the statement: “He won the Stanley Cup.” The negation of +the conditional statement is equal to +“Henrick Lundqvist played professional hockey, and he won the Stanley +Cup.” +YOUR TURN 2.27 +Write the negation of each conditional statement. +1. If Edna Mode makes a new superhero costume, then it will not include a cape. +2. If I had pancakes for breakfast, then I used maple syrup. +EXAMPLE 2.28 +Constructing the Negation of a Conditional Statement with Quantifiers +Write the negation of each conditional statement. +1. +If all cats purr, then my partner’s cat purrs. +2. +If a penguin is a bird, then some birds do not fly. +Solution +1. +The negation of the conditional statement +is the statement +The hypothesis of the conditional +statement is +: “All cats purr,” and the conclusion of the conditional statement is : “My partner’s cat purrs.” The +negation of the conclusion, +, is the statement: “My partner’s cat does not purr.” Therefore, the answer is +“All cats purr, but my partner’s cat does not purr.” +2. +The hypothesis is +: “A penguin is a bird,” and the conclusion of the conditional statement is : “Some birds do not +fly.” The negation of +is the statement: “All birds fly.” Therefore, the negation of the conditional statement is equal +to +“A penguin is a bird, and all birds fly.” +YOUR TURN 2.28 +Write the negation of each conditional statement. +1. If some people like ice cream, then ice cream makers will make a profit. +2. If Raquel cannot play video games, then nobody will play video games. +Many of the properties that hold true for number systems and sets also hold true for logical statements. The following +table summarizes some of the most useful properties for analyzing and constructing logical arguments. These +properties can be verified using a truth table. +Property +Conjunction (AND) +Disjunction (OR) +Commutative +Associative +2.6 • De Morgan’s Laws +105 + +Property +Conjunction (AND) +Disjunction (OR) +Distributive +De Morgan’s +Conditional +EXAMPLE 2.29 +Negating a Conditional Statement with a Conjunction or Disjunction +Write the negation of each conditional statement applying De Morgan’s Law. +1. +If mom needs to buy chips, then Mike had friends over and Bob was hungry. +2. +If Juan had pizza or Chris had wings, then dad watched the game. +Solution +1. +The conditional has the form “If +then +or ,” where +is “Mom needs to buy chips,” +is “Mike had friends over,” and +is “Bob was hungry.” The negation of +is +Applying De Morgan’s Law to the statement +the result is +, so our conditional statement becomes +By the distributive property for +conjunction over disjunction, this statement is equivalent to +Translating the statement +into words, the solution is: “Mom needs to buy chips and Mike did not have friends over, or Mom +needs to buy chips and Bob was not hungry.” +2. +The conditional has the form “If +or , then ,” where +is “Juan had pizza,” +is “Chris had wings,” and +is “Dad +watched the game.” The negation of +is +By the distributive property for disjunction over +conjuction, the statement is equivalent to +Translating the statement +into +words, the solution is: “Juan had pizza or dad did not watch the game, and Chris had wings or dad did not watch the +game.” +YOUR TURN 2.29 +Write the negation of each conditional statement applying De Morgan’s Law. +1. If Eric needs to replace the light bulb, then Marcus left the light on all night or Dan broke the bulb. +2. If Trenton went to school and Regina went to work, then Merika cleaned the house. +Evaluating De Morgan’s Laws with Truth Tables +In Chapter 1, you learned that you could prove the validity of De Morgan’s Laws using Venn diagrams. Truth tables can +also be used to prove that two statements are logically equivalent. If two statements are logically equivalent, you can use +the form of the statement that is clearer or more persuasive when constructing a logical argument. +The next example will prove the validity of one of De Morgan’s Laws using a truth table. The same procedure can be +applied to any two logical statement that you believe are equivalent. If the last column of the truth table is a tautology, +then the two statements are logically equivalent. +EXAMPLE 2.30 +Verifying De Morgan’s Law for Negation of a Conjunction +Construct a truth table to verify De Morgan’s Law for the negation of a conjunction, +, is valid. +Solution +Step 1: To verify any logical equivalence, you must first replace the logical equivalence symbol, +, with the biconditional +symbol, +. The statement +becomes +Step 2: Next, you create a truth table for the statement. Because we have two basic statements, +, and , the truth table +will have four rows to account for all the possible outcomes. The columns will be +, , +, +, +and +106 +2 • Logic +Access for free at openstax.org + +the biconditional statement is +T +T +T +F +F +F +F +T +T +F +F +T +F +T +T +T +F +T +F +T +T +F +T +T +F +F +F +T +T +T +T +T +Step 3: Finally, verify that the statement is valid by confirming it is a tautology. In this instance, the last column is all true. +Therefore, the statement is valid and De Morgan’s Law for the negation of a conjunction is verified. +YOUR TURN 2.30 +1. Construct a truth table to verify De Morgan’s Law for the negation of a disjunction, +, is valid. +Check Your Understanding +30. De Morgan’s Law for the negation of a conjunction states that +is logically equivalent to ___________________. +31. De Morgan’s Law for the negation of a disjunction states that +is logically equivalent to __________________. +32. The negation of the conditional statement, +, is logically equivalent to _____________. +33. +, which means the conditional statement is logically equivalent to +Apply +_______________________ to the statement +to show that the conditional statement +SECTION 2.6 EXERCISES +For the following exercises, use De Morgan’s Laws to write each statement without parentheses. +1. +2. +3. +4. +For the following exercises, use De Morgan’s Laws to write the negation of each statement in words without using the +phrase, “It is not the case that, …” +5. Sergei plays right wing and Patrick plays goalie. +6. Mario is a carpenter, or he is a plumber. +7. Luigi is a plumber, or he is not a video game character. +8. Ralph Macchio was the original Karate Kid, and karate is not for defense only. +9. Some people like broccoli, but my siblings did not like broccoli. +10. Some people do not like chocolate or all people like pizza. +For the following exercises, write each statement as a conjunction or disjunction in symbolic form by applying the +property for the negation of a conditional. +11. +12. +13. +14. +15. +16. +17. +2.6 • De Morgan’s Laws +107 + +18. +For the following exercises, write the negation of each conditional in words by applying the property for the negation of +a conditional. +19. If a student scores an 85 on the final exam, then they will receive an A in the class. +20. If a person does not pass their road test, then they will not receive their driver’s license. +21. If a student does not do their homework, then they will not play video games. +22. If a commuter misses the bus, then they will not go to work today. +23. If a racecar driver gets pulled over for speeding, then they will not make it to the track on time for the race. +24. If Rene Descartes was a philosopher, then he was not a mathematician. +25. If George Boole invented Boolean algebra and Thomas Edison invented the light bulb, then Pacman is not the +best video game ever. +26. If Jonas Salk created the polio vaccine, then his child received the vaccine or his child had polio. +27. If Billie Holiday sang the blues or Cindy Lauper sang about true colors, then John Lennon was not a Beatle. +28. If Percy Jackson is the lightning thief and Artemis Fowl is a detective, then Artemis Fowl will catch Percy Jackson. +29. If all rock stars are men, then Pat Benatar is not a rock star. +30. If Lady Gaga is a rock star, then some rock stars are women. +31. If yellow combined with blue makes green, then all colors are beautiful. +32. If leopards have spots and zebras have stripes, then some animals are not monotone in color. +For the following exercises, construct a truth table to verify that the logical property is valid. +33. +34. +35. +36. +2.7 Logical Arguments +Figure 2.14 Not all logical arguments are valid, and the ongoing fight for equal rights proves that much progress has yet +to be made. (credit: "The Steam Roller" by Library of Congress Prints and Photographs Division, public domain) +Learning Objectives +After completing this section, you should be able to: +1. +Apply the law of detachment to determine the conclusion of a pair of statements. +2. +Apply the law of denying the consequent to determine the conclusion for pairs of statements. +3. +Apply the chain rule to determine valid conclusions for pairs of true statements. +The previous sections of this chapter provide the foundational skills for constructing and analyzing logical arguments. All +logical arguments include a set of premises that support a claim or conclusion; but not all logical arguments are valid +and sound. A logical argument is valid if its conclusion follows from the premises, and it is sound if it is valid and all of its +premises are true. A false or deceptive argument is called a fallacy. Many types of fallacies are so common that they +have been named. +108 +2 • Logic +Access for free at openstax.org + +WHO KNEW? +In 1936, Dale Carnegie published his first book, titled How to Win Friends and Influence People. It was marketed as +training materials for the improvement of public speaking and negotiation skills, and the methods it presented are +still used today. Carnegie famously said, “When dealing with people, remember you are not dealing with creatures of +logic, but creatures of emotion.” +People who put forth fallacious logical arguments often take advantage of our susceptibility to emotional appeals, to +try to convince us that what they are saying is true. The study of logic helps us combat this weakness through +recognition and learning to focus on the facts and structure of the argument. +This section focuses on the two main forms that logical arguments can take. While inductive arguments attempt to draw +a more general conclusion from a pattern of specific premises, deductive arguments attempt to draw specific +conclusions from at least one or more general premises. Deductive arguments can be proven to be valid using Venn +diagrams or truth tables. +Inductive arguments generally cannot be proven to be true. They are judged as being strong or weak, but, like any +opinion, whether you believe an argument is strong or weak often depends on your knowledge of the topic being +discussed along with the evidence being provided in the premises. Hasty generalization is the name given to any fallacy +that presents a weak inductive argument. +Be careful! Premises may be true or false. If a premise is false, the claims made by the argument should be +questioned. +Law of Detachment +The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, +, and the +hypothesis, +, are true, then the conclusion +must also be true. The law of detachment is also called affirming the +hypothesis (or antecedent) and modus ponens. Symbolically, it has the form +. +Law of Detachment +Premise: +Premise: +Conclusion: +The +is read as the word, “therefore.” +Looking at the truth table for the conditional statement, the only time the conditional is true is when the hypothesis +is +also true. The only place this happens is in the first row, where +is also true, confirming that the law of detachment is a +valid argument. +T +T +T +T +F +F +F +T +T +F +F +T +2.7 • Logical Arguments +109 + +Another way to verify that the law of detachment is a valid argument is to construct a truth table for the argument +and verify that it is a tautology. +T +T +T +T +T +T +F +F +F +T +F +T +T +F +T +F +F +T +F +T +Venn diagrams may also be used to verify deductive arguments, which include conditional premises. Consider the +statement +“If you play guitar, then you are a musician.” The set of guitarists is a subset of the set of musicians, +To verify that an argument is valid using a Venn diagram, draw the Venn diagram representing all the premises in +the argument only, as shown in Figure 2.15. Then verify if the conclusion is also represented by the Venn diagram of the +premises. If it is, the argument is valid. If it is not, the argument is not valid. The set of guitarists is drawn as a subset of +the set of musicians to represent the premise, +The +represents the premise: +is true. This completes the +drawing of the premises. +Figure 2.15 +Now, examine the Venn diagram to verify if the conclusion is included in the picture. The conclusion is . Because the +is +in the set +, and +is a subset of , +is also in ; therefore, the law of detachment is a valid argument. +Remember that an argument can be valid without being true. For the argument to be proven true, it must be both +valid and sound. An argument is sound if all its premises are true. +VIDEO +Logic Part 14: Common Argument Forms like Modus Ponens and Tollens (https://openstax.org/r/ +Modus_Ponens_and_Tollens) +EXAMPLE 2.31 +Applying the Law of Detachment to Determine a Valid Conclusion +Each pair of statements represents the premises in a logical argument. Based on these premises, apply the law of +detachment to determine a valid conclusion. +1. +If Leonardo da Vinci was an artist, then he painted the Mona Lisa. Leonardo da Vinci was an artist. +2. +If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player. Michael Jordan played +for the Chicago Bulls. +3. +If all fish have gills, then clown fish have gills. All fish have gills. +Solution +1. +The premises are +If Leonardo da Vinci was an artist, then he painted the Mona Lisa, and +Leonardo da +110 +2 • Logic +Access for free at openstax.org + +Vinci was an artist. This argument has the form of the law of detachment, so, the conclusion is +Leonardo da Vinci +painted the Mona Lisa. +2. +The premises follow the form of the law of detachment, so a valid conclusion would be . The premises are +If +Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player, and +Michael Jordan +played for the Chicago Bulls. The conclusion that follows from the premises is +Michael Jordan was not a soccer +player. +3. +The premises are +If all fish have gills, then clown fish have gills, and +All fish have gills. This argument has +the form of the law of detachment, so the conclusion is +clown fish have gills. +YOUR TURN 2.31 +Each pair of statements represents the premises in a logical argument. Based on these premises, apply the law of +detachment to determine a valid conclusion. +1. If my classmate likes history, then some people like history. My classmate likes history. +2. If you do not like to read, then some people do not like reading. You do not like to read. +3. If the polygon has five sides, then it is not an octagon. The polygon has five sides. +Law of Denying the Consequent +Another form of a valid conditional argument is called the law of denying the consequent, or modus tollens. Recall, +that the conditional statement, +, is logically equivalent to the contrapositive, +So, if the conditional +statement is true, then the contrapositive statement is also true. By the law of detachment, if +is also true, then it +follows that +must also be true. Symbolically, it has the form +. +Law of Denying the Consequent +Premise: +Premise: +Conclusion: +The conditional statement can also be described as, “If antecedent, then consequent.”This is where the law of +denying the consequent gets its name. +To verify if the law of denying the consequent is a valid argument, construct a truth table for the argument, +, and verify that it is a tautology. +T +T +F +F +T +F +T +T +F +F +T +F +F +T +F +T +T +F +T +F +T +F +F +T +T +T +T +T +To verify an argument of this form using a Venn diagram, again consider the premise: +“If you play guitar, then +you are a musician.” We will change the second premise to +In this case, the +represents the premise, +So, it will be +placed inside the universal set of all people, but outside the set of musicians, as depicted in the Venn diagram in Figure +2.7 • Logical Arguments +111 + +2.16. +Figure 2.16 +Because the +is also outside the set of guitarists, the statement +follows from the premises and the argument is valid. +EXAMPLE 2.32 +Applying the Law of Denying the Consequent to Determine a Valid Conclusion +Each pair of statements represents the premises in a logical argument. Based on these premises, apply the law of +denying the consequent to determine a valid conclusion. +1. +If Leonardo da Vinci was an artist, then he painted the Mona Lisa. Leonardo da Vinci did not paint the Mona Lisa. +2. +If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player. Michael Jordan was a +soccer player. +3. +If all fish have gills, then clown fish have gills. Clown fish do not have gills. +Solution +1. +The premises are +If Leonardo da Vinci was an artist, then he painted the Mona Lisa, and +Leonardo da +Vinci did not paint the Mona Lisa. This argument has the form of the law of denying the consequent, so the +conclusion is +Leonardo da Vinci was not an artist. +2. +The premises follow the form of the law of denying the consequent, so a valid conclusion would be +. The premises +are: +If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player, and +Michael Jordan was a soccer player. The conclusion that follows from the premises is +Michael Jordan did not play +for the Chicago Bulls. +3. +The premises are +If all fish have gills, then clown fish have gills, and +Clown fish do not have gills. This +argument has the form of the law denying the consequent, so the conclusion is +Some fish do not have gills. +YOUR TURN 2.32 +Each pair of statements represents the premises in a logical argument. Based on these premises, apply the law of +denying the consequent to determine a valid conclusion. +1. If my classmate likes history, then some people like history. Nobody likes history. +2. If Homer does not like to read, then some people do not like reading. All people like reading. +3. If the polygon has five sides, then it is not an octagon. The polygon is an octagon. +Chain Rule for Conditional Arguments +The chain rule for conditional arguments is another form of a valid conditional argument. It is also called hypothetical +syllogism or the transitivity of implication. Recall that the conditional statement +can also be read as +implies . +This is where the name transitivity of implication comes from. The transitive property for numbers states that, if +and +then it follows that +The chain rule extends this property to conditional statements. If the premises of +the argument consist of two conditional statements, with the form “ +” and “ +” then it follows that +Symbolically, it has the form +. +112 +2 • Logic +Access for free at openstax.org + +Chain Rule for Conditional Arguments +Premise: +Premise: +Conclusion: +To verify the chain rule for conditional arguments, construct a truth table for the argument, +, and verify that it is a tautology. +T +T +T +T +T +T +T +T +T +T +F +T +F +F +F +T +T +F +T +F +T +F +T +T +T +F +F +F +T +F +F +T +F +T +T +T +T +T +T +T +F +T +F +T +F +F +T +T +F +F +T +T +T +T +T +T +F +F +F +T +T +T +T +T +To verify an argument of this form using a Venn diagram, again consider the premise +“If you play guitar, then you +are a musician,” but change the second premise to +“If you are a musician, then you are an artist.” In this case, the +set +of guitarists is a subset of the set +of artists, and it follows that if you are a guitarist, then you are an artist. +Therefore, the conclusion +follows from the premises and the chain rule for logical arguments is valid. See Figure +2.17. +Figure 2.17 +EXAMPLE 2.33 +Applying the Chain Rule for Conditional Arguments to Determine a Valid and Sound Conclusion +Each pair of statements represents true premises in a logical argument. Based on these premises, apply the chain rule +for conditional arguments to determine a valid and sound conclusion. +2.7 • Logical Arguments +113 + +1. +If my roommate goes to work, then my roommate will get paid. If my roommate gets paid, then my roommate will +pay their bills. +2. +If robins can fly, then some birds can fly. If some birds can fly, then we will watch birds fly. +3. +If Irma is a teacher, then Irma has a college degree. If Irma has a college degree, then Irma graduated from college. +Solution +1. +The premises are +“If my roommate goes to work, then they will get paid,” and +“If my roommate gets +paid, then my roommate will pay their bills.” This argument has the form of the chain rule for conditional +arguments, so the valid conclusion will have the form “ +” Because all the premises are true, the valid and +sound conclusion of this argument is: ���If my roommate goes to work, then my roommate will pay their bills.” +2. +The premises are +“If robins can fly, then some birds can fly,” and +“If some birds can fly, then we will +watch them fly.” This argument has the form of the chain rule for conditional arguments, so, the valid conclusion +will have the form “ +” Because all the premises are true, the valid and sound conclusion of this argument is: “If +robins can fly, then we will watch birds fly.” +3. +The premises are +(see line 1 of solution 1 and 2 above) “If Irma is a teacher, then Irma has a college degree,” +and +“If Irma has a college degree, then Irma graduated from college.” This argument has the form of the +chain rule for conditional arguments, so the valid conclusion will have the form “ +” Because all the premises +are true, the valid and sound conclusion of this argument is: “If Irma is a teacher, then Irma graduated from +college.” +YOUR TURN 2.33 +Each pair of statements represent true premises in a logical argument. Based on these premises, apply the chain +rule for conditional arguments to determine a valid and sound conclusion. +1. If my roommate does not go to work, then my roommate will not get paid. If my roommate does not get paid, +then they will not be able to pay their bills. +2. If penguins cannot fly, then some birds cannot fly. If some birds cannot fly, then we will watch the news. +3. If Marcy goes to the movies, then Marcy will buy popcorn. If Marcy buys popcorn, then she will buy water. +Check Your Understanding +34. A __________________ is a logical statement used as a fact to support the conclusion of an argument. +35. A logical argument is _______________ if its conclusion follows from the premises. +36. A logical argument that attempts to draw a more general conclusion from a pattern of specific premises is called +an _______________________ argument. +37. A _______________________ argument draws specific conclusions from more general premises. +38. Not all arguments are true. A false or deceptive argument is called a ___________________. +39. If an argument is valid and all of its premises are true, then it is considered ________________. +SECTION 2.7 EXERCISES +For the following exercises, analyze the argument and identify the form of the argument as the law of detachment, the +law of denying the consequent, the chain rule for conditional arguments, or none of these. +1. If Apple Inc. releases a new iPhone, then customers will buy it. Customers did not buy a new iPhone. Therefore, +Apple Inc. did not release a new iPhone. +2. In the animated movie Toy Story, if Paul Newman turned down the role of voicing Woody, then Tom Hanks was +chosen for the role. Tom Hanks was chosen as the voice for Woody, therefore, Paul Newman turned down the +role of voicing Woody in Toy Story. +3. +and +. +4. +and +5. +and +6. If all people are created equal, then all people are the same with respect to the law. If all people are the same +with respect to the law, then justice is blind. Therefore, if all people are created equal, then justice is blind. +114 +2 • Logic +Access for free at openstax.org + +7. If I mow the lawn, then my caregiver will pay me twenty dollars. I mowed the lawn. Therefore, my caregiver paid +me twenty dollars. +8. If Robin Williams was a comedian, then some comedians are funny. No comedians are funny. Therefore, Robin +Williams was not a comedian. +For the following exercises, each pair of statements represents the premises in a logical argument. Based on these +premises, apply the law of detachment to determine and write a valid conclusion. +9. +and +10. +and +11. If Richard Harris played Dumbledore, then Daniel Radcliffe played Harry Potter. Richard Harris played +Dumbledore. +12. If Emma Watson is an actor, then Emma Watson starred as Belle in the movie Beauty and the Beast. Emma +Watson is an actor. +13. If some Granny Smiths are available, then we will make an apple pie. Some Granny Smiths are available. +14. If Peter Rabbit lost his coat, then all rabbits must avoid Mr. McGregor’s garden. Peter Rabbit lost his coat. +For the following exercises, each pair of statements represents the premises in a logical argument. Based on these +premises, apply the law of denying the consequent to determine and write a valid conclusion. +15. If Greg and Ralph are friends, then Greg will not play a prank on Ralph. Greg played a prank on Ralph. +16. If Drogon is not a dragon, then Daenerys ruled Westeros. Daenerys did not rule Westeros. +17. +and +18. +and +19. If all dragons breathe fire, then rainwings are not dragons. Rainwings are dragons. +20. If some pirates have parrots as pets, then some parrots do not like crackers. All parrots like crackers. +For the following exercises, each pair of statements represent true premises in a logical argument. Based on these +premises, apply the chain rule for conditional arguments to determine a valid and sound conclusion. +21. +and +22. +and +23. +and +24. +and +25. If Mr. Spock is a science officer, then Montgomery Scott is an engineer. If Montgomery Scott is an engineer, then +James T. Kirk is the captain. +26. If Prince Charles is a character from Star Wars, then Luke Skywalker is not a Jedi. If Luke Skywalker is not a Jedi, +then Darth Vader is not his father. +For the following exercises, each pair of statements represent true premises in a logical argument. Based on these +premises, state a valid conclusion based on the form of the argument. +27. If my siblings drink milk out of the carton, then they will leave the carton on the counter. My siblings did not +leave the carton on the counter. +28. If my friend likes to bowl, then my partner does not like to play softball. My friend likes to bowl. +29. If mathematics is fun, then students will study algebra. If students study algebra, then they will score a 100 on +their final exam. +30. If all fleas bite and our dog has fleas, then our dog will scratch a lot. Our dog will not scratch a scratch a lot. +31. If the toddler is not tall, then they will use a stepladder to reach the cookie jar. If the toddler will use a +stepladder to reach the cookie jar, then they will drop the jar. If they drop the cookie jar, then they will not eat +any cookies. +32. If you do not like to dance, then you will not go to the club. You went to the club. +For the following exercises, use a truth table or construct a Venn diagram to prove whether the following arguments +are valid. +33. Denying the hypothesis: +and +Therefore, +34. Affirming the consequent: +and +Therefore, +35. +and +Therefore, +36. +and +Therefore, +2.7 • Logical Arguments +115 + +Chapter Summary +Key Terms +2.1 Statements and Quantifiers +• +logic +• +logical statement +• +truth values +• +symbolic form +• +negation of a logical statement +• +quantifier +• +premises +• +conclusion +• +inductive logical arguments +2.2 Compound Statements +• +compound statement +• +connective +• +conjunction +• +disjunction +• +conditional +• +hypothesis +• +conclusion +• +biconditional +• +dominance of connectives +2.3 Constructing Truth Tables +• +Truth table +• +Multiplication principle +• +Valid +2.5 Equivalent Statements +• +logically equivalent +• +tautology +• +inverse +• +converse +• +contrapositive +2.6 De Morgan’s Laws +• +Boolean logic +• +negation of a conditional +2.7 Logical Arguments +• +sound +• +fallacy +• +deductive arguments +• +law of detachment +• +law of denying the consequent +• +chain rule for conditional arguments +Key Concepts +2.1 Statements and Quantifiers +• +Logical statements have the form of a complete sentence and make claims that can be identified as true or false. +• +Logical statements are represented symbolically using a lowercase letter. +• +The negation of a logical statement has the opposite truth value of the original statement. +• +Be able to +◦ +Determine whether a sentence represents a logical statement. +116 +2 • Chapter Summary +Access for free at openstax.org + +◦ +Write and translate logical statements between words and symbols. +◦ +Negate logical statements, including logical statements containing quantifiers of all, some, and none. +2.2 Compound Statements +• +Logical connectives are used to form compound logical statements by using words such as and, or, and if …, then. +• +A conjunction is a compound logical statement formed by combining two statements with the words “and” or “but.” +If the two independent clauses are represented by +and , respectively, then the conjunction is written symbolically +as +. For the conjunction to be true, both +and +must be true. +• +A disjunction joins two logical statements with the or connective. In, logic or is inclusive. For an or statement to be +true at least one statement must be true, but both may also be true. +• +A conditional statement has the form if +, then , where +and +are logical statements. The only time the +conditional statement is false is when +is true, and +is false. +• +The biconditional statement is formed using the connective +for the biconditional statement to be true, +the true values of +and , must match. If +is true then +must be true, if +is false, then +must be false. +• +Translate compound statements between words and symbolic form. +Connective +Symbol +Name +and +but +conjunction +or +disjunction, inclusive or +not +~ +negation +if +, then implies +conditional, implication +if and only if +biconditional +• +The dominance of connectives explains the order in which compound logical statements containing multiple +connectives should be interpreted. +• +The dominance of connectives should be applied in the following order +◦ +Parentheses +◦ +Negations +◦ +Disjunctions/Conjunctions, left to right +◦ +Conditionals +◦ +Biconditionals +Figure 2.18 +2.3 Constructing Truth Tables +• +Determine the true values of logical statements involving negations, conjunctions, and disjunctions. +◦ +The negation of a logical statement has the opposite true value of the original statement. +◦ +A conjunction is true when both +and +are true, otherwise it is false. +2 • Chapter Summary +117 + +◦ +A disjunction is false when both +and +are false, otherwise it is true. +• +Know how to construct a truth table involving negations, conjunctions, and disjunctions and apply the dominance of +connectives to determine the truth value of a compound logical statement containing, negations, conjunctions, and +disjunctions. +Negation +Conjunction (AND) +Disjunction (OR) +T +F +T +T +T +T +T +T +F +T +T +F +F +T +F +T +F +T +F +F +T +T +F +F +F +F +F +F +• +A logical statement is valid if it is always true. Know how to construct a truth table for a compound statement and +use it to determine the validity of compound statements involving negations, conjunctions, and disjunctions. +2.4 Truth Tables for the Conditional and Biconditional +• +The conditional statement, if +then , is like a contract. The only time it is false is when the contract has been +broken. That is, when +is true, and +is false. +Conditional +T +T +T +T +F +F +F +T +T +F +F +T +• +The biconditional statement, +if and only if , it true whenever +and +have matching true values, otherwise it is +false. +Biconditional +T +T +T +T +F +F +F +T +F +F +F +T +• +Know how to construct truth tables involving conditional and biconditional statements. +• +Use truth tables to analyze conditional and biconditional statements and determine their validity. +118 +2 • Chapter Summary +Access for free at openstax.org + +2.5 Equivalent Statements +• +Two statements +and +are logically equivalent if the biconditional statement, +is a valid argument. That is, the +last column of the truth table consists of only true values. In other words, +is a tautology. Symbolically, +is +logically equivalent to +is written as: +• +A logical statement is a tautology if it is always true. +• +To be valid a local argument must be a tautology. It must always be true. +• +Know the variations of the conditional statement, be able to determine their truth values and compose statements +with them. +• +The converse of a conditional statement, if +then , is the statement formed by interchanging the hypothesis and +conclusion. It is the statement if +then +. +• +The inverse of a conditional statement if formed by negating the hypothesis and the conclusion of the conditional +statement. +• +The contrapositive negates and interchanges the hypothesis and the conclusion. +Conditional +Contrapositive +Converse +Inverse +T +T +F +F +T +T +T +T +T +F +F +T +F +F +T +T +F +T +T +F +T +T +F +F +F +F +T +T +T +T +T +T +• +The conditional statement is logically equivalent to the contrapositive. +• +The converse is logically equivalent to the inverse. +• +Know how to construct and use truth tables to determine whether statements are logically equivalent. +2.6 De Morgan’s Laws +• +De Morgan’s Law for the negation of a disjunction states that, +is logically equivalent to +• +De Morgan’s Law The negation of a conjunction states that, +• +Use De Morgan’s Laws to negate conjunctions and disjunctions. +• +The negation of a conditional statement, if +then +is logically equivalent to the statement +and not . Use this +property to write the negation of conditional statements. +• +Use truth tables to evaluate De Morgan’s Laws. +2.7 Logical Arguments +• +A logical argument uses a series of facts or premises to justify a conclusion or claim. It is valid if its conclusion +follows from the premises, and it is sound if it is valid, and all of its premises are true. +• +The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, +is +true and the hypothesis, +is true, then the conclusion +must also be true. +Law of Detachment +Premise: +Premise: +Conclusion: +• +Know how to apply the law of detachment to determine the conclusion of a pair of statements. +• +The law of denying the consequent is a valid form of a conditional argument that asserts that if both the +conditional, +is true and the negation of the conclusion, +is true, then the negation of the hypothesis +2 • Chapter Summary +119 + +must also be true. +Law of Denying the Consequent +Premise: +Premise: +Conclusion: +• +Know how to apply the law of denying the consequent to determine the conclusion for pairs of statements. +• +The chain rule for conditional arguments is a valid form of a conditional argument that asserts that if the premises +of the argument have the form, +and +, then it follows that +Chain Rule for Conditional Arguments +Premise: +Premise: +Conclusion: +• +Know how to apply the chain rule to determine valid conclusions for pairs of true statements. +Videos +2.1 Statements and Quantifiers +• +Logic Part 1A: Logic Statements and Quantifiers (https://openstax.org/r/Logic_Statements_and_Quantifiers) +2.2 Compound Statements +• +Logic Part 1B: Compound Statements, Connectives and Symbols (https://openstax.org/r/Compound_Statements) +2.3 Constructing Truth Tables +• +Logic Part 2: Truth Values of Conjunctions: Is an "AND" statement true or false? (https://openstax.org/r/ +Truth_Values_of_Conjunctions) +• +Logic Part 3: Truth Values of Disjunctions: Is an "OR" statement true or false? (https://openstax.org/r/ +Truth_Values_of_Disjunctions) +• +Logic Part 4: Truth Values of Compound Statements with "and", "or", and "not" (https://openstax.org/r/opL9I4tZCC0) +• +Logic Part 5: What are truth tables? How do you set them up? (https://openstax.org/r/-tdSRqLGhaw) +• +Logic Part 6: More on Truth Tables and Setting Up Rows and Column Headings (https://openstax.org/r/j3kKnUNIt6c) +2.4 Truth Tables for the Conditional and Biconditional +• +Logic Part 8: The Conditional and Tautologies (https://openstax.org/r/Conditional_and_Tautologies) +• +Logic Part 11B Biconditional and Summary of Truth Value Rules in Logic (https://openstax.org/r/omKzui0Fytk) +• +Logic Part 13: Truth Tables to Determine if Argument is Valid or Invalid (https://openstax.org/r/AQB3svnxxiw) +2.7 Logical Arguments +• +Logic Part 14: Common Argument Forms like Modus Ponens and Tollens (https://openstax.org/r/ +Modus_Ponens_and_Tollens) +Projects +Logic Gates +Logic gates are the basis for all digital circuits. +1. +Research and document the following terms: logic gate, OR gate, AND gate, and NOT gate. +2. +Construct a diagram of a NAND gate, NOR gate, and a XOR gate by using at least two of the following gates: AND, +OR, and NOT. +120 +2 • Chapter Summary +Access for free at openstax.org + +3. +Digital electronics use a 1 for true or on, and a 0 for false or off. Create a truth table documenting all possible cases +using 0s and 1s for the NAND gate, NOR gate and XOR gate. +4. +Use a truth table to explain how XOR is related to the biconditional statement. +Logical Fallacies +Fallacies are false or deceptive logical arguments. +1. +Research and document the structure of five of the following named fallacies: hasty generalization, limited choice, +false cause, appeal to popularity, appeal to emotion, appeal to authority, personal attack, gamblers' ruin, slippery +slope, and circular reasoning. +2. +Create a presentation highlighting one of the five fallacies researched in the previous question. The presentation +must include an introductory slide with the title of the fallacy and the form or structure of the argument. The second +slide must include an example of this fallacy as used in a commercial, a political cartoon or a current event or new +article. The third slide must include an explanation of why the example on slide to is a representative example of the +fallacy. The last slide must include citations for any materials used. No textbooks should be used as reference. +Careers in Logic +Lawyers, mathematicians, and computer programmers are a few of the careers that require knowledge of logic. +1. +What career are you interested in? Research how knowledge of logic applies to your chosen field of study. Then, +write a cover letter for a position in your field you'd like to apply to. In the cover letter, include how your knowledge +of logic qualifies you for the position you are applying for. If you do not think logic is important for your given career +choice, find a position where logic is an essential element of the position and complete the project by pretending +you are writing a cover letter for that job. +2 • Chapter Summary +121 + +Chapter Review +Statements and Quantifiers +Fill in the blanks to complete the following sentences. +1. The ______________ of a logical statement has the opposite truth value of the original statement. +2. _______________ are logical statements presented as the facts used to support the conclusion of a logical +argument. +Determine whether each of the following sentences represents a logical statement, also called a proposition. If it is a +logical statement, determine whether it is true or false. +3. Where is the restroom? +4. No even numbers are odd numbers. +5. +Write the negation of each following statement symbolically and in words. +6. +Pink Floyd’s album The Wall is not a rock opera. +7. +Some dogs are Labrador retrievers. +8. +Some universities are not expensive. +Draw a logical conclusion to the following arguments, and include in both one of the following quantifiers: all, some, or +none. +9. Spaghetti noodles are made with wheat, ramen noodles are made with wheat, and lo mein noodles are made +with wheat. +10. A Porsche Boxster does not have four doors, a Volkswagen Beetle does not have four doors, and a Mazda Miata +does not have four doors. +Compound Statements +Fill in the blanks to complete the following sentences. +11. ___________________ are words or symbols used to join two or more logical statement together to from a +compound statement. +12. __________________ and __________________ have equal dominance and are evaluated from left to right when no +parentheses are present in a compound logical statement. +Translate each compound statement below into symbolic form. +Given: +“Tweety Bird is a bird,” +“Bugs is a bunny,” +“Bugs says, ‘What’s up, Doc?’,” +“Sylvester is a cat,” and +“Sylvester chases Tweety Bird.” +13. If Tweety Bird is a bird, then Sylvester will not chase him. +14. Tweety Bird is a bird and Sylvester chases him if and only if Bugs says, “What’s up Doc?” +Translate the symbolic form of each compound logical statement below into words. +Given: +“Tweety Bird is a bird,” +“Bugs is a bunny,” +“Bugs says, ‘What’s up, Doc?’,” +“Sylvester is a cat,” and +“Sylvester chases Tweety Bird.” +15. +16. +For each of the following compound logical statements, apply the proper dominance of connectives by adding +parentheses to indicate the order to evaluate the statement. +17. +18. +Constructing Truth Tables +Fill in the blanks to complete the sentences. +19. A ______________ is true if at least one of its component statements is true. +20. For a ____________________ to be true, all of its component statements must be true. +Given the statements, +“No fish are mammals,” +“All lions are cats,” and +“Some birds do not lay eggs,” construct a +truth table to determine the truth value of each compound statement below. +21. +22. +23. +122 +2 • Chapter Summary +Access for free at openstax.org + +Construct a truth table to analyze all the possible outcomes of the following statements, and determine whether the +statements are valid. +24. +25. +Truth Tables for the Conditional and Biconditional +Fill in the blanks to complete the following sentences. +26. If the ________, +, of a conditional statement is true, then the conclusion, , must also be true for the conditional +statement +to be true. +27. The biconditional statement +is __________ whenever the truth value of +matches the truth value of , +otherwise it is _________. +Complete the truth tables below to determine the truth value of the proposition in the last column. +28. +F +F +T +29. +T +F +Assume the following statements are true. +“Poof is a baby fairy,” +“Timmy Turner has fairly odd parents,” +“Cosmo +and Wanda will grant Timmy’s wishes,” and +“Timmy Turner is 10 years old.” Translate each of the following +statements into symbolic form, then determine its truth value. +30. If Timmy Turner is 10 years old and Poof is not a baby fairy, then Timmy Turner has fairly odd parents. +31. Cosmos and Wanda will not grant Timmy’s wishes if and only if Timmy Turner is 10 years old or he does not +have fairly odd parents. +32. Construct a truth table to analyze all the possible outcomes and determine the validity of the following argument. +Equivalent Statements +Fill in the blanks to complete the sentences below. +33. The _________________ is logically equivalent to the inverse +34. The _________________ is logically equivalent to the conditional +Use the conditional statement, +“If Novak makes the basket, then Novak’s team will win the game," to answer the +following questions. +35. Write the conclusion of the conditional statement in words and label it appropriately. +36. Write the hypothesis of the conditional statement in words and label it appropriately. +37. Identify the following statement as the converse, inverse, or contrapositive: “If Novak does not make the +basket, then his team will not win the game.” +38. Identify the following statement as the converse, inverse, or contrapositive: “If Novak’s team wins the game, +then he made the basket.” +De Morgan’s Laws +Fill in the blanks to complete the sentences. +39. De Morgan’s Law for the negation of a disjunction states that +____________. +40. De Morgan’s Law for the negation of a conjunctions states that _____________ +41. Apply De Morgan’s Law to write the statement without parentheses: +. +42. Apply the property for the negation of a conditional to write the statement as a conjunction or disjunction: +. +43. Write the negation of the conditional statement in words: If Thomas Edison invented the phonograph, then +albums are made of vinyl, or the transistor radio was the first portable music device. +44. Construct a truth table to verify that the logical property is valid: +. +2 • Chapter Summary +123 + +Logical Arguments +Fill in the blanks to complete the sentences below. +45. The _____ ___ __________ is a valid logical argument with premises, +and +, used to support the conclusion, +46. The chain rule for conditional arguments states that the ___________________ property applies to conditional +arguments, so that: +Assume each pair of statements represents true premises in a logical argument. Based on these premises, state a valid +conclusion that is consistent with the form of the argument. +47. If the Tampa Bay Buccaneers did not win Super Bowl LV, then Tom Brady was not their quarterback. Tom Brady +was the Tampa Bay Buccaneers quarterback. +48. If +, then +and if , then +. +49. If Kamala Harris is the vice president of the United States, then Kamala Harris is the president of the U.S. +Senate. Kamala Harris is the vice president of the United States. +50. Construct a truth table or Venn diagram to prove whether the following argument is valid. If the argument is valid, +determine whether it is sound. +If all frogs are brown, then Kermit is not a frog. Kermit is a frog. Therefore, some frogs are not brown. +Chapter Test +Determine whether each of the following sentences represent a logical statement. If it is a logical statement, determine +whether it is true or false. +1. +2. Please, sit down over there. +3. All mammals lay eggs. +Write the negation of each statement below. +4. Some monkeys do not have tails. +5. +6. If the plumber does not remove the clog, then the homeowner will not pay the plumber. +Given: +Frodo is a hobbit, +Gandalf is a wizard, +Frodo and Samwise will take the ring to Mordor, and +Gollum will +help Frodo get into Mordor. +Translate the symbolic form of each compound logical statement into words. +7. +8. +Translate the written form of each compound logical statement into symbolic form. +9. Frodo and Samwise will take the ring to Mordor or Gandalf is not a wizard and Frodo is a hobbit. +10. If Gollum will not help Frodo get into Mordor, then Gandalf is not a wizard and Frodo is not a hobbit. +For each of the following compound logical statements, apply the proper dominance of connectives by adding +parentheses to indicate the order in which the statement must be evaluated. +11. +12. +13. Complete the truth table to determine the truth value of the proposition in the last column. +F +T +Given the true statements +“A right triangle has one 90-degree angle," +"The triangle is a right triangle," +" +" and +"The longest side of a triangle is +implies +must be +." Write each of the following +compound statements in symbolic form, then construct a truth table to determine the truth value of the compound +statement. +14. If a triangle is a right triangle, then it does not have one 90-degree angle or +15. The triangle is a right triangle, or a right triangle does not have a 90-degree angle, if and only if it is not the +case that the longest side of a triangle is +implies +must be +. +124 +2 • Chapter Summary +Access for free at openstax.org + +Use the conditional statement, +"If Phil Mickelson is 50 years old, then Phil Mickelson won the Player’s +Championship," to answer the following questions. +16. Write the converse statement in words. +17. If the conditional statement is true, and the hypothesis is true, what is a valid conclusion to the argument? +18. If the conditional statement is true, and the conclusion is false, what is a valid conclusion to the argument? +19. Construct a truth table to analyze all the possible outcomes and determine the validity of the following argument. +20. Construct a truth table or Venn diagram to prove whether the following argument is valid. If the argument is valid, +determine whether it is sound. +If John Mayer played MTV unplugged, then some guitars are acoustic. John Mayer played MTV unplugged. +Therefore, some guitars are acoustic. +2 • Chapter Summary +125 + +126 +2 • Chapter Summary +Access for free at openstax.org + +Figure 3.1 Encryption of computers and messages use very large prime numbers. (credit: modification of work "Jefferson +cylinder cipher (replica)" by Daderot/Wikimedia Commons, Public Domain) +Chapter Outline +3.1 Prime and Composite Numbers +3.2 The Integers +3.3 Order of Operations +3.4 Rational Numbers +3.5 Irrational Numbers +3.6 Real Numbers +3.7 Clock Arithmetic +3.8 Exponents +3.9 Scientific Notation +3.10 Arithmetic Sequences +3.11 Geometric Sequences +Introduction +Encryption is used to secure online banking, for secure online shopping, and for browsing privately using VPNs (Virtual +Private Networks). We need encryption (using prime numbers) for a secure exchange of information. For a prime +number to be useful for encryption, though, it has to be large. Encryption uses a composite number that is the product +of two very large primes. In order to break the encryption, one must determine the two primes that were used to form +the composite number. If the two prime numbers used are sufficiently large, even the fastest computer cannot +determine those two prime numbers in a reasonable amount of time. It would take a computer 300 trillion years to crack +the current encryption standard. +3 +REAL NUMBER SYSTEMS AND +NUMBER THEORY +3 • Introduction +127 + +3.1 Prime and Composite Numbers +Figure 3.2 Computers are protected using encryption based on prime numbers. (credit: “Data Security” by +Blogtrepreneur/Flickr, CC BY 2.0) +After completing this section, you should be able to: +1. +Apply divisibility rules. +2. +Define and identify numbers that are prime or composite. +3. +Find the prime factorization of composite numbers. +4. +Find the greatest common divisor. +5. +Use the greatest common divisor to solve application problems. +6. +Find the least common multiple. +7. +Use the least common multiple to solve application problems. +Encryption, which is needed for the secure exchange of information (i.e., online banking or shopping) is based on prime +numbers. Encryption uses a composite number that is the product of two very large prime numbers. To break the +encryption, the two primes that were used to form the composite number need to be determined. If the two prime +numbers used are sufficiently large, even the fastest computer cannot determine those two prime numbers in a +reasonable amount of time. It would take a computer 300 trillion years to crack the current encryption standard. +Applying Divisibility Rules +Before we begin our investigation of divisibility, we need to know some facts about important sets of numbers: +• +The counting numbers are referred to as the natural numbers. This set of numbers, +, is denoted with +the symbol ℕ . +• +Another important set of numbers is the integers. The integers are the natural numbers, along with 0, and the +negatives of the natural numbers. This set is often written as +. We denote the integers +with the symbol ℤ. +• +Notice that ℕ is a proper subset of ℤ, or, ℕ ⊂ ℤ. All the ideas of this section apply to the natural numbers, while only +some apply to all the integers. +Divisibility is when the integer +is divisible by +, if +can be written as +times another integer. Equivalently, there is no +remainder when +is divided by +. There are many occasions when separating items into equal groups comes into play +to ensure an equal distribution of whole items. For example, Francis, a preschool art teacher, has 15 students in one +class. Francis has 225 sheets of construction paper and wants to provide each student with an equal number of pieces. +To know if he will use all the construction paper, Francis is really asking if 225 can be evenly divided into 15 groups. +128 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +EXAMPLE 3.1 +Determining if a Number Divides Another Number +Determine if 36 is divisible by 4. +Solution +We could divide 36 by 4 and see if there is a remainder, or we could see if we can write 36 as 4 times another integer. If +we divide 36 by 4, we see +with no remainder. We see that 36 is divisible by 4. We can write 36 as 4 times +another integer, +. By the definition of divisibility, 36 is divisible by 4. +YOUR TURN 3.1 +1. Determine if 54 is divisible by 9. +You can quickly check if a number is divisible by 2, 3, 4, 5, 6, 9, 10, and 12. Each has an easy-to-identify feature, or rule, +that indicates the divisibility by those numbers, as shown in the following table. +Divisor +Rule +2 +Last digit is even +3 +Add the digits of the number together. If that sum is divisible by 3, then so is the original number +4 +Look at only the last two digits. If this number is divisible by 4, so is the original number +5 +Look at only the last digit. If it is a 5 or a 0, then the original number is divisible by 5 +6 +If the number passes the rule for divisibility by 2 and for 3, then the number is divisible by 6 +9 +Add the digits of the number together. If that number is divisible by 9, then so is the original number +10 +Look at only the last digit. If it is a 0, then the original number is divisible by 10 +12 +If the number passes the rule for 3 and 4, the number is divisible by 12 +EXAMPLE 3.2 +Using Divisibility Rules +Using divisibility rules, determine if 245 is divisible by 5. +Solution +Since the last digit is a 5, the number 245 is divisible by 5 because the rule states if the last digit of the number is a 5 or a +0, then the original number is divisible by 5. +YOUR TURN 3.2 +1. Using divisibility rules, determine if 45,730 is divisible by 5. +3.1 • Prime and Composite Numbers +129 + +EXAMPLE 3.3 +Using Divisibility Rules +Using divisibility rules, determine if 25,983 is divisible by 9. +Solution +The divisibility rule for 9 is when the digits of the number are added, the sum is divisible by 9. So, we calculate the sum of +the digits. +. Since 27 is divisible by 9, so is the original number 25,983. +YOUR TURN 3.3 +1. Using divisibility rules, determine if 342,887 is divisible by 9. +EXAMPLE 3.4 +Using Divisibility Rules +Can 298 coins be stacked into 6 stacks with an equal number of coins in each stack? +Solution +In order for the coins to be in equal-sized stacks, 298 would need to be divisible by 6. The divisibility rule for 6 is that the +number passes the divisibility rules for both 2 and 3. Since the last digit is even, 298 is divisible by 2. To determine if 298 +is divisible by 3, we first add the digits of the number: +. Since 19 is not divisible by 3, neither is 298. +Because 298 is not divisible by 3, it is also not divisible by 6, which means they cannot be put into 6 equal stacks of coins. +YOUR TURN 3.4 +1. Can 43,568 pieces of mail be separated into 6 bins with the same number of pieces of mail per bin? +EXAMPLE 3.5 +Using Divisibility Rules +Using divisibility rules, determine if 4,259 is divisible by 10. +Solution +The divisibility rule for 10 is that the last digit of the number is 0. Since the last digit of 4,259 is not 0, then 4,259 is not +divisible by 10. +YOUR TURN 3.5 +1. Using divisibility rules, determine if 87,762 is divisible by 10. +EXAMPLE 3.6 +Using Divisibility Rules +Using divisibility rules, determine if 936,276 is divisible by 4. +Solution +The divisibility rule for 4 is to check the last two digits of the number. If the number formed by the last two digits of the +original number is divisible by 4, then so is the original number. The last two digits make the number 76 and 76 is +130 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +divisible by 4, since +. Since 76 is divisible by 4, so is 936,276. +YOUR TURN 3.6 +1. Using divisibility rules, determine if 43,568 is divisible by 4. +VIDEO +Divisibility Rules (https://openstax.org/r/Divisibility_Rules) +Prime and Composite Numbers +Sometimes, a natural number has only two unique divisors, 1 and itself. For instance, 7 and 19 are prime. In other +words, there is no way to divide a prime number into groups with an equal number of things, unless there is only one +group, or those groups have one item per group. Other natural numbers have more than two unique divisors, such as 4, +or 26. These numbers are called composite. The number 1 is special; it is neither prime nor composite. +To determine if a number is prime or composite, you have to determine if the number has any divisors other than 1 and +itself. The divisibility rules are useful here, and can quickly show you if a number has a divisor on that list. +However, if none of those divide the number, you still have to check all other possible prime divisors. What are the prime +numbers that are possibly divisors of the number you are checking? You need only check the prime numbers up to the +square root of the number in question. For instance, if you want to know if 2,117 is prime, you need to determine if any +primes up to the square root of 2,117 (which is 46.0 when rounded to one decimal place) divide 2,117. If any of those +primes are divisors of the number in question, then the number is composite. If none of those primes work, then the +number is itself prime. +We can check divisibility with whatever tool we wish. Divisibility rules are quick for some prime divisors (2 and 5 come to +mind) but aren't quick for other values (like 11). In place of divisibility rules, we could just use a calculator. If the prime +number divides the number evenly (that is, there is no decimal or fractional part), then the number is divisible by that +prime. Table 3.1 is a quick list of the prime numbers up to 50. There are 15 prime numbers less than 50. +2 +3 +5 +7 +11 +13 +17 +19 +23 +29 +31 +37 +41 +43 +47 +Table 3.1 Prime Numbers +Less than 50 +EXAMPLE 3.7 +Determining If a Number Is Prime or Composite +Determine if 2,117 is prime or composite. +Solution +The square root of 2,117 is 46.0 (rounded to one decimal place). So, we need to check if 2,117 is divisible by any prime up +to 46. +Step 1: First we’ll use the rules of divisibility we learned earlier: +• +We can tell 2,117 is not divisible by 2, as the last digit isn't even. +• +2,117 is not divisible by 5 (the last digit isn't 0 or 5). +• +Add the digits of 2,117 to get 11, which is not divisible by 3. So, 2,117 is also not divisible by 3. +Step 2: Now we repeat the process for all the primes up to 46. +3.1 • Prime and Composite Numbers +131 + +Using a calculator, we find that 2,117 divided by the prime numbers 7, 11, 13, 17, 19, and 23 results in a remainder, a +decimal part. So, we know that 2,117 is not divisible by these prime numbers. (You should check these results yourself.) +Moving on, we check the next prime: 29. Using the calculator to divide 2,117 by 29 results in 73. Since there is no decimal +part, 2,117 is divisible by 29. +This means that 2,117 is not a prime number, but rather, a composite number. Writing 2,117 as the product of 29 and +another natural number, +. +YOUR TURN 3.7 +1. Determine if 1,429 is prime or composite. +EXAMPLE 3.8 +Determining if a Number Is Prime or Composite +Determine if 423 is prime or composite. +Solution +The square root of 423 is 20.57 (rounded to two decimal places). So, we need to check if 423 is divisible by any prime up +to 20. +Step 1: Check 2. We can tell 423 is not divisible by 2, as the last digit isn't even. +Step 2: Check 5. It is not divisible by 5 (the last digit isn't 0 or 5). +Step 3: Check 3. To check if 423 is divisible by 3, we use the divisibility rule for 3. When we take the sum of the digits of +423, the result is 9. Since 9 is divisible by 3, so is 423. +Since 423 is divisible by 3, then 423 is a composite number. Writing 423 as the product of 3 and another natural number, +. +YOUR TURN 3.8 +1. Determine if 859 is prime or composite. +EXAMPLE 3.9 +Determining if a Number Is Prime or Composite +Determine if 1,034 is prime or composite +Solution +A quick inspection of 1,034 shows it is divisible by 2 since the last digit is even, and so 1,034 is a composite number. +YOUR TURN 3.9 +1. Determine if 5,067,322 is prime or composite. +EXAMPLE 3.10 +Determining if a Number Is Prime or Composite +Determine if 2,917 is prime or composite. +132 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Solution +The square root of 2,917 is 50.01 (rounded to two decimal places). So, we need to check if 2,917 is divisible by any prime +up to 50. +Step 1: Check 2. We can tell 2,917 is not divisible by 2, as the last digit isn't even. +Step 2: Check 5. It is it divisible by 5 (the last digit isn't 0 or 5). +Step 3: Check 3. Using the divisibility rule for 3, we take the sum of the digits of 2,917, which is 19. Since 19 is not +divisible by 3, neither is 2,917. +Step 4: Check the rest of the primes up to 50 using a calculator. When 2,917 is divided by every prime number up to 50, +the result has a decimal part. +Since no prime up to 50 divides 2,917, it is a prime number. +YOUR TURN 3.10 +1. Determine if 1,477 is prime. +WHO KNEW? +ILLEGAL PRIMES +Large primes are a hot commodity. Using two very large primes (some have more than 22 million digits!) is necessary +for secure encryption. Anyone who has a new prime that is large enough can use that prime to create a new +encryption. Of course, whoever discovers a large prime could sell it to a security company. These primes are so useful +for encryption, it is necessary to protect that intellectual property. In fact, at least one prime number was declared +illegal. +VIDEO +Illegal Prime Number (https://openstax.org/r/Illegal_Prime_Number) +3.1 • Prime and Composite Numbers +133 + +PEOPLE IN MATHEMATICS +Sophie Germain +Figure 3.3 Sophie Germain (credit: “Sophie Germain at 14 years,” Illustration from histoire du socialisme, approx. +1880, Wikimedia Commons, public domain) +Born into a wealthy French family in 1776, Sophie Germain discovered and fell in love with mathematics by browsing +her father’s books. Clandestine study, hard and tenacious work, and a mathematical mindset did not lead to college, +however, as she was not allowed to attend. She did manage, through friends, to obtain problem sets and submit +brilliant solutions under the name Monsieur LaBlanc. One of her great interests was number theory, which is the +study of properties of integers. One of her theorems, titled “Sophie Germain’s Theorem,” partially solved one of the +great mathematical mysteries, Fermat’s Last Theorem. From this she discovered what are now known as Sophie +Germain Primes. A Sophie Germain Prime is a prime number that can be written in the form +, where +is a +prime number. For instance, 23 is prime: +, which is prime, so 47 is a Sophie Germain Prime. It should be +noted that +, where +is a prime number, may or may not be prime (check for +!). +Finding the Prime Factorization of Composite Numbers +Before we can start with prime factorization, let’s remind ourselves what it means to factor a number. We factor a +number by identifying two (or more) numbers that, when multiplied, result in the original number. For instance, 3 and +24, when multiplied, give 72. So, 72 can be factored into +. Notice that we could have factored the 72 differently, say +as +, or +, or even as +. +Finding the prime factorization of a composite number means writing the number as the product of all of its prime +factors. For example, +. Notice that all the numbers being multiplied on the right-hand side are +prime numbers. Sometimes prime numbers repeat themselves in the factorization. When prime factors do repeat, we +may write the prime factorization using exponents, as in +. In that equation, the 2 is raised to the 4th power. +The 4 is the exponent, and the 2 is the base. More generally, in the exponent notation +, the number represented by +is the base, and the number represented by +is the exponent. +One has to wonder if finding the prime factorization could result in different factorizations. The Fundamental Theorem +of Arithmetic tells us that there is only one prime factorization for a given natural number. +134 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Fundamental Theorem of Arithmetic +Every natural number, other than 1, can be expressed in exactly one way, apart from the arrangement, as a product of +primes. +The process of finding the prime factorization of a number is iterative, which means we do a step, then repeat it until we +cannot do the step any longer. The step we use is to identify one prime factor of the number, then write the number as +the prime factor times another factor. We repeat this step on the other, newly found, factor. This step is repeated until no +more primes can be factored from the remaining factor. This is easier to see and explain with an example. +EXAMPLE 3.11 +Finding the Prime Factorization +Find the prime factorization of 140. +Solution +Step 1: Identify a prime number that divides 140. Since 140 is even (the last digit is even), 2 divides 140. We then factor +the 2 out of the 140, giving us +. +Step 2: With the other factor, 70, find a prime factor of 70. Since 70 is also even, 2 divides 70. We factor the 2 out of the +70 and the factorization is now +. +Notice that we expressed the two factors of 2 as +. +Step 3: Look to the remaining factor, 35. The last digit of 35 is 5, so 5 is a factor of 35. We factor the 5 out of the 35. The +factorization is now +. +Step 4: Look to the remaining factor, 7. Since 7 is prime, the process is complete. +The prime factorization of 140 is +. +YOUR TURN 3.11 +1. Find the prime factorization of 90. +Factor Trees +A useful tool for helping with prime factorization is a factor tree. To create a factor tree for the natural number +(where +is not 1), perform the following steps: +Step 1: If +is prime, you're done. If +is composite, continue to the next step. +Step 2: Identify two divisors of +, call them +and +. +Step 3: Write the number +down, and draw two branches extending down (or to the right) of the number +. +Step 4: Label the end of one branch +, the other as +. See Figure 3.4. +Figure 3.4 +Step 5: If +and +are prime, the tree is complete. When a number at this step is a prime number, we refer to it as a leaf +of the tree diagram. +Step 6: If either +or +are composite, repeat Steps 2 through 4 for +and +. +Step 7: The process stops when the leaves are all prime. +Step 8: The prime factorization is then the product of all the leaves. +This is best seen in an example. +3.1 • Prime and Composite Numbers +135 + +EXAMPLE 3.12 +Finding the Prime Factorization +Find the prime factorization of 66. +Solution +Since 66 is even, 2 is a factor. +Step 1: Factor out the 2. The factorization is +. The factor tree is shown in Figure 3.5. +Figure 3.5 +Since 2 is a factor, that branch is done, and 2 is a leaf. +Step 2: The 33, though, is divisible by 3, and is the product of 3 and 11. We attach that to the factor tree (Figure 3.6). +Figure 3.6 +Since the 2, 3 and 11 are all prime, the factor tree is done. +The prime factorization of 66 is the product of the leaves, so +. The factorization is complete. +YOUR TURN 3.12 +1. Find the prime factorization of 85. +VIDEO +Using a Factor Tree to Find the Prime Factorization (https://openstax.org/r/Prime_Factorization) +EXAMPLE 3.13 +Finding the Prime Factorization +Find the prime factorization of 135. +Solution +The number 135 is divisible by 3, and so 3 is a factor of 135. +Step 1: Factor out the 3. The factorization is +. Using the factor tree (Figure 3.7), +136 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Figure 3.7 +45 is also divisible by 3. +Step 2: Factor out a 3 from 45. The other factor is 15. The factor tree is shown in Figure 3.8. +Figure 3.8 +15 is also divisible by 3. +Step 3: The factors of 15 are 3 and 5. The factor tree is shown in Figure 3.9. +Figure 3.9 +All the leaves are prime, so the process is complete. The prime factorization of 135 is +. +YOUR TURN 3.13 +1. Find the prime factorization of 280. +EXAMPLE 3.14 +Identifying Prime Factors +How many different prime factors does 10,241 have? +Solution +To know how many different prime factors 10,241 has, we need the prime factorization of 10,241. +Step 1: Use divisibility rules, to see that the number 10,241 is not divisible by 2 or by 3 (the sum of the digits is 8), or by 5. +However, it is divisible by 7. +3.1 • Prime and Composite Numbers +137 + +Step 2: After factoring the 7, the factorization is +; 1,463 is also divisible by 7. +Step 3: After factoring the 7, the factorization is +. The number 209 is not divisible by 7. +Step 4: Check the next prime number: 11; 11 does divide 1,463. +Step 5: After factoring the 11, the factorization is +. +Since 19 is prime, the prime factorization of 10,241 is complete. We see that 10,241 has three different prime factors: 7, +11, and 19. +YOUR TURN 3.14 +1. Find the number of different prime factors of 180. +VIDEO +Finding the Prime Factorization of 168 (https://openstax.org/r/Prime_Factorization_of_168) +TECH CHECK +Using Wolfram Alpha to Find Prime Factorizations +The Wolfram Alpha website (https://openstax.org/r/wolframalpha) is a powerful resource available for free to use. It is +designed using AI so that it understands natural language requests. For instance, typing the question “What is the +prime factorization of 543,390?” gets a rather quick answer of +. So, if you want to find the prime +factorization of a number, you can simply ask Wolfram Alpha to find the prime factorization of the number. +Finding the Greatest Common Divisor +Two numbers often have more than one divisor in common (all pairs of natural numbers have the common divisor 1). +When listing the common divisors, it’s often the case that the largest is of interest. This divisor is called the greatest +common divisor and is denoted GCD. It is also sometimes referred to as the greatest common factor (GCF). +For instance, 6 is the greatest common divisor of 12 and 18. We can see this by listing all the divisors of each number +and, by inspection, select the largest value that shows up in each list. +The divisors of 12 +1, 2, 3, 4, 6, 12 +The divisors of 18 +1, 2, 3, 6, 9 +It is easy to see that 6 is the largest value that appears in both lists. +EXAMPLE 3.15 +Finding the Greatest Common Divisor Using Lists +Find the greatest common divisor of 1,400 and 250 by listing all divisors of each number. +Solution +We create a list of all the divisors of 1,400 and of 250, and choose the largest one. +The divisors of 1,400 are +1, 2, 4, 5, 7, 8, 10, 14, 20, 25, 28, 35, 40, 50, 56, 70, 100, 140, 175, 200, 280, 350, 700, 1,400. +The divisors of 250 are +138 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +1, 2, 5, 10, 25, 50, 125, 250. +The largest value that appears on both lists is 50, so the greatest common divisor of 1,400 and 250 is 50. +YOUR TURN 3.15 +1. Find the greatest common divisor of 270 and 99 by listing all divisors of each number. +Listing all the divisors of the numbers in the set will always work, but for some relatively small numbers, the set of all +divisors can become pretty big, and finding them all can be a chore. Another approach to finding the greatest common +divisor is to use the prime factorization of the numbers. To do so, use the following steps: +Step 1: Find the prime factorization of the numbers. +Step 2: Identify the prime factors that appear in every number’s prime factorization. These are called the common prime +factors. +Step 3: Identify the smallest exponent of each prime factor identified in Step 2 in the prime factorizations. +Step 4: Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. The result is the greatest +common divisor. +EXAMPLE 3.16 +Finding the Greatest Common Divisor Using Prime Factorization +Find the greatest common divisor of 1,400 and 250 by using their prime factorizations. +Solution +Step 1: Find the prime factorizations of the numbers. +The prime factorization of 1,400 is +. +The prime factorization of 250 is +. +Step 2: Identify the prime factors that appear in every number’s prime factorization. +The common prime factors are 2 and 5. +Step 3: Identify the smallest exponent of each prime identified in Step 2 in the prime factorizations. +The exponent of common prime factor 2 in the prime factorization of 1,400 is 3, and in the prime factorization of 250 is +1. The smallest of those exponents is 1. +The exponent of the common prime factor 5 in the prime factorization of 1,400 is 2 and is in the prime factorization of +250 is 3. The smallest of these exponents is 2. +Step 4: Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. +This gives +. The greatest common divisor of 1,400 and 250 is +. +Notice that the answer matches the one we found in Example 3.15. +YOUR TURN 3.16 +1. Using prime factorization, determine the greatest common divisor of 36 and 128. +EXAMPLE 3.17 +Finding the Greatest Common Divisor Using Prime Factorization +Find the greatest common divisor of 600 and 784 by using their prime factorizations. +3.1 • Prime and Composite Numbers +139 + +Solution +Step 1: Find the prime factorizations of the numbers. +The prime factorization of 600 is +. +The prime factorization of 784 is +. +Step 2: Identify the prime factors that appear in every number’s prime factorization. +There is only one common prime factor, 2. +Step 3: Identify the smallest exponent of each prime from identified in Step 2 in the prime factorizations. +The exponent of 2 in the prime factorization of 600 is 3. The exponent of 2 in the prime factorization of 784 is 4. So, the +smallest exponent of 2 is 3. +Step 4: Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. +This gives +. The greatest common divisor of 600 and 784 is 8. +YOUR TURN 3.17 +1. Using prime factorization, determine the greatest common divisor of 120 and 200. +PEOPLE IN MATHEMATICS +Srinivasa Ramanujan +Figure 3.10 Srinivasa Ramanujan (credit: Srinivasa Ramanujan, photo by Konrad Jacobs/Oberwolfach Photo +Collection/public domain) +Ramanujan was born in southern India in 1887, during British rule. He was a self-taught mathematician, who, while in +high school, began working through a two-volume text of mathematical theorems and results. His work included +examination of the distribution of primes. He eventually came to the attention of British mathematician, G.H. Hardy. +During one visit, Hardy mentioned to Ramanujan that his taxicab number was 1,729, remarking that 1,729 appeared +to be a rather dull number. To which Ramanujan responded, “It is a very interesting number; it is the smallest number +expressible as the sum of two cubes in two different ways.” +140 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +TECH CHECK +Using Desmos to Find the GCD +To find the GCD of a set of numbers in Desmos, type gcd(first_number,second_number…) and Desmos will display the +GCD of the numbers, as the numbers are typed! +VIDEO +Using Desmos to Find the GCD (https://openstax.org/r/Using_Desmos_to_Find_the_GCD) +Applications of the Greatest Common Divisor +The greatest common divisor has uses that are related to other mathematics (reducing fractions) but also in everyday +applications. We’ll look at two such applications, which have very similar underlying structures. In each case, something +must divide the groups or measurements equally. +EXAMPLE 3.18 +Calculating Floor Tile Size +Suppose you have a rectangular room that is 570 cm wide and 450 cm long. You wish to cover the floor of the room with +square tiles. What’s the largest size square tile that can be used to cover this floor? +Solution +Using squares means that the length and width of the tiles are equal. To ensure we are using full tiles, the side length of +the square tiles must divide the length of the room and the width of the room. Since we want the largest square tiles, we +need the GCD of the width and length of the room or the GCD of 570 and 450. +Step 1: Find the prime factorizations of the numbers. +The prime factorization of 570 is +. +The prime factorization of 450 is +. +Step 2: Identify the prime factors that appear in every number’s prime factorization. +The common prime factors are 2, 3, and 5. +Step 3: Identify the smallest exponent of each prime identified in Step 2 in the prime factorizations. +The exponents of 2 in the prime factorizations of 570 and 450 are 1 and 1. So the smallest exponent for 2 is 1. +The exponents of 3 in the prime factorizations of 570 and 450 are 1 and 2, so the smallest exponent for 3 is 1. +The exponents of 5 in the prime factorizations of 570 and 450 are 1 and 2, so the smallest exponent for 5 is 1. +Step 4: Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. +This gives +. The GCD of 450 and 570 is 30, so the largest size square tile that can be used to cover the +floor is 30 cm by 30 cm. +YOUR TURN 3.18 +1. You are designing a brick patio made of square bricks 5 cm thick, but you need to determine the width and +length of those bricks. The patio will be 400 cm by 540 cm. What are the largest size square bricks that can be +used so that you do not need to cut any bricks? +3.1 • Prime and Composite Numbers +141 + +EXAMPLE 3.19 +Organizing Books Per Shelf +Suppose you want to organize books onto shelves, and you want the shelves to hold the same number of books. Each +shelf will only contain one genre of book. You have 24 sci-fi, 42 fantasy, and 30 horror books. How many books can go on +each shelf? +Solution +Since we want shelves that hold an equal number of books, and a shelf can only hold one genre of book, we need a +number that will equally divide 24, 42, and 30. So, we need the GCD of the number of books of each genre or the GCD of +24, 42, and 30. +Step 1: Find the prime factorizations of the numbers. +The prime factorization of 24 is +. +The prime factorization of 42 is +. +The prime factorization of 30 is +. +Step 2: Identify the prime factors that appear in every number’s prime factorization. +The common prime factors are 2 and 3. +Step 3: Identify the smallest exponent of each prime identified in Step 2 in the prime factorizations. +The smallest exponent of 2 and 3 in the factorizations is 1. +Step 4: Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. +This gives +. The GCD of 24, 42, and 30 is 6, so the largest number of books that can go on a shelf is 6. +YOUR TURN 3.19 +1. There are three gym classes. The number of students in the classes is 21, 35, and 28. What is the largest team +size that can be formed if teams from every class must have the same number of students? +VIDEO +Applying the GCD (https://openstax.org/r/Applying_the_GCD) +Finding the Least Common Multiple +The flip side to a divisor of a number is a multiple of a number. For example, 5 is a divisor of 45 and so 45 is a multiple of +5. More generally, if the number +divides the number +, then +is a multiple of +. +This drives the idea of the least common multiple of a set of numbers. A common multiple of a set of numbers is a +multiple of each of those numbers. For instance, 45 is a common multiple of 9 and 5, because 45 is a multiple of 9 (9 +divides 45) and 45 is also a multiple of 5 (5 divides 45). The least common multiple (LCM) of a set of number is the +smallest positive common multiple of that set of numbers. +There are (at least) three ways to find the LCM of a set of numbers, and we will explore two of them. One way is to create +a list of multiples of each number in the set, and then identify the smallest multiple that appears in those lists. +EXAMPLE 3.20 +Finding the Least Common Multiple Using Lists +Find the LCM of 24 and 90 by listing multiples and choosing the smallest common multiple. +Solution +Create a list of the multiples of each number. +Step 1: The first 15 multiples for 24: +142 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360. +Step 2: The first 15 multiples for 90: +90, 180, 270, 360, 450, 540, 630, 720, 810, 900, 990, 1,080, 1,170, 1,260, 1,350. +There is one multiple common to these lists, which is 360. So, 360 is the LCM of 24 and 90. +YOUR TURN 3.20 +1. Use lists to find the LCM of 12 and 15. +The second method we can use is to find the prime factorizations of the number in the set to build the LCM of the +numbers based on the prime divisors of the numbers. The LCM can be built from the prime factorization of the numbers +in the set in a similar way as when finding the greatest common divisor. Here are the steps for using the prime +factorization process for finding the LCM. +Step 1: Find the prime factorization of each number. +Step 2: Identify each prime that is present in any of the prime factorizations. +Step 3: Identify the largest exponent of each prime identified in Step 2 in the prime factorizations. +Step 4:. Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. +EXAMPLE 3.21 +Finding the Least Common Multiple Using Prime Factorization +Use the prime factorizations of 24 and 90 to identify their LCM. +Solution +Step 1: Find the prime factorization of each number. +Step 2: Identify each prime that is present in any of the prime factorizations. +The prime numbers present in the prime factorizations are 2, 3, and 5. +Step 3: Identify the largest exponent of each prime identified in Step 2 in the prime factorizations. +Prime +2 +3 +5 +Largest exponent +3 +2 +1 +Step 4: Compute the LCM by multiplying the prime factors identified in Step 2 raised to the powers identified in Step 3. +The LCM for 24 and 90 is +. +YOUR TURN 3.21 +1. Use prime factorization to find the LCM of 20 and 28. +3.1 • Prime and Composite Numbers +143 + +EXAMPLE 3.22 +Finding the Least Common Multiple Using Prime Factorization +Use the prime factorizations of 36, 66, and 250 to identify the LCM. +Solution +Step 1: Find the prime factorization of each number. +Step 2: Identify each prime that is present in any of the prime factorizations. +The prime numbers present in the prime factorizations are 2, 3, 5, and 11. +Step 3: Identify the largest exponent of each prime identified in Step 2 in the prime factorizations. +Prime +2 +3 +5 +11 +Largest exponent +2 +2 +3 +1 +Step 4: Compute the LCM by multiplying the prime factors identified in Step 2 raised to the powers identified in Step 3. +The LCM for 36, 66, and 250 is +. +YOUR TURN 3.22 +1. Use the prime factorization method to find the LCM of 150, 240, and 462. +Using lists for three or more numbers, particularly larger numbers, could take quite a bit of time. Frequently, as in this +example, the prime factorization process is much quicker. In practice, you can use either listing or prime factorization to +find the LCM. +EXAMPLE 3.23 +Finding the Least Common Multiple Using Both Methods +Find the LCM of 20, 36, and 45 using lists and prime factorization. +Solution +Step 1: Use listing. List the multiples: +20 +20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220 +36 +36, 72, 108, 144, 180, 216, 252, 288, 324, 360 +45 +45, 90, 135, 180, 225, 270, 315, 360, 405, 450, 495, 540 +The smallest value that appears on all the lists is 180, so 180 is the LCM of 20, 36, and 45. +Step 2: Find the prime factorization of each number. +144 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Step 3: Identify each prime that is present in any of the prime factorizations. +The prime numbers present in the prime factorizations are 2, 3, 5. +Step 4: Identify the largest exponent of each prime identified in Step 3 in the prime factorizations. +Prime +2 +3 +5 +Largest exponent +2 +2 +1 +Step 5: Compute the LCM by multiplying the prime factors identified in Step 3 raised to the powers identified in Step 4. +The LCM for 20, 36, and 45 is +. +Both listing and prime factorization produced the same result: the LCM is 180. +YOUR TURN 3.23 +1. Find the LCM of 18, 24, and 40 using lists and prime factorization. +VIDEO +Finding the LCM (https://openstax.org/r/Finding_the_LCM) +TECH CHECK +Using Desmos to Find the LCM +To find the LCM of a set of numbers in Desmos, type lcm(first_number,second_number…) and Desmos will display the +LCM of the numbers, as the numbers are typed! +VIDEO +Using Desmos to find the LCM (https://openstax.org/r/Using_Desmos_to_find_the_LCM) +Applications of the Least Common Multiple +Some applications of LCM involve events that repeat at fixed intervals, such as visits to a location. Other applications +involve getting things to be of equal magnitude when using parts that are different sizes (see Example 3.25, for +instance). In each case, we may be looking to determine when two processes “line up.” +EXAMPLE 3.24 +Determining Scheduling Overlap Using the Least Common Multiple +Two students, João, and Amelia, meet one day at an assisted living facility where they volunteer. João volunteers every 6 +days. Amelia volunteers every 10 days. How many days will it be until they are both volunteering on the same day again? +Solution +If we list the days that each student will volunteer, it becomes clear that we could solve this problem using the LCM of 6 +and 10. +João will be at the assisted living facility 6, 12, 18, 24, 30, 36, 42, and 48 days later. +Amelia will be at the assisted living facility 10, 20, 30, 40, 50, and 60 days later. +3.1 • Prime and Composite Numbers +145 + +The smallest number appearing on both lists is 30. João and Amelia will once more be volunteering together 30 days +later. +YOUR TURN 3.24 +1. The sun, Venus, and Jupiter all line up on a given day. Venus orbits the sun once every 255 days. Jupiter orbits the +sun every 4,330 days (we’ll ignore the decimal values of days for each orbit). How many days will it be until they +line up again? +EXAMPLE 3.25 +Determining the Minimum Height Using the LCM +A team-building exercise has teams build a house of cards as high as possible. However, the cards for different teams are +of different sizes. Team 1 uses 10 cm × 10 cm cards, while Team 2 uses 8 cm × 8 cm cards. What is the minimum height +when the two teams will be tied? Ignore the width of the cards. +Solution +This is an example where we want to put together objects with different sizes. We want to know the minimum height +when they are tied, or when the houses of cards line up the first time. The heights of the towers built using the 10 cm × +10 cm cards will be 10, 20, 30, 40, 50, and 60 cm tall. When the 8 cm × 8 cm cards are used, the tower will be 8, 16, 24, 32, +40, 48, and 56 cm tall. The smallest number appearing on both lists is 40. The first time they are tied is when the two +towers are 40 cm tall. +YOUR TURN 3.25 +1. In an Internet giveaway, every 130th person who submits a survey receives $250, and every 900th person +receives a free cell phone. How many submissions must be received for the first person to receive both prizes? +VIDEO +Application of LCM (https://openstax.org/r/Application_of_LCM) +WORK IT OUT +Prime Number Life Cycles—Cicadas +Cicadas are known to have life cycles of 13 or 17 years, which are prime numbers. Why would a prime number life +cycle be an evolutionary advantage? To figure this out, we have to explore how common multiples work with prime +numbers. +Make a conjecture regarding the LCM of a prime number and another number. Test this conjecture with a few +examples of your own making. +Check Your Understanding +1. Identify which of the following numbers are prime and which are composite. +31, 56, 213, 48, 701 +2. Find the prime factorization of 4,570. +3. Find the greatest common divisor of 410 and 144. +4. Find the least common multiple of 45 and 70. +5. You want to fill gift bags for children in the after-school program where you volunteer. You have 30 crayons, 20 +146 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +sticker sheets, and 70 bite-sized candies. If each gift bag must contain the same number of crayons, sticker sheets, +and bite-sized candies, what is the maximum number of bags that can be filled? +SECTION 3.1 EXERCISES +For the following exercises, use divisibility rules to determine if each of the following is divisible by 2. +1. 24 +2. 37 +3. 1,345,321 +For the following exercises, use divisibility rules to determine if each of the following is divisible by 3. +4. 48 +5. 210 +6. 5,345,324 +For the following exercises, use divisibility rules to determine if each of the following is divisible by 5. +7. 130 +8. 237 +9. 1,345,321 +For the following exercises, use divisibility rules to determine if each of the following is divisible by 9. +10. 48 +11. 210 +12. 5,345,325 +For the following exercises, use divisibility rules to determine if each of the following is divisible by 12. +13. 48 +14. 210 +15. 5,355,324 +16. Determine which of the following numbers are prime: {3, 27, 77, 131, 457} +17. Determine which of the following numbers are prime: {31, 97, 188, 389} +For the following exercises, find the prime factorization of the given number. +18. 12 +19. 53 +20. 72 +21. 345 +22. 938 +23. 36,068 +24. 8,211,679 +For the following exercises, find the greatest common divisor of the given set of numbers. +25. {45, 245} +26. {11, 24} +27. {56, 44} +28. {150, 600} +29. {1,746, 28,324} +30. {30, 40, 75} +31. {19, 45, 70} +32. {293, 7,298, 19,229} +33. {3,927,473, 82,709, 1,210,121} +34. Make a list of the common divisors of 12 and 18. What is the GCD of 12 and 18? Which of the other common +divisors of 12 and 18 divide the GCD? +35. Make a list of the common divisors of 20 and 84. What is the GCD? Which of the other common divisors of 20 +and 84 also divide the GCD? +36. Make a list of the common divisors of 120 and 88. What is the GCD? Which of the other common divisors of 120 +and 88 also divide the GCD? +37. Based on the answers to 34, 35, and 36, make a conjecture about the GCD of two numbers, and the other +common divisors of those numbers. +38. Rebecca wants to cut two lengths of board into equal length pieces, with no leftover piece. The two boards are +3.1 • Prime and Composite Numbers +147 + +230 cm long and 370 cm long. What is the longest length that Rebecca can cut from these boards so that all the +cut boards are the same length? +39. Yasmin is playing with her younger brother, Cameron. They are grouping Skittles by color. They have 14 green, +10 yellow, and 8 purple Skittles. Each group must have the same number of green, the same number of yellow, +and the same number of purple Skittles. What’s the maximum number of piles that Sophia can build with +Cameron? +40. Gathii is creating a tile backsplash for his kitchen. He wants to use square tiles to cover a 330 cm × 12 cm area. +What is the largest size square tile he can use to create this backsplash? +41. Deiji is designing a contest where teams will be given the same number of toothpicks, 5 oz. paper cups, and 2 +cm length pieces of string. She has 420 pieces of string, 300 paper cups, and 1,610 toothpicks. What is the +maximum number of teams she can have so that every team gets an equal number of pieces of string, paper +cups, and toothpicks? +For the following exercises, find the least common multiple of the given set of numbers. +42. {30, 40} +43. {11, 24} +44. {14, 45} +45. {200, 450} +46. {38,077, 9,088,687} +47. {36, 42, 70} +48. {7, 13, 36} +49. {4,450,864, 339,889, 157,339} +50. Benjamin and Mia both work at the Grease Fire diner, a local eatery. Benjamin has every 4th day off, and Mia +has every 6th day off. How many days pass until they have another day off together? +51. A lunar month is 30 days (rounding off). A new lunar month begins on a Saturday. How many days is it until a +lunar month begins on a Saturday again? +52. Isabella is creating a collage for a project and wants a horizontal cut in the collage. The cut will be made by +using purple strips of cloth that are 28 mm long, and yellow strips of paper that are 36 mm long. What is the +minimum length of the cut can she make using strips with those lengths? +53. Asteroids are objects that orbit the sun. The smallest distance that an asteroid gets to the sun during its orbit is +called the perihelion. Asteroids also have orbital periods, or the time it takes to go around the sun exactly one +time. The asteroid Ceres has an orbital period (number of days to circle the sun) of 1,680 days. The asteroid +Hygiea has an orbital period of 2,031 days. Suppose they are at their perihelion on the same day. How many +days will it be before Ceres and Hygiea are at their perihelion on the same day again? +148 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +3.2 The Integers +Figure 3.11 A ledger comparing assets to debts, resulting in net wealth. (credit: modification of work “Reviewing +Financial Statements” by Mary Cullen/Flickr, CC BY 2.0) +After completing this section, you should be able to: +1. +Define and identify numbers that are integers. +2. +Graph integers on a number line. +3. +Compare integers. +4. +Compute the absolute value of an integer. +5. +Add and subtract integers. +6. +Multiply and divide integers. +Positive net wealth is when the total value of a person’s assets, such as their home, their 401(k), their car, and savings +account balance, exceed that of their debts, such as car loans, mortgages, or credit card debt. However, when the total +value of debt exceeds the total value of assets, then the person has negative net wealth. Expressing the negative net +wealth as a negative number allows people to work with the positive net values and negative net values with the same +mathematical processes, and in the same applications. This section introduces the integers and operations with integers. +Defining and Identifying Integers +Extending the counting numbers to include negative numbers and zero forms the integers. Any other number that +cannot be written as +is not an integer. +EXAMPLE 3.26 +Identifying Integers +Which of the following are integers and which are not? +−3 +Is an integer, as it is the negative of a counting number +This is not written as an integer. Entering the square root of 24 in a calculator, such as desmos, the result is +4.899 (rounded off). Since this is not an integer, then +is not an integer. +36/4 +Since 36 divided by 4 is 9, and 9 is an integer, then 36/4 is an integer. +45 +Is an integer, as it is a counting number +63.9 +Is not an integer, because it is not a counting number and not the negative of a counting number. +3.2 • The Integers +149 + +2/7 +Dividing 2 by 7 results in a number less than 1, but greater than 0, so is between two consecutive integers. +So, 2/7 is not an integer. +−16.0 +Is an integer, since the decimal part is 0 +YOUR TURN 3.26 +Which of the following are integers? +1. −214 +2. 38/11 +3. +4. +5. 420/35 +Graphing Integers on a Number Line +Integers are often imagined as steps along a path. You start at 0, and going to the left is going backward, or in the +negative direction, while going to the right is going forward, or in the positive direction. A number line (Figure 3.12) +helps envision the integers. This also means that an integer gives magnitude (size) and direction (positive is to the right, +negative is to the left). Graphing an integer on the number line means placing a solid dot at the integer on the number +line. +Figure 3.12 +VIDEO +Graphing Integers on the Number Line (https://openstax.org/r/Graphing_Integers) +EXAMPLE 3.27 +Graphing Numbers on the Number Line +Graph the following on the number line: +1. +1 +2. +−4 +3. +3 +Solution +1. +Figure 3.13 +2. +Figure 3.14 +3. +Figure 3.15 +150 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +YOUR TURN 3.27 +Graph the following numbers on the number line: +1. −10 +2. 4 +3. 0 +Comparing Integers +When determining if one quantity or size is larger than another, we know it means there is more of whatever is being +discussed. In terms of positive integers, we can envision that larger integers are further to the right on the number line. +This idea applies to negative integers also. This means that +is greater than +when +is to the right of +on the number +line. We write +. When +is greater than +, we can also say that +is less than +. On the number line, +would be to +the left of +. We write +. +We need to recognize that +means the same thing as +. This can be seen on the number line in Figure 3.16. On +this number line, +is to the right of +, so +. But this means +is to the left of +, so +. +Figure 3.16 +VIDEO +Comparing Integers Using the Number Line (https://openstax.org/r/the_Number_Line) +EXAMPLE 3.28 +Comparing Integers Using a Number Line +Determine which of −6 and 4 is larger using a number line, and express that using both the greater than and the less +than notations. +Solution +To illustrate this, we use a number line (Figure 3.17). +Figure 3.17 +Since −6 is to the left of 4, then −6 is less than 4. We can write this as −6 < 4. Another way of expressing this is that 4 is +greater than −6. So we can also write +. +YOUR TURN 3.28 +1. Determine which of −38 and 27 is larger using a number line, and express that using both the greater than and +the less than notations. +EXAMPLE 3.29 +Comparing Negative Integers +Determine which of −6 and −2 is larger, and express that using both the greater than and the less than notations. +Solution +To illustrate this, we use a number line (Figure 3.18). +3.2 • The Integers +151 + +Figure 3.18 +Since −6 is to the left of −2, then −6 is less than −2. We can write this as +. +Another way of expressing this is that −2 is greater than −6. So we can also write +. +Warning: People often ignore the negative signs, and think than since 6 is greater than 2, −6 is greater than −2. To +avoid that error, remember that the greater number is to the right on the number line. +YOUR TURN 3.29 +1. Determine which of −63 and −213 is larger, and express that using both the greater than and the less than +notations. +EXAMPLE 3.30 +Comparing Integers by Quantity +Determine which of 27 and 410 is larger, and express that using both the greater than and the less than notations. +Solution +When thinking about quantity, 410 is more than 27. So, 410 is greater than 27 and 27 is less than 410. We can write this +as +or as +. +YOUR TURN 3.30 +1. Determine which of 101 and 98 is larger, and express that using both the greater than and the less than +notations. +The Absolute Value of an Integer +When talking about graphing integers on the number line, one interpretation suggests it is like walking along a path. +Negative is going to the left of 0, and positive going to the right. If you take 30 steps to the right, you are 30 steps away +from 0. On the other hand, when you take 30 steps to the left, you are still 30 steps away from 0. So, in a way, even +though one is negative and the other positive, these two numbers, 30 and −30, are equal since both are 30 steps away +from 0. The absolute value of an integer +is the distance from +to 0, regardless of the direction. The notation for +absolute value of the integer +is +. +If we think of an integer as both direction and magnitude (size), absolute value is the magnitude part. +Calculating the absolute value of an integer is very straightforward. If the integer is positive, then the absolute value of +the integer is just the integer itself. If the integer is negative, then to compute the absolute value of the integer, simply +remove the negative sign. Keeping in mind the number line as a path, when you’ve gone 10 steps to the left of 0, you +have still taken 10 steps, and the direction does not matter. +VIDEO +Evaluating the Absolute Value of an Integer (https://openstax.org/r/Absolute_Value_of_an_Integer) +152 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +EXAMPLE 3.31 +Calculating the Absolute Value of a Positive Integer +Calculate |19|. +Solution +Since the number inside the absolute value symbol is positive, the absolute value is just the number itself. So |19| = 19. +YOUR TURN 3.31 +1. Calculate |38|. +EXAMPLE 3.32 +Calculating the Absolute Value of a Negative Integer +Calculate |−435|. +Solution +Since the number inside the absolute value is negative, the absolute value removes the negative sign. So |−435| = 435. +YOUR TURN 3.32 +1. Calculate |−81|. +Adding and Subtracting Integers +You may recall having approached adding and subtracting integers using the number line from earlier in your academic +life. Adding a positive integer results in moving to the right on the number line. Adding a negative integer results in +moving to the left. Subtracting a positive integer results in a move to the left on the number line. But subtracting a +negative integer results in a move to the right. +This leads to a few adding and subtracting rules, such as: +Rule 1: Subtracting a negative is the same as adding a positive. +Rule 2: Adding two negative integers always results in a negative integer. +Rule 3: Adding two positive integers always results in a positive integer. +Rule 4: The sign when adding integers with opposite signs is the same as the integer with the larger absolute value. +These rules are good to keep in the back of your mind, as they can serve as a quick error check when you use a +calculator. +EXAMPLE 3.33 +Adding Integers +Use your calculator to calculate 4 + (−7). Explain how the answer agrees with what was expected. +Solution +Using a calculator, we find that 4 + (−7) = −3. Since we are adding integers with opposite signs, the sign of the answer +matches the sign of the integer with the larger absolute value which |−7|=7. +3.2 • The Integers +153 + +YOUR TURN 3.33 +1. Use your calculator to calculate (−18) + 11. Explain how the answer agrees with what was expected. +EXAMPLE 3.34 +Subtracting Positive Integers +Use your calculator to calculate 18 − 9. Explain how the answer agrees with what was expected. +Solution +Using a calculator, we find that 18 − 9 = 9. Since 18 was larger than 9, we expected the difference to be positive. +YOUR TURN 3.34 +1. Use your calculator to calculate 38 − 100. Explain how the answer agrees with what was expected. +EXAMPLE 3.35 +Subtracting with Negative Integers +Use your calculator to calculate 27 − (−13). Explain how the answer agrees with what was expected. +Solution +Using a calculator, we find that 27 – (−13) = 40. Since we’re subtracting a negative number, it is the same as adding a +positive, so this is the same as 27 + 13 = 40. +YOUR TURN 3.35 +1. Use your calculator to calculate 45 − (−26). Explain how the answer agrees with what was expected. +EXAMPLE 3.36 +Adding Integers with Opposite Signs +Use your calculator to calculate (−13) + 90. Explain how the answer agrees with what was expected. +Solution +Using a calculator, we find that (−13) + 90 = 77. Since we are adding integers with opposite signs, the sign of the answer +matches the sign of the integer with the larger absolute value, which is positive since 90 is positive. +YOUR TURN 3.36 +1. Use your calculator to calculate 19 + (−36). Explain how the answer agrees with what was expected. +One use of negative numbers is determining net worth, which is all the weath someone owns less all that someone +owes. Sometimes net worth is positive (which is good), and sometimes net worth is negative (which can be stressful). +EXAMPLE 3.37 +Calculating Net Worth +Jennifer is owed $50 from her friend Janice, but owes her friend Pat $87. What is Jennifer’s net worth? +154 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Solution +Net worth is the amount that one is owed minus the amount one owes. Jennifer is owed $50 but owes $87. So, her net +worth is $50 – $87 = −$37. The negative indicates that Jennifer owes more than she is owed. +YOUR TURN 3.37 +1. Christian is owed $180 from his friend Chanel, but owes his friend Jeff $91. What is Christian’s net worth? +Multiplying and Dividing Integers +Similar to addition and subtraction, the signs of the integers impact the results when multiplying and dividing integers. +The rules are fairly straightforward, but again rely on the direction on the number line. There are only two rules. +Rule 1: When multiplying or dividing two integers with the same sign, the result is positive. +Rule 2: When multiplying or dividing two integers with opposite signs, the result is negative. +Just as before, these rules can serve as a quick error check when using a calculator. +EXAMPLE 3.38 +Multiplying Positive Integers +Use your calculator to calculate 4 × 8. Explain how the answer agrees with what was expected. +Solution +Entering 4 × 8 into your calculator, the result is 32. This agrees with our expectation. The numbers have the same signs, +so the result is positive. +YOUR TURN 3.38 +1. Use your calculator to calculate 81 × 26. Explain how the answer agrees with what was expected. +EXAMPLE 3.39 +Multiplying Integers with Different Signs +Use your calculator to calculate 9 × (−10). Explain how the answer agrees with what was expected. +Solution +Entering 9 × (−10) into your calculator, the result is −90. This agrees with our expectation. The numbers have opposite +signs, so the result is negative. +YOUR TURN 3.39 +1. Use your calculator to calculate (−18) × 13. Explain how the answer agrees with what was expected. +EXAMPLE 3.40 +Dividing Integers with Different Signs +Use your calculator to calculate 400/(−25). Explain how the answer agrees with what was expected. +Solution +Entering 400/(−25) into your calculator, the result is −16. This agrees with our expectation. The numbers have opposite +3.2 • The Integers +155 + +signs, so the result is negative. +YOUR TURN 3.40 +1. Use your calculator to calculate (−116)/4. Explain how the answer agrees with what was expected. +EXAMPLE 3.41 +Dividing Negative Integers +Use your calculator to calculate −750/(−3). Explain how the answer agrees with what was expected. +Solution +Entering −750/(−3) into your calculator, the result is 250. This agrees with our expectation. The numbers have the same +signs, so the result is positive. +YOUR TURN 3.41 +1. Use your calculator to calculate (−77)/(−11). Explain how the answer agrees with what was expected. +At the end of a season, a team may wish to buy their coach an end-of season gift. It makes sense to share the cost +equally among the members. To do so, the team would need to find the average (or mean) cost per member. The +average (or mean) of a set of numbers is the sum of the numbers divided by the number values that are being +averaged. +EXAMPLE 3.42 +Finding the Average of a Set of Numbers +The daily low temperatures in Barrie, Ontario, for the week of February 14, 2021, were −20°, −12°, −15°, −23°, −17°, −13°, +and −19° degrees Celsius. What was the average daily temperature for the week of February 14, 2021, in Barrie? +Solution +Step 1: To find the average daily temperature, we first need to add the temperatures. +(−20) + (−12) + (−15) + (−23) + (−17) + (−13) + (−19) = −119 +Step 2: That sum will then be divided by 7 since we are averaging over seven days, giving −119/7 = −17. So, the average +daily temperature in Barrie, Ontario the week of February 14, 2021, was −17° Celsius. +YOUR TURN 3.42 +1. Banks and credit cards often base their interest on the average daily balance of an account, which is the average +of the balance from each day of the period. The account balance of Jada’s checking account on each day of the +week of December 13, 2020, was $1,250, $673, −$1,500, $1,000, $785, $785, and $710. What was Jada’s average +daily balance for the week of December 13, 2020? Assume Jada pays no fees for a negative balance. +Check Your Understanding +6. Identify which of the following numbers are integers: +−4, 15.2, +, +, 430 +7. Graph the following integers on the number line: 4, −2, 7. +8. Place these integers in increasing order: 4, −2, −7, 10, −13. +156 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +9. Calculate |−7|. +10. Calculate 4 − (−9). +11. Calculate +. +SECTION 3.2 EXERCISES +1. Identify all the integers in the following list: 4, −17, 8, 0.5, +, +, −300. +2. Identify all integers in the following list: −9.2, 13, −1, +, +, 567, −300. +For the following exercises, plot the integers on the same number line. +3. 4, −2, 10, 0 +4. −6, −3, 10, 1, 4 +5. −3, −10, 7, 2 +6. 2, 4, 8, −2, −5 +For the following exercises, determine if the comparison is true or false. +7. −3 < −10 +8. −10 > −3 +9. 7 > −6 +10. −6 < 7 +11. 18 < 20 +12. 20 < 18 +13. What are two numbers with an absolute value of 76? +14. What are two numbers with an absolute value 87? +15. Determine |−67|. +16. Determine |98|. +17. Determine |61|. +18. Determine |−903|. +19. What are two numbers that are 23 away from 0? +20. What are two numbers 13 away from 0? +For the following exercises, complete the indicated calculation. +21. 47 + 200 +22. 67 + (−86) +23. (−86) + 104 +24. 13 – (−54) +25. (−45) – (−26) +26. (−13) + (−102) +27. +28. +29. +30. +31. +32. +33. +34. +35. +36. +37. The daily low temperatures, in degrees centigrade, in Fargo for the week of January 17, 2021, were −9, −17, −18, +−14, −17, −19, and −11. What was the average low temperature in Fargo that week? +38. Riley collects checks for a fundraiser supporting the homeless in town. Through Venmo, they collect the +following amounts: $20, $20, $50, $75, $250, $10, $15, $65, $30, $15. What was the average donation that Riley +collected? +39. Heath has $495 in an account. They will collect two paychecks this week, one for $150 and the other for $250. +Heath also will pay three bills, one for $50, one for $110, and one for $300. After all those transactions, how +much will Heath have in their account? +40. Five diners decide to split the check evenly. The total bill comes to $475. How much does each diner owe? +3.2 • The Integers +157 + +41. There are many people who are single-issue voters, which means that they will vote for (or against!) a candidate +based on one issue and one issue only. Suppose a politician wants to earn votes based on single issues: Issue 1, +Issue 2, and Issue 3. By publicly supporting issue 1, the politician gains 127 voter but loses 154. By publicly +supporting issue 2, the politician gains 350 voters but loses 83. By publicly denouncing Issue 3, the politician +gains 306 voters but loses 158. By publicly taking those stances, what is the politician’s net gain or loss in +number of voters? +3.3 Order of Operations +Figure 3.19 Calculators may automatically apply order of operations to calculations. (credit: “Precision” by Leonid +Mamchenkov/Flickr, CC BY 2.0) +After completing this module, you should be able to: +1. +Simplify expressions using order of operations. +2. +Simplify expressions using order of operations involving grouping symbols. +Calculates else sure someone be rules expect explicit we what that needs to need make, we to that them what be +calculate to calculated first. +You probably read that sentence and couldn't make heads or tails of it. Seems like it might concern calculations, but +maybe it concerns needs? You may even be attempting to unscramble the sentence as you read it, placing words in the +order you might expect them to appear in. The reason that the sentence makes no sense is that the words don't follow +the order you expect them to follow. Unscrambled, the sentence was intended to be “To be sure that someone else +calculates when we expect them to calculate, we need rules that make explicit what needs to be calculated first.” +Similarly, when working with math expressions and equations, if we don't follow the rules for order of operations, +arithmetic expressions make no sense. Just a simple expression would be problematic if we didn't have some rules to tell +us what to calculate first. For instance, +can be calculated in many ways. You could get 5,184. Or, you +could get 80. Or, 96. The issue is that without following a set of rules for calculation, the same expression will give various +results. In case you are curious, using the appropriate order of operations, we find +. +Simplify Expressions Using Order of Operations +The order in which mathematical operations is performed is a convention that makes it easier for anyone to correctly +calculate. They follow the acronym EMDAS: +E +Exponents +M/D +Multiplication and division +A/S +Addition and Subtraction +So, what does EMDAS tell us to do? In an equation, moving left to right, we begin by calculating all the exponents first. +Once the exponents have been calculated, we again move left to right, calculating the multiplications and divisions, one +at a time. Multiplication and division hold the same position in the ordering, so when you encounter one or the other at +this step, do it. Once the multiplications and divisions have been calculated, we again move left to right, calculating the +158 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +additions and subtractions, one at a time. Additions and subtractions hold the same position in the ordering, so when +you encounter one or the other at this step, do it. (You may have previously learned the order of operations as PEMDAS, +with parentheses first; we will add that aspect later on.) We’ll explore this as we work an example. +EXAMPLE 3.43 +Using Two Order of Operations +Calculate +. +Solution +There are no exponents in this expression, so the next operations to check are multiplication and division. +Step 1: Moving left to right, the first multiplication encountered is 4 multiplied by 13. We perform that operation first. +Step 2: The only operation remaining is the subtraction. +. +So, +. +YOUR TURN 3.43 +1. Calculate +. +EXAMPLE 3.44 +Using Two Order of Operations +Calculate +. +Solution +Step 1: Moving left to right, we see there is an exponent. We calculate the exponent first. +Step 2: The only operation remaining is the multiplication. +So, +. +YOUR TURN 3.44 +1. Calculate +. +EXAMPLE 3.45 +Using Three Order of Operations +Calculate +. +Solution +Step 1: To calculate this, move left to right, and compute all the exponents first. The only exponent we see is the +squaring of the 3, so that is calculated first. +3.3 • Order of Operations +159 + +Step 2: Since the exponents are all calculated, now calculate all the multiplications and divisions moving left to right. The +only multiplication or division present is 9 times 4. +Step 3: Moving left to right, perform the additions and subtractions. There is only one such operation, 2 plus 36. +So, +. +YOUR TURN 3.45 +1. Calculate +. +Even if the expression being calculated gets more complicated, we perform the operations in the order: EMDAS. +VIDEO +Order of Operations 1 (https://openstax.org/r/Order_of_Operations_1) +EXAMPLE 3.46 +Using Eight Order of Operations +Correctly apply the order of operations to compute the following: +. +Solution +Step 1: To do so, calculate the exponents first, moving left to right. There are two occurrences of exponents in the +expression, 3 squared and 2 cubed. +Step 2: Now that the exponents are calculated, perform the multiplication and division, moving left to right. The first is +the product of 25 and 6. +Step 3: Next is the 150 divided by 10. +Step 4: Next is 15 multiplied by 9. +Step 5: Finally, multiply the 7 and 8. +As all the multiplications and divisions have been calculated, the additions and subtractions are performed, moving left +to right. +160 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +The computed value is −75. +YOUR TURN 3.46 +1. Correctly apply the order of operations to compute the following: +. +VIDEO +Order of Operations 2 (https://openstax.org/r/Order_of_Operations_2) +EXAMPLE 3.47 +Using Six Order of Operations +Correctly apply the rules for the order of operations to accurately compute the following: +. +Solution +Step 1: Calculate exponents first, moving left to right: +Step 2: Multiply and divide, moving left to right: +Step 3: Add and subtract, moving left to right: +YOUR TURN 3.47 +1. Correctly apply the rules for the order of operations to accurately compute the following: +. +EXAMPLE 3.48 +Using Order of Operations +Correctly apply the rules for the order of operations to accurately compute the following: +. +Solution +Step 1: Calculate the exponents first, moving left to right: +3.3 • Order of Operations +161 + +Step 2: Multiply and divide, moving left to right: +Step 3: Add and subtract, moving left to right: +YOUR TURN 3.48 +1. Correctly apply the rules for the order of operations to accurately compute the following: +. +Using the Order of Operations Involving Grouping Symbols +We have examined how to use the order of operations, denoted by EMDAS, to correctly calculate expressions. However, +there may be expressions where a multiplication should happen before an exponent, or a subtraction before a division. +To indicate an operation should be performed out of order, the operation is placed inside parentheses. When +parentheses are present, the operations inside the parentheses are performed first. Adding the parentheses to our list, +we now have PEMDAS, as shown below. +P +Parentheses +E +Exponents +M/D +Multiplication and division +(division is just the multiplication by the reciprocal) +A/S +Addition and subtraction +(subtraction is just the addition of the negative) +As said previously, parentheses indicate that some operation or operations will be performed outside the standard order +of operation rules. For instance, perhaps you want to multiply 4 and 7 before squaring. To indicate that the multiplication +happens before the exponent, the multiplication is placed inside parentheses: +. +This means operations inside the parentheses take precedence, or happen before other operations. Now, the first step in +calculating arithmetic expressions using the order of operations is to perform operations inside parentheses first. Inside +the parentheses, you follow the order of operation rules EMDAS. +EXAMPLE 3.49 +Prioritizing Parentheses in the Order of Operations +Correctly apply the rules for the order of operations to accurately compute the following: +162 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +. +Solution +Step 1: Perform all calculations within the parentheses before all other operations. +Step 2: Since all parentheses have been cleared, move left to right, and compute all the exponents next. +Step 3: Perform all multiplications and divisions moving left to right. +YOUR TURN 3.49 +1. Correctly apply the rules for the order of operations to accurately compute the following: +. +Be aware that there can be more than one set of parentheses, and parentheses within parentheses. When one set of +parentheses is inside another set, do the innermost set first, and then work outward. +VIDEO +Order of Operations 3 (https://openstax.org/r/Order_of_Operations_3) +EXAMPLE 3.50 +Working Innermost Parentheses in the Order of Operations +Correctly apply the rules for order of operations to accurately compute the following: +. +Solution +Step 1: Perform all calculations within the parentheses before other operations. Evaluate the innermost parentheses +first. We can work separate parentheses expressions at the same time. The innermost set of parentheses has the 2 + 5 +inside. The 3 + 8 is in a separate set of parentheses, so that addition can occur at the same time as the 2 + 5. +Step 2: Now that those parentheses have been handled, move on to the next set of parentheses. Applying the order of +operation rules inside that set of parentheses, the exponent is evaluated first, then the multiplication, and then the +addition. +Step 3: Since all parentheses have been cleared, apply the EMDAS rules to finish the calculation. +YOUR TURN 3.50 +1. Correctly apply the rules for the order of operations to accurately compute the following: +3.3 • Order of Operations +163 + +. +VIDEO +Order of Operations 4 (https://openstax.org/r/Order_of_Operations_4) +Check Your Understanding +12. Which operation has highest precedence? +13. Which is performed first, exponents or addition? +14. Calculate +. +15. What is used to indicate operations that should be performed out of order? +16. Calculate +. +SECTION 3.3 EXERCISES +1. Which operations have the lowest precedence in order of operations? +2. If many operations have the same precedence in an expression, in what order should the operations be +performed? +3. Which operations have the same precedence in order of operations? +4. After all operations in parentheses have been performed, which operations should be performed next? +For the following exercises, perform the indicated calculation. +5. +6. +7. +8. +9. +10. +11. +12. +13. +14. +15. +16. +17. +18. +19. +20. +21. +22. +23. +24. +25. +26. +27. +28. +29. +30. +31. +32. +33. +34. +164 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +35. +36. +37. +38. +39. +40. +3.4 Rational Numbers +Figure 3.20 Stock gains and losses are often represented as percentages.(credit: "stock market quotes in newspaper" by +Andreas Poike/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Define and identify numbers that are rational. +2. +Simplify rational numbers and express in lowest terms. +3. +Add and subtract rational numbers. +4. +Convert between improper fractions and mixed numbers. +5. +Convert rational numbers between decimal and fraction form. +6. +Multiply and divide rational numbers. +7. +Apply the order of operations to rational numbers to simplify expressions. +8. +Apply density property of rational numbers. +9. +Solve problems involving rational numbers. +10. +Use fractions to convert between units. +11. +Define and apply percent. +12. +Solve problems using percent. +We are often presented with percentages or fractions to explain how much of a population has a certain feature. For +example, the 6-year graduation rate of college students at public institutions is 57.6%, or 72/125. That fraction may be +unsettling. But without the context, the percentage is hard to judge. So how does that compare to private institutions? +There, the 6-year graduation rate is 65.4%, or 327/500. Comparing the percentages is straightforward, but the fractions +are harder to interpret due to different denominators. For more context, historical data could be found. One study +reported that the 6-year graduation rate in 1995 was 56.4%. Comparing that historical number to the recent 6-year +graduation rate at public institutions of 57.6% shows that there hasn't been much change in that rate. +Defining and Identifying Numbers That Are Rational +A rational number (called rational since it is a ratio) is just a fraction where the numerator is an integer and the +denominator is a non-zero integer. As simple as that is, they can be represented in many ways. It should be noted here +that any integer is a rational number. An integer, +, written as a fraction of two integers is +. +In its most basic representation, a rational number is an integer divided by a non-zero integer, such as +. Fractions may +3.4 • Rational Numbers +165 + +be used to represent parts of a whole. The denominator is the total number of parts to the object, and the numerator is +how many of those parts are being used or selected. So, if a pizza is cut into 8 equal pieces, each piece is +of the pizza. +If you take three slices, you have +of the pizza (Figure 3.21). Similarly, if in a group of 20 people, 5 are wearing hats, +then +of the people are wearing hats (Figure 3.22). +Figure 3.21 Pizza cut in 8 slices, with 3 slices highlighted +Figure 3.22 Group of 20 people, with 5 people wearing hats +Another representation of rational numbers is as a mixed number, such as +(Figure 3.23). This represents a whole +number (2 in this case), plus a fraction (the +). +166 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Figure 3.23 Two whole pizzas and one partial pizza +Rational numbers may also be expressed in decimal form; for instance, as 1.34. When 1.34 is written, the decimal part, +0.34, represents the fraction +, and the number 1.34 is equal to +. However, not all decimal representations are +rational numbers. +A number written in decimal form where there is a last decimal digit (after a given decimal digit, all following decimal +digits are 0) is a terminating decimal, as in 1.34 above. Alternately, any decimal numeral that, after a finite number of +decimal digits, has digits equal to 0 for all digits following the last non-zero digit. +All numbers that can be expressed as a terminating decimal are rational. This comes from what the decimal represents. +The decimal part is the fraction of the decimal part divided by the appropriate power of 10. That power of 10 is the +number of decimal digits present, as for 0.34, with two decimal digits, being equal to +. +Another form that is a rational number is a decimal that repeats a pattern, such as 67.1313… When a rational number is +expressed in decimal form and the decimal is a repeated pattern, we use special notation to designate the part that +repeats. For example, if we have the repeating decimal 4.3636…, we write this as +. The bar over the 36 indicates that +the 36 repeats forever. +If the decimal representation of a number does not terminate or form a repeating decimal, that number is not a rational +number. +One class of numbers that is not rational is the square roots of integers or rational numbers that are not perfect +squares, such as +and +. More generally, the number +is the square root of the number +if +. The +notation for this is +, where the symbol +is the square root sign. An integer perfect square is any integer that +can be written as the square of another integer. A rational perfect square is any rational number that can be written as a +fraction of two integers that are perfect squares. +Sometimes you may be able to identify a perfect square from memory. Another process that may be used is to factor the +number into the product of an integer with itself. Or a calculator (such as Desmos) may be used to find the square root +of the number. If the calculator yields an integer, the original number was a perfect square. +TECH CHECK +Using Desmos to Find the Square Root of a Number +When Desmos is used, there is a tab at the bottom of the screen that opens the keyboard for Desmos. The keyboard +is shown below. On the keyboard (Figure 3.24) is the square root symbol +. To find the square root of a number, +click the square root key, and then type the number. Desmos will automatically display the value of the square root as +you enter the number. +3.4 • Rational Numbers +167 + +Figure 3.24 Desmos keyboard with square root key +EXAMPLE 3.51 +Identifying Perfect Squares +Which of the following are perfect squares? +1. +45 +2. +144 +Solution +1. +We could attempt to find the perfect square by factoring. Writing all the factor pairs of 45 results in +, +and +. None of the pairs is a square, so 45 is not a perfect square. Using a calculator to find the square root of +45, we obtain 6.708 (rounded to three decimal places). Since this was not an integer, the original number was not a +perfect square. +2. +We could attempt to find the perfect square by factoring. Writing all the factor pairs of 144 results in +, and +. Since the last pair is an integer multiplied by itself, 144 is a +perfect square. Alternately, using Desmos to find the square root of 144, we obtain 12. Since the square root of 144 +is an integer, 144 is a perfect square. +YOUR TURN 3.51 +Determine if the following are perfect squares: +1. 94 +2. 441 +VIDEO +Introduction to Fractions (https://openstax.org/r/Equivalent_Fractions) +EXAMPLE 3.52 +Identifying Rational Numbers +Determine which of the following are rational numbers: +1. +2. +3. +4. +5. +Solution +1. +Since 73 is not a perfect square, its square root is not a rational number. This can also be seen when a calculator is +used. Entering +into a calculator results in 8.544003745317 (and then more decimal values after that). There is +no repeated pattern, so this is not a rational number. +2. +Since 4.556 is a decimal that terminates, this is a rational number. +168 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +3. +is a mixed number, so it is a rational number. +4. +is an integer divided by an integer, so it is a rational number. +5. +is a decimal that repeats a pattern, so it is a rational number. +YOUR TURN 3.52 +Determine which of the following are rational numbers: +1. +2. +3. +4. +5. +Simplifying Rational Numbers and Expressing in Lowest Terms +A rational number is one way to express the division of two integers. As such, there may be multiple ways to express the +same value with different rational numbers. For instance, +and +are the same value. If we enter them into a +calculator, they both equal 0.8. Another way to understand this is to consider what it looks like in a figure when two +fractions are equal. +In Figure 3.25, we see that +of the rectangle and +of the rectangle are equal areas. +Figure 3.25 Two Rectangles with Equal Areas +They are the same proportion of the area of the rectangle. The left rectangle has 5 pieces, three of which are shaded. The +right rectangle has 15 pieces, 9 of which are shaded. Each of the pieces of the left rectangle was divided equally into +three pieces. This was a multiplication. The numerator describing the left rectangle was 3 but it becomes +, or 9, as +each piece was divided into three. Similarly, the denominator describing the left rectangle was 5, but became +, or +15, as each piece was divided into 3. The fractions +and +are equivalent because they represent the same portion +(often loosely referred to as equal). +This understanding of equivalent fractions is very useful for conceptualization, but it isn’t practical, in general, for +determining when two fractions are equivalent. Generally, to determine if the two fractions +and +are equivalent, we +check to see that +. If those two products are equal, then the fractions are equal also. +EXAMPLE 3.53 +Determining If Two Fractions Are Equivalent +Determine if +and +are equivalent fractions. +3.4 • Rational Numbers +169 + +Solution +Applying the definition, +and +. So +. Also, +. Since +these values are equal, the fractions are equivalent. +YOUR TURN 3.53 +1. Determine if +and +are equivalent fractions. +That +indicates the fractions +and +are equivalent is due to some algebra. One property of natural +numbers, integers, and rational numbers (also irrational numbers) is that for any three numbers +and +with +, if +, then +. In other words, when two numbers are equal, then dividing both numbers by the same non-zero +number, the two newly obtained numbers are also equal. We can apply that to +and +, to show that +and +are +equivalent if +. +If +, and +, we can divide both sides by and obtain the following: +. We can divide out +the +on the right-hand side of the equation, resulting in +. Similarly, we can divide both sides of the equation by +and obtain the following: +. We can divide out +the on the left-hand side of the equation, resulting in +. So, +the rational numbers +and +are equivalent when +. +VIDEO +Equivalent Fractions (https://openstax.org/r/Equivalent_Fractions) +Recall that a common divisor or common factor of a set of integers is one that divides all the numbers of the set of +numbers being considered. In a fraction, when the numerator and denominator have a common divisor, that common +divisor can be divided out. This is often called canceling the common factors or, more colloquially, as canceling. +To show this, consider the fraction +. The numerator and denominator have the common factor 3. We can rewrite the +fraction as +. The common divisor 3 is then divided out, or canceled, and we can write the fraction as +. The 3s have been crossed out to indicate they have been divided out. The process of dividing out two +factors is also referred to as reducing the fraction. +If the numerator and denominator have no common positive divisors other than 1, then the rational number is in lowest +terms. +The process of dividing out common divisors of the numerator and denominator of a fraction is called reducing the +fraction. +One way to reduce a fraction to lowest terms is to determine the GCD of the numerator and denominator and divide out +the GCD. Another way is to divide out common divisors until the numerator and denominator have no more common +factors. +EXAMPLE 3.54 +Reducing Fractions to Lowest Terms +Express the following rational numbers in lowest terms: +1. +2. +3. +Solution +1. +One process to reduce +to lowest terms is to identify the GCD of 36 and 48 and divide out the GCD. The GCD of 36 +and 48 is 12. +Step 1: We can then rewrite the numerator and denominator by factoring 12 from both. +170 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Step 2: We can now divide out the 12s from the numerator and denominator. +So, when +is reduced to lowest terms, the result is +. +Alternately, you could identify a common factor, divide out that common factor, and repeat the process until the +remaining fraction is in lowest terms. +Step 1: You may notice that 4 is a common factor of 36 and 48. +Step 2: Divide out the 4, as in +. +Step 3: Examining the 9 and 12, you identify 3 as a common factor and divide out the 3, as in +. The 3 +and 4 have no common positive factors other than 1, so it is in lowest terms. +So, when +is reduced to lowest terms, the result is +. +2. +Step 1: To reduce +to lowest terms, identify the GCD of 100 and 250. This GCD is 50. +Step 2: We can then rewrite the numerator and denominator by factoring 50 from both. +. +Step 3: We can now divide out the 50s from the numerator and denominator. +So, when +is reduced to lowest terms, the result is +. +3. +Step 1: To reduce +to lowest terms, identify the GCD of 51 and 136. This GCD is 17. +Step 2: We can then rewrite the numerator and denominator by factoring 17 from both. +Step 3: We can now divide out the 17s from the numerator and denominator. +So, when +is reduced to lowest terms, the result is +. +YOUR TURN 3.54 +1. Express +in lowest terms. +VIDEO +Reducing Fractions to Lowest Terms (https://openstax.org/r/Reducing_Fractions_to_Lowest_Terms) +TECH CHECK +Using Desmos to Find Lowest Terms +Desmos is a free online calculator (https://openstax.org/r/calculator). Desmos supports reducing fractions to lowest +terms. When a fraction is entered, Desmos immediately calculates the decimal representation of the fraction. +3.4 • Rational Numbers +171 + +However, to the left of the fraction, there is a button that, when clicked, shows the fraction in reduced form. +VIDEO +Using Desmos to Reduce a Fraction (https://openstax.org/r/Using_Desmos_to_Reduce_a_Fraction) +Adding and Subtracting Rational Numbers +Adding or subtracting rational numbers can be done with a calculator, which often returns a decimal representation, or +by finding a common denominator for the rational numbers being added or subtracted. +TECH CHECK +Using Desmos to Add Rational Numbers in Fractional Form +To create a fraction in Desmos, enter the numerator, then use the division key (/) on your keyboard, and then enter +the denominator. The fraction is then entered. Then click the right arrow key to exit the denominator of the fraction. +Next, enter the arithmetic operation (+ or –). Then enter the next fraction. The answer is displayed dynamically +(calculates as you enter). To change the Desmos result from decimal form to fractional form, use the fraction button +(Figure 3.26) on the left of the line that contains the calculation: +Figure 3.26 Fraction button on the Desmos keyboard +EXAMPLE 3.55 +Adding Rational Numbers Using Desmos +Calculate +using Desmos. +Solution +Enter +in Desmos. The result is displayed as +(which is +). Clicking the fraction button to the +left on the calculation line yields +. +YOUR TURN 3.55 +1. Calculate +. +Performing addition and subtraction without a calculator may be more involved. When the two rational numbers have a +common denominator, then adding or subtracting the two numbers is straightforward. Add or subtract the numerators, +and then place that value in the numerator and the common denominator in the denominator. Symbolically, we write +this as +. This can be seen in the Figure 3.27, which shows +. +172 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Figure 3.27 Partially Shaded Rectangle +It is customary to then write the result in lowest terms. +FORMULA +If +is a non-zero integer, then +. +EXAMPLE 3.56 +Adding Rational Numbers with the Same Denominator +Calculate +. +Solution +Since the rational numbers have the same denominator, we perform the addition of the numerators, +, and then +place the result in the numerator and the common denominator, 28, in the denominator. +Once we have that result, reduce to lowest terms, which gives +. +YOUR TURN 3.56 +1. Calculate +. +EXAMPLE 3.57 +Subtracting Rational Numbers with the Same Denominator +Calculate +. +Solution +Since the rational numbers have the same denominator, we perform the subtraction of the numerators, +, and +then place the result in the numerator and the common denominator, 136, in the denominator. +Once we have that result, reduce to lowest terms, this gives +. +3.4 • Rational Numbers +173 + +YOUR TURN 3.57 +1. Calculate +. +When the rational numbers do not have common denominators, then we have to transform the rational numbers so +that they do have common denominators. The common denominator that reduces work later in the problem is the LCM +of the numerator and denominator. When adding or subtracting the rational numbers +and +, we perform the +following steps. +Step 1: Find +. +Step 2: Calculate +and +. +Step 3: Multiply the numerator and denominator of +by +, yielding +. +Step 4: Multiply the numerator and denominator of +by +, yielding +. +Step 5: Add or subtract the rational numbers from Steps 3 and 4, since they now have the common denominators. +You should be aware that the common denominator is +. For the first denominator, we have +, since we multiply and divide +by the same number. For the same reason, +. +EXAMPLE 3.58 +Adding Rational Numbers with Unequal Denominators +Calculate +. +Solution +The denominators of the fractions are 18 and 15, so we label +and +. +Step 1: Find LCM(18,15). This is 90. +Step 2: Calculate +and +. +and +. +Step 3: Multiplying the numerator and denominator of +by +yields +. +Step 4: Multiply the numerator and denominator of +by +yields +. +Step 5: Now we add the values from Steps 3 and 4: +. +This is in lowest terms, so we have found that +. +YOUR TURN 3.58 +1. Calculate +. +EXAMPLE 3.59 +Subtracting Rational Numbers with Unequal Denominators +Calculate +. +Solution +The denominators of the fractions are 25 and 70, so we label +and +. +Step 1: Find LCM(25,70). This is 350. +174 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Step 2: Calculate +and +: +and +. +Step 3: Multiplying the numerator and denominator of +by +yields +. +Step 4: Multiplying the numerator and denominator of +by +yields +. +Step 5: Now we subtract the value from Step 4 from the value in Step 3: +. +This is in lowest terms, so we have found that +. +YOUR TURN 3.59 +1. Calculate +. +VIDEO +Adding and Subtracting Fractions with Different Denominators (https://openstax.org/r/ +Adding_and_Subtracting_Fractions) +Converting Between Improper Fractions and Mixed Numbers +One way to visualize a fraction is as parts of a whole, as in +of a pizza. But when the numerator is larger than the +denominator, as in +, then the idea of parts of a whole seems not to make sense. Such a fraction is an improper +fraction. That kind of fraction could be written as an integer plus a fraction, which is a mixed number. The fraction +rewritten as a mixed number would be +. Arithmetically, +is equivalent to +, which is read as “one and 11 +twelfths.” +Improper fractions can be rewritten as mixed numbers using division and remainders. To find the mixed number +representation of an improper fraction, divide the numerator by the denominator. The quotient is the integer part, and +the remainder becomes the numerator of the remaining fraction. +EXAMPLE 3.60 +Rewriting an Improper Fraction as a Mixed Number +Rewrite +as a mixed number. +Solution +When 48 is divided by 13, the result is 3 with a remainder of 9. So, we can rewrite +as +. +YOUR TURN 3.60 +1. Rewrite +as a mixed number. +VIDEO +Converting an Improper Fraction to a Mixed Number Using Desmos (https://openstax.org/r/ +Improper_Fraction_to_Mixed_Number) +Similarly, we can convert a mixed number into an improper fraction. To do so, first convert the whole number part to a +fraction by writing the whole number as itself divided by 1, and then add the two fractions. +Alternately, we can multiply the whole number part and the denominator of the fractional part. Next, add that product to +the numerator. Finally, express the number as that product divided by the denominator. +3.4 • Rational Numbers +175 + +EXAMPLE 3.61 +Rewriting a Mixed Number as an Improper Fraction +Rewrite +as an improper fraction. +Solution +Step 1: Multiply the integer part, 5, by the denominator, 9, which gives +. +Step 2: Add that product to the numerator, which gives +. +Step 3: Write the number as the sum, 49, divided by the denominator, 9, which gives +. +YOUR TURN 3.61 +1. Rewrite +as an improper fraction. +TECH CHECK +Using Desmos to Rewrite a Mixed Number as an Improper Fraction +Desmos can be used to convert from a mixed number to an improper fraction. To do so, we use the idea that a mixed +number, such as +, is another way to represent +. If +is entered in Desmos, the result is the decimal +form of the number. However, clicking the fraction button to the left will convert the decimal to an improper fraction, +. As an added bonus, Desmos will automatically reduce the fraction to lowest terms. +Converting Rational Numbers Between Decimal and Fraction Forms +Understanding what decimals represent is needed before addressing conversions between the fractional form of a +number and its decimal form, or writing a number in decimal notation. The decimal number 4.557 is equal to +. +The decimal portion, .557, is 557 divided by 1,000. To write any decimal portion of a number expressed as a terminating +decimal, divide the decimal number by 10 raised to the power equal to the number of decimal digits. Since there were +three decimal digits in 4.557, we divided 557 by +. +Decimal representations may be very long. It is convenient to round off the decimal form of the number to a certain +number of decimal digits. To round off the decimal form of a number to +(decimal) digits, examine the ( +)st decimal +digit. If that digit is 0, 1, 2, 3, or 4, the number is rounded off by writing the number to the +th decimal digit and no +further. If the ( +)st decimal digit is 5, 6, 7, 8, or 9, the number is rounded off by writing the number to the +th digit, +then replacing the +th digit by one more than the +th digit. +EXAMPLE 3.62 +Rounding Off a Number in Decimal Form to Three Digits +Round 5.67849 to three decimal digits. +Solution +The third decimal digit is 8. The digit following the 8 is 4. When the digit is 4, we write the number only to the third digit. +So, 5.67849 rounded off to three decimal places is 5.678. +YOUR TURN 3.62 +1. Round 5.1082 to three decimal places. +176 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +EXAMPLE 3.63 +Rounding Off a Number in Decimal Form to Four Digits +Round 45.11475 to four decimal digits. +Solution +The fourth decimal digit is 7. The digit following the 7 is 5. When the digit is 5, we write the number only to the fourth +decimal digit, 45.1147. We then replace the fourth decimal digit by one more than the fourth digit, which yields 45.1148. +So, 45.11475 rounded off to four decimal places is 45.1148. +YOUR TURN 3.63 +1. Round 18.6298 to two decimal places. +To convert a rational number in fraction form to decimal form, use your calculator to perform the division. +EXAMPLE 3.64 +Converting a Rational Number in Fraction Form into Decimal Form +Convert +into decimal form. +Solution +Using a calculator to divide 47 by 25, the result is 1.88. +YOUR TURN 3.64 +1. Convert +into decimal form. +Converting a terminating decimal to the fractional form may be done in the following way: +Step 1: Count the number of digits in the decimal part of the number, labeled +. +Step 2: Raise 10 to the +th power. +Step 3: Rewrite the number without the decimal. +Step 4: The fractional form is the number from Step 3 divided by the result from Step 2. +This process works due to what decimals represent and how we work with mixed numbers. For example, we could +convert the number 7.4536 to fractional from. The decimal part of the number, the .4536 part of 7.4536, has four digits. +By the definition of decimal notation, the decimal portion represents +. The decimal number 7.4536 is +equal to the improper fraction +. Adding those to fractions yields +. +EXAMPLE 3.65 +Converting from Decimal Form to Fraction Form with Terminating Decimals +Convert 3.2117 to fraction form. +Solution +Step 1: There are four digits after the decimal point, so +. +Step 2: Raise 10 to the fourth power, +. +Step 3: When we remove the decimal point, we have 32,117. +Step 4: The fraction has as its numerator the result from Step 3 and as its denominator the result of Step 2, which is the +3.4 • Rational Numbers +177 + +fraction +. +YOUR TURN 3.65 +1. Convert 17.03347 to fraction form. +The process is different when converting from the decimal form of a rational number into fraction form when the +decimal form is a repeating decimal. This process is not covered in this text. +Multiplying and Dividing Rational Numbers +Multiplying rational numbers is less complicated than adding or subtracting rational numbers, as there is no need to find +common denominators. To multiply rational numbers, multiply the numerators, then multiply the denominators, and +write the numerator product divided by the denominator product. Symbolically, +. As always, rational +numbers should be reduced to lowest terms. +FORMULA +If +and +are non-zero integers, then +. +EXAMPLE 3.66 +Multiplying Rational Numbers +Calculate +. +Solution +Multiply the numerators and place that in the numerator, and then multiply the denominators and place that in the +denominator. +This is not in lowest terms, so this needs to be reduced. The GCD of 120 and 525 is 15. +YOUR TURN 3.66 +1. Calculate +. +VIDEO +Multiplying Fractions (https://openstax.org/r/Multiplying_Fractions) +As with multiplication, division of rational numbers can be done using a calculator. +EXAMPLE 3.67 +Dividing Decimals with a Calculator +Calculate 3.45 ÷ 2.341 using a calculator. Round to three decimal places if necessary. +Solution +Using a calculator, we obtain 1.473729175565997. Rounding to three decimal places we have 1.474. +178 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +YOUR TURN 3.67 +1. Calculate +using a calculator. Round to three decimal places, if necessary. +Before discussing division of fractions without a calculator, we should look at the reciprocal of a number. The reciprocal +of a number is 1 divided by the number. For a fraction, the reciprocal is the fraction formed by switching the numerator +and denominator. For the fraction +, the reciprocal is +. An important feature for a number and its reciprocal is that +their product is 1. +When dividing two fractions by hand, find the reciprocal of the divisor (the number that is being divided into the other +number). Next, replace the divisor by its reciprocal and change the division into multiplication. Then, perform the +multiplication. Symbolically, +. As before, reduce to lowest terms. +FORMULA +If +and +are non-zero integers, then +. +EXAMPLE 3.68 +Dividing Rational Numbers +1. +Calculate +. +2. +Calculate +. +Solution +1. +Step 1: Find the reciprocal of the number being divided by +. The reciprocal of that is +. +Step 2: Multiply the first fraction by that reciprocal. +The answer, +is not in lowest terms. The GCD of 140 and 126 is 14. Factoring and canceling gives +. +2. +Step 1: Find the reciprocal of the number being divided by, which is +. The reciprocal of that is +. +Step 2: Multiply the first fraction by that reciprocal: +The answer, +, is not in lowest reduced form. The GCD of 28 and 40 is 4. Factoring and canceling gives +. +YOUR TURN 3.68 +1. Calculate +. +2. Calculate +. +VIDEO +Dividing Fractions (https://openstax.org/r/Dividing_Fractions) +Applying the Order of Operations to Simplify Expressions +The order of operations for rational numbers is the same as for integers, as discussed in Order of Operations. The order +of operations makes it easier for anyone to correctly calculate and represent. The order follows the well-known acronym +PEMDAS: +3.4 • Rational Numbers +179 + +P +Parentheses +E +Exponents +M/D +Multiplication and division +A/S +Addition and subtraction +The first step in calculating using the order of operations is to perform operations inside the parentheses. Moving down +the list, next perform all exponent operations moving from left to right. Next (left to right once more), perform all +multiplications and divisions. Finally, perform the additions and subtractions. +EXAMPLE 3.69 +Applying the Order of Operations with Rational Numbers +Correctly apply the rules for the order of operations to accurately compute +. +Solution +Step 1: To calculate this, perform all calculations within the parentheses before other operations. +Step 2: Since all parentheses have been cleared, we move left to right, and compute all the exponents next. +Step 3: Now, perform all multiplications and divisions, moving left to right. +YOUR TURN 3.69 +1. Correctly apply the rules for the order of operations to accurately compute +. +EXAMPLE 3.70 +Applying the Order of Operations with Rational Numbers +Correctly apply the rules for the order of operations to accurately compute +. +Solution +To calculate this, perform all calculations within the parentheses before other operations. Evaluate the innermost +parentheses first. We can work separate parentheses expressions at the same time. +Step 1: The innermost parentheses contain +. Calculate that first, dividing after finding the common denominator. +Step 2: Calculate the exponent in the parentheses, +. +180 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Step 3: Subtract inside the parentheses is done, using a common denominator. +Step 4: At this point, evaluate the exponent and divide. +Step 5: Add. +Had this been done on a calculator, the decimal form of the answer would be 4.0232 (rounded to four decimal places). +YOUR TURN 3.70 +1. Correctly apply the rules for the order of operations to accurately compute +. +VIDEO +Order of Operations Using Fractions (https://openstax.org/r/Operations_Using_Fractions) +Applying the Density Property of Rational Numbers +Between any two rational numbers, there is another rational number. This is called the density property of the rational +numbers. +Finding a rational number between any two rational numbers is very straightforward. +Step 1: Add the two rational numbers. +Step 2: Divide that result by 2. +The result is always a rational number. This follows what we know about rational numbers. If two fractions are added, +then the result is a fraction. Also, when a fraction is divided by a fraction (and 2 is a fraction), then we get another +fraction. This two-step process will give a rational number, provided the first two numbers were rational. +EXAMPLE 3.71 +Applying the Density Property of Rational Numbers +Demonstrate the density property of rational numbers by finding a rational number between +and +. +3.4 • Rational Numbers +181 + +Solution +To find a rational number between +and +: +Step 1: Add the fractions. +Step 2: Divide the result by 2. Recall that to divide by 2, you multiply by the reciprocal of 2. The reciprocal of 2 is +, as +seen below. +So, one rational number between +and +is +. +We could check that the number we found is between the other two by finding the decimal representation of the +numbers. Using a calculator, the decimal representations of the rational numbers are 0.363636…, 0.473484848…, and +0.5833333…. Here it is clear that +is between +and +. +YOUR TURN 3.71 +1. Demonstrate the density property of rational numbers by finding a rational number between +and +. +Solving Problems Involving Rational Numbers +Rational numbers are used in many situations, sometimes to express a portion of a whole, other times as an expression +of a ratio between two quantities. For the sciences, converting between units is done using rational numbers, as when +converting between gallons and cubic inches. In chemistry, mixing a solution with a given concentration of a chemical +per unit volume can be solved with rational numbers. In demographics, rational numbers are used to describe the +distribution of the population. In dietetics, rational numbers are used to express the appropriate amount of a given +ingredient to include in a recipe. As discussed, the application of rational numbers crosses many disciplines. +EXAMPLE 3.72 +Mixing Soil for Vegetables +James is mixing soil for a raised garden, in which he plans to grow a variety of vegetables. For the soil to be suitable, he +determines that +of the soil can be topsoil, but +needs to be peat moss and +has to be compost. To fill the raised +garden bed with 60 cubic feet of soil, how much of each component does James need to use? +Solution +In this example, we know the proportion of each component to mix, and we know the total amount of the mix we need. +In this kind of situation, we need to determine the appropriate amount of each component to include in the mixture. For +each component of the mixture, multiply 60 cubic feet, which is the total volume of the mixture we want, by the fraction +required of the component. +Step 1: The required fraction of topsoil is +, so James needs +cubic feet of topsoil. Performing the multiplication, +James needs +(found by treating the fraction as division, and 120 divided by 5 is 24) cubic feet of +topsoil. +Step 2: The required fraction of peat moss is also +, so he also needs +cubic feet, or +cubic feet +of peat moss. +Step 3: The required fraction of compost is +. For the compost, he needs +cubic feet. +YOUR TURN 3.72 +1. Ashley wants to study for 10 hours over the weekend. She plans to spend half the time studying math, a quarter +182 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +of the time studying history, an eighth of the time studying writing, and the remaining eighth of the time +studying physics. How much time will Ashley spend on each of those subjects? +EXAMPLE 3.73 +Determining the Number of Specialty Pizzas +At Bella’s Pizza, one-third of the pizzas that are ordered are one of their specialty varieties. If there are 273 pizzas +ordered, how many were specialty pizzas? +Solution +One-third of the whole are specialty pizzas, so we need one-third of 273, which gives +, found by +dividing 273 by 3. So, 91 of the pizzas that were ordered were specialty pizzas. +YOUR TURN 3.73 +1. Danny, a nutritionist, is designing a diet for her client, Callum. Danny determines that Callum’s diet should be +30% protein. If Callum consumes 2,400 calories per day, how many calories of protein should Danny tell Callum +to consume? +VIDEO +Finding a Fraction of a Total (https://openstax.org/r/Finding_Fraction_of_Total) +Using Fractions to Convert Between Units +A common application of fractions is called unit conversion, or converting units, which is the process of changing from +the units used in making a measurement to different units of measurement. +For instance, 1 inch is (approximately) equal to 2.54 cm. To convert between units, the two equivalent values are made +into a fraction. To convert from the first type of unit to the second type, the fraction has the second unit as the +numerator, and the first unit as the denominator. +From the inches and centimeters example, to change from inches to centimeters, we use the fraction +. If, on the +other hand, we wanted to convert from centimeters to inches, we’d use the fraction +. This fraction is multiplied by +the number of units of the type you are converting from, which means the units of the denominator are the same as the +units being multiplied. +EXAMPLE 3.74 +Converting Liters to Gallons +It is known that 1 liter (L) is 0.264172 gallons (gal). Use this to convert 14 liters into gallons. +Solution +We know that 1 liter = 0.264172 gal. Since we are converting from liters, when we create the fraction we use, make sure +the liter part of the equivalence is in the denominator. So, to convert the 14 liters to gallons, we multiply 14 by +. Notice the gallon part is in the numerator since we’re converting to gallons, and the liter part is in the +denominator since we are converting from liters. Performing this and rounding to three decimal places, we find that 14 +liters is +. +YOUR TURN 3.74 +1. One mile is equal to 1.60934 km. Convert 200 miles to kilometers. Round off the answer to three decimal places. +3.4 • Rational Numbers +183 + +EXAMPLE 3.75 +Converting Centimeters to Inches +It is known that 1 inch is 2.54 centimeters. Use this to convert 100 centimeters into inches. +Solution +We know that 1 inch = 2.54 cm. Since we are converting from centimeters, when we create the fraction we use, make +sure the centimeter part of the equivalence is in the denominator, +. To convert the 100 cm to inches, multiply 100 +by +. Notice the inch part is in the numerator since we’re converting to inches, and the centimeter part is in the +denominator since we are converting from centimeters. Performing this and rounding to three decimal places, we obtain +. This means 100 cm equals 39.370 in. +YOUR TURN 3.75 +1. It is known that 4 quarts equals 3.785 liters. If you have 25 quarts, how many liters do you have? Round off to +three decimal places. +VIDEO +Converting Units (https://openstax.org/r/Converting_Units) +Defining and Applying Percent +A percent is a specific rational number and is literally per 100. +percent, denoted +%, is the fraction +. +EXAMPLE 3.76 +Rewriting a Percentage as a Fraction +Rewrite the following as fractions: +1. +31% +2. +93% +Solution +1. +Using the definition and +, 31% in fraction form is +. +2. +Using the definition and +, 93% in fraction form is +. +YOUR TURN 3.76 +Rewrite the following as fractions: +1. 4% +2. 50% +EXAMPLE 3.77 +Rewriting a Percentage as a Decimal +Rewrite the following percentages in decimal form: +1. +54% +2. +83% +Solution +1. +Using the definition and +, 54% in fraction form is +. Dividing a number by 100 moves the decimal two +184 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +places to the left; 54% in decimal form is then 0.54. +2. +Using the definition and +, 83% in fraction form is +. Dividing a number by 100 moves the decimal two +places to the left; 83% in decimal form is then 0.83. +YOUR TURN 3.77 +Rewrite the following percentages in decimal form: +1. 14% +2. 7% +You should notice that you can simply move the decimal two places to the left without using the fractional definition of +percent. +Percent is used to indicate a fraction of a total. If we want to find 30% of 90, we would perform a multiplication, with 30% +written in either decimal form or fractional form. The 90 is the total, 30 is the percentage, and 27 (which is +) is +the percentage of the total. +FORMULA +of +items is +. The +is referred to as the total, the +is referred to as the percent or percentage, and the +value obtained from +is the part of the total and is also referred to as the percentage of the total. +EXAMPLE 3.78 +Finding a Percentage of a Total +1. +Determine 40% of 300. +2. +Determine 64% of 190. +Solution +1. +The total is 300, and the percentage is 40. Using the decimal form of 40% and multiplying we obtain +. +2. +The total is 190, and the percentage is 64. Using the decimal form of 64% and multiplying we obtain +. +YOUR TURN 3.78 +1. Determine 25% of 1,200. +2. Determine 53% of 1,588. +In the previous situation, we knew the total and we found the percentage of the total. It may be that we know the +percentage of the total, and we know the percent, but we don't know the total. To find the total if we know the +percentage the percentage of the total, use the following formula. +FORMULA +If we know that +% of the total is +, then the total is +. +3.4 • Rational Numbers +185 + +EXAMPLE 3.79 +Finding the Total When the Percentage and Percentage of the Total Are Known +1. +What is the total if 28% of the total is 140? +2. +What is the total if 6% of the total is 91? +Solution +1. +28 is the percentage, so +. 28% of the total is 140, so +. Using those we find that the total was +. +2. +6 is the percentage, so +. 6% of the total is 91, so +. Using those we find that the total was +. +YOUR TURN 3.79 +1. What is the total if 25% of the total is 30? +2. What is the total if 45% of the total is 360? +The percentage can be found if the total and the percentage of the total is known. If you know the total, and the +percentage of the total, first divide the part by the total. Move the decimal two places to the right and append the +symbol %. The percentage may be found using the following formula. +FORMULA +The percentage, +, of +that is +is +. +EXAMPLE 3.80 +Finding the Percentage When the Total and Percentage of the Total Are Known +Find the percentage in the following: +1. +Total is 300, percentage of the total is 60. +2. +Total is 440, percentage of the total is 176. +Solution +1. +The total is 300; the percentage of the total is 60. Calculating yields 0.2. Moving the decimal two places to the right +gives 20. Appending the percentage to this number results in 20%. So, 60 is 20% of 300. +2. +The total is 440; the percentage of the total is 176. Calculating yields 0.4. Moving the decimal two places to the right +gives 40. Appending the percentage to this number results in 40%. So, 176 is 40% of 440. +YOUR TURN 3.80 +Find the percentage in the following: +1. Total is 1,000, percentage of the total is 70. +2. Total is 500, percentage of the total is 425. +Solve Problems Using Percent +In the media, in research, and in casual conversation percentages are used frequently to express proportions. +Understanding how to use percent is vital to consuming media and understanding numbers. Solving problems using +percentages comes down to identifying which of the three components of a percentage you are given, the total, the +percentage, or the percentage of the total. If you have two of those components, you can find the third using the +methods outlined previously. +186 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +EXAMPLE 3.81 +Percentage of Students Who Are Sleep Deprived +A study revealed that 70% of students suffer from sleep deprivation, defined to be sleeping less than 8 hours per night. If +the survey had 400 participants, how many of those participants had less than 8 hours of sleep per night? +Solution +The percentage of interest is 70%. The total number of students is 400. With that, we can find how many were in the +percentage of the total, or, how many were sleep deprived. Applying the formula from above, the number who were +sleep deprived was +; 280 students on the study were sleep deprived. +YOUR TURN 3.81 +1. Riley has a daily calorie intake of 2,200 calories and wants to take in 20% of their calories as protein. How many +calories of protein should be in their daily diet? +EXAMPLE 3.82 +Amazon Prime Subscribers +There are 126 million users who are U.S. Amazon Prime subscribers. If there are 328.2 million residents in the United +States, what percentage of U.S. residents are Amazon Prime subscribers? +Solution +We are asked to find the percentage. To do so, we divide the percentage of the total, which is 126 million, by the total, +which is 328.2 million. Performing this division and rounding to three decimal places yields +. The decimal is +moved to the right by two places, and a % sign is appended to the end. Doing this shows us that 38.4% of U.S. residents +are Amazon Prime subscribers. +YOUR TURN 3.82 +1. A small town has 450 registered voters. In the primaries, 54 voted. What percentage of registered voters in that +town voted in the primaries? +EXAMPLE 3.83 +Finding the Percentage When the Total and Percentage of the Total Are Known +Evander plays on the basketball team at their university and 73% of the athletes at their university receive some sort of +scholarship for attending. If they know 219 of the student-athletes receive some sort of scholarship, how many student- +athletes are at the university? +Solution +We need to find the total number of student-athletes at Evander’s university. +Step 1: Identify what we know. We know the percentage of students who receive some sort of scholarship, 73%. We also +know the number of athletes that form the part of the whole, or 219 student-athletes. +Step 2: To find the total number of student-athletes, use +, with +and +. Calculating with those values +yields +. +So, there are 300 total student-athletes at Evander’s university +3.4 • Rational Numbers +187 + +YOUR TURN 3.83 +1. A store declares a deep discount of 40% for an item, which they say will save $30. What was the original price of +the item? +Check Your Understanding +17. Identify which of the following are rational numbers. +18. Express +in lowest terms. +19. Calculate +and express in lowest terms. +20. Convert 0.34 into fraction form. +21. Convert +into a mixed number. +22. Calculate +and express in lowest terms. +23. Calculate +. +24. Identify a rational number between +and +. +25. Convert +into a mixed number. +26. Lina decides to save +of her take-home pay every paycheck. Her most recent paycheck was for $882. How much +will she save from that paycheck? +27. Determine 38% of 600. +28. A microchip factory has decided to increase its workforce by 10%. If it currently has 70 employees, how many new +employees will the factory hire? +SECTION 3.4 EXERCISES +For the following exercises, identify which of the following are rational numbers. +1. 4.598 +2. +3. +For the following exercises, reduce the fraction to lowest terms +4. +5. +6. +7. +8. +For the following exercises, do the indicated conversion. If it is a repeating decimal, use the correct notation. +9. Convert +to a mixed number. +10. Convert +to a mixed number. +11. Convert +to an improper fraction. +12. Convert +to an improper fraction. +13. Convert +to decimal form. +14. Convert +to decimal form. +15. Convert +to decimal form. +16. Convert +to decimal form. +17. Convert 0.23 to fraction form and reduce to lowest terms. +188 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +18. Convert 3.8874 to fraction form and reduce to lowest terms. +For the following exercises, perform the indicated operations. Reduce to lowest terms. +19. +20. +21. +22. +23. +24. +25. +26. +27. +28. +29. +30. +31. +32. +33. Find a rational number between +and +34. Find a rational number between +and +. +35. Find two rational numbers between +and +. +36. Find three rational numbers between +and +. +37. Convert 24% to fraction form and reduce completely. +38. Convert 95% to fraction form and reduce completely. +39. Convert 0.23 to a percentage. +40. Convert 1.22 to a percentage. +41. Determine 30% of 250. +42. Determine 75% of 600. +43. If 25% of a group is 41 members, how many members total are in the group? +44. If 80% of the total is 60, how much is in the total? +45. 13 is what percent of 20? +46. 80 is what percent of 320? +47. Professor Donalson’s history of film class has 60 students. Of those students, +say their favorite movie genre is +comedy. How many of the students in Professor Donalson’s class name comedy as their favorite movie genre? +48. Naia’s dormitory floor has 80 residents. Of those, +play Fortnight for at least 15 hours per week. How many +students on Naia’s floor play Fortnight at least 15 hours per week? +49. In Tara’s town there are 24,000 people. Of those, +are food insecure. How many people in Tara’s town are +food insecure? +50. Roughly +of air is nitrogen. If an enclosure holds 2,000 liters of air, how many liters of nitrogen should be +expected in the enclosure? +51. To make the dressing for coleslaw, Maddie needs to mix it with +mayonnaise and +apple cider vinegar. If +Maddie wants to have 8 cups of dressing, how many cups of mayonnaise and how many cups of apple cider +vinegar does Maddie need? +52. Malika is figuring out their schedule. They wish to spend +of their time sleeping, +of their time studying and +going to class, +of their time at work, and +of their time doing other activities, such as entertainment or +exercising. There are 168 hours in a week. How many hours in a week will Malika spend: +a. +Sleeping? +b. +Studying and going to class? +c. +Not sleeping? +53. Roughly 20.9% of air is oxygen. How much oxygen is there in 200 liters of air? +54. 65% of college students graduate within 6 years of beginning college. A first-year cohort at a college contains +400 students. How many are expected to graduate within 6 years? +3.4 • Rational Numbers +189 + +55. A 20% discount is offered on a new laptop. How much is the discount if the new laptop originally cost $700? +56. Leya helped at a neighborhood sale and was paid 5% of the proceeds. If Leya is paid $171.25, what were the +total proceeds from the neighborhood sale? +57. Unit Conversion. 1 kilogram (kg) is equal to 2.20462 pounds. Convert 13 kg to pounds. Round to three decimal +places, if necessary. +58. Unit Conversion. 1 kilogram (kg) is equal to 2.20462 pounds. Convert 200 pounds to kilograms. Round to three +decimal places, if necessary. +59. Unit Conversion. There are 12 inches in a foot, 3 feet in a yard, and 1,760 yards in a mile. Convert 10 miles to +inches. To do so, first convert miles to yards. Next, convert the yards to feet. Last, convert the feet to inches. +60. Unit Conversion. There are 1,000 meters (m) in a kilometer (km), and 100 centimeters (cm) in a meter. Convert +4 km to centimeters. +61. Markup. In this exercise, we introduce the concept of markup. The markup on an item is the difference +between how much a store sells an item for and how much the store paid for the item. Suppose Wegmans (a +northeastern U.S. grocery chain) buys cereal at $1.50 per box and sells the cereal for $2.29. +a. +Determine the markup in dollars. +b. +The markup is what percent of the original cost? Round the percentage to one decimal place. +62. In this exercise, we explore what happens when an item is marked up by a percentage, and then marked down +using the same percentage. +Wegmans purchases an item for $5 per unit. The markup on the item is 25%. +a. +Calculate the markup on the item, in dollars. +b. +What is the price for which Wegmans sells the item? This is the price Wegmans paid, plus the markup. +c. +Suppose Wegmans then offers a 25% discount on the sale price of the item (found in part b). In dollars, +how much is the discount? +d. +Determine the price of the item after the discount (this is the sales price of the item minus the discount). +Round to two decimal places. +e. +Is the new price after the markup and discount equal to the price Wegmans paid for the item? Explain. +63. Repeated Discounts. In this exercise, we explore applying more than one discount to an item. +Suppose a store cuts the price on an item by 50%, and then offers a coupon for 25% off any sale item. We will +find the price of the item after applying the sale price and the coupon discount. +a. +The original price was $150. After the 50% discount, what is the price of the item? +b. +The coupon is applied to the discount price. The coupon is for 25%. Find 25% of the sale price (found in +part a). +c. +Find the price after applying the coupon (this is the value from part a minus the value from part b). +d. +The total amount saved on the item is the original price after all the discounts. Determine the total amount +saved by subtracting the final price paid (part c) from the original price of the item. +e. +Determine the effective discount percentage, which is the total amount saved divided by the original price +of the item. +f. +Was the effective discount percentage equal to 75%, which would be the 50% plus the 25%? Explain. +Converting Repeating Decimals to a Fraction +It was mentioned in the section that repeating decimals are rational numbers. To convert a repeating decimal to a +rational number, perform the following steps: +Step 1: Label the original number +. +Step 2: Count the number of digits, +, in the repeating part of the number. +Step 3: Multiply +by +, and label this as +. +Step 4: Determine +. +Step 5: Calculate +. If done correctly, the repeating part of the number will cancel out. +Step 6: If the result from Step 5 has decimal digits, count the number of decimal digits in the number from Step 5. +Label this +. +Step 7: Remove the decimal from the result of Step 5. +Step 8: Add +zeros to the end of the number from Step 4. +Step 9: Divide the result from Step 7 by the result from Step 8. This is the fraction form of the repeating decimal. +64. Convert +to fraction form. +65. Convert +to fraction form. +66. Convert +to fraction form. +67. Convert +to fraction form. +190 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +3.5 Irrational Numbers +Figure 3.28 The Pythagoreans were a philosophical sect of ancient Greece, often associated with mathematics. (credit: +Fedor Andreevich Bronnikov (1827-1902) “Hymn of the Pythagoreans to the Rising Sun,” 1877, oil on canvas/Wikimedia, +public domain) +Learning Objectives +After completing this section, you should be able to: +1. +Define and identify numbers that are irrational. +2. +Simplify irrational numbers and express in lowest terms. +3. +Add and subtract irrational numbers. +4. +Multiply and divide irrational numbers. +5. +Rationalize fractions with irrational denominators. +The Pythagoreans were a philosophical sect in ancient Greece. Their philosophy included reincarnation and purifying the +mind through the study and contemplation of mathematics and science. One of their principles was the cosmos is ruled +by order, specifically mathematics and music. They even held mystic beliefs about specific numbers and figures. For +example, the number 1 was associated with the mind and essence. Four represented justice, as it is the first product of +two even numbers. Most famously, though, is the association with the Pythagorean Theorem, which states that in a right +triangle, the sum of the squares of the shorter sides of the triangle (the legs) equals the square of the longer side (the +hypotenuse). Even the ancient Egyptians used this relationship, as triangles with side measures 3, 4, and 5 were often +used in surveying following the flooding of the Nile. +There is a story of a Pythagorean, Hippasus, discovering that not all numbers could be expressed as fractions. In other +words, not all numbers were rational numbers. The story ends with Hippasus, who shared this, or in some versions +discovered it, put to death by drowning for sharing this fact, that not all quantities could be expressed as the ratio of two +natural numbers. +As colorful as that story may be, it is most likely false, as there are no contemporary sources to corroborate it. But it does +seem to mark the discovery that not all quantities or measures were fractions of numbers. And so, irrational numbers +were discovered. +VIDEO +The Philosophy of the Pythagoreans (https://openstax.org/r/Philosophy_of_Pythagoreans) +Defining and Identifying Numbers That Are Irrational +We defined rational numbers in the last section as numbers that could be expressed as a fraction of two integers. +Irrational numbers are numbers that cannot be expressed as a fraction of two integers. Recall that rational numbers +could be identified as those whose decimal representations either terminated (ended) or had a repeating pattern at +some point. So irrational numbers must be those whose decimal representations do not terminate or become a +repeating pattern. +3.5 • Irrational Numbers +191 + +One collection of irrational numbers is square roots of numbers that aren’t perfect squares. +is the square root of the +number +, denoted +, if +. The number +is the perfect square of the integer +if +. The rational number +is a perfect square if both +and +are perfect squares. +One method of determining if an integer is a perfect square is to examine its prime factorization. If, in that factorization, +all the prime factors are raised to even powers, the integer is a perfect square. Another method is to attempt to factor +the integer into an integer squared. It is possible that you recognize the number as a perfect square (such as 4 or 9). Or, +if you have a calculator at hand, use the calculator to determine if the square root of the integer is an integer. +EXAMPLE 3.84 +Identifying Perfect Squares +Determine which of the following are perfect squares. +1. +45 +2. +81 +3. +4. +Solution +1. +The prime factorization of 45 is +. Since the 5 is not raised to an even power, 45 is not a perfect square. +2. +The prime factorization of 81 is +. All the prime factors are raised to even powers, so 81 is a perfect square. +3. +We must determine if both the numerator and denominator of +are perfect squares for the rational number to be +a perfect square. The numerator is 9, and as mentioned above, 9 is a perfect square (it is 3 squared). Now we check +the prime factorization of the denominator, 28, which is +. Since 7 is not raised to an even power, 28 is not +a perfect square. Since the denominator is not a perfect square, +is not a perfect square. +4. +We must determine if both the numerator and denominator of +are perfect squares for the rational number to +be a perfect square. The numerator is 144. The prime factorization of 144 is +. Since all the prime +factors of 144 are raised to even powers, 144 is a perfect square. Now we check the prime factorization of the +denominator, 400, which is +. Since all the prime factors of 400 are raised to even powers, 400 is a +perfect square. Since the numerator and denominator of +are perfect squares, +is a perfect square. +YOUR TURN 3.84 +Determine which of the following are perfect squares. +1. 36 +2. 27 +3. +4. +TECH CHECK +Using Desmos to Determine if a Number Is a Perfect Square +Desmos may be used to determine if a number is a perfect square by using its square root function. When Desmos is +opened, there is a tab in the lower left-hand corner of the Desmos screen. This tab opens the Desmos keypad, shown +in Figure 3.29. +192 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Figure 3.29 Desmos keyboard with square root key circles +There you find the key for the square root, which is circled in Figure 3.29. To find the square root of a number, click the +square root key, which begins a calculation, and then enter the value for which you want a square root. If the result is +an integer, then the number is a perfect square. +VIDEO +Using Desmos to Find the Square Root of a Number (https://openstax.org/r/square_root_of_a_number) +Another collection of irrational numbers is based on the special number, pi, denoted by the Greek letter +, which is the +ratio of the circumference of the diameter of the circle (Figure 3.30). +Figure 3.30 Circle with radius, diameter, and circumference labeled +Any multiple or power of +is an irrational number. +Any number expressed as a rational number times an irrational number is an irrational number also. When an irrational +number takes that form, we call the rational number the rational part, and the irrational number the irrational part. It +should be noted that a rational number plus, minus, multiplied by, or divided by any irrational number is an irrational +number. +EXAMPLE 3.85 +Identifying Irrational Numbers +Identify which of the following numbers are irrational. +1. +2. +3. +4. +Solution +1. +35 can be factored as +, showing that 35 is not the square of an integer or a rational number. This mean its +square root is an irrational number. +2. +Since +is a decimal with a repeating pattern, it is rational, so it is not an irrational number. +3. +. Since 121 is the square of an integer, its square root is a rational number. +4. +Since +is a multiple of pi, it is irrational. In this case, the rational part of the number is 4, while the irrational part is +. +3.5 • Irrational Numbers +193 + +YOUR TURN 3.85 +Identify which of the following numbers are irrational. +1. +2. +3. +4. +WHO KNEW? +Euler-Mascheroni Constant +Determining if a number is rational or irrational is not trivial. There are numbers that defied such classification for +quite a long time. One such is the Euler-Mascheroni constant. The Euler-Mascheroni constant is used in mathematics, +and is primarily associated with the natural logarithm, which is a mathematical function. The constant has been +around since around 1790. However, it was unknown if this constant was rational or irrational until 2013, at which +point it was proven to be irrational. +Simplifying Square Roots and Expressing Them in Lowest Terms +To simplify a square root means that we rewrite the square root as a rational number times the square root of a +number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square +root. +The number inside the square root symbol is referred to as the radicand. So in the expression +the number +is +referred to as the radicand. +Before discussing how to simplify a square root, we need to introduce a rule about square roots. The square root of a +product of numbers equals the product of the square roots of those number. Written symbolically, +. +FORMULA +For any two numbers +and +, +. +Using this formula, we can factor an integer inside a square root into a perfect square times another integer. Then the +square root can be applied to the perfect square, leaving an integer times the square root of another integer. If the +number remaining under the square root has no perfect square factors, then we’ve simplified the irrational number into +lowest terms. To simplify the irrational number into lowest terms when +is an integer: +Step 1: Determine the largest perfect square factor of +, which we denote +. +Step 2: Factor +into +. +Step 3: Apply +. +Step 4: Write +in its simplified form, +. +When a square root has been simplified in this manner, +is referred to as the rational part of the number, and +is +referred to as the irrational part. +EXAMPLE 3.86 +Simplifying a Square Root +Simplify the irrational number +and express in lowest terms. Identify the rational and irrational parts. +194 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Solution +Begin by finding the largest perfect square that is a factor of 180. We can do this by writing out the factor pairs of 180: +Looking at the list of factors, the perfect squares are 4, 9, and 36. The largest is 36, so we factor the into +. +In the formula, +and +. Apply +. +The simplified form of +is +. In this example, the 6 is the rational part, and the +is the irrational part. +YOUR TURN 3.86 +1. Simplify the irrational number +and express in lowest terms. Identify the rational and irrational parts. +VIDEO +Simplifying Square Roots (https://openstax.org/r/Simplifying_Square_Roots) +EXAMPLE 3.87 +Simplifying a Square Root +Simplify the irrational number +and express in lowest terms. Identify the rational and irrational parts. +Solution +Begin by finding the largest perfect square that is a factor of 330. We can do this by writing out the factor pairs of 330: +Looking at the list of factors, there are no perfect squares other than 1, which means +is already expressed in +lowest terms. In this case, 1 is the rational part, and +is the irrational part. Though we could write this as +, +but the product of 1 and any other number is just the number. +YOUR TURN 3.87 +1. Simplify the irrational number +and express in lowest terms. Identify the rational and irrational parts. +EXAMPLE 3.88 +Simplifying a Square Root +Simplify the irrational number +and express in lowest terms. Identify the rational and irrational parts. +Solution +Begin by finding the largest perfect square that is a factor of 2,548. We can do this by writing out the factor pairs of +2,548: +Looking at the list of factors, the perfect squares are 4, 49, and 196. The largest is 196, so we factor the 2,548 into +. In the formula, +and +. Apply +. +The simplified form of +is +. In this example, 14 is the rational part, and +is the irrational part. +3.5 • Irrational Numbers +195 + +YOUR TURN 3.88 +1. Simplify the irrational number +. +VIDEO +Simplifying Square Roots (https://openstax.org/r/Simplifying_Square_Roots) +Adding and Subtracting Irrational Numbers +Just like any other number we’ve worked with, irrational numbers can be added or subtracted. When working with a +calculator, enter the operation and a decimal representation will be given. However, there are times when two irrational +numbers may be added or subtracted without the calculator. This can happen only when the irrational parts of the +irrational numbers are the same. +To add or subtract two irrational numbers that have the same irrational part, add or subtract the rational parts of the +numbers, and then multiply that by the common irrational part. +FORMULA +Let our first irrational number be +, where +is the rational and +the irrational parts. +Let the other irrational number be +, where +is the rational and +the irrational parts. +Then +. +EXAMPLE 3.89 +Subtracting Irrational Numbers with Similar Irrational Parts +If possible, subtract the following irrational numbers without using a calculator. If this is not possible, state why. +Solution +Since these two irrational numbers have the same irrational part, +, we can subtract without using a calculator. The +rational part of the first number is 3. The rational part of the second number is 8. Using the formula yields +. +YOUR TURN 3.89 +1. If possible, subtract the following irrational numbers without using a calculator. If this is not possible, state why. +EXAMPLE 3.90 +Adding Irrational Numbers with Similar Irrational Parts +If possible, add the following irrational numbers without using a calculator. If this is not possible, state why. +Solution +Since these two irrational numbers have the same irrational part, +, the addition can be performed without using a +calculator. The rational part of the first number is 35. The rational part of the second number is 17. Using the formula +yields +. +196 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +YOUR TURN 3.90 +1. If possible, add the following irrational numbers without using a calculator. If this is not possible, state why. +EXAMPLE 3.91 +Subtracting Irrational Numbers with Different Irrational Parts +If possible, subtract the following irrational numbers without using a calculator. If this is not possible, state why. +Solution +The two numbers being subtracted do not have the same irrational part, so the operation cannot be performed. +YOUR TURN 3.91 +1. If possible, subtract the following irrational numbers without using a calculator. If this is not possible, state why. +Multiplying and Dividing Irrational Numbers +Just like any other number that we’ve worked with, irrational numbers can be multiplied or divided. When working with a +calculator, enter the operation and a decimal representation will be given. Sometimes, though, you may want to retain +the form of the irrational number as a rational part times an irrational part. The process is similar to adding and +subtracting irrational numbers when they are in this form. We do not need the irrational parts to match. Even though +they need not match, they do need to be similar, such as both irrational parts are square roots, or both irrational parts +are multiples of pi. Also, if the irrational parts are square roots, we may need to reduce the resulting square root to +lowest terms. +When multiplying two square roots, use the following formula. It is the same formula presented during the discussion of +simplifying square roots. +FORMULA +For any two positive numbers +and +, +. +When dividing two square roots, use the following formula. +FORMULA +For any two positive numbers +and +, with +not equal to 0, +. +To multiply or divide irrational numbers with similar irrational parts, do the following: +Step 1: Multiply or divide the rational parts. +Step 2: If necessary, reduce the result of Step 1 to lowest terms. This becomes the rational part of the answer. +Step 3: Multiply or divide the irrational parts. +Step 4: If necessary, reduce the result from Step 3 to lowest terms. This becomes the irrational part of the answer. +Step 5: The result is the product of the rational and irrational parts. +3.5 • Irrational Numbers +197 + +EXAMPLE 3.92 +Dividing Irrational Numbers with Similar Irrational Parts +Perform the following operations without a calculator. Simplify if possible. +1. +2. +. +Solution +1. +In this division problem, +, notice that the irrational parts of these numbers are similar. They are both +square roots, so follow the steps given above. +Step 1: Divide the rational parts. +Step 2: If necessary, reduce the result of Step 1 to lowest terms. The 3 and 8 have no common factors, so +is +already in lowest terms. +Step 3: Divide the irrational parts. +Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand can be reduced, which yields +. +Step 5: The result is the product of the rational and irrational parts, which is +. +2. +In this division problem, +, notice that the irrational parts of these numbers are similar. They are +both square roots, so follow the steps given above. +Step 1: Divide the rational parts. +Step 2: If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal so +will not be reduced. +Step 3: Divide the irrational parts. +Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand can be reduced, which yields +. +Step 5: The result is the product of the rational and irrational parts, which is +. +YOUR TURN 3.92 +Perform the following operations without a calculator. Simplify if possible. +1. +2. +EXAMPLE 3.93 +Multiplying Irrational Numbers with Similar Irrational Parts +Perform the following operations without a calculator. Simplify if possible. +1. +2. +198 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Solution +1. +In this multiplication problem, +, notice that the irrational parts of these numbers are similar. +They are both square roots. Follow the process above. +Step 1: Multiply the rational parts. +Step 2: If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal and +will not be reduced. +Step 3: Multiply the irrational parts. +Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand is 36, which is the square of 6. The +irrational part reduces to +. +Step 5: The result is the product of the rational and irrational parts, which is +. +Notice that sometimes multiplying or dividing irrational numbers can result in a rational number. +2. +In this multiplication problem, +, notice that the irrational parts of these numbers are the same, +. Follow +the process above. +Step 1: Multiply the rational parts. +Step 2: If necessary, reduce the result of Step 1 to lowest terms. That result is an integer. +Step 3: Multiply the irrational parts. +Step 4: If necessary, reduce the result from Step 3 to lowest terms. This cannot be reduced. +Step 5: The result is the product of the rational and irrational parts, which is +. +YOUR TURN 3.93 +Perform the following operations without a calculator. Simplify if possible. +1. +2. +Rationalizing Fractions with Irrational Denominators +Fractions often represent that some amount is being equally divided into some number of parts. But to conceptualize a +fraction in that manner, the denominator needs to be an integer. An irrational number in the denominator interferes with +that interpretation of a fraction. Fractions that have denominators that are just the square root of an integer can be +altered into fractions with integer denominators using a process called rationalizing the denominator. The process +relies on the following property of square roots: +and the following property of fractions: +for any +non-zero number . +Using these two properties, when a fraction has a square root in the denominator, we can eliminate that square root. +Multiply the numerator and denominator by that square root from the denominator, +. Then apply +to the denominator, yielding +. Notice that there is no longer a square root in the +denominator, which allows for interpreting the fraction as dividing a whole into equal parts. +VIDEO +Rationalizing the Denominator (https://openstax.org/r/Rationalizing_Denominator) +3.5 • Irrational Numbers +199 + +EXAMPLE 3.94 +Rationalizing the Denominator +Rationalize the denominator of the following: +1. +2. +Solution +1. +The square root in the denominator is +. In order to rationalize the denominator of +, we need to multiply the +numerator and denominator by +and simplify. +The square root is in simplified form, so the final answer is +. +2. +The square root in the denominator is +. +Step 1: In order to rationalize the denominator of +, we need to multiply the numerator and denominator by +and simplify. +Step 2: The 60 under the square root can be factored into the following factor pairs: +Step 3: The largest square factor of 60 is 4, so we simplify the +in the numerator into +. We also cancel any +common factors. +This is completely simplified. +YOUR TURN 3.94 +Rationalize the denominator of the following: +1. +2. +There are occasions when the denominator is irrational but is the sum of two numbers where one or both involve square +roots. For instance, +. The process used earlier required that the denominator was the square root of a number +and would not work here. However, this type of denominator can be rationalized. In order to rationalize such a +denominator, we will multiply the numerator and denominator of the fraction by the conjugate of the denominator. The +conjugate of +is +. We say that +and +are conjugate numbers. +So, the conjugate of +is just +. But why is this of interest? The reason is because it leads to the +difference of squares formula, which is used to factor the difference of two squares. Or, for our purposes, in reverse it +allows us to eliminate a square root. +FORMULA +For any two numbers, +and +, +. +Looking at that formula, you should see that the two factors on the right-hand side of the equals sign are conjugates of +200 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +one another. So, for our purposes, we’re interested in +. This tells us that when we multiply +by its conjugate, we get +squared minus +squared, or +. But how is this useful? Let’s return to the fraction above, +. The denominator is +. Its conjugate is +. According to the formula, and letting +and +, +we see that +. But +is just 3. That means the product is +or 13. This no +longer has a square root. We use this to rationalize the denominator. +We will also need the distributive property of numbers. +FORMULA +For any three numbers +, +, and , +. This is called the distributive property. +EXAMPLE 3.95 +Rationalizing the Denominator Using Conjugates +Rationalize the denominator of +. +Solution +Step 1: We recognize that the denominator is the sum of two numbers where one or both involve square roots. This +means the conjugate can be used to remove the square root from the denominator. +Step 2: To do so, we multiply the numerator and the denominator each by the conjugate of the denominator. Since the +denominator is +, the conjugate we will use is +. +Step 3: The conjugate is multiplied by the numerator and the denominator. +Step 4: Remembering how a number times its conjugate works, this becomes +. +Step 5: In the numerator, we apply the distributive property. Using it yields +. +Step 6: Notice that the denominator no longer contains a square root. It has been rationalized. If desired, this can then +be written as a rational number minus an irrational number, by recalling that +. +Applying that to the answer, we have +. +Step 7: With a bit of cancellation, this reduces to +. +YOUR TURN 3.95 +1. Rationalize the denominator of +. +VIDEO +Rationalizing the Denominator (https://openstax.org/r/Rationalizing_Denominator) +3.5 • Irrational Numbers +201 + +Check Your Understanding +29. Simplify the following square root: +. +30. Perform the following operation: +. +31. Perform the following operation: +. +32. Rationalize the denominator of the following: +. +SECTION 3.5 EXERCISES +1. Identify which of the following numbers are irrational: +, 4.33, +, +2. Identify which of the following numbers are irrational: +, +, +, +For the following exercises, simplify the square root by expressing it in lowest terms. +3. +4. +5. +6. +7. +8. +9. +10. +11. +12. +For the following exercises, perform the arithmetic operations without a calculator, if possible. If it is not possible, state +why. +13. +14. +15. +16. +17. +18. +19. +20. +21. +22. +23. +24. +25. +26. +For the following exercises, rationalize the denominators of the fractions, and then simplify. +27. +28. +29. +30. +202 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +31. +32. +33. Determine the conjugate of +. +34. Determine the conjugate of +. +35. Determine the conjugate of +. +36. Determine the conjugate of +. +37. Find the product of +and its conjugate. +38. Find the product of +and its conjugate. +For the following exercises, rationalize the denominator of the fraction, and then simplify the fraction. +39. +40. +41. +42. +3.6 Real Numbers +Figure 3.31 Quick mental math involves using the known properties of real numbers. +Learning Objectives +After completing this section, you should be able to: +1. +Define and identify numbers that are real numbers. +2. +Identify subsets of the real numbers. +3. +Recognize properties of real numbers. +Have you ever been impressed by the speed at which someone can do math in their head? Most of us at one time or +another have witnessed a person speed through mental math, an impressive feat that often bests calculators. One such +person is Neelkantha Bhanu Prakash. As of September 20, 2020, he is considered the world’s fastest human calculator. +He currently holds four world records. How does someone do that, though? Have they memorized lots of arithmetic +facts? Are they simply brilliant? +The answer isn't simple so much as it is about knowledge. Real numbers behave in some very regular ways, following +rules that can be learned. In this section, those rules are explored. +Watch the video of Arthur Benjamin’s TED Talk to learn about another mathematician with remarkable mental abilities. +VIDEO +Arthur Benjamin TED Talk, Faster Than a Calculator (https://openstax.org/r/ +Arthur_Benjamin_TED_talk,_Faster_than_a_Calculator) +3.6 • Real Numbers +203 + +Defining and Identifying Real Numbers +Real numbers are the rational and irrational numbers combined. The real numbers represent the collection of all +physical distances that exist, along with 0 and the negatives of those physical distances. For example, if you take a +measure of three units, and divide that distance into eight (8) equal lengths, the distance you have formed is +units. +Also, if you draw a right triangle (a triangle with one angle equal to 90 degrees) with one side length of 1, and the other +side length of 3, the long side of the triangle will have length +units, as shown in Figure 3.32. +Figure 3.32 Right triangle +Of course, if we name something the real numbers, there must be numbers that aren't real. Otherwise, they’d just be +called the numbers. One such not real number, one that cannot be a length, is +. It is part of a collection of numbers +called the complex numbers, it is denoted with the letter . As an extension, the square root of any negative number is +not a real number, but instead a complex number. +To determine if a number is real, check to see if there are any negatives under a square root or any +. If there are any +present, the number is not real. +EXAMPLE 3.96 +Identifying Real Numbers +Determine if each of the following are real numbers: +1. +2. +3. +Solution +1. +is a real number, as there are no negatives under the square roots, nor is there any factor of . +2. +is a rational number, and so it is a real number. +3. +is not a real number, as there is a negative number under the square root. +YOUR TURN 3.96 +Determine if each of the following are real numbers: +1. +2. +3. +Identifying Subsets of Real Numbers +The real numbers were built out of pieces, including integers, rational numbers, and irrational numbers. As such, the real +numbers have named subsets, as shown in the table below. +204 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Set Name +Set +Symbol +Set Description +Natural +Numbers +ℕ +The counting numbers +Whole Numbers +The counting numbers and 0 +Integers +ℤ +The natural numbers, their negatives, and 0 +Rational +Numbers +ℚ +Fractions of integers +Irrational +Numbers +ℙ +Numbers that cannot be written as a fraction of integers +Real Numbers +ℝ +The union of the rational and irrational numbers, all possible physical lengths, and +their negatives +When we categorize numbers using these sets, we use the smallest set that they belong to. For instance, −7 is an integer, +and a rational number, and a real number. The smallest set to which −7 belongs is integer, so we’d say it belongs to the +integers. +We can also represent the relationships between the different sets of real numbers using set notation. All natural +numbers are integers, but there are integers that are not natural numbers, so ℕ +ℤ . Similarly, every integer is a +rational number, but there are rational numbers that are not integers, so ℤ +ℚ . The same is true of the rational +numbers and the real numbers, so ℚ +ℝ . +There is no agreed-upon symbol for the irrational numbers. If we represent the irrationals as the set +, we should note +that the following are true: ℚ +ℝ and ℚ +. Recall that this means the irrationals are the complement of the +rational numbers in the universal set of real numbers. +EXAMPLE 3.97 +Categorizing Numbers +Identify all subsets of the real numbers to which the following real numbers belong: +1. +2. +3. +Solution +1. +is a natural number, integer, and rational number. +2. +is a rational number. +3. +is an irrational number. +YOUR TURN 3.97 +Identify all subsets of the real numbers to which the following real numbers belong: +1. +2. +3. +3.6 • Real Numbers +205 + +EXAMPLE 3.98 +Categorizing Numbers within a Venn Diagram +Place the following numbers correctly in the Venn diagram (Figure 3.33). +Figure 3.33 +Solution +Since +is irrational, it belongs in the real numbers, but outside the rational numbers (Figure 3.34). +Figure 3.34 +Since −10 is an integer, it belongs in the integers but outside the natural numbers (Figure 3.35). +Figure 3.35 +Since +is a rational number, it belongs in the rational numbers but not in the integers (Figure 3.36). +206 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Figure 3.36 +Since 41 is a natural number, it belongs in the natural numbers circle (Figure 3.37). +Figure 3.37 +Since +is a rational number, it belongs in the rational numbers but not in the integers (Figure 3.38). +Figure 3.38 +Since +is irrational, it belongs in the real numbers, but outside the rational numbers (Figure 3.39). +Figure 3.39 +YOUR TURN 3.98 +1. Place the following numbers correctly into the Venn diagram. +3.6 • Real Numbers +207 + +VIDEO +Identifying Sets of Real Numbers (https://openstax.org/r/Sets_of_Real_Numbers) +Recognizing Properties of Real Numbers +The real numbers behave in very regular ways. These behaviors are called the properties of the real numbers. Knowing +these properties helps when evaluating formulas, working with equations, or performing algebra. Being familiar with +these properties is helpful in all settings where numbers are used and manipulated. For example, when multiplying +, you could multiply the 4 and 25 first. If you know that product is 100, it makes the multiplication easier. +The table below is a partial list of properties of real numbers. +Property +Example +In Words +Distributive property +Multiplication distributes across addition +Commutative property of addition +Numbers can be added in any order +Commutative property of +multiplication +Numbers can be multiplied in any order +Associative property of addition +Doesn't matter which pair of numbers is added +first +Associative property of +multiplication +Doesn't matter which pair of numbers is +multiplied first +Additive identity property +Any number plus 0 is the number +Multiplicative identity property +Any number times one is the number +Additive inverse property +Every number plus its negative is 0 +Multiplicative inverse property +, provided +Every non-zero number times its reciprocal is 1 +208 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +The names of the properties are suggestive. The commutative properties, for example, suggest commuting, or +moving. Associative properties suggest which items are associated with others, or if order matters in the computation. +The distributive property addresses how a number is distributed across parentheses. +EXAMPLE 3.99 +Identifying Properties of Real Numbers +In each of the following, identify which property of the real numbers is being applied. +1. +2. +3. +Solution +1. +Here, the pair of numbers that is added first is switched. This is the associative property of addition. +2. +Here, a number is multiplied by its reciprocal, resulting in 1. This is the multiplicative inverse property. +3. +Here, the order in which numbers are added is switched. This is the commutative property of addition. +YOUR TURN 3.99 +In each of the following, identify which property of the real numbers is being applied. +1. +2. +Using these properties to perform arithmetic quickly relies on spotting easy numbers to work with. Look for numbers +that add to a multiple of 10, or multiply to a multiple of 10 or 100. +EXAMPLE 3.100 +Using Properties of Real Numbers in Calculations +Use properties of the real numbers and mental math to calculate the following: +1. +2. +3. +Solution +1. +Notice that +, so that becomes the multiplication to do first. Use the commutative property of +multiplication to change the order of the numbers being multiplied. +2. +Notice that +, so that becomes the addition to do first. Use the commutative property of addition to +change the order in which the numbers are added. +3. +Notice that +. Using that, the problem can be changed to +. That, however, doesn't look easy at all. +But +. Using the distributive property, we rewrite and expand this as +. The last step is subtraction, so the final answer is +1,584. So, multiplying by 99 is the same as multiplying by 100, and then subtracting the other number once. +YOUR TURN 3.100 +1. Use properties of real numbers and mental math to calculate the following: +3.6 • Real Numbers +209 + +Check Your Understanding +33. Which of the following are real numbers: +? +34. Indicate which of the sets are subsets of the others: +, +, +, +. +35. Which property is demonstrated here: +? +SECTION 3.6 EXERCISES +For the following exercises, identify each number as a natural number, an integer, a rational number, or a real number. +1. +2. +3. +4. +5. +6. +7. +8. +For the following exercises, correctly place the numbers in the Venn diagram. +9. +10. +For the following exercises, identify the property of real numbers that is being illustrated. +11. +12. +13. +14. +15. +16. +17. +18. +19. +20. +21. +22. +23. +24. +25. +26. +27. +28. +210 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +For the following exercises, use properties of real numbers and mental math to calculate the expression. +29. +30. +31. +32. +33. +34. +35. +36. +37. +38. +39. +40. +3.7 Clock Arithmetic +Figure 3.40 If a credit card number is entered incorrectly, error checking algorithms will often catch the mistake. (credit: +modification of work “Senior couple at home checking finance on credit card from above” by Nenad Stojkovic/Flickr, CC +BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Add, subtract, and multiply using clock arithmetic. +2. +Apply clock arithmetic to calculate real-world applications. +Online shopping requires you to enter your credit card number, which is then sent electronically to the vendor. Using an +ATM involves sliding your bank card into a reader, which then reads, sends, and verifies your card. Swiping or tapping for +a purchase in a brick–and-mortar store is how your card sends its information to the machine, which is then +communicated to the store’s computer and your credit card company. This information is read, recorded, and +transferred many times. Each instance provides one more opportunity for error to creep into the process, a misrecorded +digit, transposed digits, or missing digits. Fortunately, these card numbers have a built-in error checking system that +relies on modular arithmetic, which is often referred to as clock arithmetic. In this section, we explore clock, or modular, +arithmetic. +VIDEO +Determining the Day of the Week for Any Date in History (https://openstax.org/r/ +Determining_the_Day_of_the_Week_for_Any_Date_in_History) +Adding, Subtracting, and Multiplying Using Clock Arithmetic +When we do arithmetic, numbers can become larger and larger. But when we work with time, specifically with clocks, the +numbers cycle back on themselves. It will never be 49 o’clock. Once 12 o’clock is reached, we go back to 1 and repeat the +numbers. If it's 11 AM and someone says, “See you in four hours,” you know that 11 AM plus 4 hours is 3 PM, not 15 AM +(ignoring military time for now). Math worked on the clock, where numbers restart after passing 12, is called clock +arithmetic. +3.7 • Clock Arithmetic +211 + +Clock arithmetic hinges on the number 12. Each cycle of 12 hours returns to the original time (Figure 3.41). Imagine +going around the clock one full time. Twelve hours pass, but the time is the same. So, if it is 3:00, 14 hours later and two +hours later both read the same on the clock, 5:00. Adding 14 hours and adding 2 hours are identical. As is adding 26 +hours. And adding 38 hours. +Figure 3.41 Clock showing 3:00 with arrow going around the clock one full time, or 12 hours +What do 2, 14, 26, and 38 have in common in relation to 12? When they are divided by 12, they each have a remainder of +2. That's the key. When you add a number of hours to a specific time on the clock, first divide the number of hours being +added by 12 and determine the remainder. Add that remainder to the time on the clock to know what time it will be. +A good visualization is to wrap a number line around the clock, with the 0 at the starting time. Then each time 12 on the +number line passes, the number line passes the starting spot on the clock. This is referred to as modulo 12 arithmetic. +Even though the process says to divide the number being added by 12, first perform the addition; the result will be the +same if you add the numbers first, and then divide by 12 and determine the remainder. +In general terms, let +be a positive integer. Then +modulo 12, written ( +mod 12), is the remainder when +is divided by +12. If that remainder is +, we would write +(mod 12). +Caution: 12 mod 12 is 0. So, if a mod 12 problem ends at 0, that would be 12 on the clock. +EXAMPLE 3.101 +Determining the Value of a Number modulo 12 +Find the value of the following numbers modulo 12: +1. +34 +2. +539 +3. +156 +Solution +To determine the value of a number modulo 12, divide the number by 12 and record the remainder. +1. +To find the value 34 modulo 12: +Step 1: Determine the remainder when 34 is divided by 12 using long division. The largest multiple of 12 that is less +than or equal to 34 is 24, which is the product of 12 and 2. +Step 2: Performing the subtraction yields 10. +Since that subtraction resulted in a number less than 12, that is the remainder, 10. The value of 34 modulo 12 is 10, +or 34 = 10 (mod 12). +2. +To find the value 539 modulo 12: +Step 1: Determine the remainder when 539 is divided by 12 using long division. We first look to the first two digits of +539, 53. The largest multiple of 12 that is less than or equal to 53 is 48, which is the product of 12 and 4. +212 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Step 2: Performing the subtraction results in 5. +Step 3: Now, the 9 is brought down. +Step 4: The largest multiple of 12 that is less than or equal to 59 is once more 48 itself, which is +. +Step 5: Finishing the process, the 48 is subtracted from the 59, yielding 11. +We've used all the digits of 539, and the last subtraction resulted in a number less than 12, so that number, 11, is the +remainder. The value of 539 modulo 12 is 11, or, 539 = 11 (mod 12). +3. +To find the value 156 modulo 12: +Step 1: Determine the remainder when 156 is divided by 12 using long division. We first look to the first two digits of +156, 15. The largest multiple of 12 that is less than or equal to 15 is 12 itself, which is the product of 12 and 1. +Step 2: Performing the subtraction results in 3. +Step 3: Now, the 6 is brought down. +Step 4: The largest multiple of 12 that is less than or equal to 36 is 36 itself, which is +. +3.7 • Clock Arithmetic +213 + +Step 5: Finishing the process, the 36 is subtracted from the 36, yielding 0. +We've used all the digits of 156, and the last subtraction resulted in a number less than 12, so that number, 0, is the +remainder. The value of 156 modulo 12 is 0, or, 156 = 0 (mod 12). +We should note here that, had we been speaking of time, the 0 would be interpreted as 12:00. +YOUR TURN 3.101 +Find the value of the following numbers modulo 12. +1. 93 +2. 387 +TECH CHECK +Using Desmos to Determine the Value of a Number module 12 +Desmos may be used to determine the value of a number modulo 12. It is flexible enough to find the value of a +number modulo of any other integer you want. To determine the value of +modulo 12, type mod( ,12) into Desmos. +The result will be displayed immediately. This can be used to find 539 modulo 12, as shown in the Figure 3.42. +Figure 3.42 Display of 539 modulo 12 +Clock arithmetic is modulo 12 arithmetic but applied to time. As time is divided into 12 hours that repeat a cycle, we +use modulo 12 for clock arithmetic. +VIDEO +Clock Arithmetic (https://openstax.org/r/Clock_Arithmetic) +EXAMPLE 3.102 +Adding with Clock Arithmetic +If it's 3:00, what time will it be in 89 hours? +Solution +To find that future time, we may determine the value of 89 (mod 12), either by long division or by using a calculator, such +as Desmos. Then add the result to 3:00. Entering mod(89,12) in Desmos results in 5. Adding 5 hours, which was 89 +214 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +(mod12), to 3:00 results in 8:00. +YOUR TURN 3.102 +1. If it is 9:00 now, what time will it be in 43 hours? +Subtracting time on the clock works in much the same way as addition. Find the value of the number of hours being +subtracted modulo 12, then subtract that from the original time. +EXAMPLE 3.103 +Subtracting with Clock Arithmetic +If it is 4:00 now, what time was it 67 hours ago? +Solution +To find that past time, we may determine the value of 67 (mod 12), either by long division or by using a calculator, such +as Desmos. Then subtract the result to 4:00. Entering mod(67,12) in Desmos results in 7. Subtracting 7 hours from 4:00 +results in ‒3:00. We know, though, that time is not represented with negative times. This value, ‒3:00, indicates three +hours before 12:00, which is 9:00. So, 67 hours before 4:00 was 9:00. We see this in the Figure 3.43. +Figure 3.43 Clock showing 7 hours subtracted from 4:00 +YOUR TURN 3.103 +1. If it is 7:00 now, what time was it 34 hours ago? +Recall that clock arithmetic was referred to as modulo 12 arithmetic. Multiplying in modulo 12 also relies on the +remainder when dividing by 12. To multiply modulo 12 is just to multiply the two numbers, and then determine the +remainder when divided by 12. +EXAMPLE 3.104 +Multiplying modulo 12 +What is the product of 11 and 45 modulo 12? +Solution +We begin by multiplying 11 and 45, which is 495. Next, we find 495 modulo 12, either by dividing the result by 12 to +determine the remainder, or by using a calculator. Entering mod(495,12) in Desmos yields 3. Had long division been +used, the remainder would be 3. So +modulo 12. +YOUR TURN 3.104 +1. What is the product of 4 and 19 modulo 12? +3.7 • Clock Arithmetic +215 + +Calculating Real-World Applications with Clock Arithmetic +EXAMPLE 3.105 +Applying Clock Arithmetic +Suppose it is 3:00, and you decide to check your email every 5 hours. What time will it be when you check your email the +ninth time? +Solution +If you check your email every 5 hours nine times, that ninth check will occur 45 hours after 3:00, which is an addition of +45 hours to 3:00. So, we find 45 modulo 12, which is 9. Nine hours after 3:00 is 12:00. It will be 12:00 when you check +your email the ninth time. +YOUR TURN 3.105 +1. You have agreed to text your friend every 3 hours while driving across the country. You began your trip at 8 AM. +What time will it be when you text your friend the 15th time? +Clock arithmetic processes can be applied to days of the week. Every 7 days the day of the week repeats, much like every +12 hours the time on the clock repeats. The only difference will be that we work with remainders after dividing by 7. In +technical terms, this is referred to as modulo 7. More generally, let +be a positive integer. Then +modulo 7, written +mod 7, is the remainder when +is divided by 7. If that value is +, we may write +(mod 7). +EXAMPLE 3.106 +Applying Clock Arithmetic to Days of the Week +Your family has a cat, and no one wants to empty the litter box. However, it has to be done daily. The six of you agree to +take turns, so everyone has to empty the litter box every 6 days. You empty the box on a Thursday. What day will you +empty the box for the 10th time? +Solution +The first time you emptied the litter box was on a Thursday. So,the 10th time you empty the litter box will be 9 times +later (you've already had your first turn, so 9 turns left!). This will happen 54 (9 times 6) days later. Finding the value of 54 +modulo 7, using division to determine the remainder or using a calculator to find the value of 54 modulo 7 gives the +answer 5. Five days after a Thursday is Tuesday. +YOUR TURN 3.106 +1. Your family shares the cooking duties in the home. You've agreed to prepare the meal every 5 days. The last time +you prepared dinner was a Tuesday. What day of the week will it be after you've prepared the meals 20 more +times? +Check Your Understanding +For the following exercises, use clock arithmetic to perform the following: +36. +37. +38. +39. Calene calls her mother every fourth day. She calls on a Monday. What day of the week will it be on Calene's eighth +time calling after that? +SECTION 3.7 EXERCISES +1. Explain what modulo 12 means. +216 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +2. Explain what modulo 7 means. +3. What is 75 modulo 12? +4. What is 139 modulo 12? +5. What is 38 modulo 7? +6. What is 83 modulo 7? +For the following exercises, use clock arithmetic (mod 12), to perform the indicated calculation. +7. +8. +9. +10. +11. +12. +13. +14. +15. +16. +17. It is 8:00. What time will it be in 70 hours? +18. It is a Thursday. What day of the week will it be in 100 days? +19. It is Monday. What day of the week will it be in 58 days? +20. It is 3:00. What time of the day will it be in 150 hours? +21. It is 6:00. What time was it 34 hours ago? +22. It is 2:00. What time was it 100 hours ago? +23. A trucker passes through Kokomo, Indiana, once every 9 days. They come through Kokomo on a Wednesday. +What day of the week will the driver pass through Kokomo after 8 more visits? +24. Jason checks his email every 5 hours. He checks it at 6 PM one day. What time of the day will it be when he +checks his mail the 50th time after that 6 PM check? +25. Mickey gets a new prescription of a drug that she needs to take every day. The prescription is for 250 days. She +takes the first pill of the new bottle on a Friday. What day of the week will her prescription run out? +26. Zainab visits the nursing home every 5 days. She visits on a Sunday. What day of the week will it be when she +visits it for the 7th time after that? +27. Micaela has to check in with her boss every 14 hours. If she checks in at 3:00, what time will it be when she +checks in the 10th time after that? +28. Tracy has an alarm set for every 4 hours. It goes off at 3:00. What time will it be when the alarm goes off the +20th time after that? +29. Dejan must check his blood sugar every 5 hours. He checks his blood sugar at 4:00. What time will it be when +Dejan checks their his sugar the 40th time after that? +30. Latanjana is in the hospital, where her blood pressure is checked every 3 hours. If her blood pressure is +checked at 5:00, what time will it be when her blood pressure is checked the 13th time after that? +Months come in twelves, just as hours do. This means that months can be calculated using modulo 12, just like hours. +For the following exercises, calculate what month it will be for each exercise. +31. Micaela works for a sprinkler maintenance company and runs a routine check on the Harris's sprinkler system +every third month. If Micaela checks the system in an April, what month will it be when Micaela returns to the +Harris's for the 11th time after that? +32. Dana runs a half marathon every 5 months. She runs one in a May. What month will it be when she runs her 8th +marathon after that? +3.7 • Clock Arithmetic +217 + +3.8 Exponents +Figure 3.44 Astronomical distances are written using exponents. (credit: “Our Solar System (Artist's Concept)” by NASA/ +Jet Propulsion Laboratory-Caltech/Public Domain) +Learning Objectives +After completing this section, you should be able to: +1. +Apply the rules of exponents to simplifying expressions. +Sometimes, we look for shorthand when writing or expressing something that simply takes too long. The use of LOL and +tl;dr. This shorthand only works if everyone reading the shorthand knows what it stands for. Using exponents is a similar +instance. Writing out a long string of a number times itself over and over takes too much time, and eventually one would +forget how many of the value has been written or read. For example, +. There has to be a shorter and more efficient +way to write 8 times itself 1, 2, 3….hmmmm, 19 times. +And that’s the role that exponents play in mathematics. They are shorthand for multiplying a number by itself a number +of times. Without it, calculations would become a mess and we’d have to write a lot more. +Applying the Rules of Exponents to Simplify Expressions +Squaring a number is multiplying it by itself, and has that name because it is the area of a square with that side length. +Cubing a number is finding the volume of a cube with that length of sides. That’s why we refer to +as five squared, or +as ten cubed. Exponents represent that multiplication. +Let’s remind ourselves of the terminology associated with exponents and what exponents represent. Suppose you want +to multiply a number, let’s label that number +, by itself some number of times. Let’s label the number of times +. We +denote that as +. We say +raised to the +th power. When we write or see +, we call the 7 the base and we call 5 the +exponent. What it represents is 7 multiplied by itself 5 times. This means exponents are used as a shorthand for +repeated multiplications, where we write +. We would write +and say seven to the fifth power. +VIDEO +Exponential Notation (https://openstax.org/r/Exponential_Notation) +The definitions of base and exponent make it possible to understand the exponent rules. +Product Rule for Exponents +The first rule we examine is the product rule, +. This rule means that when we multiply a base raised to a +power times the same base to another power, the result is the base raised to the sum of the powers. To demonstrate, +consider +. If we apply the product rule to that we get +. This can be tested by looking at the +multiplications that are represented. The +is 9 times itself 3 times, while +is 9 times itself 5 times. Substituting those +into +we see +, which is what the formula told us would happen. +Caution: The product rule only applies when the bases are the same. If the bases are different, we do not apply this +rule. +218 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +FORMULA +If a number, +, raised to a power, +, is then multiplied by +raised to another power, +, the result is +. +EXAMPLE 3.107 +Using the Product Rule for Exponents +If possible, use the product rule to simplify the following: +1. +2. +Solution +1. +We can apply the product rule to simplify the expression because the bases are the same and we are multiplying. +2. +Since the bases are not the same (one is 5, the other 8), this cannot be simplified using the product rule for +exponents. +YOUR TURN 3.107 +If possible, use the product rule to simplify the following: +1. +2. +These rules can be applied to unknowns too. +EXAMPLE 3.108 +Using the Product Rule for Exponents of Unknowns +Use the product rule to simplify +. +Solution +The bases are the same, and we are multiplying, so we apply the multiplication rule to simplify the expression. +YOUR TURN 3.108 +1. Use the product rule to simplify +. +Quotient Rule for Exponents +The next rule we examine is the quotient, or division, rule. +FORMULA +When a number, +, raised to a power, +, is divided by +raised to another power, +, then the result is +. +This rule means that when we divide a base raised to a power by the same base to another power, the result is the base +raised to the difference of the powers. To demonstrate, consider +. If we apply the quotient rule to that, we get +. This can be tested by looking at the division that is represented. Remember, +is 14 multiplied +3.8 • Exponents +219 + +to itself 13 times, while +is 14 multiplied to itself 6 times. Substituting those into +gives the following: +We see here that there are a LOT of fours to be divided out. +What remains is 4 to the 7th power, +. +All of the work above confirmed what the formula told us would be the result. +Caution: The quotient rule only applies when the bases are the same. If the bases are different, we do not apply this +rule. +EXAMPLE 3.109 +Using the Quotient Rule for Exponents +Use the quotient rule to simplify +. +Solution +We can apply the quotient rule to simplify the expression since the bases are the same and we are dividing. +YOUR TURN 3.109 +1. Use the quotient rule to simplify +. +VIDEO +Product and Quotient Rule for Exponents (https://openstax.org/r/Product_and_Quotient_Rule_for_Exponents) +A natural consequence of the quotient rule is what it means to raise a non-zero number to the zeroth power. Let’s look at +the simplification when the exponents are equal. +We know that a number divided by itself is 1, so +. From that is must be that +. This provides the rule +for a number raised to the power 0: +. +FORMULA +If you have a non-zero number +, then +. +Distributive Rule for Exponents +The next rule we look to is a distributive rule for exponents. +220 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +FORMULA +If you have a product, +, and raise it to an exponent, +, then +. +This means that when we have two numbers multiplied together, and that is raised to a power, it is the same as raising +each of the numbers to the same power first, then multiplying. For example, +. This can be explained +using the definition of exponents and multiplying all the factors. +We may change the order in which numbers are multiplied. This is the commutative property of the real numbers. This +can be written as +. Using exponents, that shortens to +. +This also works in the other direction, +. Read this way, if we have one base raised to an exponent, and +another base raised to the same exponent, we can multiply the bases and raise that product to the shared exponent. For +instance, +. +Caution: The exponent distributive rule, +, only works if the exponents are the same. +EXAMPLE 3.110 +Using the Distributive Rule for Exponents +Use the exponent distributive rule to expand +. +Solution +Applying the distributive rule to the product, we get +. +YOUR TURN 3.110 +1. Use the exponent distributive rule to expand +. +EXAMPLE 3.111 +Using the Distributive Rule for Exponents +Use the exponent distributive rule to expand +. +Solution +Applying the distributive rule to the product, we get +. +YOUR TURN 3.111 +1. Use the exponent distributive rule to expand +. +This distribution also works for quotients. A fraction raised to an exponent equals the numerator raised to the exponent +divided by the denominator raised to the exponent. For example, +. Demonstrating this is similar to the +previous rule. +FORMULA +When you have a fraction, +, raised to an exponent, +, then +. +3.8 • Exponents +221 + +EXAMPLE 3.112 +Using the Distributive Rule for Exponents with Fractions +Use the exponent distributive rule to expand the following: +1. +2. +Solution +1. +Applying the distributive rule to the quotient, we get +. +2. +Applying the distributive rule to the quotient, we get +. +YOUR TURN 3.112 +Use the exponent distributive rule to expand the following: +1. +2. +VIDEO +Fraction Raised to a Power (https://openstax.org/r/Fraction_Raised_to_a_Power) +Power Rule +In the previous two sets of rules, we’ve seen exponents applied to products and quotients. Now we look to exponents +applied to other exponents. For example, +. This can be explained by examining what the outer +exponent does. We raise +to the fourth power, so we multiply +by itself 4 times, +. Now if +we apply the product rule for exponents, this becomes +. +FORMULA +If you raise a non-zero base, say +, to an exponent +, and raise that to another exponent, +, you get the base raised +to the product of the exponents, which is +. +EXAMPLE 3.113 +Raising an Exponent to an Exponent +Expand the following: +1. +2. +Solution +1. +Using the power rule of exponents, +. +2. +Using the power rule of exponents, +. +YOUR TURN 3.113 +Expand the following: +222 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +1. +2. +Negative Exponent Rule +Up until now, we’ve only looked at positive exponents. The last exponent rule we look at is what negative exponents +represent. Recall the quotient rule: +. What would happen if the exponent in the denominator was larger +than that in the numerator? For example, +. If we apply the quotient rule, we obtain +. We need to +make sense of that negative exponent. To do so, we can expand the quotient and see what happens: +. When we divide out common factors, only two factors of 4 are left in the denominator, as we see +here: +. Using exponent notation, this is +. Since +and +represent the same number, +, they are equal. This +demonstrates how negative exponents are defined. +FORMULA +provided that +. +Similarly, +. +EXAMPLE 3.114 +Eliminating Negative Exponents +Convert the following to expressions with no negative exponent: +1. +2. +3. +Solution +1. +Using the negative exponent rule on the +and multiplying, +. +2. +Using the negative exponent rule on the +and multiplying, +. +3. +Begin by rewriting the expression as +. Apply the negative exponent rule to +in the expression, +which becomes +, which has no negative exponents. +YOUR TURN 3.114 +Convert the following to expressions with no negative exponent: +1. +2. +EXAMPLE 3.115 +Eliminating Denominators by Using Negative Exponents +Use negative exponents to rewrite the following expressions with no denominator: +1. +2. +3.8 • Exponents +223 + +Solution +1. +Rewrite the expression +as +. Then use the definition of negative exponents to rewrite the +as +. +Last, multiply, yielding +. +2. +Rewrite the expression +as +. Then use the definition of negative exponents to rewrite the +as +. +Last, multiply, yielding +. +YOUR TURN 3.115 +Use negative exponents to rewrite the following expressions with no denominator: +1. +2. +The table below shows a summary of the exponent rules from this section. +Rule +Example +In Words +Product Rule +A base raised to a power, times the same based raised to +another power, is the base raised to the sum of the powers. +Quotient Rule +A base raised to a power, divided by the same based raised to +another power, is the base raised to the difference of the +powers. +Zero Power Rule +provided that +Any non-zero number raised to the zeroth power equals 1. +Distributive Rule, +Multiplication +Exponents distribute across multiplication. +Distributive Rule, +Division +Exponents distribute across division. +Power Rule +A base raised to a power, raised to another power, is the base +raised to the first power times the second power. +Negative Exponent Rule +provided that +A base raised to a negative exponent is 1 divided by the base +raised to the positive power, and vice versa. +These rules often occur in tandem with each other, but it requires that you carefully apply the rules. +EXAMPLE 3.116 +Simplifying Expressions Using Exponent Rules +Simplify the following: +224 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +1. +2. +Solution +1. +Step 1: To simplify this, we start by distributing the power 5 across the quotient: +Step 2: We distribute the power 5 in the numerator across that multiplication: +Step 3: We apply the power rule where indicated: +2. +Step 1: To simplify this, we start by distributing the power 6 across the quotient: +Step 2: We distribute the power 5 in the numerator across that multiplication: +Step 3: We apply the power rule where indicated: +YOUR TURN 3.116 +Simplify the following: +1. +2. +VIDEO +Simplifying Expressions with Exponents (https://openstax.org/r/Simplifying_Expressions_with_Exponents) +Check Your Understanding +40. Simplify +. +41. Simplify +. +42. Simplify +. +43. Simplify +. +3.8 • Exponents +225 + +44. Simplify +. +SECTION 3.8 EXERCISES +For the following exercises, simplify the expression. +1. +2. +3. +4. +5. +6. +7. +8. +9. +10. +11. +12. +13. +14. +15. +16. +17. +18. +19. +20. +21. +22. +23. +24. +25. +26. +27. +28. +29. +For the following exercises, rewrite the expression without a denominator. +30. +31. +32. +33. +For the following exercises, rewrite the expression without negative exponents. +34. +35. +226 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +36. +37. +3.9 Scientific Notation +Figure 3.45 Calculations in the sciences often involve numbers in scientific notation form. +Learning Objectives +After completing this section, you should be able to: +1. +Write numbers in standard or scientific notation. +2. +Convert numbers between standard and scientific notation. +3. +Add and subtract numbers in scientific notation. +4. +Multiply and divide numbers in scientific notation. +5. +Use scientific notation in computing real-world applications. +The amount of information available on the Internet is simply incomprehensible. One estimate for the amount of data +that will be on the Internet by 2025 is 175 Zettabytes. A single zettabyte is one billion trillion. Written out, it is +1,000,000,000,000,000,000,000. One estimate is that we’re producing 2.5 quintillion bytes of data per day. A quintillion is +a trillion trillion, or, written out, 1,000,000,000,000,000,000. To determine how many days it takes to increase the amount +of information that is on the Internet by 1 zettabyte, divide these two numbers, a zettabyte being +1,000,000,000,000,000,000,000, and 2.5 quintillion, being 2,500,000,000,000,000,000, shows it takes 400 days to generate +1 zettabyte of information. But doing that calculation is awkward with a calculator. Keeping track of the zeros can be +tedious, and a mistake can easily be made. +On the other end of the scale, a human red blood cell has a diameter of 7.8 micrometers. One micrometer is one +millionth of a meter. Written out, 7.8 micrometers is 0.0000078 meters. Smaller still is the diameter of a virus, which is +about 100 nanometers in diameter, where a nanometer is a billionth of a meter. Written out, 100 nanometers is +0.0000001 meters. To compare that to engineered items, a single transistor in a computer chip can be 14 nanometers in +size (0.000000014 meters). Smaller yet is the diameter of an atom, at between 0.1 and 0.5 nanometers. +Sometimes we have numbers that are incredibly big, and so have an incredibly large number of digits, or sometimes +numbers are incredibly small, where they have a large number of digits after the decimal. But using those +representations of the names of the sizes makes comparing and computing with these numbers problematic. That’s +where scientific notation comes in. +Writing Numbers in Standard or Scientific Notation Form +When we say that a number is in scientific notation, we are specifying the form in which that number is written. That +form begins with an integer with an absolute value between 1 and 9, then perhaps followed the decimal point and then +some more digits. This is then multiplied by 10 raised to some power. When the number only has one non-zero digit, the +scientific notation form is the digit multiplied by 10 raised to an exponent. When the number has more than one non- +zero digit, the scientific notation form is a single digit, followed by a decimal, which is then followed by the remaining +digits, which is then multiplied by 10 to a power. +The following numbers are written in scientific notation: +The following numbers are not written in scientific notation: +3.9 • Scientific Notation +227 + +because it isn't multiplied by 10 raised to a power +because the absolute value of −50.053 is not at least 1 and less than 10 +because 41.7 is not at least 1 and less than 10 +because 0.036 is not at least one and less than 10 +EXAMPLE 3.117 +Identifying Numbers in Scientific Notation +Which of the following numbers are in scientific notation? If the number is not in scientific notation, explain why it is not. +1. +2. +3. +Solution +1. +The number +is in scientific notation because the absolute value of −9.67 is at least 1 and less than 10. +2. +The number +is not in scientific notation because 145 is not at least 1 and less than 10. +3. +The number +is not in scientific notation form. Even though it is at least 1 but less than 10, it is not multiplied by +10 raised to a power. +YOUR TURN 3.117 +Which of the following numbers are in scientific notation? If the number is not in scientific notation, explain why it is +not. +1. +2. +3. +Some numbers are so large or so small that it is impractical to write them out fully. Avogadro’s number is important in +chemistry. It represents the number of units in 1 mole of any substance. The substance many be electrons, atoms, +molecules, or something else. Written out, the number is: 602,214,076,000,000,000,000,000. Another example of a +number that is impractical to write out fully is the length of a light wave. The wavelength of the color blue is about +0.000000450 to 0.000000495 meters. Such numbers are awkward to work with, and so scientific notation is often used. +We need to discuss how to convert numbers into scientific notation, and also out of scientific notation. +Recall that multiplying a number by 10 adds a 0 to the end of the number or moves the decimal one place to the right, as +in +or +. And if you multiply by 100, it adds two zeros to the end of the number or moves +the decimal two places to the right, and so on. For example, +and +. +Multiplying a number by 1 followed by some number of zeros just adds that many zeros to the end of the number or +moves the decimal place that many places to the right. Numbers written as 1 followed by some zeros are just powers of +10, as in +, +, +, etc. Generally, +. +We can use this to write very large numbers. For instance, Avogadro’s number is 602,214,076,000,000,000,000,000, which +can be written as +. The multiplication moves the decimal 23 places to the right. +Similarly, when we divide by 10, we move the decimal one place to the left, as in +. If we divide by 100, we +move the decimal two places to the left, as in +. In general, when you divide a number by a 1 followed by +zeros, you move the decimal +places to the left, as in +. This denominator could be written as +. If we use that in the expression and allow for negative exponents, rewrite the number as +. With this, we can write division by a 1 followed by +zeros as multiplication by +10 raised to +. +Using that information, we can demonstrate how to convert from a number in standard form into scientific notation +form. +228 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Case 1: The number is a single-digit integer. +In this case, the scientific notation form of the number is +. +Case 2: The absolute value of the number is less than 1. +Follow the process below. +• +Step 1: Count the number of zeros between the decimal and the first non-zero digit. Label this +. +• +Step 2: Starting with the first non-zero digit of the number, write the digits. If the number was negative, include the +negative sign. +• +Step 3: If there is more than one digit, place the decimal after the first digit from Step 2. +• +Step 4: Multiply the number from Step 3 by +. +Case 3: The absolute value of the number is 10 or larger. +Follow the process below. +• +Step 1: Count the number of digits that are to the left of the decimal point. Label this +. +• +Step 2: Write the digits of the number without the decimal place, if one was present. If the number was negative, +include the negative sign. +• +Step 3: If there is more than one digit, place the decimal point after the first digit. +• +Step 4: Multiply the number from Step 3 by +. +EXAMPLE 3.118 +Writing a Number in Scientific Notation +Write the following numbers in scientific notation form: +1. +428.9 +2. +−0.00000981 +3. +8 +Solution +1. +Since the absolute value of 428.9 is 10 or larger, so we use the process from Case 3, above. +Step 1: There are three digits to the left of the decimal point, so +. +Step 2: Write the digits of the number without the decimal place, which is 4289. +Step 3: Since there is more than one digit, place the decimal point after the first digit. We now have 4.289. +Step 4: Since +, we multiply 4.289 by 10 raised to the second power, +. +The scientific notation form of 428.9 is +. +2. +Since the absolute value of −0.00000981 is less than 1, we use the process from Case 2. +Step 1: The number of zeros between the decimal and the first non-zero digit is 5, so +. +Step 2: We write the non-zero digits, including the negative sign, yielding −981. +Step 3: The decimal gets placed to the right of the first digit, resulting in −9.81. +Step 4: Since +, we multiply −9.81 by 10 raised to the fourth power, +. +The scientific notation form of −0.00000981 is +. +3. +Since 8 is a single-digit integer, apply Case 1. The scientific notation form of 8 is +. +YOUR TURN 3.118 +Write the following numbers in scientific notation form: +1. −38300 +2. 0.0045 +3. 1 +3.9 • Scientific Notation +229 + +When we write numbers in scientific notation form, we can manipulate the representation of the number by moving the +decimal around, and making an appropriate change to the exponent of the 10. For instance, let’s look at +. +If we wanted to move the decimal one place to the left, we’d have to increase the power of 10, as shown here: +. Since we moved the decimal one to the left, we balance that with moving the +exponent up by one. Similarly, if we move the decimal one place to the right, we have to balance that by moving the +exponent one to the left, or subtracting one from the exponent, as shown here: +. +Generally, for a number in the form +: +• +If you move the decimal to the left by +digits, you increase the exponent by +. +• +If you move the decimal to the right by +digits, you decrease the exponent by +digits. +EXAMPLE 3.119 +Increasing the Exponent +Change +by moving the decimal two places to the left. +Solution +Since we are moving the decimal to the left by two places, we increase the exponent of 10 by 2, so that the exponent is +now 7. This gives us +. +YOUR TURN 3.119 +1. Change +by moving the decimal four places to the left. +EXAMPLE 3.120 +Decreasing the Exponent +Change +by moving the decimal five places to the right. +Solution +Since we are moving the decimal to the right by five places, we decrease the exponent of 10 by 5, so that exponent is +now −3. This give us +. +YOUR TURN 3.120 +1. Change +by moving the decimal two places to the right. +Converting Numbers from Scientific Notation to Standard Form +In the previous section, converting a number from standard form to scientific notation was explored. Now, we explore +converting from scientific notation back into standard form. Doing so involves moving the decimal according to the +power of the 10. The decimal is moved a number of steps equal to the exponent of the 10. As demonstrated previously, +when the exponent of the 10 is negative, the decimal is moved to the left and when the exponent of the 10 is positive, +the decimal is moved to the right. +EXAMPLE 3.121 +Converting from Scientific Notation to Standard Form +Convert the following into standard form: +1. +2. +Solution +1. +Since the exponent is positive, the decimal moves nine places to the right, so +is +. +230 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +2. +Since the exponent is negative, the decimal moves eight places to the left, so +is +. +YOUR TURN 3.121 +Convert the following into standard form: +1. +2. +VIDEO +Converting from Standard Form to Scientific Notation Form (https://openstax.org/r/ +Converting_from_Standard_Form_to_Scientific_Notation_Form) +Converting from Scientific Notation Form to Standard Form (https://openstax.org/r/ +Converting_from_Scientific_Notation_Form_to_Standard_Form) +TECH CHECK +Scientific Notation on a Calculator +Most scientific and graphing calculators come with the ability to directly convert from standard form to scientific +notation. On the TI-83, it is accessed through the MODE menus. For a commonly used, free phone scientific calculator, +the calculator can be forced to work in scientific notation mode through its settings. +Some calculators, such as the Desmos online calculator, display scientific notation as a number times 10 to a power as +you’ve seen in this section. However, some calculators indicate scientific notation by replacing the +with an E (or +EE) followed by the exponent. For example, Figure 3.46 shows what you may see on a TI-84. +Figure 3.46 Calculator screens +Adding and Subtracting Numbers in Scientific Notation +To add or subtract numbers in scientific notation, the numbers first need to have the same exponent for the 10s. It is +possible to add the following since the powers of 10 match: +Notice that the number parts were added, but the exponent part remained the same. This is due to the distributive +property of the real numbers. The +is factored from the two terms, as shown: +Numbers in scientific notation can be added or subtracted directly using a calculator. Simply enter the values in scientific +form and set your calculator to display scientific notation. +3.9 �� Scientific Notation +231 + +EXAMPLE 3.122 +Adding and Subtracting Numbers in Scientific Notation with the Same Powers of 10 +Calculate the following: +1. +2. +Solution +1. +Since the powers of 10 match, we use the distributive property of real numbers to factor 10−3 from the numbers. We +then add the number parts separately to get 4.806. +2. +Since the powers of 10 match, we use the distributive property of real numbers to factor 108 from the numbers. We +then subtract the number parts separately to get 5.76. +YOUR TURN 3.122 +Calculate the following: +1. +2. +Adding and subtracting in scientific notation is straightforward when the exponents are the same. There are two issues +that can arise. The first issue is what to do if after adding or subtracting the result is not in scientific notation. +EXAMPLE 3.123 +Correcting an Answer to Scientific Notation After Adding or Subtracting +Calculate the following: +1. +2. +Solution +1. +Since the powers of 10 match, we add the number parts and multiply that by +: +. +However, +is not in scientific notation because the absolute value of 15.53 is more than 10. To put this +number in scientific notation, the decimal needs to move one to the left. To balance that move, the power of 10 +must be increased by 1. So, the answer in scientific notation is +. +2. +Since the powers of 10 match, we add the number parts: +However, +is not in scientific notation because it is less than 1. To put it in scientific notation, the +decimal needs to move one to the right. To balance that move, the power of 10 must be decreased by 1. So, the +answer in scientific notation is +. +YOUR TURN 3.123 +Calculate the following: +1. +2. +The second issue that might be encountered when adding or subtracting is that the powers of 10 do not match. In that +case, one of the numbers must be changed so that the powers of 10 match. It is easiest to make the smaller power of 10 +232 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +larger to match the other power of 10. +For example, to perform the following, +, we’d change the +so that the power of 10 is 5. To +do so, we need to increase the power of 10 and move the decimal in the number part two places to the left. That would +alter +into +. We would use +in the addition problem, so that the exponents match, +allowing the addition to occur. +The steps to take when the exponents of the 10s are not equal are: +Step 1: Increase the smaller exponent to equal the larger exponent. Label the amount increased as +. +Step 2: For the number with the smaller power of 10, move the decimal point of the number part to the left +places. +Step 3: Perform the addition or subtraction. +Step 4: If the result is not in scientific notation, adjust the number to be in scientific notation. +EXAMPLE 3.124 +Adding Numbers in Scientific Notation with Different Powers of 10 +Calculate the following: +Solution +Step 1: The lower exponent is 4. To make this equal to the larger exponent, we increased it by 1. +Step 2: Since the smaller exponent was increased by 1, move the decimal one to the left, so the addition become +. +Step 3: Now add the numbers, +Step 4: The result is in scientific notation, so no additional adjustment is necessary. +YOUR TURN 3.124 +1. Calculate the following: +EXAMPLE 3.125 +Subtracting Numbers in Scientific Notation with Different Powers of 10 +Calculate the following: +Solution +Step 1: The lower exponent is −15 and the larger is −13. To make −15 equal to the larger exponent, we increased it by 2. +Step 2: Since the smaller exponent increased by 2, move the decimal two to the left. The subtraction changes to +. +Step 3: Subtract the numbers, +. +Step 4: The result is in scientific notation, so no additional adjustment is necessary. +YOUR TURN 3.125 +1. Calculate the following: +3.9 • Scientific Notation +233 + +Multiplying and Dividing Numbers in Scientific Notation +Multiplying and dividing numbers in scientific notation is somewhat easier than adding or subtracting, because the +exponents of the 10s do not have to match. However, it is much more likely that the result will not be in scientific +notation, and so that will have to be adjusted at the end. Generally, we multiply or divide the number parts of the two +values, and then apply exponent rules to the 10 raised to the powers. +To multiply two numbers in scientific notation: +Step 1: Multiply the number parts. +Step 2: Add the exponents of the 10s. +Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2. +Step 4: If the number is not in scientific notation, adjust it appropriately. +EXAMPLE 3.126 +Multiplying Numbers in Scientific Notation +Calculate the following: +1. +2. +Solution +1. +Step 1: Multiply the number parts to get +. +Step 2: Add the exponents of the 10s to get +. +Step 3: The result is then +. +Step 4: This number is already in scientific notation, so no additional adjustment is necessary, +. +2. +Step 1: Multiply the number parts to get +. +Step 2: Add the exponents of the 10s to get +. +Step 3: The result then is +. +Step 4:Since the number is not in scientific notation, it must be adjusted. To put +into scientific notation, +the decimal moves one to the left, so the exponent would be increased by 1, giving +. +YOUR TURN 3.126 +Calculate the following: +1. +2. +VIDEO +Multiplying Numbers in Scientific Notation (https://openstax.org/r/Multiplying_Numbers_in_Scientific_Notation) +Dividing Numbers in Scientific Notation +To divide two numbers that are in scientific notation: +Step 1: Divide the number parts. +234 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Step 2: Subtract the exponent of the denominator from the exponent of the numerator. +Step 3: The answer is the result from Step 1 times 10 raised to the result from Step 2. +Step 4: If the number is not in scientific notation, adjust it appropriately. +EXAMPLE 3.127 +Dividing Numbers in Scientific Notation +Calculate the following: +1. +2. +Solution +1. +Step 1: Divide the number parts to get +. +Step 2: Subtract the exponent of the denominator from the exponent of the numerator to get +. +Step 3: The result is then +. +Step 4: This number is already in scientific notation, so no adjustment is necessary. +2. +Step 1: Divide the number parts to get +. +Step 2: Subtract the exponent of the denominator from the exponent of the numerator to get +. +Step 3: The result then is +. +Step 4: Since this number is not in scientific notation, it must be adjusted. To put +into scientific notation, +the decimal needs to move one to the right, so the exponent is decreased by 1, giving +. +YOUR TURN 3.127 +Calculate the following: +1. +2. +VIDEO +Dividing Numbers in Scientific Notation (https://openstax.org/r/Dividing_Numbers_in_Scientific_Notation) +Using Scientific Notation in Computing Real-World Applications +As noted at the start of this section, scientific notation is useful when the standard representation of a number is +awkward or impractical, which occurs when the numbers being used are extremely large or extremely small. For +example, Venus is 67,667,000 miles from the sun. In scientific notation, this is +. Planetary and galaxy +distances is one set of numbers that is easier to express using scientific notation. +EXAMPLE 3.128 +Calculating Distances +How much farther from the sun is Earth compared to Venus if Venus is +miles from the sun and Earth is +miles from the sun? +3.9 • Scientific Notation +235 + +Solution +To determine how much farther Earth is compared to Venus, we’d subtract the distances. +. +So, Earth is +miles farther from the sun than Venus. +YOUR TURN 3.128 +1. Earlier we saw that a single transistor in a computer chip 0.000000014 meters, or +m, in size, and that +the diameter of an atom could be 0.2 nanometers, or +m in size. How much larger is the transistor than +the atom? +EXAMPLE 3.129 +Calculating Probability +The probability of winning the Mega Millions lottery is published as +. The probability of being hit by +lightning is approximated to be +. How many times more likely are you to be hit by lightning than win the Mega +Millions? +Solution +To find out how many times more likely you are to be hit by lightning, divide the probability of being hit by lightning by +the probability of winning the Mega Millions. +Step 1: Divide the number parts to get = +(rounded to the fourth digit). +Step 2: Subtract the exponent of the denominator from the exponent of the numerator to get +. +Step 3: The result then is +. +Step 4: Since this number is not in scientific notation, it must be adjusted. To put +into scientific notation, +the decimal needs to move one place to the right, so the exponent is decreased by 1, giving +. +You are +, or 605.2, times more likely to be hit by lightning than you are to win the Mega Millions. +YOUR TURN 3.129 +1. Mercury is about +miles from the sun. Neptune is about +miles from the sun. How many +times further is Neptune from the sun than Mercury? +EXAMPLE 3.130 +Calculating Time and Length +Sometimes it is entertaining to determine the time it takes for something to happen. Fingernails grow about +km per minute. How many kilometers long would fingernails be after +minutes? +Solution +To find the length of the fingernails after the specified time, we multiply their rate of growth and the time they’ve grown. +So, after +minutes, the fingernails would be +km long. To put this in perspective, +km is a +millimeter, and +minutes is about 4.16 days. So, after about 4.16 days, fingernails have grown about 4.8 +millimeters. +236 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +YOUR TURN 3.130 +1. There are approximately +grains of sand in a cubic meter. If the number of grains of sand on the +Australian coastline is roughly +grains, roughly how many cubic meters of sand is there on the +Australian coastline? +EXAMPLE 3.131 +Calculating Data Generated +As mentioned in the opening to this section, it is estimated that we’re producing 2.5 quintillion bytes of data per day. A +good estimate is that there are 7.674 billion people on the planet. Convert both of those numbers to scientific notation, +and then determine how much data is being generated per person each day. +Solution +Written in standard form, 2.5 quintillion is 2,500,000,000,000,000,000. Changing that to scientific notation, move the +decimal 18 places, so 2.5 quintillion bytes = +bytes. Writing 7.647 billion in scientific notation would be +because a billion is 1,000,000,000 = +. So, to find out how much data is being produced daily per person, +we would divide these two numbers. +In standard form, that’s 327,000,000 bytes per person, so 327 million bytes of data daily are being produced per person. +YOUR TURN 3.131 +1. Humans collectively exhale approximately +pounds of carbon dioxide per year. There are +approximately +humans currently living on Earth. How many pounds of carbon dioxide does a single +human, on average, exhale per year? +VIDEO +Application of Scientific Notation (https://openstax.org/r/Application_of_Scientific_Notation) +What Numbers Could Be Considered “Too Big” or “Too Small”? +One wonders when the numbers we represent become too large or small for consideration. Perhaps the following +examples put limits on what is meaningful. The number of particles in the known universe has been estimated at +particles. The smallest distance that has been measured is +, though the theoretical smallest +measurable value is +. The distance across the universe is +. Considering what those numbers +represent, the extreme largest and extreme smallest, they might be numbers that constrain what we should reasonably +be expected to deal with. +Check Your Understanding +45. Write 0.00456 in scientific notation. +46. Write +in standard form. +47. Calculate +. +48. Calculate +. +49. Calculate +. +50. Calculate +. +51. The distance from Earth to the moon is +inches. The thickness of a dollar bill is +inches. How +many dollar bills must be stacked so the pile reaches the moon? +3.9 • Scientific Notation +237 + +SECTION 3.9 EXERCISES +For the following exercises, convert numbers to scientific notation. +1. +2. +3. +4. +For the following exercises, convert numbers to standard form. +5. +6. +7. +8. +For the following exercises, the numbers are not in scientific notation. Convert them to scientific notation. +9. +10. +11. +12. +For the following exercises, make the conversions required. +13. The distance from the sun to the star Polaris is about 3,056,000,000,000,000 km. Express that distance in +scientific notation. +14. The distance from us to the next-closest galaxy is about 662,000,000,000,000,000 km. Express that distance in +scientific notation. +15. The mass of a grain of sand is about +g. Convert that mass to standard form. +16. The diameter of a cell is about +m. Convert that diameter to standard form. +17. The equatorial circumference of Earth is approximately +km. Convert that circumference to standard +form. +18. The straight-line distance from Buffalo, NY, to Buenos Aires, Argentina, is approximately +m. Convert +that distance to standard form. +19. The mass of a proton is approximately +kg. Convert that mass to standard form. +20. The diameter of a housefly egg is approximately +m. Convert that diameter to standard form. +21. The tallest building in the world, the Burj Khalifa in Dubai, stands at 829.8 m tall. Convert that height to +scientific notation. +22. Using the rings of the shell, the age of an Icelandic clam is 507 years. Express that age in scientific notation. +Calculate the following: +23. +24. +25. +26. +27. +28. +29. +30. +31. +32. +33. +34. +35. +36. +37. +38. +39. +40. +41. +42. +238 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +43. +44. +45. +46. +47. +48. +49. +50. +For the following exercises, apply your understanding of scientific notation to real-world applications. +51. When stretched out, a strand of human DNA is, on average, +cm. One centimeter, or 1 cm, is +km. Determine the average length of a strand of human DNA in kilometers. +52. One approximation of the average number of cells in the human body is +cells (30 trillion!!!). If the DNA +of each cell were stretched out and laid end to end, what would be the total length of the DNA in km? Use your +answer from Exercise 51 for the length, in kilometers, of DNA. +53. The equatorial circumference of Earth is approximately +km. Use the answer from Exercise 52 to +determine the number of times that the stretched out human DNA would encircle Earth. +54. The average stride length of a 1.651 m tall woman is +meters. If such a person could walk from +Buffalo, NY, to Buenos Aires, Argentina, in a straight line, how many steps would that person need to take? See +Exercise 18 for the distance from Buffalo, NY, to Buenos Aires, Argentina. +3.10 Arithmetic Sequences +Learning Objectives +After completing this section, you should be able to: +1. +Identify arithmetic sequences. +2. +Find a given term in an arithmetic sequence. +3. +Find the +th term of an arithmetic sequence. +4. +Find the sum of a finite arithmetic sequence. +5. +Use arithmetic sequences to solve real- world applications +As we saw in the previous section, we are adding about 2.5 quintillion bytes of data per day to the Internet. If there are +550 quintillion bytes of data today, then there will be 552.5 quintillion bytes tomorrow, and 555 quintillion bytes in 2 +days. This is an example of an arithmetic sequence. There are many situations where this concept of fixed increases +comes into play, such as raises or table arrangements. +Identifying Arithmetic Sequences +A sequence of numbers is just that, a list of numbers in order. It can be a short list, such as the number of points earned +on each assignment in a class, such as {10, 10, 8, 9, 10, 6, 10}. Or it can be a longer list, even infinitely long, such as the +list of prime numbers. For example, here’s a sequence of numbers, specifically, the squares of the first 12 natural +numbers. +{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144} +Each value in the sequence is called a term. Terms in the list are often referred to by their location in the sequence, as in +the +th term. For the sequence {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}, the first term of the sequence is 1, the fourth +term is 16, and so on. In the sequence of assignment scores {10, 10, 8, 9, 10, 6, 10}, the first term is 10 and the third term +is 8 (Figure 3.47). +Figure 3.47 Sequence showing first, second, and fifth terms +The notation we use with sequences is a letter, which represents a term in the sequence, and a subscript, which indicates +3.10 • Arithmetic Sequences +239 + +what place the term is in the sequence. For the sequence {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}, we will use the +letter +as a value in the sequence, and so +would be the term in the sequence at the fifth position. That number is 25, +so we can write +. +In this section, we focus on a special kind of sequence, one referred to as an arithmetic sequence. Arithmetic +sequences have terms that increase by a fixed number or decrease by a fixed number, called the constant difference +(denoted by +), provided that value is not 0. This means the next term is always the previous term plus or minus a +specified, constant value. Another way to say this is that the difference between any consecutive terms of the sequence +is always the same value. +To see a constant difference, look at the following sequence: {7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87}. Figure 3.48 +illustrates that each term of the sequence is the previous term plus 8. Eight is the constant difference here. +Figure 3.48 Sequence of numbers with 8 added to each term +EXAMPLE 3.132 +Identifying Arithmetic Sequences +Determine if the following sequences are arithmetic sequences. Explain your reasoning. +1. +2. +3. +Solution +1. +In the sequence +, every term is the previous term plus 3. The ellipsis indicates that the +pattern continues, which means keep adding 3 to the previous term to get the new term. Therefore, this is an +infinite arithmetic sequence. +2. +In the sequence +, terms increase by various amounts, for instance from term 1 to term 2, +the sequence increases by 20, but from term 2 to term 3 the sequence increases by 40. So, this is not an arithmetic +sequence. +3. +In the sequence +, every term is the previous term minus 6, so this is an +arithmetic sequence. +YOUR TURN 3.132 +Determine if the following sequences are arithmetic sequences. Explain your reasoning. +1. +2. +3. +Arithmetic sequences can be expressed with a formula. When we know the first term of an arithmetic sequence, which +we label +, and we know the constant difference, which is denoted +, we can find any other term of the arithmetic +sequence. The formula for the +term of an arithmetic sequence is +. +FORMULA +If we have an arithmetic sequence with first term +and constant difference +, then the +term of the arithmetic +sequence is +. +Let’s examine the formula with this arithmetic sequence: +. In this sequence +and +. The table below shows the values calculated. +240 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +, Place in Sequence +, +Term +Value of Term +Term Written as +1 +4 +2 +7 +3 +10 +4 +13 +5 +16 +We can see how the +term can be directly calculated. In this sequence, the formula is +where the first +term, +, is 4 and the constant difference +is 3. We can then determine the +term of this sequence: +. +EXAMPLE 3.133 +Calculating a Term in an Arithmetic Sequence +Identify +and +for the following arithmetic sequence. Use this information to determine the +term. +Solution +Inspecting the sequence shows that +and +. We use those values in the formula, with +. +YOUR TURN 3.133 +1. Identify +and +for the following arithmetic sequence. Use this information to determine the +term. +VIDEO +Arithmetic Sequences (https://openstax.org/r/Arithmetic_Sequences) +If we know two terms of the sequence, it is possible to determine the general form of an arithmetic sequence, +. +FORMULA +If we have the th term of an arithmetic sequence, +, and the th term of the sequence, +, then the constant +difference is +and the first term of the sequence is +. +3.10 • Arithmetic Sequences +241 + +EXAMPLE 3.134 +Determining First Term and Constant Difference Using Two Terms +A sequence is known to be arithmetic. Two of its terms are +and +. Use that information to find the +constant difference, the first term, and then the +term of the sequence. +Solution +To find the constant difference, use +. The location of the terms is given by the subscript of the two +terms, +and +. So, the constant difference can be calculated as such: +. +The constant difference of 4 is then used to find +. +. +So +and +. +With this information, the +term can be found. +. +The +term is +. +YOUR TURN 3.134 +1. A sequence is known to be arithmetic. Two of the terms are +and +. Use that information to +find the constant difference and the first term. Then determine the +term of the sequence. +VIDEO +Finding the First Term and Constant Difference for an Arithmetic Sequence (https://openstax.org/r/ +Finding_the_First_Term_and_Constant_Difference_for_an_Arithmetic_Sequence) +Finding the Sum of a Finite Arithmetic Sequence +Sometimes we want to determine the sum of the numbers of a finite arithmetic sequence. The formula for this is fairly +straightforward. +FORMULA +The sum of the first +terms of a finite arithmetic sequence, written +, with first and last term +and +, respectively, +is +. +EXAMPLE 3.135 +Finding the Sum of a Finite Arithmetic Sequence +What is the sum of the first 60 terms of an arithmetic sequence with +and +? +Solution +The formula requires the first and last terms of the sequence. The first term is given, +. The +term is needed. +Using the formula +provides the value for the +term. +. +Applying the formula +provides the sum of the first 60 terms. +242 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +. +The sum of the first 60 terms is 4,695. +YOUR TURN 3.135 +1. What is the sum of the first 101 terms of an arithmetic sequence with +and +? +VIDEO +Finding the Sum of a Finite Arithmetic Sequence (https://openstax.org/r/ +Finding_the_Sum_of_a_Finite_Arithmetic_Sequence) +Using Arithmetic Sequences to Solve Real-World Applications +Applications of arithmetic sequences occur any time some quantity increases by a fixed amount at each step. For +instance, suppose someone practices chess each week and increases the amount of time they study each week. The first +week the person practices for 3 hours, and vows to practice 30 more minutes each week. Since the amount of time +practicing increases by a fixed number each week, this would qualify as an arithmetic sequence. +EXAMPLE 3.136 +Applying an Arithmetic Sequence +Jordan has just watched The Queen’s Gambit and decided to hone their skills in chess. To really improve at the game, +Jordan decides to practice for 3 hours the first week, and increase their time spent practicing by 30 minutes each week. +How many hours will Jordan practice chess in week 20? +Solution +Jordan’s practice scheme is an arithmetic sequence, as it increases by a fixed amount each week. The first week there are +3 hours of practice. This means +. Jordan increases the time spent practicing by 30 minutes, or half an hour, each +week. This means +. Using those values, and that we want to know the amount of time Jordan will study in week +20, we determine the time in week 20 using +. +So, Jordan will practice 12.5 hours in week 20. +YOUR TURN 3.136 +1. Christina decides to save money for after graduation. Christina starts by setting aside $10. Each week, Christina +increases the amount she saves by $5. How much money will Christina save in week 52? +EXAMPLE 3.137 +Finding the Sum of a Finite Arithmetic Sequence +Let’s check back in on Jordan. Recall, Jordan had just watched The Queen’s Gambit and decided to hone their skills, +practicing for 3 hours the first week, and increasing the time spent practicing by 30 minutes each week. How many hours +total will Jordan have practiced chess after 30 weeks of practice? +Solution +To calculate the total amount of time that Jordan practiced, we need to use +. The formula requires the +first and last terms of the sequence. Since Jordan practiced 3 hours in the first week, the first term is +. Because we +want the total practice time after 30 weeks, we need the +term. Because the constant difference is +, the +3.10 • Arithmetic Sequences +243 + +term is +. +Applying the formula +provides the sum of the first 30 terms. +. +This means that Jordan practiced a total of 615 hours after 30 weeks. +YOUR TURN 3.137 +1. In a theater, the first row has 24 seats. Each row after that has 2 more seats. How many total seats are there if +there are 40 rows of seat in the theater? +WHO KNEW? +The Fibonacci Sequence +Not all sequences are arithmetic. One special sequence is the Fibonacci sequence, which is the sequence that has as +its first two terms 1 and 1. Every term thereafter is the sum of the previous two terms. The first nine terms of the +Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, and 34. +This sequence is found in nature, architecture, and even music! In nature, the Fibonacci sequence describes the +spirals of sunflower seeds, certain galaxy spirals, and flower petals. In music, the band Tool used the Fibonacci +sequence in the song “Lateralus.” The Fibonacci sequence even relates to architecture, as it is closely related to the +golden ratio. +VIDEO +Fibonacci Sequence and “Lateralus” (https://openstax.org/r/Fibonacci_Sequence_and_Lateralus) +Check Your Understanding +52. Is the following an arithmetic sequence? Explain. +{3, 6, 9, 15, 25, 39, 90} +53. What is the 7th term of the following sequence? +{1, 5, 7, 100, 4, -17, 8, 100, 19, 7.6, 345} +54. In an arithmetic sequence, the first term is 10 and the constant difference is 4.5. What is the 135th term? +55. If the eighth term of an arithmetic sequence is 35 and the 40th term is 131, what is the constant difference and the +first term of the sequence? +56. What is the sum of the first 100 terms of the arithmetic sequence with first term 4 and constant difference 7? +57. A new marketing firm began with 30 people in its survey group. The firm adds 4 people per day. How many people +will be in their survey group after 100 days? +SECTION 3.10 EXERCISES +For the following exercises, determine if the sequence is an arithmetic sequence. +1. +2. +3. +4. +5. +6. +244 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +7. +8. +For the following exercises, the sequences given are arithmetic sequences. Determine the constant difference for each +sequence. Verify that each term is the previous term plus the constant difference. +9. +10. +11. +12. +13. +14. +For the following exercises, the first term and the constant difference of an arithmetic sequence is given. Using that +information, determine the indicated term of the sequence. +15. +, +, find +. +16. +, +, find +. +17. +, +, find +. +18. +, +, find +. +19. +, +, find +. +20. +, +, find +. +21. +, +, find +. +22. +, +, find +. +For the following exercises, two terms of an arithmetic sequence are given. Using that information, identify the first +term and the constant difference. +23. +, +24. +, +25. +, +26. +, +27. +, +28. +, +For the following exercises, the first term and the constant difference is given for an arithmetic sequence. Use that +information to find the sum of the first +terms of the sequence, +. +29. +, +, calculate +30. +, +, calculate +. +31. +, +, calculate +. +32. +, +, calculate +. +33. +, +, calculate +. +34. +, +, calculate +. +For the following exercises, apply your knowledge of arithmetic sequences to these real-world scenarios. +35. A collection is taken up to support a family in need. The initial amount in the collection is $135. Everyone places +$20 in the collection. When the 35th person puts their $20 in the collection, how much is present in the +collection? +36. There are 50 songs on a playlist. Every minute, 3 more songs are added to the playlist. How many songs are on +the playlist after 40 minutes have passed? +37. One genre on Netflix has 1,000 shows. Every week, 20 shows are added to that genre. After 15 weeks, how +many shows are in that genre? +38. A new local band has 10 people come to their first show. News of the band spreads afterwards. Each week, 4 +more people attend their show than the previous week. After 50 weeks, how many people are at their show? +39. The Jester Comic book store is going out of business and is taking in no new inventory. Its inventory is currently +13,563 titles. Each day after, they sell or give away 250 titles. After 15 days, how many titles are left? +40. Jasmyn has decided to train for a marathon. In week one, Jasmyn runs 5 miles. Each week, Jasmyn increased the +running distance by 2 miles. How many miles will Jasmyn run in week 13 of the training schedule? +41. A 42-gallon bathtub sits with 14 gallons in it. The faucet is turned on and is now being filled at the rate of 2.2 +gallons per minute, but is draining slowly, at 1.8 gallons per minute. After 20 minutes, how many gallons are in +the tub? +42. A trained diver is 250 feet deep. The diver is nearly out of air and needs to surface. However, the diver can only +comfortably ascend 30 feet per minute. How deep is the diver after ascending for 5 minutes? +3.10 • Arithmetic Sequences +245 + +43. Jaclyn, an investor, begins a start-up to revitalize homes in South Bend, Indiana. She begins with $10,000, +making her investor 1. Each investor that joins will invest $500 more than the previous investor. How much does +the 50th investor invest in the project? With that 50th investor, what is the total amount invested in the project? +44. Jasmyn has decided to train for a marathon. In week one, Jasmyn runs 5 miles. Each week, Jasmyn increased the +running distance by 2 miles. After training for 14 weeks, how many total miles will Jasmyn have run? +45. The base of a pyramidal structure has 144 blocks. Each level above has 5 fewer blocks than the previous level. +How many total blocks are there if the pyramidal structure has 25 levels? +46. As part of a deal, a friend tells you they will give you $10 on day 1, $20 on day 2, $30 on day 3, for all 30 days of +a month. At the end of that month, what is the total amount your friend has given you? +3.11 Geometric Sequences +Figure 3.49 Savings grows in a geometric sequence. (credit: modification of “A big part of financial freedom is having +your heart and mind free from worry about the what-ifs of life. – Suze Orman” by Morgan/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify geometric sequences. +2. +Find a given term in a geometric sequence. +3. +Find the +th term of a geometric sequence. +4. +Find the sum of a finite geometric sequence. +5. +Use geometric sequences to solve real-world applications. +One of the concerns when investing is the doubling time, which is length of time it takes for the value of the investment +to be twice, or double, that of its starting value. A shorter doubling times means the investment gets bigger, sooner. For +example, if you invest $200 in an account with an 8-year doubling time, then in 8 years the value of the account will be +double the starting amount, or +. After another 8 years (for a total of 16 years) the investment would be +twice its value after the first 8 years, or +. Every 8 years, the investment would double +again, so after the third 8-year period, the investment would be worth +. This process exhibits +exponential growth, an application of geometric sequences, which is explored in this section. +Identifying Geometric Sequences +We know what a sequence is, but what makes a sequence a geometric sequence? In an arithmetic sequence, each term +is the previous term plus the constant difference. So, you add a (possibly negative) number at each step. In a geometric +sequence, though, each term is the previous term multiplied by the same specified value, called the common ratio. In +the sequence +the common ratio is 2. To see the difference between an +arithmetic sequence and geometric sequence, examine these two sequences (Figures 3.52 and 3.53). +Figure 3.50 Arithmetic sequence +Each term in this arithmetic sequence is the previous term plus 5. +246 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +Figure 3.51 Geometric sequence +Each term in this geometric sequence is the previous term times 2. +In the sequence +, the numbers get big fairly quickly, and stay positive. However, +that’s not always the case with geometric sequences. Depending on the value of the common ratio, the terms could +increase each time (like in the one shown in Figure 3.51), or the terms can get smaller each time, or the terms can +alternate between positive and negative values. It all depends on the value of the common ratio, . +Consider this geometric sequence: +Each term is the previous term times 5, which means the common ratio is 5. This common ratio is larger than 1, and so +the terms increase each time. Now, look at this geometric sequence: +Each term is the previous term times −3, and the sign of the terms alternate from positive to negative. Then, there’s this +geometric sequence: +Each term is the previous term times +, and the terms decrease each time. What we should take away from these three +examples is if the common ratio is a positive number larger than 1, then the sequence increases. If the common ratio is a +negative number, then the sign of the terms alternates between positive and negative. If the common ratio is between 0 +and 1, then the terms decrease. +Two special cases of geometric sequences are when the constant ratio is 1 and when the common ratio is 0. When the +constant ratio is 1, every term of the sequence is the same, as in +. This is referred to as a constant +sequence. When the constant ratio is 0, the first term can be any number, but every term after the first term is 0, as in +. +EXAMPLE 3.138 +Identifying Geometric Sequences +For each sequence, determine if the sequence is a geometric sequence. If so, identify the common ratio. +1. +2. +3. +Solution +1. +In the sequence +, the jump from 5 to 20 is a multiplication by 4, as is the next +jump to 80, and the next to 320. Each term is 4 times the previous term. Since each term is 4 times the previous, this +is a geometric sequence. The common ratio is 4. +2. +In the sequence +, notice that 6 is −3 times −2. The jump from 6 to −12 is another +multiplication by negative. So, if this is a geometric sequence, each term should be the previous term times −2. But +the change from 24 to 11 is not a multiplication by −2, This means the sequence is not a geometric sequence. +3. +In the sequence +, the change from 4 to 2 is a multiplication by +, as is the next jump, from 2 to +1, as is the next from 1 to +. Each term is +times the previous term. Since each term is +times the previous, this is +a geometric sequence. The common ratio is +. +3.11 • Geometric Sequences +247 + +YOUR TURN 3.138 +For each sequence, determine if the sequence is a geometric sequence. If so, identify the common ratio. +1. +2. +3. +As with arithmetic sequences, the first term of a geometric sequence is labeled +. The number that is multiplied by each +term is called the common ratio and is denoted . So, if the first term is known, +, and the common ratio is known, , +then the +term, +, can be calculated with the formula +. +FORMULA +The +th term of the geometric sequence, +, with first term +and common ratio , is +. +Return to the sequence +. We observe that the first term is 3, so +. We also +found that the common ratio is 2, so +. The table below shows how any term can be calculated using just +and . +, Place in Sequence +,Term +Value of Term +Term Written as +1 +3 +2 +6 +3 +12 +4 +24 +5 +48 +EXAMPLE 3.139 +Determining the Value of a Specific Term in a Geometric Sequence +In the following geometric sequences, determine the indicated term of the geometric sequence with a given first term +and common ratio. +1. +Determine the +term of the geometric sequence with +and +. +2. +Determine the +term of the geometric sequence with +and +. +Solution +1. +Using +with +, +, and +, we calculate +. +The +term of the geometric sequence with +and +is +. +2. +Using +with +, +, and +, we calculate +. +248 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +YOUR TURN 3.139 +In the following geometric sequences, determine the indicated term of the geometric sequence with a given first +term and common ratio. +1. Determine the +term of the geometric sequence with +and +. +2. Determine the +term of the geometric sequence with +and +. +VIDEO +Geometric Sequences (https://openstax.org/r/Geometric_Sequences) +Finding the Sum of a Finite Geometric Sequence +As with arithmetic sequences, it is possible to add the terms of the geometric sequence. Like arithmetic sequences, the +formula for the finite sum of the terms of a geometric sequence has a straightforward formula. +FORMULA +The sum of the first +terms of a finite geometric sequence, written +, with first term +and common ratio , is +provided that +. +EXAMPLE 3.140 +Calculating the Sum of a Finite Geometric Sequence +1. +What is the sum of the first 13 terms of the geometric sequence with first term +and common ratio +? +2. +What is the sum of the first 7 terms of the geometric sequence with first term +and common ratio +? +Solution +1. +Using +, +, and +, we find that the sum is: +The sum of the first 13 terms of this geometric sequence is 1,328,600. +2. +Using +, +, and +, we find that the sum is: +The sum of the first 7 terms of this geometric sequence is +. +YOUR TURN 3.140 +1. What is the sum of the first 10 terms of the geometric sequence with first term +and common ratio +? +2. What is the sum of the first 6 terms of the geometric sequence with first term +and common ratio +? +Using Geometric Sequences to Solve Real-World Applications +Geometric sequences have a multitude of applications, one of which is compound interest. Compound interest is +3.11 • Geometric Sequences +249 + +something that happens to money deposited into an account, be it savings or an individual retirement account, or IRA. +The interest on the account is calculated and added to the account at regular intervals. This means the interest that was +earned later gains its own interest. This allows the money to grow faster. If that interest is added every month, we say it +is compounded monthly. If the interest is added daily, then we say it is compounded daily. The amount of money that is +deposited into the account is called the principal and is denoted +. The account earns money on that principal. The +amount it earns is a percentage of the money in the account. The interest rate, expressed as a decimal, is denoted . +FORMULA +If you deposit +dollars in an account that earns interest compounded yearly, then the amount in the account, +, +after +years is calculated with the formula: +. This is a geometric sequence, with constant ratio +and first term +. +EXAMPLE 3.141 +Calculating Interest Compounded Yearly +Daryl deposits $1,000 in an account earning +interest compounded yearly. How much money is in the account after 25 +years? +Solution +Using +with +, +, and +, we find that +. After 25 years, there is +in the account. +YOUR TURN 3.141 +1. Sophia deposited $4,000 in an account that earns 5.5% interest compounded yearly. After 20 years, Sophia +withdrew all the money in the account to pay for her child’s college. How much money was in the account when +Sophia withdrew the money? +Another application of geometric sequences is exponential growth. This arises in biology quite frequently, especially in +relation to bacterial cultures, but also with other organism population models. In bacterial cultures, the time it takes the +population to double is often recorded. This time to double is the same, regardless of how big the population gets. So, if +the population doubles after 3 hours, it doubles again after another 3 hours, and again after another 3 hours, and so on. +Put into geometric sequence language, it has a common ratio of 2. +EXAMPLE 3.142 +Doubling a Bacterial Culture +When Escherichia coli (E. coli) is in a broth culture at 37°C, the population of E. coli doubles in number with 30 organisms, +how many E. coli bacteria are present in the culture after 16 hours? +Solution +Since the population is doubling every 20 minutes, this is a geometric sequence situation with common ratio +. The +culture begins with 30 organisms, so +. The time,16 hours, is 48 twenty-minute periods, so we’re looking for the +48th term in the sequence. Using these values in the geometric sequence formula gives +. +So, after 16 hours, the culture contains +E. coli organisms. That’s more than 4,000 trillion bacteria. +YOUR TURN 3.142 +1. When Streptococcus lactis (S. lactis) is in a milk culture at 37°C, the population of S. lactis doubles in number +250 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +every 30 minutes. If the culture began with 15 organisms, how many S. lactis bacteria are present in the culture +after 20 hours? +EXAMPLE 3.143 +Applying the Sum of a Finite Geometric Sequence +A player places one grain of rice on the first square of a chess board. On the second square, the player places 2 grains of +rice. On the third square, the player places 4 grains of rice. On each successive square of the board, the player doubles +the number of grains of rice placed on the chess board. When the player places the last rice on the 64th square, how +many total grains of rice have been placed on the board? +Solution +Since the number of grains of rice is doubled at each step, this is a geometric sequence with first term +and +common ratio +. Rice is placed on 64 total squares, so we want the sum of the first 64 terms. Using this information +and the formula, the total number of grains of rice on the board will be: +That’s a 20-digit number! +YOUR TURN 3.143 +1. You have a square 1 meter on each side. You begin by coloring one half of the square blue. Then you color half +the remaining area blue. Then you color half the remaining area blue once more. At each step, you color half the +remaining area. What is the total area you have colored blue after performing this process 15 times? +VIDEO +Sum of a Finite Geometric Sequence (https://openstax.org/r/Sum_of_a_Finite_Geometric_Sequence) +Check Your Understanding +58. Is the following a geometric sequence? Explain. +{3, 6, 12, 24, 48, 96, 192} +59. Find the common ratio of the geometric sequence {3, −30, 300, −3,000, …}. +60. In a geometric sequence, the first term is 10 and the common ratio is 1.5. What is the 15th term? +61. What is the sum of the first 100 terms of the geometric sequence with first term 4 and common ratio 0.3? +62. $15,000 is deposited in an account the yields 4.2% interest compounded annually. How much is in the account +after 17 years? +SECTION 3.11 EXERCISES +For the following exercises, determine if the sequence is a geometric sequence. +1. +2. +3. +4. +5. +6. +7. +8. +For the following exercises, the sequences given are geometric sequences. Determine the common ratio for each. +3.11 • Geometric Sequences +251 + +Verify that each term is the previous term times the common ratio. +9. +10. +11. +12. +13. +14. +For the following exercises, the first term and the common ratio of a geometric sequence is given. Using that +information, determine the indicated term of the sequence. +15. +, +, find +. +16. +, +, find +. +17. +, +, find +. +18. +, +, find +. +19. +, +, find +. +20. +, +, find +. +21. +, +, find +. +22. +, +, find +. +23. +, +, find +. +24. +, +, find +. +For the following exercises, the first term and the common ratio is given for a geometric sequence. Use that +information to find the sum of the first +terms of the sequence, +. +25. +, +, calculate +. +26. +, +, calculate +. +27. +, +, calculate +. +28. +, +, calculate +. +29. +, +, calculate +. +30. +, +, calculate +. +31. +, +, calculate +. +32. +, +, calculate +. +For the following exercises, apply your understanding of geometric sequences to real-world applications. +33. Lactobacilius acidophilus (L. acidophilus) is a bacterium that grows in milk. In optimal conditions, its population +doubles every 26 minutes. If a culture starts with 20 L. acidophilus bacteria, how many bacteria will there be +after 390 minutes? Hint: This means the 26-minute time period has occurred 15 times. +34. Bacillus megaterium (B. megaterium) is a bacterium that grows in sucrose salts. In optimal conditions, its +population doubles every 25 minutes. If a culture starts with 30 B. megaterium bacteria, how many bacteria will +there be after 1,000 minutes? Hint: This means the 25-minute time period has occurred 40 times. +35. Alex and Jill deposit $4,000 in an account bearing 5% interest compounded yearly. If they do not deposit any +more money in that account, how much will it be worth in 30 years? +36. Kerry and Megan deposit $6,000 dollars in and account bearing 4% compounded yearly. If they do not deposit +any more money in that account, how much will be in the account after 40 years? +37. You decide to color a square that measures 1 m on each side in a very particular manner. You first cut the +square in half vertically. You color one side of the square with purple. On the side of the square that was not +colored, you draw a line dividing that region horizontally exactly in half. You color the lower half blue. Now, you +cut the remaining quarter of the square precisely in half with a vertical line. You color the left side red. You +repeat this process 12 times. After you color that 12th piece, what is the total area you have colored? +38. Consider the geometric sequence with first term 0.9 and common ratio of 0.1. What is the sum of the first 5 +terms? +39. Repeat Exercise 38, for the sum of the first 10 terms. +For the following questions, recall that the formula for interest compounded yearly is +, where +is the +amount in the account after +years, +is the initial amount deposited, and +is the interest rate per year. However, if the +account is compounded monthly, the formula changes to +. +40. Returning to Kerry and Megan (Exercise 36), what would their account be worth if their account was +compounded monthly? +41. Returning to Alex and Jill (Exercise 35), what would their account be worth if their account was compounded +monthly? +252 +3 • Real Number Systems and Number Theory +Access for free at openstax.org + +42. Imagine your family tree. You have two parents. Your parents have two parents: your grandparents. And so on. +How many great-great-great-great-grandparents do you have? Hint: This would be six generations back. +43. Imagine your family tree. You have two parents. Your parents have two parents: your grandparents. And so on. +How many great (20 times) grandparents do you have? Hint: This would be 22 generations back. +3.11 • Geometric Sequences +253 + +Chapter Summary +Key Terms +3.1 Prime and Composite Numbers +• +natural numbers +• +factor of a number +• +multiple of a number +• +prime number +• +composite number +• +prime factorization +• +greatest common divisor (GCD) +• +least common multiple (LCM) +3.2 The Integers +• +integer +• +absolute value +• +average of a set of numbers +3.3 Order of Operations +• +order of operations +• +PEMDAS +3.4 Rational Numbers +• +density property of rational numbers +• +improper fraction +• +lowest terms +• +mixed number +• +rational number +• +repeating decimal +• +terminating decimal +3.5 Irrational Numbers +• +conjugate numbers +• +difference of squares +• +irrational numbers +• +lowest terms +• +rationalize the denominator +3.6 Real Numbers +• +complex number +• +imaginary number +• +real number +3.7 Clock Arithmetic +• +clock arithmetic +• +modulo 7 +• +modulo 12 +3.8 Exponents +• +base +• +exponent +3.9 Scientific Notation +• +scientific notation +• +standard notation +254 +3 • Chapter Summary +Access for free at openstax.org + +3.10 Arithmetic Sequences +• +sequence +• +term of a sequence +• +arithmetic sequence +• +first term +• +constant difference +3.11 Geometric Sequences +• +geometric sequence +• +common ratio +3.11 Geometric Sequences +• +common ratio +• +geometric sequence +Key Concepts +3.1 Prime and Composite Numbers +• +The natural numbers can be categorized as 1, prime numbers, and composite numbers. +• +Prime numbers have as their only factors 1 and themselves. +• +Composite numbers have at least three distinct factors. +• +Composite numbers can be written in their prime factorization form, which is found by repeatedly factoring prime +factors from the number. +• +The greatest common divisor (GCD) of a set of numbers is the largest integer that divides all of the numbers in the +set. The prime factorizations of the numbers can be used to identify the greatest common divisor. +• +The least common multiple (LCM) of a set of numbers is the smallest integer that is divisible by all of the numbers in +the set. The prime factorizations of the numbers can be used to identify the least common multiple. +• +There are various ways that the GCD and LCM are applied. +3.2 The Integers +• +A set of numbers that can be built from the natural numbers are the integers, which consist of the natural numbers, +zero (0), and the negatives of the natural numbers. +• +Integers are often graphed on a number line, which helps display the relative positions and values of those +numbers. +• +The number line can be used to visualize when one integer is larger than or smaller than another integer. +• +Arithmetic operations with integers are similar to the operations with natural numbers, except that the sign +(positive or negative) of the numbers will determine the sign (positive or negative) of the result. +3.3 Order of Operations +• +Establishing shared rules on which arithmetic operations are calculated first is necessary. Without them, different +people may find different values for the same expression. +• +The highest precedence is with expressions in parentheses. This allows parts of an expression to be calculated in an +order different than the basic order of operations. +• +The lowest precedence is addition and subtraction, as they are the basis for all other calculations. +• +Multiplication and division have precedence over addition and subtraction, as they are representations of repeated +addition or subtraction. +• +Exponents have precedence over multiplication and division, as they represent repeated multiplication and division. +3.4 Rational Numbers +• +Rational numbers are fractions of integers, and can always be written as an integer divided by an integer. +• +The numerator and denominator of a fraction may have common factors. In such cases, the fraction can be reduced +by canceling common factors. When the numerator and denominator of a fraction have no common factors, the +fraction is said to be reduced. +• +An improper fraction is one with a numerator larger than the denominator. Such a fraction can be rewritten as an +integer plus a proper fraction. This is called a mixed number. +• +Using division and remainder, an improper fraction may be written as a mixed number. +• +A mixed number can be converted to an improper fraction by reversing the process for changing an improper +fraction to a mixed number. +3 • Chapter Summary +255 + +• +The arithmetic operations or addition, subtraction, multiplication and division can all be performed on rational +numbers. +• +Addition and subtraction of rational numbers can be performed after a common denominator has been identified, +and the fractions have been converted to forms having the common denominator. +• +Multiplication and division of rational numbers can be performed without regard to common denominators. +• +Between any two rational numbers, there is always another rational number. This is the density property of the +rational numbers. +3.5 Irrational Numbers +• +Irrational numbers are numbers that cannot be written as an integer divided by another integer. One example is pi, +denoted +. Another collection of irrational numbers are natural numbers that are not perfect squares. +• +Some irrational numbers can be written as a rational part multiplied by an irrational part. If two irrational numbers +have the same irrational parts, they can be added or subtracted. +• +When irrational numbers are similar, on can multiply and divide the numbers without a calculator. +• +Since +, and +, products and quotients of square roots can be determined. +• +Because +and +, it is possible to simplify square root expressions so the radicand +contains no perfect square factors. +• +When a fraction has an irrational number as its denominator, it is possible to convert the denominator into a +rational number using its conjugate. Doing so involves multiplying the numerator and denominator by the +conjugate of the denominator, and then applying the difference of squares formula. +• +With a single square root term +• +Using conjugate numbers for two term denominators +3.6 Real Numbers +• +Real numbers is the collection of all rational and irrational numbers. Conceptually, it is the collection of all values +that can be represented on a number line, or, as a length along with sign. +• +The subsets of the real numbers include the natural numbers, integers, rational numbers and irrational numbers. +The natural numbers are a subset of the integers, which is a subset of the rational numbers. The rational and +irrational numbers are disjoint sets. +• +The real numbers, due to order of operation rules and that performing arithmetic operations on real number always +results in a real number, have arithmetic properties that apply in all cases. There include the distributive property, +the commutative property, and the associative property. Also, every real number has an additive inverse and, except +for zero (0), have a multiplicative inverse. +3.7 Clock Arithmetic +• +Clock arithmetic uses the idea that after 12 o’clock comes 1 o’clock. For clock arithmetic, this means that every time +12 is passed in an arithmetic process, the next number is 1, not 13. +• +To determine the clock result of an arithmetic operation, divide the final result by 12 and keep the remainder. If the +remainder is 0, then the time is 12 o’clock. +• +Clock arithmetic is technically called modulo 12 arithmetic. To perform modulo 12 arithmetic, calculate the +expression, then divide the result by 12. The modulo 12 result is the remainder. +• +Days, in our system, pass in groups of seven. To calculate in day arithmetic, modulo 7 is used. To perform modulo 7 +arithmetic, calculate the expression, then divide the result by 7. The modulo 7 result is the remainder. +3.8 Exponents +• +Exponents are used to express multiplying a number by itself a number of times. The number being multiplied by +itself is the base. The number of times it is multiplied by itself is the exponent, which is often referred to as the +power. +• +Understanding that exponents represent repeated multiplication of a base makes it possible to establish some rules +for combining exponential expressions, using the product rule, the quotient rule, and the power rule. Additionally, it +allows us to formulate distributive rules for exponents. +• +Any non-zero number raised to the 0th power is 1. This makes the definition of the 0th power consistent with the +division rule for exponents. +• +For consistency, negative exponents represent the reciprocal of the base raised to the power, so that +, +provided that +. +256 +3 • Chapter Summary +Access for free at openstax.org + +3.9 Scientific Notation +• +Some numbers are so large or so small that writing the number out is clumsy and make it difficult to determine the +true size of the number. Scientific notation makes the number more readable and make the relative size of the +number immediately apparent. +• +A number written in scientific notation is a number at least 1 and smaller than 10 multiplied by 10 raised to an +exponent. Converting between scientific notation and standard notation involves correctly applying multiplication +and division by powers of 10, which in practice equates to understanding how moving the decimal point of a +number impacts the exponent of 10. +• +Adding and subtracting numbers in base 10 requires the exponent of 10 in each number be the same. Once the +numbers are converted to have the same exponent with the ten, then the numbers are added or subtracted as +indicated, with the power of 10 remaining the same. If the result is not in scientific notation (for instance, the +number has exceeded 10), then then number must be converted into scientific notation. +• +Multiplying and dividing numbers in scientific notation is done by multiplying or dividing the number parts, then +multiplying or dividing the 10 raised to the power parts, then multiplying those two results. If the new number is not +in scientific notation, then the result must be converted into scientific notation. +3.10 Arithmetic Sequences +• +A sequence is a list of numbers. Any individual number in that list, or sequence, is a term of the sequence. A specific +term of a sequence is denoted by the sequence symbol with a subscript indicating where the term in the sequence +is. +• +A special form of a sequence is an arithmetic sequence. Each arithmetic sequence is determined by its first term and +its constant difference. Any term in an arithmetic sequence is determined by adding the constant difference to the +preceding term. +• +If the first term and the constant difference of an arithmetic sequence are known, then any term of the sequence +can be found directly. +• +Because arithmetic sequences follow such a strict pattern, the sum of the first +terms of an arithmetic sequence +can be determined with the formula +. +3.11 Geometric Sequences +• +A special form of a sequence is a geometric sequence. Each geometric sequence is determined by its first term and +its constant ratio. Any term in a geometric sequence is determined by multiplying the constant ratio to the +preceding term. +• +If the first term and the constant ratio of a geometric sequence are known, then any term of the sequence can be +found directly. +• +Because geometric sequences follow such a strict pattern, the sum of the first +terms of a geometric sequence can +be determined with the formula +. +• +Finding the sum of a finite geometric sequence +• +Applying arithmetic sequences +3.11 Geometric Sequences +• +Geometric sequence. +• +Finding an arbitrary term in a geometric sequence. +• +Constant ratio. +• +Finding the sum of a finite geometric sequence. +• +Applying arithmetic sequences. +Videos +3.1 Prime and Composite Numbers +• +Divisibility Rules (https://openstax.org/r/Divisibility_Rules) +• +Illegal Prime Number (https://openstax.org/r/Illegal_Prime_Number) +• +Using a Factor Tree to Find the Prime Factorization (https://openstax.org/r/ +Using_a_Factor_Tree_to_Find_the_Prime_Factorization) +• +Finding the Prime Factorization of 168 (https://openstax.org/r/Finding_the_Prime_Factorization_of_168) +• +Using Desmos to find the GCD (https://openstax.org/r/Using_Desmos_to_find_the_GCD) +• +Applying the GCD (https://openstax.org/r/Applying_the_GCD) +3 • Chapter Summary +257 + +• +Finding the LCM (https://openstax.org/r/Finding_the_LCM) +• +Using Desmos to find the LCM (https://openstax.org/r/Using_Desmos_to_find_the_LCM) +• +Application of LCM (https://openstax.org/r/Application_of_LCM) +3.2 The Integers +• +Graphing Integers on the Number Line (https://openstax.org/r/Graphing_Integers_on_the_Number_Line) +• +Comparing Integers Using the Number Line (https://openstax.org/r/Comparing_Integers_Using_the_Number_Line) +• +Evaluating the Absolute Value of an Integer (https://openstax.org/r/Evaluating_the_Absolute_Value_of-an-Integer) +3.3 Order of Operations +• +Order of Operations 1 (https://openstax.org/r/Order_of_Operations_1) +• +Order of Operations 2 (https://openstax.org/r/Order_of_Operations_2) +• +Order of Operations 3 (https://openstax.org/r/Order_of_Operations_3) +• +Order of Operations 4 (https://openstax.org/r/Order_of_Operations_4) +3.4 Rational Numbers +• +Introduction to Fractions (https://openstax.org/r/Introduction_to_Fractions) +• +Equivalent Fractions (https://openstax.org/r/Equivalent_Fractions) +• +Reducing Fractions to Lowest Terms (https://openstax.org/r/Reducing_Fractions_to_Lowest_Terms) +• +Using Desmos to Reduce a Fraction (https://openstax.org/r/Using_Desmos_to_Reduce_a_Fraction) +• +Adding and Subtracting Fractions with Different Denominators (https://openstax.org/r/ +Adding_and_Subtracting_Fractions) +• +Converting an Improper Fraction to a Mixed Number Using Desmos (https://openstax.org/r/ +Improper_Fraction_to_Mixed_Number) +• +Multiplying Fractions (https://openstax.org/r/Multiplying_Fractions) +• +Dividing Fractions (https://openstax.org/r/Dividing_Fractions) +• +Order of Operations Using Fractions (https://openstax.org/r/Operations_Using_Fractions) +• +Finding a Fraction of a Total (https://openstax.org/r/Finding_Fraction_of_Total) +• +Converting Units (https://openstax.org/r/Converting_Units) +3.5 Irrational Numbers +• +The Philosophy of the Pythagoreans (https://openstax.org/r/Philosophy_of_Pythagoreans) +• +Using Desmos to Find the Square Root of a Number (https://openstax.org/r/square_root_of_a_number) +• +Simplifying Square Roots (https://openstax.org/r/Simplifying_Square_Roots) +• +Rationalizing the Denominator (https://openstax.org/r/Rationalizing_Denominator) +3.6 Real Numbers +• +Properties of the Real Numbers 1 (https://openstax.org/r/Properties_of_the_Real_Numbers_1) +• +Properties of the Real Numbers 2 (https://openstax.org/r/Properties_of_the_Real_Numbers_2) +• +Properties of the Real Numbers 3 (https://openstax.org/r/Properties_of_the_Real_Numbers_3) +3.6 Real Numbers +• +Arthur Benjamin TED talk, Faster than a Calculator (https://openstax.org/r/ +Arthur_Benjamin_TED_talk,_Faster_than_a_Calculator) +• +Identifying Sets of Real Numbers (https://openstax.org/r/Sets_of_Real_Numbers) +• +Properties of the Real Numbers #1 (https://openstax.org/r/Properties_of_the_Real_Numbers_1) +• +Properties of the Real Numbers #2 (https://openstax.org/r/Properties_of_the_Real_Numbers_2) +• +Properties of the Real Numbers #3 (https://openstax.org/r/Properties_of_the_Real_Numbers_3) +3.7 Clock Arithmetic +• +Determining the Day of the Week for Any Date in History (https://openstax.org/r/ +Determining_the_Day_of_the_Week_for_Any_Date_in_History) +• +Clock Arithmetic (https://openstax.org/r/Clock_Arithmetic) +3.8 Exponents +• +Exponential Notation (https://openstax.org/r/Exponential_Notation) +• +Product and Quotient Rule for Exponents (https://openstax.org/r/Product_and_Quotient_Rule_for_Exponents) +• +Fraction Raised to a Power (https://openstax.org/r/Fraction_Raised_to_a_Power) +258 +3 • Chapter Summary +Access for free at openstax.org + +• +Simplifying Expressions with Exponents (https://openstax.org/r/Simplifying_Expressions_with_Exponents) +3.9 Scientific Notation +• +Converting from Standard Form to Scientific Notation Form (https://openstax.org/r/ +Converting_from_Standard_Form_to_Scientific_Notation_Form) +• +Converting from Scientific Notation Form to Standard Form (https://openstax.org/r/ +Converting_from_Scientific_Notation_Form_to_Standard_Form) +• +Multiplying Numbers in Scientific Notation (https://openstax.org/r/Multiplying_Numbers_in_Scientific_Notation) +• +Dividing Numbers in Scientific Notation (https://openstax.org/r/Dividing_Numbers_in_Scientific_Notation) +• +Application of Scientific Notation (https://openstax.org/r/Application_of_Scientific_Notation) +3.10 Arithmetic Sequences +• +Arithmetic Sequences (https://openstax.org/r/Arithmetic_Sequences) +• +Finding the First Term and Constant Difference for an Arithmetic Sequence (https://openstax.org/r/ +Finding_the_First_Term_and_Constant_Difference_for_an_Arithmetic_Sequence) +• +Finding the Sum of a Finite Arithmetic Sequence (https://openstax.org/r/ +Finding_the_Sum_of_a_Finite_Arithmetic_Sequence) +• +Fibonacci Sequence and “Lateralus” (https://openstax.org/r/Fibonacci_Sequence_and_Lateralus) +3.11 Geometric Sequences +• +Geometric Sequences (https://openstax.org/r/Geometric_Sequences) +• +Sum of a Finite Geometric Sequence (https://openstax.org/r/Sum_of_a_Finite_Geometric_Sequence) +Formula Review +3.4 Rational Numbers +3.5 Irrational Numbers +3.6 Real Numbers +3.8 Exponents +, provided that +3 • Chapter Summary +259 + +, provided that +3.10 Arithmetic Sequences +3.11 Geometric Sequences +3.11 Geometric Sequences +Projects +Encryption began at least as far back as the Roman Empire. During the reign of Caesar, a particular cypher was used, +fittingly named the Caesar Cypher. This encryption process granted the Romans a great tactical advantage. Even if a +message was intercepted, it would not make sense to the person intercepting the message. +Find four instances when encryption was used and cracked over the course of history. +The Golden Ratio in Art and Architecture +The golden ratio has been used in art and architecture as far back as ancient Greece (possibly further). It also appears in +South America (Incan architecture). Find five instances of the use of the golden ratio in art or architecture and describe +its use in each of those instances. +Your Budget +Budgeting either is, or will shortly be, an important aspect of your life. Managing money well reduces stress in your life, +and provides space for planning for future expenses, such as vacations or home improvements. +Imagine your life 10 years from now. Estimate your monthly income. Identify expenses you will encounter monthly +(mortgage or rent, car payment, insurance, entertainment, etc.). Decide on an amount you plan to save monthly (this is +treated as an expense). Create a spreadsheet with those values. Record your monthly net income (your income minus +your expenses). Determine how much money you will have saved over the course of 5 years (ignore interest). Write a +reflection on your anticipated financial health. +Estimating Pi +The value of pi is the ratio of the circumference of a circle to the diameter of the circle. It is also equal to the ratio of the +area of the circle to the square of the radius of the square. +Research three ways to physically estimate pi. +Estimate pi using all three processes you found. +Present your process and solutions in class. +Design Your Own Shift Cypher +A cypher is a message written in such a way as to mask its contents. Changing a message into its cypher form is called +encryption. Decryption or deciphering is the process of changing a cyphertext message back into the original (legible) +message. One process of encryption is to scramble the letters, symbols, and punctuation of a message according to a +mathematical rule. One rule that could be used for such a cypher is addition in a chosen modulus. In this project, you will +create such a cypher, encrypt a message, and then decrypt the message. +260 +3 • Chapter Summary +Access for free at openstax.org + +Step 1: Choose the letters, symbols, and punctuation marks you want to allow in your messages. This should include at +least the uppercase letters and a space character. This is your character set. +Step 2: Count the number of characters you will use. Label this number +. +Step 3: Pair each character of your character set an integer from 0 to +. Do not assign more than one character to +an integer. +Step 4: Choose an integer between 1 and +. This will be the number used to create the cypher. Label this number . +Step 5: Write a message using your character set. +Step 6: Replace every character in your message by the integer with which it was paired in Step 3. +Step 7: For every number, +, from Step 7, perform the addition +(mod +). +Step 8: Replace every number found in Step 7 with the character with which it was paired in Step 3. This is your +cyphertext. +To decrypt your cyphertext, reverse the steps above. +Step 1: Replace the cyphertext characters with the paired values. +Step 2: For each value +, perform the subtraction +(mod +). +Step 3: Replace the numbers from Step 2 with their paired characters from the character set. +The message is then deciphered. +Design Your Own Cypher Using Multiplication +A cypher is a message written in such a way as to mask its contents. Changing a message into its cypher form is called +encryption. Decryption or deciphering is the process of changing a cyphertext message back into the original (legible) +message. One process of encryption is to scramble the letters, symbols, and punctuation of a message according to a +mathematical rule. One rule that could be used for such a cypher is multiplication in a chosen modulus. In this project, +you will create such a cypher, encrypt a message, then decrypt the message. +Step 1: Choose the letters, symbols, and punctuation marks you want to allow in your messages. This should include at +least the uppercase letters and a space character. This is your character set. +Step 2: Count the number of characters you will use. Label this number +. +Step 3: Pair each character of your character set an integer from 0 to +. Do not assign more than one character to +an integer. +Step 4: Choose an integer, labeled , between 1 and +so that +. This will be the number used to create +the cypher. +Step 5: Write a message using your character set. +Step 6: Replace every character in your message by the integer with which it was paired in Step 3. +Step 7: For every number, +, from Step 6, perform the multiplication +(mod +). +Step 8: Replace every number found in Step 7 with the character with which it was paired in Step 3. This is your +cyphertext. +Before beginning to decrypt in this cypher, you need to know the multiplicative inverse of the value you chose as s. +Step 1: The multiplicative inverse of +is the number that, when multiplied by +in your modulus, equals 1. To find this, +you will have to multiply +and every number between 2 and ( +) until the product is 1 (mod +). Once this number is +found, the message can be decrypted. Call this number . +Step 2: To decrypt your cyphertext, replace the cyphertext characters with the paired values. +Step 3: For each of the value, +, perform the multiplication +. +Step 4: Replace the numbers from Step 3 with their paired characters from the character set. +The message is then deciphered. +3 • Chapter Summary +261 + +Chapter Review +Prime and Composite Numbers +1. Identify which of the following numbers are prime, composite, or neither: +201, 34, 17, 1, 37. +2. Find the prime factorization of 500. +3. Find the greatest common divisor of 80 and 340. +4. Find the greatest common divisor of 30, 40, and 70. +5. Find the least common multiple of 45 and 60. +6. Bella and JJ volunteer at the zoo. Bella volunteers every 8 days, while JJ volunteers every 14 days. How many days +pass between days they volunteer together? +The Integers +7. Identify all the integers in the following list: +. +8. Plot the following on the same number line: +. +9. What two numbers have absolute value of 18? +10. Calculate +. +11. Calculate +. +12. Calculate +. +13. Six students rent a house together. The total monthly rent (including heat and electricity) is $3,120. If they all pay +an equal amount, how much does each student pay? +Order of Operations +14. Calculate +. +15. Calculate +. +16. +17. +18. +Rational Numbers +19. Reduce +to lowest terms. +20. Convert +to a mixed number and reduce to lowest terms. +21. Convert +to decimal form. +22. Convert +to decimal form. +23. Calculate +and reduce to lowest terms. +24. Compute +and reduce to lowest terms. +25. Determine 30% of 400. +26. 18 is what percent of 40? +27. In Professor Finnegan’s Science Fiction course, there are 60 students. Of those, 15% say they’ve read A Hitchhiker’s +Guide to the Galaxy. How many of the students have read that book? +Irrational Numbers +28. Simplify the square root by expressing it in lowest terms: +. +29. Calculate +without a calculator. If not possible, explain why. +262 +3 • Chapter Summary +Access for free at openstax.org + +30. Calculate +without a calculator. If not possible, explain why. +31. Rationalize the denominator of +, and simplify the fraction. +32. Find the conjugate of +and find the product of +and its conjugate. +33. Rationalize the denominator of +and simplify the fraction. +Real Numbers +34. Identify the numbers of the following list as a natural number, an integer, a rational number, or a real number: +, −0.43, 18, −43, +. +35. Identify the property of real numbers illustrated here: +. +36. Identify the property of real numbers illustrated here: +. +37. Identify the property of real numbers illustrated here: +. +38. Identify the property of real numbers illustrated here: +. +39. Use mental math to calculate +. +Clock Arithmetic +40. Determine 74 modulo 9. +41. Use clock arithmetic to calculate +. +42. Use clock arithmetic to calculate +. +43. It is Wednesday. What day of the week will it be in 44 days? +44. It is 4:00. What time will it be in 100 hours? +45. Security guards with Acuriguard submit a report on campus activity every 4 days. If they make a report on a +Monday, what day of the week will it be after 10 more reports? +Exponents +46. Use exponent rules to simplify +. +47. Use exponent rules to simplify +. +48. Use exponent rules to simplify +. +49. Use exponent rules to simplify +. +50. Use exponent rules to simplify +. +51. Use exponent rules to simplify +. +52. Rewrite +without a denominator. +53. Rewrite +without negative exponents. +Scientific Notation +54. Convert +to scientific notation. +55. Convert +to standard notation. +56. Convert +to scientific notation. +57. Calculate +. +58. Calculate +. +59. The Sextans Dwarf Spheroidal Galaxy has diameter 8,400 light years (ly). Express this in Scientific notation. +60. The Reticulum II Galaxy has diameter +light years (ly), while the Andromeda Galaxy has a diameter of +3 • Chapter Summary +263 + +. How many times bigger is the Andromeda Galaxy compared to the Reticulum II Galaxy? +Arithmetic Sequences +61. Determine the common difference of the following sequence: {19, 13, 7, 1, -5, …}. +62. Find the 25th term, +, of the arithmetic sequence with +and +. +63. Find the first term and the common difference of the arithmetic sequence with 8th term +and 15th term +. +64. Find the sum of the first 30 terms, +, for the arithmetic sequence with first term +and common +difference +. +65. Jem makes a stack of 5 pennies. Each day, Jem adds three pennies to the stack. How many pennies are in the stack +after 10 days? +Geometric Sequences +66. Determine the common ratio of the following geometric sequence: {6, 18, 54, 162, …}. +67. Find the 6th term, +, of the geometric sequence with +and +. +68. Find the sum of the first 12 terms, +, for the geometric sequence with first term +and common ratio +. +69. Carolann and Tyler deposit $8,500 in an account bearing 5.5% interest compounded yearly. If they do not deposit +any more money in that account, how much will be in the account after 15 years? +70. The total number of ebooks sold in 2013 was 242 million ( +). Each year, the number of ebooks sold has +declined by 3% ( +). How many ebooks were sold between 2013 and 2022? +Chapter Test +1. Find the prime factorization of 300. +2. Tiles will be used to cover an area that is 650 cm × 1,200 cm. What is the largest size square tile that can be used so +that all the tiles used are full tiles? +3. Calculate the following: +. +4. What does PEMDAS stand for? +5. David’s tax return is for $1,560. He decides to spend 20% of that return. How much does David spend? +6. Convert +into a mixed number. +7. Convert +to a fraction of integers. +8. Simplify the following square root: +. +9. What is the conjugate of +? +10. Rationalize the denominator of +. +11. What two sets of numbers comprise the real numbers? +12. Which property of the real numbers is shown: +? +13. Georita believes it will take 30 hours for her and her family to drive to their vacation. If they leave at 2:00, what +time should they arrive (ignore AM/PM)? +14. Suppose a bacterium in the gut has a generation time (time to divide) of 16 hours. If it first divides at 4:00, what +time will it be when they divide the 80th time afterward? +15. Simplify +. +16. The moon is +m from Earth. A dollar bill has a thickness of +m. If dollar bills could be +stacked perfectly, how many would it take to reach the moon? +17. A single long table can seat 8 people, 3 on each side and 1 on each end. If a second table is added to the first, end +264 +3 • Chapter Summary +Access for free at openstax.org + +to end, then 14 people can sit at the table, 6 per side and 1 at each end. Adding another table adds another 6 +people. How many people can sit at a table made by placing 10 of these tables end to end? +18. The first row of a theater seats 25 people. Each following row seats 2 more people. If there are 80 rows in the +theater, how many people, total, can sit in the theater? +19. Alice deposits $2,500 in a bond yielding 6% interest compounded annually. How much is the bond worth in 20 +years? +20. What is the 15th term of a geometric sequence with first term 5 and common ratio 3? +3 • Chapter Summary +265 + +266 +3 • Chapter Summary +Access for free at openstax.org + +Figure 4.1 Different cultures developed different ways to record quantity. (credit: modification of work "Tally sticks from +the Swiss Alps" by Sandstein, Swiss Alpine Museum permanent collection/Wikimedia Commons, CC BY 3.0) +Chapter Outline +4.1 Hindu-Arabic Positional System +4.2 Early Numeration Systems +4.3 Converting with Base Systems +4.4 Addition and Subtraction in Base Systems +4.5 Multiplication and Division in Base Systems +Introduction +Right now, almost all cultures use the familiar Hindu-Arabic numbering system, which uses the digits 0, 1, 2, 3, 4, 5, 6, 7, +8, and 9 along with place values based on powers of ten. This is a relatively recent development. The system didn’t +develop until the 6th or 7th century C.E. and took some time to spread across the world, which means other cultures at +other times had to develop their own methods of counting and recording quantity. Being different cultures and different +times means there were significant differences in counting systems. Cultures needed to count and measure time for +agriculture and for religious observations. It was needed for trade. Some languages had words only for one, two, and +many. Other cultures developed more complex ways to represent quantity, with the Oksapmin people of New Guinea +using an astonishing 27 words for their system. +Representing these quantities in a recorded form likely began with a simple marking system, where one scratch on a +stick or bone represented one of whatever was being counted. We still see this today with tally marks. These systems use +repeated symbols to represent more than one. We also have systems where different symbols represent different +quantities but still use some repetition, such as in Roman numerals. +Other systems were devised that rely on place values, like the Hindu-Arabic system in use today. Place value systems +needed a zero, though, and weren't immediately recognized and took time to develop. And within these positional +systems there is variation. Some systems counted in twenties, others in tens, and some in a mix (adding another reason +to visit Hawaii). Even now, though we all use and think using tens, computers are designed to work in groups of two, +which requires a different perspective on numbering. +In this chapter, we explore different numbering systems and grouping systems, eventually discussing base 2, the +language of computers. +4 +NUMBER REPRESENTATION AND +CALCULATION +4 • Introduction +267 + +4.1 Hindu-Arabic Positional System +Figure 4.2 This manuscript is an early example of Hindu numerals. (credit: modification of work “Bakshali manuscript”, +Bodleian Libraries/ University of Oxford, public domain) +Learning Objectives +After completing this section, you should be able to: +1. +Evaluate an exponential expression. +2. +Convert a Hindu-Arabic numeral to expanded form. +3. +Convert a number in expanded form to a Hindu-Arabic numeral. +The modern system of counting and computing isn’t necessarily natural. That different symbols are used to indicate +different quantities or amounts is a relatively new invention. Simple marking by scratches or dots, one for each item +being counted, was the norm long into human history. The modern system doesn’t use repeated symbols to indicate +more than one of a thing. It uses the place of a digit in a numeral to determine what that digit represents. A numeral is a +symbol used to represent a number. A number is an abstract idea that represents quantity or amount. +Being clear about the difference between numeral and number is important. Just like a person can be called by various +names, such as brother, father, husband, uncle, they are all representing the same person, John Smith. The person John +Smith is the number, and the names brother, father, husband, and uncle are the numerals. +WHO KNEW? +Hindu-Arabic Numerals +The numerals we currently use are referred to as Hindu-Arabic numerals, although they have changed as time has +passed. Early forms of the numerals for 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 began in India, and passed through Persia to the +Middle East. Place value was also employed in the early systems of India. Once this system was in north Africa and the +Middle East, it spread to Europe, eventually replacing Roman numerals. Over time, the original symbols transformed +into our modern ones. Read this article for another perspective on how the symbols began (based on the moon!) +(https://openstax.org/r/myindiamyglory.com/2018/09/05). +The system we use for counting and computing uses place values based on powers of 10. In this section, we review +exponents and our positional system. +Evaluating Exponential Expressions +Most modern numerical systems depend on place values, where the quantity represented depends not only on the digit, +but also on where the digit is in the number. The place value is a power of some specific number, which means most +numbering systems are actually exponential expressions. An exponential expression is any mathematical expression +that includes exponents. So, evaluating such an expression means performing the calculation. In this chapter, we will be +using exponents that are positive integer values. Before we do so, let’s remind ourselves about exponents and what they +represent. Suppose you want to multiply a number. Let’s label that number +, by itself some number of times. Let’s label +the number of times +. We denote that as +. We say +, or the base, raised to the +th power, or the exponent. For +example, if we are multiplying 13 by itself eight times, we write +and say 13 to the eighth power. +When computing exponential expressions, we should be careful to remember the order of operations. Using the order of +operation rules, calculations inside the parentheses are done first, then exponents are calculated, then multiplication +268 +4 • Number Representation and Calculation +Access for free at openstax.org + +and division calculations are performed, and then addition and subtraction. +VIDEO +Exponential Notation (https://openstax.org/r/Exponential_Notation) +EXAMPLE 4.1 +Evaluating an Exponential Expression +Evaluate the following exponential expressions. +1. +2. +3. +Solution +1. +To evaluate, or calculate, this expression, we use order of operations, which means the exponents are done first, +then multiplications, and then additions. +2. +To evaluate the expression, we use the order of operations, which means the exponents are done first, then the +multiplications, then the additions. Remember that any base raised to the exponent 0 is 1. +3. +To evaluate the expression, we use the order of operations, which means the exponents are done first, then the +multiplications, and then the additions. Remember that any base raised to the exponent 0 is 1. +YOUR TURN 4.1 +Evaluate the following exponential expressions. +1. +2. +3. +Converting Hindu-Arabic Numerals to Expanded Form +When you see the number 738, and you speak the number out loud, what do you say? You probably said “seven hundred +thirty-eight” while wondering what point could possibly be made by asking this. What you didn’t say was “seven, and +three, and eight.” A pre-K student might say that. Which should make you wonder, why? +The reason is that you’ve been taught place values, or the positions of digits in a number that determine the values of +those digits. You know that in a three-digit number, the first digit is hundreds, the second digit is tens, and the last digit +is ones. These place values rely on powers of 10, which makes this system a base 10 system. +This sense of place value is what makes our system of numbers so useful. You’ve also been taught the Hindu-Arabic +numeration system. This system, which uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, and also employs place value based +on powers of 10, is in use today. +Writing a number using these place values is writing them in expanded form. For a number with +digits, the expanded +form is the first digit times 10 raised to one less than +, plus each following digit times 10 raised to one less than the +previous power of 10. For example, the number 738 would be written as +. +What about a four-digit number, like 5,825? Out loud, we’d say five thousand, seven hundred twenty-five. In expanded +form, it would be +. Notice that the largest exponent is one less than the number +of digits, and that the exponents go down by one as we move through the number. +4.1 • Hindu-Arabic Positional System +269 + +PEOPLE IN MATHEMATICS +Aryabhata of Kusumapura and Brahmagupta +The Hindu-Arabic numeral system developed in India, and Aryabhata of Kusumapura is credited with the place value +notation in the 5th century. However, the system wasn’t as complete as it could be, until. roughly a century later, when +Brahmagupta introduced the symbol for 0. The 0 is necessary to indicate that a given place value has been skipped, +as in 4,098. In 4,098, the 102 power is skipped. Without such a symbol, 4,098 and 498 look similar. The value of both +the place value notation and the introduction of the symbol 0 cannot be overstated, for math and the sciences. +EXAMPLE 4.2 +Writing a Number in Expanded Form +Write the following in expanded form. +1. +563 +2. +4,821 +3. +903,786 +Solution +1. +Step 1: Since there are three digits in 563, +is 3. So, this is the first digit times 10 raised to the power of 2, so we +start with +. +Step 2: Then we add the next digit, 6, multiplied by 10 to a power one less than the previous, at which point we have +. +Step 3: Finally, the last digit is multiplied by 10 to the zeroth power and added to the previous. This results in +. +2. +Step 1: Since there are four digits in 4,821, +is 4. We multiply the first digit, 4, by 10 raised to the power of 3, which +is +. +Step 2: Then we add the next digit, 8, multiplied by 10 to a power one less than the previous, at which point we have +. +Step 3: We continue to the next digit, lowering the exponent of 10 by one. Now we have +. +Step 4: Finally, the last digit is multiplied by 10 to the zeroth power and added to the previous. This results in +. +3. +Since there are six digits in 903,786, +is 6. So, we begin the process with 9 times 10 raised to the 5th power and +continue through the numbers, reducing the exponent of 10 by one each time. This results in +. +YOUR TURN 4.2 +Write the following in expanded form. +1. 924 +2. 1,279 +3. 4,130,045 +Converting Numbers in Expanded Form to Hindu-Arabic Numerals +Converting from expanded form back into a Hindu-Arabic numeral is the reverse process of expanding a number, and is +equivalent to evaluating the exponential expression. +270 +4 • Number Representation and Calculation +Access for free at openstax.org + +EXAMPLE 4.3 +Converting a Number from Expanded Form to a Hindu-Arabic Numeral +Convert the following into Hindu-Arabic numerals. +1. +2. +3. +Solution +1. +Evaluating the expression results in: +2. +Evaluating the expression results in: +3. +Evaluating the expression results in: +YOUR TURN 4.3 +Convert the following to Hindu-Arabic Numerals. +1. +2. +3. +Check Your Understanding +1. What is meant by a place value system? +2. Evaluate the following exponential expression: +. +3. Express the following number in expanded form: 45,209. +4. What number provides the value of a digit in our system of numeration? +5. How are numerals and numbers different? +6. Express as a Hindu-Arabic number: +. +SECTION 4.1 EXERCISES +1. What does it mean for a system to be a place value system? +2. In the system we use today, what number are the place values based on? +3. How are numerals and numbers different? +4. What relates numerals to numbers? +For the following exercises, evaluate the exponential expression. +5. +6. +7. +8. +9. +10. +11. +12. +13. +4.1 • Hindu-Arabic Positional System +271 + +14. +For the following exercises, express the Hindu-Arabic number in expanded form. +15. 13 +16. 25 +17. 82 +18. 99 +19. 131 +20. 408 +21. 651 +22. 3,901 +23. 5,098 +24. 12,430 +For the following exercises, express the expanded number as a Hindu-Arabic number. +25. +26. +27. +28. +29. +30. +31. +32. +33. +34. +272 +4 • Number Representation and Calculation +Access for free at openstax.org + +4.2 Early Numeration Systems +Figure 4.3 Babylonians used clay tablets for writing and record keeping. (credit: modification of work by Osama Shukir +Muhammed Amin FRCP(Glasg), CC BY 4.0 International) +Learning Objectives +After completing this section, you should be able to: +1. +Understand and convert Babylonian numerals to Hindu-Arabic numerals. +2. +Understand and convert Mayan numerals to Hindu-Arabic numerals. +3. +Understand and convert between Roman numerals and Hindu-Arabic numerals. +Each culture throughout history had to develop its own method of counting and recording quantity. The system used in +Australia would necessarily differ from the system developed in Babylon that would, in turn, differ from the system +developed in sub-Saharan Africa. These differences arose due to cultural differences. In nearly all societies, knowing the +difference between one and two would be useful. But it might not be useful to know the difference between 145 and +167, as those quantities never had a practical use. For example, a shepherd likely didn't manage more than 100 sheep, +so quantities larger than 100 might never have been encountered. This can even be seen in our use of the term few, +which is an inexact quantity that most would agree means more than two. However, as societies became more complex, +as commerce arose, as military bodies developed, so did the need for a system to handle large numbers. No matter the +system, the issues of representing multiple values and how many symbols to use had to be addressed. In this section, we +explore how the Babylonians, Mayans, and Romans addressed these issues. +Understand and Convert Babylonian Numerals to Hindu-Arabic Numerals +The Babylonians used a mix of an additive system of numbers and a positional system of numbers. An additive +system is a number system where the value of repeated instances of a symbol is added the number of times the symbol +appears. A positional system is a system of numbers that multiplies a “digit” by a number raised to a power, based on the +position of the “digit.” +The Babylonian place values didn’t use powers of 10, but instead powers of 60. They didn’t use 60 different symbols +though. For the value 1, they used the following symbol: +For values up to 9, that symbol would be repeated, so three would be written as +4.2 • Early Numeration Systems +273 + +To represent the quantity 10, they used +For 20, 30, 40, and 50, they repeated the symbol for 10 however many times it was needed, so 40 would be written +When they reached 60, they moved to the next place value. The complete list of the Babylonian numerals up to 59 is in +Table 4.1. +Table 4.1 Babylonian Numerals +You can see how Babylonians repeated the symbols to indicate multiples of a value. The number 6 is 6 of the symbol for +1 grouped together. The symbol for 30 is three of the symbols for 10 grouped together. However, their system doesn’t go +past 59. To go past 59, they used place values. As opposed to the Hindu-Arabic system, which was based on powers of +10, the Babylonian positional system was based on powers of 60. You should also notice there is no symbol for 0, which +has some impact on the number system. Since the Babylonian number system lacked a 0, they didn’t have a placeholder +when a power of 60 was absent. Without a 0, 101, 110, and 11 all look the same. However, there is some evidence that +the Babylonians left a small space between "digits" where we would use a 0, allowing them to represent the absence of +that place value. To summarize, the Babylonian system of numbers used repeating a symbol to indicate more than +one, used place values, and lacked a 0. +WHO KNEW? +Invention of 0 +The idea of 0 is not a natural one. Most cultures failed to recognize the need for a 0. If someone asked a farmer in 300 +B.C.E. how many cows they had, but they had none, they would not answer "zero." They’d say “I don’t have any” and +be done with it. It wasn’t until roughly 3 B.C.E. that 0 appeared in Mesopotamia. It was independently discovered (or +invented!) in the Mayan culture around 4 C.E. it made its appearance in India in the 400s C.E., and began to spread at +274 +4 • Number Representation and Calculation +Access for free at openstax.org + +that point. It wasn’t developed earlier mostly because positional systems were not yet fully developed. Once +positional systems arose, the need to represent a missing power had to be addressed. +So how do we convert from Babylonian numbers to Hindu-Arabic numbers? To do so, we need to use the symbols from +Table 4.1, and then place values based on powers of 60. If you have +digits in the Babylonian number, you multiply the +first “digit” by 60 raised to one less than the number of “digits.” You then continue through the “digits,” multiplying each +by 60 raised to a power that is one smaller. However, be careful of spaces, since they represent a zero in that place. +EXAMPLE 4.4 +Converting Two-Digit Babylonian Numbers to Hindu-Arabic Numbers +Convert the Babylonian number +into a Hindu-Arabic number. +Solution +has two digits: +and +Step 1: So the first symbol, +represents 4 in the Babylonian system. This is multiplied by 60 to the first power (just as would happen in a two digit +number), which gives us +. +Step 2: The next symbol is +which represents 27 in the Babylonian system. This is multiplied by 60 raised to 0, which gives +. +Step 3: Calculating that yields +. So the Babylonian number +equals 267 in the Hindu-Arabic number system. +YOUR TURN 4.4 +1. Convert the Babylonian number +into a Hindu-Arabic number. +4.2 • Early Numeration Systems +275 + +EXAMPLE 4.5 +Converting Three-Digit Babylonian Numbers to Hindu-Arabic Numbers +Convert the Babylonian number +into a Hindu-Arabic number. +Solution +has three digits: +and +and +Step 1: So the first symbol, +represents 13 in the Babylonian system. This is multiplied by 60 to the second power (since there are 3 digits), which +gives us +. +Step 2: The next symbol is +which represents 8 in the Babylonian system, is multiplied by 60 raised to the first power, which gives us +. +Step 3: The last digit is +representing 54, which is multiplied by 60 raised to 0, which gives +. +Step 4: Calculating that yields +. +So, the Babylonian number +equals 47,334 in the Hindu-Arabic number system. +276 +4 • Number Representation and Calculation +Access for free at openstax.org + +YOUR TURN 4.5 +1. Convert the Babylonian number +into a Hindu-Arabic number. +EXAMPLE 4.6 +Converting Four-Digit Babylonian Numbers to Hindu-Arabic Numbers +Convert the Babylonian number +into a Hindu-Arabic number. +Solution +It appears that +has three digits, but there is a space in between +and +Remember, the Babylonian system has no 0, it instead employs a space where we expect a zero. This means this is a four +digit number. +Step 1: The first symbol, +represents 12 in the Babylonian system. This is multiplied by 60 to the third power since there are four digits, which gives +us +. +Step 2: The next symbol is a blank, which for us is a 0, representing +, giving us +. +Step 3: The next symbol is +which represents 42 in the Babylonian system, is multiplied by 60 raised to the first power, which gives us +. +Step 4: The last Babylonian digit, +represents 39 in the Babylonian system. This is multiplied by 60 raised to 0, which gives +. +Step 5: Calculating that yields +4.2 • Early Numeration Systems +277 + +So the Babylonian number +equals 2,594,559 in the Hindu-Arabic number system. +YOUR TURN 4.6 +1. Convert the Babylonian number +into a Hindu-Arabic number. +WHO KNEW? +The Legacy of Babylonian System +The Babylonian system can still be seen today. An hour is 60 minutes, and a minute is 60 seconds. Additionally, when +measuring angles in degrees, each degree can be split into 60 minutes (1/60th of a degree) and 60 seconds (1/60th of +a minute). +VIDEO +Converting Between Babylonian and Hindu-Arabic Numbers (https://openstax.org/r/Babylonian_to_Hindu- +Arabic_Numbers) +278 +4 • Number Representation and Calculation +Access for free at openstax.org + +Understand and Convert Mayan Numerals to Hindu-Arabic Numerals +The Mayans employed a positional system just as we do and the Babylonians did, but they based their position values on +powers of 20 and they had a dedicated symbol for zero. Similar to the Babylonians, the Mayans would repeat symbols to +indicate certain values. A single dot was a 1, two dots were a 2, up to four dots. Then a five was a horizontal bar. The +horizontal bars could be used three times, since the fourth horizontal bar would make a 20, which was a new position in +the number. The 0 was a special picture, which appears like a turtle lying on its back. The shell would then be "empty," so +maybe that’s why the symbol was 0. The complete list is provided in Table 4.2. Another feature of Mayan numbers was +that they were written vertically. The powers of 20 increased from bottom to top. +Table 4.2 Mayan Numerals +To summarize, the Mayan system of numbers used repeating symbol to indicate more than one, used place values, and +employed a 0. So how do we convert from Mayan numbers to Hindu-Arabic numbers? To do so, we need to use the +symbols from Table 4.2 and then place values based on powers of 20. If you have +digits in the Mayan number, you +multiply the first “digit” by 20 raised to one less than the number of “digits.” You then continue through the “digits,” +multiplying each by 20 raised to a power that is one smaller than the previous power. Fortunately, there is an explicit 0, +so there is no ambiguity about numbers like 110, 101, and 11. +EXAMPLE 4.7 +Converting Two-Digit Mayan Numbers to Hindu-Arabic Numbers +Convert the Mayan number +into a Hindu-Arabic number. +Solution +4.2 • Early Numeration Systems +279 + +has two digits: +and +Step 1: So, the first symbol, +represents 15 in the Mayan system. This is multiplied by 20 to the first power, which gives us +. +Step 2: The next symbol is +which represents 9 in the Mayan system. This is multiplied by 20 raised to 0, which gives +. +Step 3: Calculating that yields +. So +equals 309 in the Hindu-Arabic number system. +YOUR TURN 4.7 +1. Convert the Mayan number into a Hindu-Arabic number. +EXAMPLE 4.8 +Converting Three-Digit Mayan Numbers to Hindu-Arabic Numbers +Convert the Mayan number +into a Hindu-Arabic number. +Solution +has three digits: +280 +4 • Number Representation and Calculation +Access for free at openstax.org + +and +and +Step 1: So the first symbol, +represents 6 in the Mayan system. This is multiplied by 20 to the second power (since there are 3 digits), which gives us +. +Step 2: The next symbol is +which represents 8 in the Mayan system, is multiplied by 20 raised to the first power, which gives us +. +Step 3: The last digit is +representing 4, which is multiplied by 20 raised to 0, which gives +. +Step 4: Calculating that yields +. So +the Mayan number +equals 2,564 in the Hindu-Arabic number system. +YOUR TURN 4.8 +1. Convert the Mayan number into a Hindu-Arabic number. +EXAMPLE 4.9 +Converting Four-Digit Mayan Numbers to Hindu-Arabic Numbers +Convert the Mayan number +4.2 • Early Numeration Systems +281 + +into a Hindu-Arabic number. +Solution +has four digits, so the first power of 20 that is used is 3. +Step 1: The first symbol, +represents 8 in the Mayan system. This is multiplied by 20 to the third power (since there are four digits), which gives us +. +Step 2: The next symbol is +which is a 0, representing +, giving us +. +Step 3: The next symbol is +which represents 16 in the Mayan system, is multiplied by 20 raised to the first power, which gives us +. +Step 4: The last Mayan digit, +represents 5 in the Mayan system. This is multiplied by 20 raised to 0, which gives +. +Step 5: Calculating that yields +So the Mayan number +282 +4 • Number Representation and Calculation +Access for free at openstax.org + +equals 64,325 in the Hindu-Arabic number system. +YOUR TURN 4.9 +1. Convert the Mayan number into a Hindu-Arabic number. +WHO KNEW? +The Mayan Calendar +The Mayans used this base 20 system for everyday situations. But their culturally important, and extremely accurate, +calendar system used a slightly different system. For their calendars, they used a system where the place values were +1, 20, then 20*18, then 20*18*18. The reason for this is 20*18 is 360, which is closer to the number of days in a year. +Had they used a purely base 20 system for their calendar, they’d be very far off with 400 days in a year. +Three hundred sixty days still left the Mayans a bit short, as there are 365 days in a year (ignoring leap years). The +Mayan calendar also included 5 days, called Wayeb days, which brings their calendar to 365 days. As it happens, +Wayeb is the Mayan god of misfortune, so these 5 days were considered the bad luck days. +VIDEO +Converting Mayan Numbers to Hindu-Arabic Numbers (https://openstax.org /r/Mayan_to_Hindu-Arabic_Numbers) +Understand and Convert Between Roman Numerals and Hindu-Arabic Numerals +The Mayan and Babylonian systems shared two features, one of which we are familiar with (place value) and one that we +don’t use (repeated symbols). The Roman system of numbers used repeated symbols, but does not employ a place +value. It also lacks a 0. The Roman system is built on the following symbols in Table 4.3. +Roman Numeral +Hindu-Arabic Value +I +1 +V +5 +X +10 +L +50 +Table 4.3 Roman Numerals +4.2 • Early Numeration Systems +283 + +Roman Numeral +Hindu-Arabic Value +C +100 +D +500 +M +1,000 +Table 4.3 Roman Numerals +As in the Mayan and Babylonian systems, a symbol may be repeated to indicate a larger value. However, at 4, they did +not use IIII. They instead used IV. Since the I came before the V, the number stands for “one before five.” A similar +process was used for 9, which was written IX, or “one before ten.” The value 40 was written XL, or “ten before fifty,” while +49 was written XLIX, or “forty plus nine.” +The following are the rules for writing and reading Roman numerals. +• +The representations for bigger values precede those for smaller values. +• +Up to three symbols may be grouped together; for example, III for 3, or XXX for 30, or CC for 200. +• +A larger value followed by a smaller value indicated addition; for example, VII for 7, XIII for 13, LV for 55, and MCC +for 1200. +• +I can be placed before V to indicate 4, or before X, to indicate 9. These are the only ways I is used as a subtraction. +• +X can be placed before L to indicate 40, and before C to indicate 90. These are the only ways X is used as a +subtraction. +• +C can be placed before D to indicate 400, and before M to indicate 900. These are the only ways C is used as a +subtraction. +• +If multiple symbols are used, and a subtraction involving that symbol, the subtraction part comes after the multiple +symbols. For example, XXIX for 29 and CCXC for 290. +WHO KNEW? +Legacy of Roman Numerals +The Roman numbering system is still used today in some situations. Many cornerstones of buildings have the year +written in Roman numerals. Movie titles often represent the year the movie was produced as Roman numerals. The +most recognizable might be that the Super Bowl is numbered using Roman numerals. +EXAMPLE 4.10 +Converting Roman Numerals to Hindu-Arabic Numbers +Convert the following Roman numerals into Hindu-Arabic numerals. +1. +XXVII +2. +XXXIV +3. +MMCMXLVIII +Solution +1. +The numeral XXVII begins with two X’s, which is then followed by a V. So, the two X’s combine to be 20. The V is +followed by two I’s, so the V indicates the addition of 5. The two I’s that follow indicate addition of two. That ends the +symbols, so the value is 20 plus 5 plus 2, or 27 in Hindu-Arabic numerals. +2. +The numeral XXXIV begins with three X’s, which is then followed by an I. So, the three X’s combine to be 30. The I is +followed by a V, which indicates 4. That ends the symbols, so the value is 30 plus 4, or 34 in Hindu-Arabic numerals. +3. +The numeral MMCMXLVIII begins with two M’s, which is then followed by a C. So, the two M’s combine to make +2000. The C is followed by an M, which indicates 900. The CM is followed by XL, which indicates 40. The L is followed +by V, which indicates 5. The V is followed by three I’s, indicating 3. Adding those values yields 2,948. +284 +4 • Number Representation and Calculation +Access for free at openstax.org + +YOUR TURN 4.10 +Convert the following Roman numerals into Hindu-Arabic numerals. +1. LXXVII +2. CCXL +3. MMMCDXLVII +VIDEO +Converting From Roman Numbers to Hindu-Arabic Numbers (https://openstax.org/r/Roman_to_Hindu- +Arabic_Numbers) +Of course, we can convert from Hindu-Arabic numerals, to Roman numerals, too. +EXAMPLE 4.11 +Converting Hindu-Arabic Numbers to Roman Numerals +Convert the following Hindu-Arabic numerals into Roman numerals. +1. +38 +2. +94 +3. +846 +4. +2,987 +Solution +1. +Thirty is represented as three X’s, and the 8 is represented with VIII, so 38 in Roman numerals is XXXVIII. +2. +Ninety is represented by XC, and four is represented by IV, so 94 in Roman numerals is XCIV. +3. +The number is less than 900 and more than 500, so the first symbol to be used is D, which is 500. To get to 800, we +need 300 more, which is represented with three C’s. Forty is represented with XL, and the six. The Roman numerals +are DCCCXLVI. +4. +The two thousand is represented by two M’s. The 900 is represented by CM. The 80 is represented by LXXX (50 plus +30). Finally, the 7 is represented by VII. We have that 2,987 in Roman numerals is MMCMLXXXVII. +YOUR TURN 4.11 +Convert the following Hindu-Arabic numerals into Roman numerals. +1. 27 +2. 49 +3. 739 +4. 3,647 +VIDEO +Converting From Hindu-Arabic Numbers to Roman Numbers (https://openstax.org/r/Hindu- +Arabic_to_Roman_Numbers) +Check Your Understanding +7. What is the place value for Babylonian numerals? +8. What place value is used in the Mayan numeration system? +9. What place value is used for Roman numerals? +10. Convert the Babylonian numeral +into a Hindu-Arabic numeral. +4.2 • Early Numeration Systems +285 + +11. Convert the Mayan numeral into a Hindu-Arabic numeral. +12. Convert the Roman numeral CCXLVII into a Hindu-Arabic numeral. +13. Convert 479 into a Roman numeral. +SECTION 4.2 EXERCISES +For the following exercises, convert the Babylonian numeral into a Hindu-Arabic numeral. +1. +2. +3. +4. +5. +6. +7. +8. +For the following exercises, express the Mayan numeral as a Hindu-Arabic numeral. Use the common system, which is +based on powers of 20 only. +9. +10. +11. +12. +13. +14. +15. +16. +286 +4 • Number Representation and Calculation +Access for free at openstax.org + +For the following exercises, express the Roman numeral as a Hindu-Arabic numeral. +17. VII +18. XI +19. IX +20. XXIV +21. MCXLII +22. CXXII +23. DCCXLIV +24. MCMLIX +For the following exercises, express the Hindu-Arabic numeral as a Roman numeral. +25. 8 +26. 14 +27. 27 +28. 94 +29. 274 +30. 487 +31. 936 +32. 2,481 +33. What uses a place value system for numbers: Roman, Babylonian, Egyptian, Greek? +34. What uses an additive system: Roman, Mayan, Egyptian, Greek? +35. What uses a 0: Roman, Mayan, Egyptian, Greek? +4.3 Converting with Base Systems +Figure 4.4 Computers use Base 2, which only uses 0's and 1's, to represent quantity. (credit: modification of work +“IMAG0933” by yvanhou/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Convert another base to base 10. +2. +Write numbers in different base systems. +3. +Convert base 10 to other bases. +4. +Determine errors in converting between bases. +In our system of numbers, we use base 10, but using base 10 was not a given within other systems. There were other +systems that used bases other than 10, as we saw with the Mayans and the Babylonians. The base 10 system comes +down to grouping objects in sets of 10, but grouping in sets of 10 only happens if the culture values grouping by that +4.3 • Converting with Base Systems +287 + +many. We feel 10 is natural because we have 10 fingers. There are other systems using other grouping values, such as 4 +or 20. +One good reason for examining other bases is to remind ourselves how we had to learn arithmetic when we were +young, memorizing rules for our base 10 system. We had to learn why those arithmetic rules made sense, such as why +and +. Another good reason for learning other base systems is due to computers; their circuitry +instead uses base 2. +In this section, we explore other base systems and how to convert between them. +Conversion of Another Base into Base 10 and Other Bases +We saw in Hindu-Arabic Positional System that our Hindu-Arabic system uses base 10, which is a system using place +values of digits that depend on powers of 10 (or, are based on powers of 10). We’ve already worked with bases other than +base 10: The Babylonian system was base 60, while the Mayan system was base 20. +To explore how our base 10 system is used, answer the following question: What’s the following quantity: 4,572? You +probably said four thousand five hundred seventy-two (no, there is no “and” between hundred and seventy). But why do +you think that 4 means four thousand? A very young person when learning their numbers might say that’s a four five +seven and two. But you added the context of thousands to the four. Why? +Place value, that’s why. You learned early on that where the numeral was gave it different meanings. Ten thousands, +thousands, hundreds, tens, and ones. So, you translate that symbol string (4,572) into “four thousand five hundred +seventy-two.” As we saw in Hindu-Arabic Positional System, expanding a Hindu-Arabic number involved writing the +number using each digit times its appropriate power of 10. So, we could write 4,572 as +. +One possible reason we use base 10 is that we have 10 fingers, and in the cultures where the Hindu-Arabic system +developed, that became the standard. Other cultures may have used other ways of organizing numbers, perhaps using +20 by including toes, or using 60 because 60 has many divisors. Mathematically though, base 10 is an awkward base to +work in since 10 has limited divisors. But we think it is easy and simple because that’s what we’ve been taught to use. +Using a base 10 system means we need 10 symbols to make our numbering system work: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. +Now imagine that we all only had 6 fingers instead of 10 and our counting system was based on those 6 fingers. We +would be counting in groups of 6, not groups of 10. How would this change how we work with quantity? +First, we’d need only six symbols. Let’s use 0, 1, 2, 3, 4, 5. Second, our place values would be based on powers of 6, not +powers of 10. For instance, the number 3,024 in base 6 would be +. That is how you can +translate a base 6 number into a base 10 number. When we calculate that expression we get +. +This means the base 6 number 3,024 is equal to the base 10 number 664. +From now on, if we are using a base 6 number, we will follow it with the subscript 6, like the following: 3,0246 means the +number is in base 6. +A base 10 number gets no subscript (it’s the standard). So, 3,024 is a base 10 number. A base 13 number would be +4,67213. +So, a base 6 system uses only the symbols 0, 1, 2, 3, 4, and 5. Also, the place values use powers of 6. However, we still +don’t know how to count in base 6. In order to do so, we’d have to know how to represent the quantities larger than five +in base 6. Let’s review how our base 10 system works by counting from 0 to 100, which shows how larger values are +represented. +288 +4 • Number Representation and Calculation +Access for free at openstax.org + +In writing the base 10 numbers, you start with these first 10 values: +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +But you’ve run out of symbols. So, we use two digits: +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +The 1 out front means you’ve run out of digits one time. +But now you’ve run out twice. Continuing with those numbers gives: +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +And so on, +30 +31 +32 +33 +34 +etc.… +Eventually, you hit the 90s, +90 +91 +92 +93 +94 +95 +96 +97 +98 +99 +And you’ve run out of the digits again! So, we say we’ve run out of digits in the tens place one time, hence: +100 +101 +102 +103 +104 +105 +106 +107 +108 +109 +That’s the pattern we use in base 10. We write out the symbols until we’ve used all the symbols, then add a digit in front +that counts how many times we’ve used the digits. Knowing the numbers, or being able to count higher and higher, is +necessary to understand how all the arithmetic works, as it all goes back to counting. +The counting pattern is the same for any other base, including base 6. So, let’s start: +0 +1 +2 +3 +4 +5 +But we’ve run out of symbols! Just like in base 10, we use a second digit, where the first digit will tell us we’ve run out of +symbols one time. +10 +11 +12 +13 +14 +15 +And we use the same pattern: +20 +21 +22 +23 +24 +25 +30 +31 +32 +33 +34 +35 +40 +41 +42 +43 +44 +45 +50 +51 +52 +53 +54 +55 +4.3 • Converting with Base Systems +289 + +But we’ve run out of symbols for that front digit. So, we indicate it the same way as in base 10…by adding a third digit in +front, indicating we’ve run out of symbols once in the second place: +100 +101 +102 +103 +104 +105 +110 +111 +etc. +The symbol pattern is the same, but truncated. We only use the six symbols. So that is how we represent base 6. Being +able to write out these numbers is important when working with addition in the base. +When using a base larger than 10, though, we need more symbols. Instead of creating new symbols, we use capital +letters, with A representing the digit for "10," B representing the digit for "11," and so on. +EXAMPLE 4.12 +Determining Digits of a Base with Less Than 10 Digits +What are the digits used for base 7? +Solution +Since this is base 7, we need only 7 symbols: 0, 1, 2, 3, 4, 5, 6. +YOUR TURN 4.12 +1. What are the digits for base 4? +EXAMPLE 4.13 +Determining Digits of a Base with More Than 10 Digits +What are the digits used for base 14? +Solution +Since this is base 14, we need 14 symbols. We don’t have single character numbers for 10, 11, 12, and 13, so, in a fit of +inspired creativity, we use capital letters A, B, C, D to represent those quantities. So, the digits in base 14 are 0, 1, 2, 3, 4, +5, 6, 7, 8, 9, A, B, C, D. +YOUR TURN 4.13 +1. What are the digits used for base 12? +WHO KNEW? +Using Base 12 +As mentioned in the text, working in base 10 is mathematically awkward. Ten has only two natural number divisors: 2 +and 5. This means dividing into groups is not easy. However, 12, or a dozen, has more divisors: 2, 3, 4, and 6. The +Dozenal Society recognizes this more mathematically pleasant detail. It advocates for a switch to using base 12 for +numbers. Their argument is based on the divisibility of the number 12. But has there ever been a society that used +such a system? The answer is yes. A dialect of the Gwandara language in Nigeria uses the base 12 system. It is +unlikely, though, that the Dozenal Society will achieve their goal, as the base 10 system is so entrenched in our +society. +290 +4 • Number Representation and Calculation +Access for free at openstax.org + +EXAMPLE 4.14 +Converting from One Base into Another +Convert 3,6017 into base 10. +Solution +In base 7, the place values are powers of 7. Since there are four digits, the highest power of 7 that is used is 3. This yields +. +VIDEO +Convert Base 7 to Base 10 (https://openstax.org/r/Convert_Base_7_to_Base_10) +YOUR TURN 4.14 +1. Convert 4216 into base 10. +EXAMPLE 4.15 +Converting from Base 14 to Base 10 +Convert 4B714 into base 10. +Solution +In base 14, the place values are powers of 14. Since there are three digits, the highest power of 14 is 2. Also recall that in +base 14, 10 is represented by A, 11 is represented by B, 12 is represented by C, and 13 is represented by D. Using that, we +convert to base 10: +. +YOUR TURN 4.15 +1. Convert A3C14 into base 10. +EXAMPLE 4.16 +Converting from Base 12 to Base 10 +Convert A1612 into base 10. +Solution +In base 12, the place values are powers of 12. Since there are three digits, the highest power of 12 is 2. Also recall that in +base 12, 10 is represented by A and 11 is represented by B. Using that, we convert to base 10: +. +YOUR TURN 4.16 +1. Convert 5AB12 into base 10. +4.3 • Converting with Base Systems +291 + +EXAMPLE 4.17 +Converting from Base 2 to Base 10 +Convert 10112 into base 10. +Solution +In base 2, the place values are powers of 2. Since there are four digits, the highest power of 2 is 3. Using that, we convert +to base 10: +. +YOUR TURN 4.17 +1. Convert 110112 into base 10. +WHO KNEW? +Before Napoleon +Before Napoleon’s France, which adopted the base 10 system, a modified base 12 system was often used in Europe. +Twelve is easily divisible into groups of 2, 3, 4, and 6, which makes it easier to work with. Even our numbering system +retains a bit of this. You have likely noticed that we use the words thirteen, fourteen, fifteen, and so on to indicate 10 +and 3, 10 and 4, 10 and 5, and so one. Even the 20s reinforce this idea, as in twenty-one, and twenty-two. However, +two numbers don’t follow this pattern, namely 11 and 12. If they followed the same rules, they’d be one teen and two +teen. We even have a special word for 12; that is, a dozen. However, etymologically speaking, the words eleven and +twelve are likely derived by referencing the number 10. These two numbers may date back to the Old English words +endleofan and twelf, which can be traced back further to ain lif and twa lif. The word lif here may be the base word for +“to leave.” This would suggest ain lif is one left after 10, and twa lif is two left after 10, or, 11 and 12. +EXAMPLE 4.18 +Writing Numbers in Base Systems Other Than Base 10 +Write the numbers in base 7 up to 1007. +Solution +Step 1: Using the patterns we indicated earlier, we begin with the first seven digits. +0, 1, 2, 3, 4, 5, 6 +Step 2: Since we’ve run out of digits, we start with 10, indicating we’ve run out of symbols once. +10, 11, 12, 13, 14, 15, 16 +Step 3: Continuing in the same way, we get: +20, 21, 22, 23, 24, 25, 26 +30, 31, 32, 33, 34, 35, 36 +40, 41, 42, 43, 44, 45, 46 +50, 51, 52, 53, 54, 55, 56 +60, 61, 62, 63, 64, 65, 66 +Now, all the digits have been used in the leading digits. Since the digits have all been used in that leading digit, we use +100, as in base 10. +100 +292 +4 • Number Representation and Calculation +Access for free at openstax.org + +YOUR TURN 4.18 +1. Write the numbers of base 4 up to 1004. +EXAMPLE 4.19 +Writing Numbers in Bases with More Than 10 Symbols +Write the numbers in base 14 up to 10014. +Solution +Step 1: Using the patterns we indicated earlier, we begin with the first 14 digits. +0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D +Step 2: Since we’ve run out of digits, we start with 10, indicating we’ve run out of symbols once. +10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D +Step 3: Continuing in the same way, we get: +20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B, 2C, 2D +30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 3A, 3B, 3C, 3D +40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 4A, 4B, 4C, 4D +50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 5A, 5B, 5C, 5D +60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 6A, 6B, 6C, 6D +70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 7A, 7B, 7C, 7D +80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 8A, 8B, 8C, 8D +90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 9A, 9B, 9C, 9D +A0, A1, A2, A3, A4, A5, A6, A7, A8, A9, AA, AB, AC, AD +B0, B1, B2, B3, B4, B5, B6, B7, B8, B9, BA, BB, BC, BD +C0, C1, C2, C3, C4, C5, C6, C7, C8, C9, CA, CB, CC, CD +D0, D1, D2, D3, D4, D5, D6, D7, D8, D9, DA, DB, DC, DD +100 +YOUR TURN 4.19 +1. Write the numbers of base 12 up to 1004. +Base 2 is important in the digital age, as it is the system used by computers. It is the simplest base to work with, but has +the drawback that the numbers in base 2 may use many, many digits. In Addition and Subtraction in Base Systems and +Multiplication and Division in Base Systems, we will look at base 2 in each situation. +EXAMPLE 4.20 +Writing Numbers in Base 2 +Write the numbers in base 2 up to 1002. +Solution +Base 2 uses only two symbols: 0 and 1. Following the pattern established previously, the numbers in base 2 up to 1002 +are 0, 1, 10, 11, and 100. +4.3 • Converting with Base Systems +293 + +YOUR TURN 4.20 +1. Write the numbers in base 3 up to 1003. +WHO KNEW? +Early Hawaiian Numeration System +Before the British arrived in Hawaii, people there used a system that combined two different bases. Objects were +initially grouped into collections of four, and a collection of four was referred to as kauna. A person could have two +kauna and three “ones” (in Hindu-Arabic, 11). Or they could have eight kauna and one “ones” (In Hindu-Arabic, 33). +However, sets of four were grouped in collections of 10. A set of 10 kauna was ka’au. The collections of ka’au were +grouped by 10 also. Which meant that 10 ka’au (this is 40 in Hindu-Arabic) would be lau (or 400 in Hindu-Arabic). What +this shows is that the Hawaiian culture developed a system that used base 4 combined with base 10. +Conversion of Base 10 into Another Base +Converting from base 10 into another base uses repeated division, recording the remainder at each step. Then, the +number in the new base is the remainder starting from the last remainder found. To be accurate in what we’re saying, +we need to remind ourselves of some terminology associated with division. When integers are divided, the one being +divided is the dividend, and the one that is dividing the dividend is the divisor. The quotient is the largest natural +number that can be multiplied by the divisor where the product is less than the dividend. +When the integer +is divided by the integer +, +is called the dividend and +is the divisor. +To convert a base 10 number +into base +, we divide +by +, recording the remainder. Then we divide the quotient from +that step by the base +, and record the remainder again. We continue this process until the quotient is 0. Then, the base +number has digits that start with the last remainder and use each remainder in reverse order. +EXAMPLE 4.21 +Converting from Base 10 into a Lower Base +Convert 298 to base 6. +Solution +We divide 298 by 6, and record the remainder. Then we divide the quotient from that step by 6, and record the remainder +again. We continue this process until the quotient is 0. Then, the base 6 number has digits that start with the last +remainder and use each remainder in reverse order. +Step 1: When we divide 298 by 6, we get +. The quotient is 49 and the remainder is 4. +Step 2: Now we divide the quotient, 49, by 6. This gives +. The quotient is 8 and the remainder is 1. +Step 3: Repeating, we get +. The quotient is 1 and the remainder is 2. +Step 4: Finally, we perform the operation on the quotient 1, +giving us a quotient of 0 and a remainder of 1. +Step 5: The base 6 number has digits equal to the remainders in reverse order, 12146. So, 298 in base 10 when converted +to base 6 is 12146. +YOUR TURN 4.21 +1. Convert 693 to base 7. +294 +4 • Number Representation and Calculation +Access for free at openstax.org + +VIDEO +Converting from Base 10 to Another Base (https://openstax.org/r/Base_10_to_Another_Base) +EXAMPLE 4.22 +Converting from Base 10 into a Higher Base +Convert 45,134 to base 13. +Solution +We divide 45,134 by 13, and record the remainder. Then we divide the quotient from that step by 13, and record the +remainder again. We continue this process until the quotient is 0. Then, the base 13 number has digits that start with the +last remainder and use each remainder in reverse order. It is at this step that we’ll convert to the base 13 digits, which +are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C. +Step 1: When we divide 45,134 by 13, we get +. The quotient is 3,471 and the remainder is 11. +Step 2: Now we divide the quotient, 3,471, by 13. This gives +. The quotient is 267 and the remainder is 0. +Step 3: Repeating, we get +. The quotient is 20 and the remainder is 7. +Step 4: Again, and we get +. The quotient is 1 and the remainder is 7. +Step 5: Finally, we get +, with quotient 0 and a remainder 1. +Step 6: The base 13 number has digits equal to the remainders in reverse order, which were 1, 7, 7, 0, and 11. The 11 is +written as B in base 13. So, 45,134 in base 10 when converted to base 13 is 1770B13. +YOUR TURN 4.22 +1. Convert 9,275 to base 12. +EXAMPLE 4.23 +Converting from Base 10 into Base 2 +Convert 100 to base 2. +Solution +Following the pattern above: +Step 1: We divide 100 by 2, and record the remainder. +Step 2: Then we divide the quotient from that step by 2, and record the remainder again. +Step 3: We continue this process until the quotient is 0. +Step 4: Following this process, the remainders are, in order, 0, 0, 1, 0, 0, 1, 1. Writing those in reverse order gives the +number in base 2, 11001002. +Notice that 100 in base 2 took seven digits. +4.3 • Converting with Base Systems +295 + +YOUR TURN 4.23 +1. Convert 137 to base 2. +Converting from Hindu-Arabic Numbers to Mayan Numbers +To convert from a Hindu-Arabic number to a Mayan number involves two distinct processes. First, the number must be +converted to base 20, using the process described and demonstrated previously. Next, that base 20 number has to be +written using Mayan numerals. For reference, the Mayan numerals and their values are below. +EXAMPLE 4.24 +Converting from Base 10 into the Mayan System +Convert the following into Mayan numbers. +1. +51 +2. +653 +Solution +1. +The Mayan system is base 20, so we must use 20 in the process from above. The first division has a quotient of 2 +and remainder of 11. The 11 serves as the “ones” digit. Dividing that quotient, 2, by 20 has a quotient of 0 with a +remainder of 2. The 2 becomes the “twenties” digit of the number. So, in base 20, the number would be 2 followed +by 11. The Mayan symbols for 2 and 11 are +and +. Writing these vertically, with the “ones” digit on top, as +appropriate for Mayan numbers, results in: +2. +The Mayan system is base 20, so we must use 20 in the process from above. The first division, 673 divided by 20, has +a quotient of 32 and remainder of 13. Dividing that quotient, 32, by 20 has a quotient of 1 with a remainder of 12. +Dividing that quotient, 1, by 20 has a quotient of 0 and a remainder of 1. Since there are three remainders here, this +is a three-digit number. The 1 is the “20-squared” digit, the 12 is the “twenties” digit, and the 13 is the “ones” digit. +So, in base 20, the number would be 1 followed by 12 followed by 13.. The Mayan symbols for 1, 12 and 13 are +, +, and +. Writing these vertically, as appropriate for Mayan numbers, would result in: +296 +4 • Number Representation and Calculation +Access for free at openstax.org + +YOUR TURN 4.24 +Convert the following into Mayan numbers. +1. 137 +2. 2,171 +WHO KNEW? +Other Languages, Other Bases +There have been base systems that use bases other than 10. Some bases used were 20, 12, and 27! Visit this site to +see more on the languages that used other bases (https://openstax.org/r/number-systems_other_languages). +Errors in Converting Between Bases +There are some common errors that are made when converting between bases. Often, it comes down to using an +“illegal” symbol in the new base. +EXAMPLE 4.25 +Detecting an Illegal Symbol When Converting Between Bases +A base 10 number is converted to base 7 and the result was 20817. Was an error committed? How do you know? +Solution +The result has the digit 8 in it. In base 7, 8 is an illegal symbol. Based on that, an error was committed. +YOUR TURN 4.25 +1. A base 10 number is converted to base 4 and the result was 37024. Was an error committed? How do you know? +When converting from base 10 to another base, an illegal symbol will be used if a mistake was made in the division +process used to find the number in the new base. Since the digits are based on the remainders, any remainder that is an +illegal symbol would indicate an error. +EXAMPLE 4.26 +Detecting an Error in Division When Converting Between Bases +When changing from base 10 to base 8, the division process resulted in the following remainders: 1, 0, 9, 2, 4. Was an +error committed? How do you know? +Solution +The remainders include 9, which in base 8 is an illegal symbol. +YOUR TURN 4.26 +1. When changing from base 10 to base 6, the division process resulted in the following remainders: 5, 0, 0, 10. Was +an error committed? How do you know? +4.3 • Converting with Base Systems +297 + +Another possible way to detect an error in converting between bases is to count the number of digits. When converting +from a higher base to a lower base, the number of digits cannot get smaller. Similarly, when converting from a lower +base to a higher base, the number of digits cannot get bigger. So, if a base 10 number is converted to a base 3 number, +the number of digits in the new base 3 numbers cannot be less than the number of digits in the base 10 number. +Similarly, if a base 7 number is converted to base 10, the number of digits in the base 10 number cannot be more than +the number of digits in the original base 7 number. +EXAMPLE 4.27 +Detecting an Error in Number of Digits When Converting Between Bases +A five-digit base 10 number is converted to a base 5 number. The base 5 number has four digits. Was an error +committed? How do you know? +Solution +Since 10 is larger than 5, the base 5 number cannot have less digits than the base 10 number. Since it did, we know an +error has been made. +YOUR TURN 4.27 +1. A six-digit base 12 number is converted into a base 10 number. The base 10 number has five digits. Was an error +committed? How do you know? +Check Your Understanding +14. In base 25, how many symbols would be necessary? +15. In base 18, what would the place value of the 4 be in the number 34818 be? +16. Convert 23045 into base 10. +17. When counting in base 9, what number would follow 389? +18. Convert 329 into base 8. +19. Convert ABC14 into base 10. +20. How do you know a mistake was made when converting from base 10 to base 4 and the result is 1524? +SECTION 4.3 EXERCISES +1. What does it mean when we say a number is written in base 7? +2. What does it mean when we say a number is written in base 12? +3. How many symbols are there in a base 3 system? What are they? +4. How many symbols are there in a base 15 system? What are they? +5. List the numbers, up to 100, in the base 5 system. +6. List the numbers, up to 100, in the base 2 system. +For the following exercises, convert the number into a base 10 number. +7. 145 +8. 216 +9. 123 +10. 345 +11. 148 +12. 789 +13. 3B12 +14. 2416 +15. 1012 +16. 4A714 +298 +4 • Number Representation and Calculation +Access for free at openstax.org + +17. 8049 +18. 1010012 +19. 32236 +20. 14367 +21. 8A0BD15 +22. 1102023 +23. 100A412 +24. 10100001012 +For the following exercises, convert the base 10 number into the given base. +25. 12 into base 4 +26. 25 into base 2 +27. 43 into base 12 +28. 153 into base 5 +29. 203 into base 2 +30. 431 into base 4 +31. 543 into base 12 +32. 1,023 into base 2 +33. 2,876 into base 4 +34. 1,765 into base 5 +35. 1,993 into base 7 +36. 2,000 into base 2 +37. 4,368 into base 12 +38. 12,562 into base 16 (Hint: Base 16 uses the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.) +39. When converting from base 10 to base 6, a student finds the following remainders: 3, 1, 3, 6. How do you know +that a mistake was made? +40. When converting from base 10 to base 4, a student finds the following remainders: 0, 0, 3, 7, 2. How do you know a +mistake was made? +41. Suppose a base 12 number is converted into a base 10 number, and one of the digits is A. Was an error +committed? How do you know? +42. Suppose a base 10 number is converted into a base 5 number and one of the digits is 6. Was an error committed? +How do you know? +43. Suppose a base 2 number is converted into a base 10 number, and the base 10 number has more digits that the +base 2 number. Was an error committed? How do you know? +44. Suppose a base 16 number is converted to base 2, and the base 2 number has fewer digits than the base 16 +number. Was an error committed? How do you know? +For the following exercises, convert the Hindu-Arabic number into a Mayan number. +45. 25 +46. 71 +47. 400 +48. 723 +The Babylonian system used base 60. To convert from Hindu Arabic numbers into Babylonian numbers, the process for +converting from base 10 to a different base would be done first. Then, the results found in the conversion process would +be changed to Babylonian numerals. This process is similar to the one for Mayan numbers. +The Babylonian system used base 60. To convert from Hindu-Arabic numbers into Babylonian numbers, the process for +converting from base 10 to a different base would be done first. Then, the results found in the conversion process +would be changed to Babylonian numerals. This process is similar to the one for Mayan numbers. For the following +exercises, convert the Hindu-Arabic number to a Babylonian number. +49. 67 +50. 135 +51. 781 +52. 10,952 +4.3 • Converting with Base Systems +299 + +4.4 Addition and Subtraction in Base Systems +Figure 4.5 All information in computers is represented by 0's and 1's, including quantity, which means computers use +Base 2 for arithmetic. (credit: modification of work “Magnifying glass and binary code” by Marco Verch Professional +Photographer/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Add and subtract in bases 2–9 and 12. +2. +Identify errors in adding and subtracting in bases 2–9 and 12. +Once we decide on a system for counting, we need to establish rules for combining the numbers we’re using. This begins +with the rules for addition and subtraction. We are familiar with base 10 arithmetic, such as +or +. How +does that change if we instead use a different base? A larger base? A smaller one? In particular, computers use base 2 for +all number representation. When your calculator adds or subtracts, multiplies or divides, it uses base 2. This is because +the circuitry recognizes only two things, high current and low current, which means the system is uses only has two +symbols. Which is what base 2 is. +In this section, we use addition and subtraction in bases other than 10 by referencing the processes of base 10, but +applied to a new base system. +Addition in Bases Other Than Base 10 +Now that we understand what it means for numbers to be expressed in a base other than 10, we can look at arithmetic +using other bases, starting with addition. When you think back to when you first learned addition, it is very likely you +learned the addition table. Once you knew the addition table, you moved on to addition of numbers with more than one +digit. The same process holds for addition in other bases. We begin with an addition table, and then move on to adding +numbers with two or more digits. +We worked with base 6 earlier, and have the numbers in base 6 up to 1006. Using that table of values, we can create the +base 6 addition table. +Here’s the beginning of the base 6 addition table: ++ +0 +1 +2 +3 +4 +5 +0 +0 +1 +2 +3 +4 +5 +1 +1 +2 +3 +4 +5 +? +2 +2 +3 +4 +5 +? +? +3 +3 +4 +5 +? +? +? +300 +4 • Number Representation and Calculation +Access for free at openstax.org + +4 +4 +5 +? +? +? +? +5 +5 +? +? +? +? +? +Many of the cells are not filled out. The ones filled in are values that never get past 5, which is the largest legal symbol in +base 6, so they are acceptable symbols. But what do we do with 5 + 3 in base 6? We can’t represent the answer as “8” +since “8” is not a symbol available to us. Let’s go back to the list of numbers we have for base 6. +0 +1 +2 +3 +4 +5 +10 +11 +12 +13 +14 +15 +20 +21 +22 +23 +24 +25 +30 +31 +32 +33 +34 +35 +40 +41 +42 +43 +44 +45 +50 +51 +52 +53 +54 +55 +So, what is 5 + 1 equal to in base 6? Well, start at the 5, and jump ahead one step. You land on 10. +This means that, in base 6, 5 + 1 = 10. +So, what is 5 + 2 in base 6? Well, 5 + 2 = 5 + 1 + 1, so 10 + 1…jump one more space and you land on 11. So, 5 + 2 = 11 in +base 6. +4.4 • Addition and Subtraction in Base Systems +301 + +And so it goes. Using that process, stepping one more along the list, we can fill in the remainder of the base 6 addition +table (Table 4.4). ++ +0 +1 +2 +3 +4 +5 +0 +0 +1 +2 +3 +4 +5 +1 +1 +2 +3 +4 +5 +10 +2 +2 +3 +4 +5 +10 +11 +3 +3 +4 +5 +10 +11 +12 +4 +4 +5 +10 +11 +12 +13 +5 +5 +10 +11 +12 +13 +14 +Table 4.4 Base 6 Addition Table +With this table, and with our understanding of “carrying the one,” we can then use the addition table to do addition in +base 6 for numbers with two or more digits, using the same processes you learned for addition when you did it by hand. +EXAMPLE 4.28 +Adding in Base 6 +Calculate 2516 + 1336. +Solution +Step 1: Let’s set up the addition using columns. +2 +5 +1 ++ +1 +3 +3 +Step 2: Let’s do the one’s place first. According to the base 6 addition table (Table 4.4), 1 + 3 = 4. +2 +5 +1 ++ +1 +3 +3 +4 +302 +4 • Number Representation and Calculation +Access for free at openstax.org + +Step 3: Now, we do the “tens” place (it’s really the sixes place). According to the base 6 addition table (Table 4.4), we have +5 + 3 = 12. So, like in base 10, we use the 2 and carry the 1. +1 +2 +5 +1 ++ +1 +3 +3 +2 +4 +Step 4: Now the “hundreds” place (really, thirty-sixes place). There, we have 1 + 2 + 1 = 3 + 1 = 4. +1 +2 +5 +1 ++ +1 +3 +3 +4 +2 +4 +So, 2516 + 1336 = 4246. +As you can see, the process is the same as when you learned base 10 addition, just a different symbol set. +YOUR TURN 4.28 +1. Calculate 4536 + 3456. +EXAMPLE 4.29 +Creating an Addition Table for a Base Lower Than 10 +1. +Create the addition table for base 7. +2. +Create the addition table for base 2. +Solution +1. +We begin with the table below. ++ +0 +1 +2 +3 +4 +5 +6 +0 +0 +1 +2 +3 +4 +5 +6 +1 +1 +2 +3 +4 +5 +6 +2 +2 +3 +4 +5 +6 +3 +3 +4 +5 +6 +4 +4 +5 +6 +4.4 • Addition and Subtraction in Base Systems +303 + +5 +5 +6 +6 +6 +In base 7, the number that follows 6 is 10 (since we’ve run out of symbols!). So, 67 + 17 = 107. Once that is +established, 67 + 27 will be two numbers past 6, which is 11 in base 7. ++ +0 +1 +2 +3 +4 +5 +6 +0 +0 +1 +2 +3 +4 +5 +6 +1 +1 +2 +3 +4 +5 +6 +10 +2 +2 +3 +4 +5 +6 +11 +3 +3 +4 +5 +6 +4 +4 +5 +6 +5 +5 +6 +6 +6 +10 +11 +Continuing, we can fill in the rows as we would in base 10, but being aware that we are working in base 7 (Table 4.5). ++ +0 +1 +2 +3 +4 +5 +6 +0 +0 +1 +2 +3 +4 +5 +6 +1 +1 +2 +3 +4 +5 +6 +10 +2 +2 +3 +4 +5 +6 +10 +11 +3 +3 +4 +5 +6 +10 +11 +12 +4 +4 +5 +6 +10 +11 +12 +13 +5 +5 +6 +10 +11 +12 +13 +14 +6 +6 +10 +11 +12 +13 +14 +15 +Table 4.5 Base 7 Addition Table +2. +We revisit base 2 here. Begin with the table: +304 +4 • Number Representation and Calculation +Access for free at openstax.org + ++ +0 +1 +0 +0 +1 +1 +1 +Table 4.6 +Base 2 +Addition +Table +In base 2, the number that follows 1 is 10 (since we’ve run out of symbols!). So, 12 + 12 = 102. The complete table for +base two then is below. ++ +0 +1 +0 +0 +1 +1 +1 +10 +This demonstrates that the rules necessary for base 2 addition are as small as possible: four rules. +YOUR TURN 4.29 +1. Create the addition table for base 4. +To summarize the creation of the addition tables for a given base, do the following. +Step 1: Set up the table. +Step 2: Fill in all the additions that use the “legal” symbols for the base. The diagonal that goes from upper left to lower +right that is immediately next to the filled boxes all get the value 10, regardless of base. +Step 3: Enter the values that are in the “teens.” This can all be done on one table without creating multiple copies of +previously done work. +EXAMPLE 4.30 +Adding in Base 7 +Calculate 5367 + 4337. +Solution +Step 1: Let’s set up the addition using columns. +5 +3 +6 ++ +4 +3 +3 +Step 2: Let’s do the one’s place first. According to the base 7 addition table in the solution for Example 4.29, 6 + 3 = 12. +We will carry the 1. +4.4 • Addition and Subtraction in Base Systems +305 + +1 +5 +3 +6 ++ +4 +3 +3 +2 +Step 3: Now, we do the “tens” place (it’s really the sevens place). According to the base 7 addition table in the solution for +Example 4.29, we have 1 + 3 + 3 = 10. So, like in base 10, we use the 0 and carry the 1. +1 +5 +3 +6 ++ +4 +3 +3 +0 +2 +Step 4: Now the “hundreds” place (really, forty-ninths place). There, we have 1 + 5 + 4 = 6 + 4 = 13. +1 +5 +3 +6 ++ +4 +3 +3 +1 +3 +0 +2 +So, 5367 + 3337 = 13027. +YOUR TURN 4.30 +1. Calculate 4617 + 1427. +As seen previously, when performing addition in another base, set up the problem exactly as you would for addition in +base 10. At each step, check the addition table for the base. As in base 10 addition, move right to left, adding down the +columns using the rules in the addition table. When necessary and just as in base 10, be sure to carry the 1. +EXAMPLE 4.31 +Creating an Addition Table for a Base Higher Than 10 +Create the addition table for base 12. +Solution +Step 1: Recall, in base 12, the symbol set is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, and B. So, the addition table begins as shown +below. +306 +4 • Number Representation and Calculation +Access for free at openstax.org + ++ +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +1 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +2 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +3 +3 +4 +5 +6 +7 +8 +9 +A +B +4 +4 +5 +6 +7 +8 +9 +A +B +5 +5 +6 +7 +8 +9 +A +B +6 +6 +7 +8 +9 +A +B +7 +7 +8 +9 +A +B +8 +8 +9 +A +B +9 +9 +A +B +A +A +B +B +B +Step 2: The diagonal immediately to the right of the filled in boxes is where the 10 goes for this base. ++ +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +1 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +10 +2 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +10 +3 +3 +4 +5 +6 +7 +8 +9 +A +B +10 +4 +4 +5 +6 +7 +8 +9 +A +B +10 +5 +5 +6 +7 +8 +9 +A +B +10 +6 +6 +7 +8 +9 +A +B +10 +7 +7 +8 +9 +A +B +10 +8 +8 +9 +A +B +10 +9 +9 +A +B +10 +4.4 • Addition and Subtraction in Base Systems +307 + +A +A +B +10 +B +B +10 +Step 3: Using the pattern we’re familiar with, and counting in base 12, we can fill in the other cells. ++ +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +1 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +10 +2 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +10 +11 +3 +3 +4 +5 +6 +7 +8 +9 +A +B +10 +11 +12 +4 +4 +5 +6 +7 +8 +9 +A +B +10 +11 +12 +13 +5 +5 +6 +7 +8 +9 +A +B +10 +11 +12 +13 +14 +6 +6 +7 +8 +9 +A +B +10 +11 +12 +13 +14 +15 +7 +7 +8 +9 +A +B +10 +11 +12 +13 +14 +15 +16 +8 +8 +9 +A +B +10 +11 +12 +13 +14 +15 +16 +17 +9 +9 +A +B +10 +11 +12 +13 +14 +15 +16 +17 +18 +A +A +B +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +B +B +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +1A +Table 4.7 Base 12 addition table +Notice that the lower-right entry is 1A12, as this is the number one past 1912. +YOUR TURN 4.31 +1. Create the addition table for base 14. +EXAMPLE 4.32 +Adding in Base 12 +Calculate 3A712 + 9BA12. +Solution +Step 1: Using the process established in the earlier addition problem, set up the columns. +308 +4 • Number Representation and Calculation +Access for free at openstax.org + +3 +A +7 ++ +9 +B +A +Step 2: Using the rules from the base 12 addition table in the solution for Example 4.31, and being careful to carry the 1 +when necessary, we get the following: +1 +1 +3 +A +7 ++ +9 +B +A +1 +1 +A +5 +The ones that were carried are located over the columns. +So, 3A712 + 9BA12= 11A512. +YOUR TURN 4.32 +1. Calculate 4B312 + B0612. +EXAMPLE 4.33 +Adding in Base 2 +We again return to base 2, the base used by computers. Calculate 10012 + 110112. +Solution +Step 1: Using the process established in the earlier addition problem, set up the columns. +1 +0 +0 +1 ++ +1 +1 +0 +1 +1 +Step 2: Using the rules from the base 2 addition table in the solution for Example 4.29, and being careful to carry the 1 +when necessary (and shown at the top of the grid), we get the following: +1 +1 +1 +1 +0 +0 +1 ++ +1 +1 +0 +1 +1 +1 +0 +0 +1 +0 +0 +Step 3: Calculate 10012 + 110112 = 1001002. +So, 10012 + 110112 = 1001002. +4.4 • Addition and Subtraction in Base Systems +309 + +YOUR TURN 4.33 +1. Calculate 1011112 + 11000112. +Subtraction in Bases Other Than Base 10 +Subtraction in bases other than base 10 follow the same processes as base 10 subtraction, but, as with addition, using +the addition table for the base. +EXAMPLE 4.34 +Subtracting in Base 6 +Calculate 526 − 346. +Solution +Step 1: Let’s set up the subtraction using columns. +5 +2 +− +3 +4 +Step 2: Just as we might do in base 10, we borrow a 1 from the 5 for the ones digit. +12 +− +3 +4 +Step 3: Referring to the base 6 addition table (Table 4.4), we see that 4 + 4 = 12, so 126 − 46 is 46. +12 +− +3 +4 +4 +Step 4: Now we deal with the “tens” (really, sixes) digit, 46 − 36, which equals 16 according to the base 6 addition table +(Table 4.4). +4 +12 +− +3 +4 +1 +4 +So, 526 − 346 = 146. +YOUR TURN 4.34 +1. Calculate 1156 − 436. +310 +4 • Number Representation and Calculation +Access for free at openstax.org + +EXAMPLE 4.35 +Subtracting in Base 12 +Calculate A1712 − 4B312. +Solution +Step 1: Let’s set up the subtraction using columns. +A +1 +7 +− +4 +B +3 +Step 2: Even in base 12, 712 − 312 = 412. +A +1 +7 +− +4 +B +3 +4 +Step 3: Moving to the “tens” digit, we have 112 − B12. Since 1 is less than B in base 12, we need to borrow a 1 from the A, +just as we would for subtraction in base 10. +9 +11 +7 +− +4 +B +3 +4 +Step 4: According to the base 12 addition table in the solution for Example 4.31, B12 + 212 = 1112, so 1112 − B12 = 212. +9 +11 +7 +− +4 +B +3 +2 +4 +Step 5: Finally, we deal with the “hundreds” digit. According to the base 12 addition table in the solution for Example +4.31, 412 + 512 = 912, so 912 − 412 = 512. +9 +11 +7 +− +4 +B +3 +5 +2 +4 +So, A1712 − 4B312 = 52412. +4.4 • Addition and Subtraction in Base Systems +311 + +YOUR TURN 4.35 +1. Calculate 71612 − 4AB12. +Errors When Adding and Subtracting in Bases Other Than Base 10 +Errors when computing in bases other than 10 often involve applying base 10 rules or symbols to an arithmetic problem +in a base other than base 10. The first type of error is using a symbol that is not in the symbol set for the base. For +instance, if a 9 shows up when working in base 7, you know an error has happened because 9 is not a legal symbol in +base 7. +EXAMPLE 4.36 +Identifying an Illegal Symbol in Arithmetic in a Base Other Than Base 10 +Explain the error in the following calculation: +Solution +Since the problem is in base 6, the symbol set available is 0, 1, 2, 3, 4 and 5. The 9 in the answer is clearly not a legal +symbol for base 6. Looking back to the base 6 addition table (Table 4.4), we see that +. Correcting the error, +we see the sum is +. +YOUR TURN 4.36 +1. Explain the error in the following calculation and correct the problem: +The second type of error is using a base 10 rule when the numbers are not in base 10. For instance, if you are working in +base 13, then 913 + 913 is not 1813, even though 18 is the correct answer in base 10. +EXAMPLE 4.37 +Identifying an Arithmetic Error in a Base Other Than Base 10 +Explain the error in the following calculation, and correct the error: +Solution +If this problem was a base 10 problem, this would be the correct answer. However, in base 12, 9 + 6 is not 15, but is +instead 13. To correct this error, carefully use the addition table for base 12. If properly used, the correct answer would +be +, as seen below: +8 +9 ++ +7 +6 +1 +4 +3 +YOUR TURN 4.37 +1. Explain the error in the following calculation, and correct the error: +312 +4 • Number Representation and Calculation +Access for free at openstax.org + +Check Your Understanding +21. Determine the addition table for base 8. +22. Compute 246 + 536. +23. Compute 358 − 268. +24. Compute 3B14 + 4514. +25. Compute A412 − 9B12. +26. How do you know an error has occurred in a base 8 addition question if the answer obtained was 288? +27. What is one common error made in calculating in base 14? +SECTION 4.4 EXERCISES +For the following exercises, create the addition table for the given base. +1. base 5 +2. base 3 +3. base 16 (Hint: Use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.) +4. base 2 +For the following exercises, perform the indicated base 6 operation. +5. 46 + 36 +6. 146 + 256 +7. 316 + 36 +8. 436 + 346 +9. 5326 + 236 +10. 2546 + 1436 +11. 206 − 36 +12. 236 − 56 +For the following exercises, perform the indicated base 12 operation. +13. 512 + 612 +14. 312 + A12 +15. 3412 + 712 +16. 7612 + B12 +17. 5912 + 1A12 +18. A112 + 3612 +19. 5312 − 912 +20. 2B12 − 712 +21. Explain two ways to detect an error in arithmetic in bases other than base 10. +22. Explain the error in the following calculation: 2813 + 4713 = 7513. +23. Explain the error in the following calculation: 367 + 237 = 597. +24. In base 10 addition, there are 100 addition rules plus a rule for carrying a 1. How many addition rules are there for +base 6? +25. In base 10 addition, there are 100 addition rules plus a rule for carrying a 1. How many addition rules are there for +base 14? +26. In base 10 addition, there are 100 addition rules plus a rule for carrying a 1. How many addition rules are there for +base 2? +For the following exercises, use the addition table that you created from Exercise 4 to perform the indicated base 2 +operations. +27. 1012 + 1112 +28. 10112 + 100112 +29. 111112 + 111112 +30. 10101012 + 10101012 +4.4 • Addition and Subtraction in Base Systems +313 + +For the following exercises, use the addition table that you created from Exercise 3 to perform the indicated base 16 +operations. +31. 2916 + 3816 +32. 4D16 + 8916 +33. 92716 + 43816 +34. BFA16 − 78E16 +For the following exercises, tell how you know an error was committed without performing the operation in the given +base. +35. +36. +4.5 Multiplication and Division in Base Systems +Figure 4.6 The processes for multiplication and division are the same for arithmetic in any bases. (credit: modification of +work “NCTR Intern Claire Boyle” by Danny Tucker/U.S. Food and Drug Administration, Public Domain) +Learning Objectives +After completing this section, you should be able to: +1. +Multiply and divide in bases other than 10. +2. +Identify errors in multiplying and dividing in bases other than 10. +Just as in Addition and Subtraction in Base Systems, once we decide on a system for counting, we need to establish rules +for combining the numbers we’re using. This includes the rules for multiplication and division. We are familiar with those +operations in base 10. How do they change if we instead use a different base? A larger base? A smaller one? +In this section, we use multiplication and division in bases other than 10 by referencing the processes of base 10, but +applied to a new base system. +Multiplication in Bases Other Than 10 +Multiplication is a way of representing repeated additions, regardless of what base is being used. However, different +bases have different addition rules. In order to create the multiplication tables for a base other than 10, we need to rely +on addition and the addition table for the base. So let’s look at multiplication in base 6. +Multiplication still has the same meaning as it does in base 10, in that +is 4 added to itself six times, +. +So, let’s apply that to base 6. It should be clear that 0 multiplied by anything, regardless of base, will give 0, and that 1 +multiplied by anything, regardless of base, will be the value of “anything.” +Step 1: So, we start with the table below: +* +0 +1 +2 +3 +4 +5 +0 +0 +0 +0 +0 +0 +0 +314 +4 • Number Representation and Calculation +Access for free at openstax.org + +1 +0 +1 +2 +3 +4 +5 +2 +0 +2 +4 +3 +0 +3 +4 +0 +4 +5 +0 +5 +Step 2: Notice +is there. But we didn’t hit a problematic number there (4 works fine in both base 10 and base 6). +But what is +? If we use the repeated addition concept, +. According to the base 6 addition +table (Table 4.4), +. So, we add that to our table: +* +0 +1 +2 +3 +4 +5 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +2 +0 +2 +4 +10 +3 +0 +3 +10 +4 +0 +4 +5 +0 +5 +Step 3: Next, we need to fill in +. Using repeated addition, +(if we use our base 6 +addition rules). So, we add that to our table: +* +0 +1 +2 +3 +4 +5 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +2 +0 +2 +4 +10 +12 +3 +0 +3 +10 +4 +0 +4 +12 +5 +0 +5 +Step 4: Finally, +. And so we add that to our table: +* +0 +1 +2 +3 +4 +5 +0 +0 +0 +0 +0 +0 +0 +4.5 • Multiplication and Division in Base Systems +315 + +1 +0 +1 +2 +3 +4 +5 +2 +0 +2 +4 +10 +12 +14 +3 +0 +3 +10 +4 +0 +4 +12 +5 +0 +5 +14 +Step 5: A similar analysis will give us the remainder of the entries. Here is +demonstrated: +. +This is done by using the addition rules from Addition and Subtraction in Base Systems, namely that +, and +then applying the addition processes we’ve always known, but with the base 6 table (Table 4.4). In the end, our +multiplication table is as follows: +* +0 +1 +2 +3 +4 +5 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +2 +0 +2 +4 +10 +12 +14 +3 +0 +3 +10 +13 +20 +23 +4 +0 +4 +12 +20 +24 +32 +5 +0 +5 +14 +23 +32 +41 +Table 4.8 Base 6 Multiplication Table +Notice anything about that bottom line? Is that similar to what happens in base 10? +To summarize the creation of a multiplication in a base other than base 10, you need the addition table of the base with +which you are working. Create the table, and calculate the entries of the multiplication table by performing repeated +addition in that base. The table needs to be drawn only the one time. +EXAMPLE 4.38 +Creating a Multiplication Table for a Base Lower Than 10 +Create the multiplication table for base 7. +Solution +Step 1: Let’s apply the process demonstrated and outlined above to find the base 7 multiplication table. It should be +clear that 0 multiplied by anything, regardless of base, will give 0, and that 1 multiplied by anything, regardless of base, +will be the value of “anything.” So, we start with the table below: +* +0 +1 +2 +3 +4 +5 +6 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +6 +316 +4 • Number Representation and Calculation +Access for free at openstax.org + +2 +0 +2 +4 +6 +3 +0 +3 +6 +4 +0 +4 +5 +0 +5 +6 +0 +6 +Step 2: Notice +is there. But we didn’t hit a problematic number there (4 works fine in both base 10 and base 6). +The same is true for +and +, which equal 6. But what is +? If we use the repeated addition concept, +. According to the base 7 addition table in the solution for Example 4.29, +. So, we +add that to our table: +* +0 +1 +2 +3 +4 +5 +6 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +6 +2 +0 +2 +4 +6 +11 +3 +0 +3 +6 +4 +0 +4 +11 +5 +0 +5 +6 +0 +6 +Step 3: Next, we need to fill in +. Using repeated addition, +if we use our +base 7 addition rules. So, we add that to our table: +* +0 +1 +2 +3 +4 +5 +6 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +6 +2 +0 +2 +4 +6 +11 +13 +3 +0 +3 +6 +4 +0 +4 +11 +5 +0 +5 +13 +6 +0 +6 +4.5 • Multiplication and Division in Base Systems +317 + +Step 4: Finally, +. And so we add that to our table: +* +0 +1 +2 +3 +4 +5 +6 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +6 +2 +0 +2 +4 +6 +11 +13 +15 +3 +0 +3 +6 +4 +0 +4 +11 +5 +0 +5 +13 +6 +0 +6 +15 +Step 5: A similar analysis will give us the remainder of the entries. Here is +demonstrated: +This is done by using the addition rules from Addition and Subtraction in Base Systems, namely that +and +then applying the addition processes we’ve always known, but with the base 7 table in the solution for Example 4.29. +Using those addition rules, the rest of the table is given below: +* +0 +1 +2 +3 +4 +5 +6 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +6 +2 +0 +2 +4 +6 +11 +13 +15 +3 +0 +3 +6 +12 +15 +21 +24 +4 +0 +4 +11 +15 +22 +26 +33 +5 +0 +5 +13 +21 +26 +34 +42 +6 +0 +6 +15 +24 +33 +42 +51 +YOUR TURN 4.38 +1. Create the multiplication table for base 4. +EXAMPLE 4.39 +Creating a Multiplication Table for a Base Higher Than 10 +Create the multiplication table for base 12. +318 +4 • Number Representation and Calculation +Access for free at openstax.org + +Solution +Let’s apply the repeated addition to base 12. Here is +demonstrated: +This is done by using the addition rules from Addition and Subtraction in Base Systems, namely that +and then applying the addition processes we’ve always known, but with the base 12 table in the solution for Example +4.31. Using those addition rules, the rest of the table is given below: +* +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +2 +0 +2 +4 +6 +8 +A +10 +12 +14 +16 +18 +1A +3 +0 +3 +6 +9 +10 +13 +16 +19 +20 +23 +26 +29 +4 +0 +4 +8 +10 +14 +18 +20 +24 +28 +30 +34 +38 +5 +0 +5 +A +13 +18 +21 +26 +2B +34 +39 +42 +47 +6 +0 +6 +10 +16 +20 +26 +30 +36 +40 +46 +50 +56 +7 +0 +7 +12 +19 +24 +2B +36 +41 +48 +53 +5A +65 +8 +0 +8 +14 +20 +28 +34 +40 +48 +54 +60 +68 +74 +9 +0 +9 +16 +23 +30 +39 +46 +53 +60 +69 +76 +83 +A +0 +A +18 +26 +34 +42 +50 +5A +68 +76 +84 +92 +B +0 +B +1A +29 +38 +47 +56 +65 +74 +83 +92 +A1 +Table 4.9 Base 12 Multiplication Table +YOUR TURN 4.39 +1. Create the multiplication table for base 14. +The multiplication table in base 2 below is as minimal as the addition table in the solution for Table 4.6. Since the product +of 1 with anything is itself, the following multiplication table is found. +* +0 +1 +0 +0 +0 +1 +0 +1 +Table 4.10 +Base 2 +Multiplication +Table +4.5 • Multiplication and Division in Base Systems +319 + +As with the addition table, we can use the multiplication tables and the addition tables to perform multiplication of two +numbers in bases other than base 10. The process is the same, with the same carry rules and placeholder rules. +EXAMPLE 4.40 +Multiplying in a Base Lower Than 10 +1. +Calculate +. +2. +Calculate +. +Solution +1. +Step 1: Use the base 6 multiplication table (Table 4.8) and, when necessary, the base 6 addition table (Table 4.4). +Set up this calculation using columns: +4 +5 +x +2 +4 +Step 2: Multiply the 1s digits, 5 and 4, using the base 6 multiplication table (Table 4.8). There we see the result is 326. +So, we enter the 2 and carry the 3. +3 +4 +5 +x +2 +4 +2 +Step 3: So, now we multiply the 4 and the 4, then add the 3 (just as you would do if multiplying two base 10 +numbers!). +(from the base 6 table [Table 4.8]), then +. So, we enter the 31. +3 +4 +5 +x +2 +4 +3 +1 +2 +Step 4: Now we move on to the 2 in the “tens” place in the bottom value. We multiply the 26 and the 56, and we get +146. So, we enter the 4 and carry the 1. +320 +4 • Number Representation and Calculation +Access for free at openstax.org + +Step 5: Next up, we multiply the 2 and the 4, and then add 1. This gives us +. We enter those on that +second line. +1 +4 +5 +x +2 +4 +3 +1 +2 +1 +3 +4 +0 +Step 6: Now we add down the columns. +1 +4 +5 +x +2 +4 +1 +3 +1 +2 +1 +3 +4 +0 +2 +0 +5 +2 +Step 7: The 3 and the 3 add to 10 in base 6, so we enter the 0 and carry the 1. We now have the result: +. +2. +Step 1: Use the base 2 multiplication table (Table 4.10) and, when necessary, the base 2 addition table in the +solution for Example 4.29. Set up this calculation using columns: +1 +0 +1 +x +1 +1 +0 +Step 2: Using the pattern established above, and the processes from multiplication from base 10, we find the +following: +1 +0 +1 +x +1 +1 +0 +0 +0 +0 +1 +0 +1 +1 +0 +1 +Step 3: Adding down the columns results in the following: +4.5 • Multiplication and Division in Base Systems +321 + +1 +0 +1 +x +1 +1 +0 +0 +0 +0 +1 +0 +1 +1 +0 +1 +1 +1 +1 +1 +0 +So, +. +YOUR TURN 4.40 +1. Calculate +. +2. Calculate +. +Summarizing the process of multiplying two numbers in different bases, the multiplication table is referenced. Using +that table, the multiplication is carried out in the same manner as it is in base 10. The addition rules for the base will also +be referenced when carrying a 1 or when adding the results for each digit’s multiplication line. +EXAMPLE 4.41 +Multiplying in a Base Higher Than 10 +Calculate +. +Solution +Step 1: Use the base 12 multiplication table in the solution for Example 4.39 and, when necessary, the base 12 addition +table in the solution for Table 4.7. Set up this calculation using columns: +3 +A +× +7 +4 +Step 2: First, the 4 is multiplied by 3A, resulting in the first line. +3 +A +× +7 +4 +1 +3 +4 +Step 3: Now we move on to the 7 in the “tens” place in the bottom value. +322 +4 • Number Representation and Calculation +Access for free at openstax.org + +5 +3 +A +x +7 +4 +1 +3 +4 +2 +2 +A +0 +Step 4: Now we add down the columns. +3 +A +x +7 +4 +1 +3 +4 +2 +2 +A +0 +2 +4 +1 +4 +Step 5: The 3 and the A add to 11 in base 12, so we enter the 1 and carry the 1. +We now have the result: +. +YOUR TURN 4.41 +1. Calculate +. +Division in Bases Other Than 10 +Just as with the other operations, division in a base other than 10, the process of division in a base other than 10 is the +same as the process when working in base 10. For instance, +because, we know that +. So, for many +division problems, we are simply looking to the multiplication table to identify the appropriate multiplication rule. +EXAMPLE 4.42 +Dividing with a Base Other Than 10 +1. +Calculate +. +2. +Calculate +Solution +1. +Looking at the multiplication table for base 6 (Table 4.8), we see that +. Using that, we know that +. +2. +Looking at the multiplication table for base 12 in the solution for Example 4.39, we see that +. +Using that, we know that +. +YOUR TURN 4.42 +1. Calculate +4.5 • Multiplication and Division in Base Systems +323 + +2. Calculate +. +Errors in Multiplying and Dividing in Bases Other Than Base 10 +The types of errors encountered when multiplying and dividing in bases other than base 10 are the same as when +adding and subtracting. They often involve applying base 10 rules or symbols to an arithmetic problem in a base other +than base 10. The first type of error is using a symbol that is not in the symbol set for the base. +EXAMPLE 4.43 +Identifying an Illegal Symbol in a Base Other Than Base 10 +Explain the error in the following calculation, and determine the correct answer: +Solution +Since the problem is in base 6, the symbol set available is 0, 1, 2, 3, 4, and 5. The 8 in the answer is clearly not a legal +symbol for base 6. Looking back to the base 6 multiplication table (Table 4.8), we see that +. +YOUR TURN 4.43 +1. Explain the error in the following calculation and determine the correct answer: +The second type of error is using a base 10 rule when the numbers are not in base 10. For instance, in base 17, +would be incorrect, even though in base 10, +. That rule doesn’t apply in base 17. +EXAMPLE 4.44 +Identifying an Error in Arithmetic in a Base Other Than Base 10 +Explain the error in the following calculation. Determine the correct answer: +Solution +If this problem was a base 10 problem, this would be the correct answer. However, in base 12, +is not 56, but is +instead 48. To correct this error, carefully use the multiplication table for base 12 (Table 4.9). If properly used, the correct +answer would be +. +YOUR TURN 4.44 +1. Explain the error in the following calculation. Determine the correct answer: +Check Your Understanding +28. To create the multiplication table for a given base, what should be used? +29. What are the differences between multiplying in base 10 and multiplying in a different base? +30. When dividing in a base other than base 10, what table is referenced? +31. Compute +. +32. Compute +. +33. How do you know an error has occurred in a base 5 multiplication question if the answer obtained was 285? +34. What are two common ways to determine an error is committed when computing in s base other than base 10? +324 +4 • Number Representation and Calculation +Access for free at openstax.org + +SECTION 4.5 EXERCISES +For the following exercises, create the multiplication table for the given base. +1. base 5 +2. base 3 +3. base 16 (Hint: Use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.) +4. base 2 +For the following exercises, perform the indicated base 6 operation. +5. +6. +7. +8. +9. +10. +11. +12. +For the following exercises, perform the indicated base 12 operation. +13. +14. +15. +16. +17. +18. +19. +20. +21. Explain two ways to detect an error in arithmetic in bases other than base 10. +22. Explain the error in the following calculation: +23. Explain the error in the following calculation: +. +24. In base 10 multiplication, there are 100 multiplication rules plus a rule for carrying a number. How many +multiplication rules are there for base 6? +25. In base 10 multiplication, there are 100 multiplication rules plus a rule for carrying a number. How many +multiplication rules are there for base 14? +26. In base 10 multiplication, there are 100 multiplication rules plus a rule for carrying a number. How many +multiplication rules are there for base 2? +27. Consider the answers from Exercise 24 and Exercise 26. Which base do you think would be more efficient: base 10, +base 6, or base 2? +For the following exercises, use the multiplication table that you created from Exercise 4 to perform the indicated base +2 operations. +28. +29. +30. +31. +32. Convert +and +to base 10. Then multiply those base 10 numbers. Next, convert the answer you got +for Exercise 31 to base 10. Do these numbers match? +For the following exercises, use the multiplication table that you created from Exercise 3 to perform the indicated base +16 operations. +33. +34. +35. +36. +For the following exercises, explain how you know an error was committed without performing the operation in the +4.5 • Multiplication and Division in Base Systems +325 + +given base. +37. +38. +326 +4 • Number Representation and Calculation +Access for free at openstax.org + +Chapter Summary +Key Terms +4.1 Hindu-Arabic Positional System +• +numeral +• +number +• +exponential expression +• +base +• +exponent +• +place value +• +base 10 system +• +Hindu-Arabic numeration system +• +expanded form +4.2 Early Numeration Systems +• +additive system of numbers +• +positional system of numbers +• +Babylonian system of numbers +• +Mayan system of numbers +• +Roman system of numbers +4.3 Converting with Base Systems +• +base 10 +• +remainder +• +dividend +• +divisor +• +quotient +Key Concepts +4.1 Hindu-Arabic Positional System +• +Exponents are used to represent repeated multiplication of a base. +• +In arithmetic, exponents are computed before multiplication, division, addition, and subtraction. Computing an +exponent is done by multiplying the base by itself the number of times equal to the exponent. +• +The system of numbers currently used is the Hindu-Arabic system. Digits in this system take on values based on +their place in the number. The place values are determined by multiplying the digit by 10 raised to the appropriate +power. +• +The expanded form of a Hindu-Arabic number is the sum of each digit times 10 raised to the exponent for that place +value. +4.2 Early Numeration Systems +• +Historically, there have been many systems for numbering. One system is an additive system, in which symbols are +repeated to express larger numbers. Another system is a positional system, in which the digits and their positions +determine the quantity being represented. +• +The Babylonian system was a combination of a positional and additive system. It used 60 as its base. Using that in +the positional system makes it possible to convert between Babylonian and Hindu-Arabic numbers. +• +The Mayan system was a combination of a positional and additive system. It used 20 as its base. Using that in the +positional system makes it possible to convert between Mayan and Hindu-Arabic numbers. +• +The Roman system was an additive system. Knowing what each symbol represents makes it possible to convert +between Roman and Hindu-Arabic numbers. +4.3 Converting with Base Systems +• +The system we use is the base 10 system. Base 10 is not the only base that can be used. To use another base, one +could start with a list of numbers in that base. +• +To indicate that a number is written in a base other than 10, a subscript is appended to the end of the number. That +subscript indicates the base for the number. +• +Numbers written in a base smaller than 10 use the same symbols as base 10. However, when using bases larger +4 • Chapter Summary +327 + +than 10, the symbols A, B, C, … are used to represent digits larger than 9. +• +To convert from a number written in a base other than 10 into a base 10 number, the number is written in expanded +form and then that expression is computed. +• +To convert a number from base 10 into another base, the base 10 number is repeatedly divided by the new base. +The remainders when performing these divisions become the digits for the number in the new base. +• +Common errors can be detected when performing base conversions. +4.4 Addition and Subtraction in Base Systems +• +Addition tables for bases other than 10 can be built using the same processes that are used in base 10, including +using a number line. +• +Addition in bases other than base 10 use the same processes as addition in base 10, but use the addition table for +that base. +• +Subtraction in bases other than base 10 use the same processes as subtraction in base 10, but use the addition +table for that base. +4.5 Multiplication and Division in Base Systems +• +Multiplication tables for bases other than 10 can be built using the same processes that are used in base 10, +including using repeated addition and the addition table for the base. +• +Multiplication in bases other than base 10 use the same processes as multiplication in base 10, but use the +multiplication table for that base. +• +Basic division in bases other than base 10 use the same processes as basic division in base 10, where the missing +factor process is used. +Videos +4.1 Hindu-Arabic Positional System +• +Exponential Notation (https://openstax.org/r/Exponential_Notation) +4.2 Early Numeration Systems +• +Converting Between Babylonian and Hindu-Arabic numbers (https://openstax.org/r/Babylonian_to_Hindu- +Arabic_Numbers) +• +Converting Mayan Numbers to Hindu-Arabic Numbers (https://openstax.org /r/Mayan_to_Hindu-Arabic_Numbers) +• +Converting From Roman Numbers to Hindu-Arabic Numbers (https://openstax.org/r/Roman_to_Hindu- +Arabic_Numbers) +• +Converting From Hindu-Arabic Numbers to Roman Numbers (https://openstax.org/r/Hindu- +Arabic_to_Roman_Numbers) +4.3 Converting with Base Systems +• +Convert Base 7 to Base 10 (https://openstax.org/r/Convert_Base_7_to_Base_10) +• +Converting from Base 10 to Another Base (https://openstax.org/r/Base_10_to_Another_Base) +Projects +Additive Systems +Go online. Google “additive number systems.”What system comes up? +• +Describe the additive system you found. +Using Google, identify three more additive systems of numbers. +• +Compare and contrast the systems you found. For instance, how many times can a symbol be used before a new +symbol is used. +• +Identify three situations where additive systems are still used. +Computers and Bases +Use Google to determine what base computers use. +Were other bases attempted for use in computers? +Determine why the base used in computers is appropriate. +Determine how the base used in computers is related to the circuitry in computers. +328 +4 • Chapter Summary +Access for free at openstax.org + +Determine how Boolean logic and the base used in computers are related, and might be identical. +There is research into using quibits in computers. Find out what quibits are and how can they improve computing speed. +Cultures Using Base Systems Other Than 10 +Using Google, find three cultures, other than Babylonian or Mayan, that use base systems other than 10. +• +Tell what base is used for each system. +• +If possible, determine why the culture used that base system. +• +Choose one of those systems. Explain that base system. Be sure to address whether the system is additive, place- +value based, a blend of the two, and if it employs a zero. +History of Zero +Using any resources available to you, determine the history of 0 in at least three different numbering systems. Address +at least when and why such a development occurred and why a 0 is vital to the use of a positional system. +Numbering Systems from Other Global Regions +Using any resources available to you, find at least three numbering systems from sub-Saharan Africa, Australia, China, or +the Pacific Islands. Explore if they are positional or additive systems (or combinations!), the terminology of the system, if +they used a 0, and what base they employed (if positional). +4 • Chapter Summary +329 + +Chapter Review +Hindu-Arabic Positional System +1. What is the base of 57? +2. What is the exponent of 57? +3. Compute +. +4. Convert the Hindu-Arabic number into expanded form: 4,201. +5. Convert the expression to a Hindu-Arabic numeral: +. +Early Numeration Systems +6. Which systems—Hindu-Arabic, Roman, Mayan, or Babylonian—are additive systems? +7. Which systems—Hindu-Arabic, Roman, Mayan, or Babylonian—are positional systems? +8. Which systems—Hindu-Arabic, Roman, Mayan, or Babylonian—use a 0? +9. In the Babylonian system, what are the place values based on? +10. In the Mayan system, what are the place values based on? +11. Convert the Babylonian numeral to a Hindu-Arabic numeral. +12. Convert the Mayan numeral to a Hindu-Arabic numeral. +13. Convert the Roman numeral MMCDXLVII into a Hindu-Arabic numeral. +14. Convert the Hindu-Arabic numeral 394 to a Roman numeral. +Converting with Base Systems +15. List the numbers from 0 to 100 in base 5. +16. In base 8, what is the place value of the 3 in the number 6388? +17. How many symbols are needed for a base 17 system? +18. What does it mean for a number to be in base 6? +19. What symbols are used in a base 12 system? +20. When converting from a base 10 number to a base 2 number, would the number of digits decrease? +21. Convert 3115 to base 10. +22. Convert 4512 to base 10. +23. Convert 10012 to base 10. +24. Convert 459 to base 8. +25. Convert 1198 to base 12. +26. Convert 38 to base 2. +27. When converting from base 10 to base 4, the result obtained was 1424. How can you tell an error was made? +Addition and Subtraction in Base Systems +28. Create the addition table for base 5. +29. How many addition rules are there for a base 7 system? +30. Calculate 345 + 445. +31. Calculate A712 + 8812. +330 +4 • Chapter Summary +Access for free at openstax.org + +32. Calculate 5416 − 2336. +33. Calculate 5B12 − 1A12. +34. When adding in base 8, the result 9118 is found. How do we know a mistake was made? +Multiplication and Division in Base Systems +35. What is the process for creating the multiplication table for a base other than 10? +36. Create the multiplication table for base 7. +37. How many multiplication rules are there in a base 3 system? +38. Calculate +. +39. Calculate +. +40. Calculate +. +41. Calculate +. +42. When multiplying +, the result 140 is found. How do we know a mistake was made? +Chapter Test +1. Expand the Hindu-Arabic numeral 5,789. +2. Evaluate the expression +. +3. Rewrite +in Hindu-Arabic form. +4. Convert the Babylonian numeral to a Hindu-Arabic numeral. +5. Convert the Mayan number to a Hindu Arabic numeral. +6. What base system did the Babylonians use? +7. Which system—Roman, Babylonian, Mayan—used place values? +8. Convert the Roman numeral MDXLVII to a Hindu-Arabic numeral. +9. How many symbols are needed for a base 9 system? +10. For a system in a base larger than 10, what symbols are used as digits representing more than 10? +11. Convert 1328 to a base 10 number. +12. List the numbers in base 4 up to 1004. +13. Convert 74 to a base 12 number. +14. Create the addition table for base 4. +15. Calculate +. +16. Calculate 4B112 − 2A612. +17. When calculating +, a student obtains 766 . How do you know an error was made? +18. Create the multiplication table for base 4. +19. Calculate +. +20. Calculate +. +4 • Chapter Summary +331 + +332 +4 • Chapter Summary +Access for free at openstax.org + +Figure 5.1 In these algebraic equations, the +represents different numbers. (credit: Larissa Chu, CC BY 4.0) +Chapter Outline +5.1 Algebraic Expressions +5.2 Linear Equations in One Variable with Applications +5.3 Linear Inequalities in One Variable with Applications +5.4 Ratios and Proportions +5.5 Graphing Linear Equations and Inequalities +5.6 Quadratic Equations with Two Variables with Applications +5.7 Functions +5.8 Graphing Functions +5.9 Systems of Linear Equations in Two Variables +5.10 Systems of Linear Inequalities in Two Variables +5.11 Linear Programming +Introduction +The jump from arithmetic to algebra can be a difficult one for many students. Many students struggle with the idea that +mathematics can include situations that aren’t static and do change. In elementary arithmetic, a situation such as: +is a static situation and will yield the answer of 8 every time. However, a situation such as: +can yield many different answers because the answer depends on what amount (number) that +represents. Since the +value of +can vary (represent different values), it is known as a variable. +Algebra is useful to better model real life situations. In the first equation shown, +can only model situations +where you add those two numbers together. For example, if your uncle gives you five dollars and your aunt gives you +three dollars, then you will always receive eight dollars. The second equation +can model more complex +situations. For example, you wish to buy a game that costs $38 but you only have three dollars. Your uncle will pay you +five dollars an hour to work for him. If you’ve worked five hours, have you earned enough money? If not, how many +hours will you have to work? +Algebra and algebraic thinking open up a world of possibilities that arithmetic alone cannot do. +5 +ALGEBRA +5 • Introduction +333 + +5.1 Algebraic Expressions +Figure 5.2 Two college graduates! (credit: modification of work UC Davis College of Engineering/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Convert between written and symbolic algebraic expressions and equations. +2. +Simplify and evaluate algebraic expressions. +3. +Add and subtract algebraic expressions. +4. +Multiply and divide algebraic expressions. +Algebraic expressions are the building blocks of algebra. While a numerical expression (also known as an arithmetic +expression) like +can represent only a single number, an algebraic expression such as +can represent many +different numbers. This section will introduce you to algebraic expressions, how to create them, simplify them, and +perform arithmetic operations on them. +Algebraic Expressions and Equations +Xavier and Yasenia have the same birthday, but they were born in different years. This year Xavier is 20 years old and +Yasenia is 23, so Yasenia is three years older than Xavier. When Xavier was 15, Yasenia was 18. When Xavier will be 33, +Yasenia will be 36. No matter what Xavier’s age is, Yasenia’s age will always be 3 years more. +In the language of algebra, we say that Xavier's age and Yasenia's age are variable and the 3 is a constant. The ages +change, or vary, so age is a variable. The 3 years between them always stays the same or has the same value, so the age +difference is the constant. In algebra, letters of the alphabet are used to represent variables. The letters most often +used for variables are +, +, +, +, +, and . Suppose we call Xavier's age +. Then we could use +to represent Yasenia's +age, as shown in the table below. +Xavier’s Age +Yasenia’s Age +15 +18 +20 +23 +33 +36 +To write algebraically, we need some symbols as well as numbers and variables. The symbols for the four basic +arithmetic operations: addition, subtraction, multiplication, and division are summarized in Table 5.1, along with words +we use for the operations and the result. +334 +5 • Algebra +Access for free at openstax.org + +Operation +Notation +Say: +The result is… +Addition +plus +The sum of +and +Subtraction +− +minus +The difference of +and +Multiplication +• +, ( )( ), ( ) , +( ), +, +times +The product of +and +Division +÷ +, +/ +divided by +The quotient of +and +Table 5.1 Symbols for Operations +In algebra, the cross symbol +is normally not used to show multiplication because that symbol could cause +confusion. For example, does +mean +(three times +) or +(three times +times +)? To make it clear, +use +or parentheses for multiplication. +We perform these operations on two numbers. When translating from symbolic form to words, or from words to +symbolic form, pay attention to the words of or and to help you find the numbers. +• +The sum of 5 and 3 means add 5 plus 3, which we write as +. +• +The difference of 9 and 2 means subtract 9 minus 2, which we write as +. +• +The product of 4 and 8 means multiply 4 times 8, which we can write as +. +• +The quotient of 20 and 5 means divide 20 by 5, which we can write as +. +EXAMPLE 5.1 +Translating from Algebra to Words +Translate the following algebraic expressions from algebra into words. +1. +2. +3. +4. +Solution +1. +According to Table 5.1, this could be translated as 12 plus 14 OR the sum of 12 and 14. +2. +According to Table 5.1, this could be translated as 30 times 5 OR the product of 30 and 5. +3. +According to Table 5.1, this could be translated as 64 divided by 8 OR the quotient of 64 and 8. +4. +According to Table 5.1, this could be translated as +minus +OR the difference of +and +YOUR TURN 5.1 +Translate the following algebraic expressions from algebra into words. +1. +2. +3. +4. +EXAMPLE 5.2 +Translating from Words to Algebra +Translate the following phrases from words into algebraic expressions. +1. +The difference of 47 and 19 +5.1 • Algebraic Expressions +335 + +2. +72 divided by 9 +3. +The sum of +and +4. +13 times 7 +Solution +1. +According to Table 5.1, these words could be translated as +. +2. +According to Table 5.1, these words could be translated as +. +3. +According to Table 5.1, these words could be translated as +. +4. +According to Table 5.1, these words could be translated as +. +YOUR TURN 5.2 +Translate the following phrases from words into algebraic expressions. +1. 43 plus 67 +2. The product of 45 and 3 +3. The quotient of 45 and 3 +4. 89 minus 42 +What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is +incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football +player was running very fast” is a sentence. A sentence has a subject and a verb. In algebra, we have expressions and +equations. Example 5.1 and Example 5.2 used expressions. An expression is like an English phrase. Notice that the +English phrases do not form a complete sentence because the phrase does not have a verb. The following table has +examples of expressions, which are numbers, variables, or combinations of numbers and variables using operation +symbols. +Expression +Words +English Phrase +3 plus 5 +The sum of three and five +minus one +The difference of +and one +6 times 7 +The product of six and seven +divided by +The quotient of +and +EXAMPLE 5.3 +Translating from an English Phrase to an Expression +Translate the following phrases from words into algebraic expressions. +1. +Seven more than a number +. +2. +A number +times itself. +3. +Six times a number +, plus two more. +4. +The cost of postage is a flat rate of 10 cents for every parcel, plus 34 cents per ounce +. +Solution +1. +2. +or +3. +4. +336 +5 • Algebra +Access for free at openstax.org + +YOUR TURN 5.3 +Translate the following phrases from words into algebraic expressions. +1. Twenty less than a number +. (Hint: you have a number +and you want 20 less than it.) +2. Add two to a number +, then multiply it by six. +3. A number +to the third power minus five. +4. A plumber charges $60 per hour +, plus a $40 flat fee for every job. +An equation is two expressions linked with an equal sign (the symbol =). When two quantities have the same value, we +say they are equal and connect them with an equal sign. When you read the words the symbols represent in an +equation, you have a complete sentence in English. The equal sign gives the verb. So, +is read “ +is equal to +.” The +following table has some examples of equations. +Equation +English Sentence +The sum of three and five is equal to eight. +minus one equals fourteen. +The product of six and seven is equal to forty-two. +is equal to fifty-three. +plus nine is equal to two times +minus three. +EXAMPLE 5.4 +Translating from an English Sentence to an Equation +Translate the following sentences from words into algebraic equations. +1. +Two times +is 6. +2. +plus 2 is equal to +times 3. +3. +The quotient of 35 and 7 is 5. +4. +Sixty-seven minus +is 56. +Solution +1. +2. +3. +4. +YOUR TURN 5.4 +Translate the following sentences from words into algebraic equations. +1. Five times +is 50. +2. Half of a number +is 30. +3. The difference of three times a number +and 7 is 2. +4. Two times +plus 7 is 21. +5.1 • Algebraic Expressions +337 + +WHO KNEW? +The Use of Variables +French philosopher and mathematician René Descartes (1596–1650) is usually given credit for the use of the letters +, +, and +to represent unknown quantities in algebra. He introduced these ideas in his publication of La Geometrie, +which was printed in 1637. In this publication, he also used the letters +, +, and +to represent known quantities. There +is a (possibly fictitious) story that, when the book was being printed for the first time, the printer began to run short +of the last three letters of the alphabet. So the printer asked Descartes if it mattered which of +, +, or +were used for +the mathematical equations in the book. Descartes decided it made no difference to him; so the printer decided to +use +predominantly for the mathematics in the book, because the letters +and +would occur more often in the +body of the text (written in French) than the letter +would! This might explain why the letter +is still used today as +the most common variable to represent unknown quantities in algebra. +Simplifying and Evaluating Algebraic Expressions +To simplify an expression means to do all the math possible. For example, to simplify +we would first multiply +to get 8 and then add 1 to get 9. We have introduced most of the symbols and notation used in algebra, but now +we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in +different values. Consider +. Do you add first or multiply first? Do you get different answers? +Add first: +Multiply first: +Which one is correct? +Early on, mathematicians realized the need to establish some guidelines when performing arithmetic operations to +ensure that everyone would get the same answer. Those guidelines are called the order of operations and are listed in +the table below. +Step 1: Parentheses and Other +Grouping Symbols +Simplify all expressions inside the parentheses or other grouping symbols, +working on the innermost parentheses first. +Step 2: Exponents +Simplify all expressions with exponents. +Step 3: Multiplication and +Division +Perform all multiplication and division in order from left to right. These +operations have equal priority. +Step 4: Addition and Subtraction +Perform all addition and subtraction in order from left to right. These operations +have equal priority. +You may have heard about Please Excuse My Dear Aunt Sally or PEMDAS. Be careful to notice in Steps 3 and 4 in the +table above that multiplication and division, as well as addition and subtraction, happen in order from LEFT to +RIGHT. It is possible, for example, to have PEDMAS or PEMDSA. The PEMDAS trick can be misleading if not fully +understood! +EXAMPLE 5.5 +Making a Numerical Equation True Using the Order of Operations +Use parentheses to make the following statements true. +1. +2. +3. +4. +338 +5 • Algebra +Access for free at openstax.org + +Solution +1. +Add the parentheses around the +. Then you have +. +2. +Add the parentheses around the +. Then you have +. +3. +Add the parentheses around the +. Then you have +. +4. +Add the parentheses around the +. Then you have +. +YOUR TURN 5.5 +Use parentheses and the order of operations to make each equation true. +1. +2. +3. +4. +In the last example, we simplified expressions using the order of operations. Now we'll evaluate some +expressions—again following the order of operations. To evaluate an expression means to find the value of the +expression when the variable is replaced by a given number. +EXAMPLE 5.6 +Evaluating and Simplifying an Expression +1. +Evaluate +when +. +2. +Evaluate +when +. +Solution +1. +To evaluate, let +in the expression, and then simplify: +. +2. +To evaluate, let +in the expression, and then simplify: +. +YOUR TURN 5.6 +1. Evaluate +when +. +2. Evaluate +when +. +Operations of Algebraic Expressions +Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more +variables. Examples of terms are 7, +, 5 +, 9 , and +. The constant that multiplies the variable is called the coefficient. +Think of the coefficient as the number in front of the variable. Consider the algebraic expressions 5 +, which has a +coefficient of 5, and 9 , which has a coefficient of 9. If there is no number listed in front of the variable, then the +coefficient is 1 since +. +Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they +are like terms. If there are like terms in an expression, you can simplify the expression by combining the like terms. We +add the coefficients and keep the same variable. +EXAMPLE 5.7 +Adding Algebraic Expressions +Add +. +Solution +Step 1: Add the terms in any order and get the same result (think: +) and drop the parentheses: +5.1 • Algebraic Expressions +339 + +Step 2: Group like terms together: +Step 3: Combine the like terms: +YOUR TURN 5.7 +1. Add +. +EXAMPLE 5.8 +Subtracting Algebraic Expressions +Subtract +. +Solution +Step 1: Distribute the negative inside the parentheses (think: +, which is the correct +answer). You cannot just drop the parentheses (for example, +, which is not correct as we have +already verified the answer is 3): +Step 2: Group like terms together: +Step 3: Combine the like terms: +YOUR TURN 5.8 +1. Subtract +. +Before looking at multiplying algebraic expressions we look at the Distributive Property, which says that to multiply a +sum, first you multiply each term in the sum and then you add the products. For example, +can also be solved as +. If we use a variable, then +. +We can extended this example to +, which can +also be solved as +. If we use variables, then +. +FORMULA +Distributive Property: +340 +5 • Algebra +Access for free at openstax.org + +EXAMPLE 5.9 +Simplifying an Expression Using the Order of Operations +Simplify each expression. +1. +2. +3. +4. +5. +Solution +1. +2. +3. +4. +5. +YOUR TURN 5.9 +Simplify each expression. +1. +2. +3. +4. +5. +EXAMPLE 5.10 +Multiplying Algebraic Expressions +Multiply +. +Solution +Step 1: Use the Distributive Property: +Step 2: Multiply: +Step 3: Combine the like terms: +YOUR TURN 5.10 +1. Multiply +. +You may have heard the term FOIL which stands for: First, Outer, Inner, Last. FOIL essentially describes a way to use +the Distributive Property if you multiply a two-term expression by another two-term expression, but FOIL only works +in that specific situation. For example, suppose you have a two-term expression multiplied by a three-term +expression, such as +. What terms qualify as inner terms and what terms qualify as outer terms? In +5.1 • Algebraic Expressions +341 + +this particular situation, FOIL cannot possibly work; the multiplication of +should yield six terms, +where FOIL is designed to only give you four! The Distributive Property works regardless of how many terms there +are. FOIL can be misleading and applied inappropriately if not fully understood! +EXAMPLE 5.11 +Dividing Algebraic Expressions +Divide +. +Solution +Divide EACH term by 4 : +YOUR TURN 5.11 +1. Divide +. +Be careful how you divide! Sometimes students incorrectly divide only one term on top by the bottom term. For +example, +might turn into +if done incorrectly. When we divide expressions, EACH +term is divided by the divisor. So, +If you forget, it is always a good +idea to check these rules by creating an example using numerical expressions. For example, +. +Dividing each term on top by 3 would yield +, which is the correct answer. +However, if you just divided the 9 on top by the 3 on the bottom, getting +, this does not +result in the correct answer. +342 +5 • Algebra +Access for free at openstax.org + +PEOPLE IN MATHEMATICS +Al-Khwarizmi +Figure 5.3 Al-Khwarizmi +Abu Ja’far Muhammad ibn Musa Al-Khwarizmi was born around 780 AD, probably in or around the region of +Khwarizm, which is now part of modern-day Uzbekistan. For most of his adult life, he worked as a scholar at the +House of Wisdom in Baghdad, Iraq. He wrote many mathematical works during his life, but is probably most famous +for his book Al-kitab al-muhtasar fi hisab al-jabr w’al’muqabalah, which translates to The Condensed Book on the +Calculation of al-Jabr (completion) and al’muqabalah (balancing). The word al-jabr would eventually become the word +we use to describe the topic that he was writing about in this book: algebra. From another book of his, with the Latin +title Algoritmi de numero Indorum (Al-Khwarizmi on the Hindu Art of Reckoning ), our word algorithm is derived. In +addition to writing on mathematics, Al-Khwarizmi wrote works on astronomy, geography, the sundial, and the +calendar. +In 2012, Andrew Hacker wrote an opinion piece in the New York Times Magazine suggesting that teaching algebra in +high school was a waste of time. Keith Devlin, a British mathematician, was asked to comment on Hacker's article by his +students in his Stanford University Continuing Studies course "Mathematics: Making the Invisible Visible" +(https://openstax.org/r/Making_the_Invisible_Visible) on iTunes University. Devlin concludes that Hacker was displaying +his ignorance of what algebra is. +VIDEO +Q&A: Why We Teach Algebra (https://openstax.org/r/Teach_Algebra) +Check Your Understanding +1. Juliette is 2 inches taller than her friend Vivian. Which algebraic equations represent their height? Use +for +Juliette’s height and +for Vivian’s height. +2. Which options represent algebraic expressions? +5.1 • Algebraic Expressions +343 + +3. Which expression equals 10 ? +4. Using the expression +, when a certain number is put in for +, the result is 50. What is the value of +? +5. Which expression equals +? Hint: Use the Distributive Property. +6. Given the expression +, the Distributive Property allows it to be rewritten as: +7. Given the two algebraic expressions +and +, the solution is +. What +mathematical operation was performed on the two algebraic expressions? +8. Given the two algebraic expressions +and +, the solution is +. What mathematical +operation was performed on the two algebraic expressions? +SECTION 5.1 EXERCISES +For the following exercises, translate from algebra to words. +1. +2. (10)( ) +3. +4. +5. +For the following exercises, translate from words to algebra. +6. 15 divided by 3. +7. The sum of 13 and 13. +8. 120 minus 12. +9. The product of 5 and 4. +10. The sum of double +and 5. +For the following exercises, translate from an English phrase to an expression. +11. Three times +minus 7. +12. +divided by 2; then add 4. +13. +squared minus 3. +14. A rental car company charges $0.15 per mile +, plus a $40 flat fee for the rental. +15. A parking garage in New York City charges $20 for the first hour, then $5 per hour +. +For the following exercises, use parentheses to make the statements true. +16. +17. +18. +344 +5 • Algebra +Access for free at openstax.org + +19. +20. +For the following exercises, evaluate and simplify the expression. +21. +when +22. +when +23. +when +24. +when +25. +when +26. +when +27. +when +28. Yasenia is 3 years older than Xavier. How old is Yasenia when Xavier is 18 years old? +29. A rental car company charges $0.15 per mile +, plus a $40 flat fee for the rental. What is the cost of the car +rental if one drives 100 miles? +30. A parking garage in New York City charges $20 for the first hour, then $5 per hour +. What is the cost of parking +for 10 hours? +For the following exercises, perform the indicated operation for the expressions. +31. Add +. +32. Add +. +33. Subtract +. +34. Subtract +. +35. Multiply +. +36. Multiply +. +37. Multiply +. +38. Multiply +. +39. +. +40. +. +5.1 • Algebraic Expressions +345 + +5.2 Linear Equations in One Variable with Applications +Figure 5.4 Most gyms have a monthly membership fee. (credit: modification of work "Morning PT after the Holidays +2021" by Fort Drum & 10th Mountain Division (LI)/Flickr, Public Domain Mark 1.0) +Learning Objectives +After completing this section, you should be able to: +1. +Solve linear equations in one variable using properties of equations. +2. +Construct a linear equation to solve applications. +3. +Determine equations with no solution or infinitely many solutions. +4. +Solve a formula for a given variable. +In this section, we will study linear equations in one variable. There are several real-world scenarios that can be +represented by linear equations: taxi rentals with a flat fee and a rate per mile; cell phone bills that charge a monthly fee +plus a separate rate per text; gym memberships with a monthly fee plus a rate per class taken; etc. For example, if you +join your local gym at $10 per month and pay $5 per class, how many classes can you take if your gym budget is $75 per +month? +Linear Equations and Applications +Solving any equation is like discovering the answer to a puzzle. The purpose of solving an equation is to find the value or +values of the variable that makes the equation a true statement. Any value of the variable that makes the equation true +is called a solution to the equation. It is the answer to the puzzle! There are many types of equations that we will learn to +solve. In this section, we will focus on a linear equation, which is an equation in one variable that can be written as +where +and +are real numbers and +, such that +is the coefficient of +and +is the constant. +To solve a linear equation, it is a good idea to have an overall strategy that can be used to solve any linear equation. In +the Example 5.12, we will give the steps of a general strategy for solving any linear equation. Simplifying each side of the +equation as much as possible first makes the rest of the steps easier. +EXAMPLE 5.12 +Solving a Linear Equation Using a General Strategy +Solve +346 +5 • Algebra +Access for free at openstax.org + +Solution +Step 1: Simplify each side of +the equation as much as +possible. +Use the Distributive Property. Notice that each side +of the equation is now simplified as much as +possible. +Step 2: Collect all variable +terms on one side of the +equation. +Nothing to do; all +-terms are on the left side. +Step 3: Collect constant terms +on the other side of the +equation. +To get constants only on the right, add 29 to each +side. +Simplify. +Step 4: Make the coefficient of +the variable term equal to 1. +Divide each side by 7. +Simplify. +Step 5: Check the solution. +Let +Subtract. +Check: +YOUR TURN 5.12 +1. Solve +In Example 5.12, we used both the addition and division property of equations. All the properties of equations are +summarized in table below. Basically, what you do to one side of the equation, you must do to the other side of the +equation to preserve equality. +Operation +Property +Example +Addition +If +Then +Subtraction +If +Then +5.2 • Linear Equations in One Variable with Applications +347 + +Operation +Property +Example +Multiplication +If +Then +Division +If +Then +for +Be careful to multiply and divide every term on each side of the equation. For example, +is solved by +multiplying BOTH sides of the equation by 3 to get +which gives +. Using parentheses will +help you remember to use the distributive property! A division example, such as +, can be solved +by dividing BOTH sides of the equation by 3 to get +which then will lead to +. +EXAMPLE 5.13 +Solving a Linear Equation Using Properties of Equations +Solve +. +Solution +Step 1: Simplify each side. +Step 2: Collect all variables on one side. +Step 3: Collect constant terms on one side. +Step 4: Make the coefficient of the variable 1. Already done! +Step 5: Check. +YOUR TURN 5.13 +1. Solve +. +348 +5 • Algebra +Access for free at openstax.org + +WHO KNEW? +Who Invented the Symbol for Equals ? +Before the creation of a symbol for equality, it was usually expressed with a word that meant equals, such as aequales +(Latin), esgale (French), or gleich (German). Welsh mathematician and physician Robert Recorde is given credit for +inventing the modern sign. It first appears in writing in The Whetstone of Witte, a book Recorde wrote about algebra, +which was published in 1557. In this book, Recorde states, "I will set as I do often in work use, a pair of parallels, or +Gemowe (twin) lines of one length, thus: ===, because no two things can be more equal." Although his version of the +sign was a bit longer than the one we use today, his idea stuck and "=" is used throughout the world to indicate +equality in mathematics. +In Algebraic Expressions, you translated an English sentence into an equation. In this section, we take that one step +further and translate an English paragraph into an equation, and then we solve the equation. We can go back to the +opening question in this section: If you join your local gym at $10 per month and pay $5 per class, how many classes can +you take if your gym budget is $75 per month? We can create an equation for this scenario and then solve the equation +(see Example 5.15). +EXAMPLE 5.14 +Constructing a Linear Equation to Solve an Application +The Beaudrie family has two cats, Basil and Max. Together, they weigh 23 pounds. Basil weighs 16 pounds. How much +does Max weigh? +Solution +Let += Basil’s weight and += Max’s weight. +We also know that Basil weighs 16 pounds so: +Steps 1 and 2: +Since both sides are simplified, the variable is on one side of the equation, we start in Step 3 and collect the constants on +one side: +Step 3: +Step 4: is already done so we go to Step 5: +Step 5: +Basil weighs 16 pounds and Max weighs 7 pounds. +YOUR TURN 5.14 +1. Sam and Henry are roommates. Together, they have 68 books. Sam has 26 books. How many books does Henry +have? +5.2 • Linear Equations in One Variable with Applications +349 + +EXAMPLE 5.15 +Constructing a Linear Equation to Solve Another Application +If you join your local gym at $10 per month and pay $5 per class, how many classes can you take if your gym budget is +$75 per month? +Solution +If we let += number of classes, the expression +would represent what you pay per month if each class is $5 and +there’s a $10 monthly fee per class. $10 is your constant. If you want to know how many classes you can take if you have +a $75 monthly gym budget, set the equation equal to 75. Then solve the equation +for +. +Steps 1 and 2: +Step 3: +Step 4: +Step 5: +The solution is 13 classes. You can take 13 classes on a $75 monthly gym budget. +YOUR TURN 5.15 +1. On June 7, 2021, the national average price for regular gasoline was $3.053 per gallon. If Aiko fills up his car with +16 gallons, how much is the total cost? Round to the nearest cent. +EXAMPLE 5.16 +Constructing an Application from a Linear Equation +Write an application that can be solved using the equation +. Then solve your application. +Solution +Answers will vary. Let’s say you want to rent a snowblower for a huge winter storm coming up. If += the number of days +you rent a snowblower, then the expression +represents what you pay if, for each day, it costs $50 to rent the +snowblower and there is a $35 flat rental fee. $35 is the constant. To find out how many days you can rent a snowblower +for $185, set the expression equal to 185. Then solve the equation +for +. +Steps 1 and 2: +350 +5 • Algebra +Access for free at openstax.org + +Step 3: +Step 4: +Step 5: +The equation is +and the solution is 3 days. You can rent a snowblower for 3 days on a $185 budget. +YOUR TURN 5.16 +1. Write an application that can be solved using the equation +. Then solve your application. +Linear Equations with No Solutions or Infinitely Many Solutions +Every linear equation we have solved thus far has given us one numerical solution. Now we'll look at linear equations for +which there are no solutions or infinitely many solutions. +EXAMPLE 5.17 +Solving a Linear Equation with No Solution +Solve +. +Solution +Step 1: Simplify each side. +Step 2: Collect all variables to one side. +The variable +disappeared! When this happens, you need to examine what remains. In this particular case, we have +, which is not a true statement. When you have a false statement, then you know the equation has no solution; +there does not exist a value for +that can be put into the equation that will make it true. +YOUR TURN 5.17 +1. Solve +. +5.2 • Linear Equations in One Variable with Applications +351 + +EXAMPLE 5.18 +Solving a Linear Equation with Infinitely Many Solutions +Solve +. +Solution +Step 1: +Step 2: +As with the previous example, the variable disappeared. In this case, however, we have a true statement ( +). When +this occurs we say there are infinitely many solutions; any value for +will make this statement true. +YOUR TURN 5.18 +1. Solve +Solving a Formula for a Given Variable +You are probably familiar with some geometry formulas. A formula is a mathematical description of the relationship +between variables. Formulas are also used in the sciences, such as chemistry, physics, and biology. In medicine they are +used for calculations for dispensing medicine or determining body mass index. Spreadsheet programs rely on formulas +to make calculations. It is important to be able to manipulate formulas and solve for specific variables. +To solve a formula for a specific variable means to isolate that variable on one side of the equal sign with a coefficient of +1. All other variables and constants are on the other side of the equal sign. To see how to solve a formula for a specific +variable, we will start with the distance, rate, and time formula. +EXAMPLE 5.19 +Solving for a Given Variable with Distance, Rate, and Time +Solve the formula +for . This is the distance formula where += distance, += rate, and += time. +Solution +Divide both sides by : +YOUR TURN 5.19 +1. Solve the formula +for . This formula is used to calculate simple interest +, for a principal +, invested at a +rate , for +years. +VIDEO +Solving for a Variable in an Equation (https://openstax.org/r/Solving_for_a_variable) +352 +5 • Algebra +Access for free at openstax.org + +EXAMPLE 5.20 +Solving for a Given Variable in the Area Formula for a Triangle +Solve the formula +for +. This is the area formula of a triangle where += area, += base, and += height. +Solution +Step 1: Multiply both sides by 2. +Step 2: Divide both sides by +. +YOUR TURN 5.20 +1. Solve the formula +for +. This formula is used to calculate the volume +of a right circular cone with +radius +and height +. +WORK IT OUT +Using Algebra to Understand Card Tricks +You will need to perform this card trick with another person. Before you begin, the two people must first decide which +of the two will be the Dealer and which will be the Partner, as each will do something different. Once you have +decided upon that, follow the steps here: +Step 1: Dealer and Partner: Take a regular deck of 52 cards, and remove the face cards and the 10s. +Step 2: Dealer and Partner: Shuffle the remaining cards +Step 3: Dealer and Partner: Select one card each, but keep them face down and don’t look at them yet. +Step 4: Dealer: Look at your card (just the Dealer!). Multiply its value by 2 (Aces = 1). +Step 5: Dealer: Add 2 to this result. +Step 6: Dealer: Multiply your answer by 5. +Step 7: Partner: Look at your card. +Step 8: Partner: Calculate: 10 - your card, and tell this information to the dealer. +Step 9: Dealer: Subtract the value the Partner tells you from your total to get a final answer. +Step 10: Dealer: verbally state the final answer. +Step 11: Dealer and Partner: Turn over your cards. Now, answer the following questions +1. +Did the trick work? How do you know? +2. +Why did this occur? In other words, how does this trick work? +Check Your Understanding +9. Is the solution strategy used in solving the linear equation correct? If it is correct, show the final step (check the +solution). If it is not correct, explain why. +5.2 • Linear Equations in One Variable with Applications +353 + +10. Is the solution strategy used in solving the linear equation correct? If it is correct, show the final step (check the +solution). If it is not correct, explain why. +11. Is the solution strategy used in solving the linear equation correct? If it is correct, show the final step (check the +solution). If it is not correct, explain why. +For the following exercises, use this scenario: The Nice Cab Company charges a flat rate of $3.00 for each fare, plus +$1.70 per mile. A competing taxi service, the Enjoyable Cab Company, charges a flat rate of $5.00 for each fare, plus +$1.60 per mile. +12. Using the variable +for number of miles, write the equation that would allow you to find the total fare +using the Nice Cab Company. +13. It is 22 miles from the airport to your hotel. What would be your total fare using the Nice Cab Company? +14. Using the variable +for number of miles, write the equation that would allow you to find the total fare +using the Enjoyable Cab Company. +15. Using the same 22-mile trip from the airport to the hotel, how much would the total fare be for using the +Enjoyable Cab Company? +16. Based on the cost of each cab ride, which cab company should you use for the trip from the airport to the +hotel? Why? +17. After solving the linear equation +, Nancy says there is no solution. Luis believes +there are infinitely many solutions. Who is right? +18. The conversion formula between the Fahrenheit temperature scale and the Celsius temperature scale is given by +this formula: +, where +is the temperature in degrees Celsius and +is the temperature in degrees +Fahrenheit. What is the correct formula when solved for +? +a. +b. +c. +d. +19. To find a temperature on the Kelvin temperature scale, add 273 degrees to the temperature in Celsius. Which +formula illustrates this? +a. +b. +c. +354 +5 • Algebra +Access for free at openstax.org + +d. +20. Using the information from exercise 18 and exercise 19, which conversion formula would you use to find degrees +Kelvin when given degrees Fahrenheit? +a. +b. +c. +d. +21. There is a fourth temperature scale, although it is not used much today. The Rankin temperature scale varies from +the Fahrenheit scale by about 460 degrees. So given a temperature in Fahrenheit, add 460 degrees to get the +temperature in Rankin. Which formula represents a formula to find degrees Rankin when given degrees Celsius? +a. +b. +c. +d. +SECTION 5.2 EXERCISES +For the following exercises, solve the linear equations using a general strategy. +1. +2. +3. +4. +5. +For the following exercises, solve the linear equations using properties of equations. +6. +7. +8. +9. +10. +For the following exercises, construct a linear equation to solve an application. +11. It costs $0.55 to mail one first class letter. Construct a linear equation and solve to find how much it costs to +mail 13 letters. +12. Normal yearly snowfall at the local ski resort is 12 inches more than twice the amount it received last season. +The normal yearly snowfall is 62 inches. Construct a linear equation and solve to find what the snowfall was last +season. +13. Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was three more than +twice the number of notebooks. He bought seven textbooks. Construct a linear equation and solve to find how +many notebooks he bought. +14. Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is +eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. Construct a linear +equation and solve to find how many crossword puzzles he did. +15. Laurie has $46,000 invested in stocks. The amount invested in stocks is $8,000 less than three times the amount +invested in bonds. Construct a linear equation and solve to find how much Laurie invested in bonds. +For the following exercises, construct an application from a linear equation. +16. +. +17. 0.36 for +. +18. +. +19. +for +. +20. +. +For the following exercises, state whether each equation has exactly one solution, no solution, or infinitely many +solutions. +21. +5.2 • Linear Equations in One Variable with Applications +355 + +22. +23. +24. +25. +26. +For the following exercises, solve the given formula for the specified variable. +27. Solve the formula +for +. +28. Solve the formula +for +. +29. Solve the formula +for +. +30. Solve the formula +for +. +31. Solve the formula +for +. +32. Solve the formula +for +. +33. Solve +for +. +34. Solve the formula +for +. +35. Solve the formula: +for +. +5.3 Linear Inequalities in One Variable with Applications +Figure 5.5 These poll results, showing a margin of error at 4 percent, are an example of a real-world scenario that can be +represented by linear inequalities. +Learning Objectives +After completing this section, you should be able to: +1. +Graph inequalities in one variable. +2. +Solve linear inequalities in one variable. +3. +Construct a linear inequality to solve applications. +In this section, we will study linear inequalities in one variable. Inequalities can be used when the possible values +(answers) in a certain situation are numerous, not just a few, or when the exact value (answer) is not known but it is +known to be within a range of possible values. There are many real-world scenarios that can be represented by linear +inequalities. For example, consider the survey of the mayoral election in Figure 5.5 Surveys and polls are usually +conducted with only a small group of people. The margin of error indicates a range of how the actual group of voters +would vote given the results of the survey. This range can be expressed using inequalities. +Another example involves college tuition. Say a local community college charges $113 per credit hour. You budget $1,500 +for tuition this fall semester. What are the number of credit hours that you could take this fall? Since this answer could be +many different values, it can be expressed as an inequality. +Graphing Inequalities on the Number Line +In Algebraic Expressions, we introduced equality and the +symbol. In this section, we look at inequality and the symbols +, +, +, and +. The table below summarizes the symbols and their meaning. +356 +5 • Algebra +Access for free at openstax.org + +Symbol +Meaning +less than +greater than +less than or equal to +greater than or equal to +Suppose you had the inequality statement +. What possible number or numbers would make the inequality +true? If you are thinking, " +could be 4," that's correct, but +could also be 5, 6, 37, 1 million, or even 3.001. The number +of solutions is infinite; any number greater than 3 is a solution to the inequality +. +Rather than trying to list all possible solutions, we show all the solutions to the inequality +on the number line. All +the numbers to the right of 3 on the number line are shaded, to show that all numbers greater than 3 are solutions. At +the number 3 itself, an open parenthesis is drawn, since the number 3 is not part of the solutions of +. +We can also represent inequalities using interval notation. There is no upper end to the solution to this inequality. In +interval notation, we express +as +. The symbol +is read as "infinity." Infinity is not an actual number. Figure +5.6 shows both the number line and the interval notation for +. +Figure 5.6 The inequality +is graphed on this number line and written in interval notation. +We used the left parenthesis symbol to show that the endpoint of the inequality is not included. Parentheses are used +when the endpoints are not included as a possible answer to the inequality. The notation for inequalities on a number +line and in interval notation use the same symbols to express the endpoints of intervals. +The inequality +means all numbers less than or equal to 1. To illustrate that solution on a number line, we first put a +bracket at +; brackets are used when the endpoint is included. We then shade in all the numbers to the left of 1, to +show that all numbers less than one are solutions. There is no lower end to those numbers. We write +in interval +notation as +. The symbol +is read as "negative infinity." Figure 5.7 shows both the number line and interval +notation for +. +Figure 5.7 The inequality +is graphed on this number line and written in interval notation. +Figure 5.8 summarizes the general representations in both number line form and interval notation of solutions for +, +, +, and +. +5.3 • Linear Inequalities in One Variable with Applications +357 + +Figure 5.8 Summary of representations in number line form and interval notation. +EXAMPLE 5.21 +Graphing an Inequality +Graph the inequality +and write the solution in interval notation. +Solution +Shade to the right of +to show all the numbers greater than +, and put a bracket at +to show that the numbers are +greater than or equal to +(Figure 5.9) +Figure 5.9 +Write in interval notation starting at +with a bracket to show that +is included in the solution and then infinity +because the solution includes all the numbers greater than or equal to +: +YOUR TURN 5.21 +1. Graph the inequality +and write the solution in interval notation. +EXAMPLE 5.22 +Graphing a Compound Inequality +Graph the inequality +and +and write the solution in interval notation. +Solution +Step 1: Graph +(Figure 5.10). +Figure 5.10 +Step 2: Graph +(Figure 5.11). +Figure 5.11 +Step 3: Graph both on the same number line and think of where the solutions are to BOTH inequalities Figure 5.12. This +358 +5 • Algebra +Access for free at openstax.org + +will be where BOTH are shaded. +Figure 5.12 +Step 4: Write the solution in interval notation: +YOUR TURN 5.22 +1. Graph the inequality +and +and write the solution in interval notation. +WHO KNEW? +Where Did the Inequality Symbols Come From? +The first use of the +symbol to represent "less than" and +to represent "greater than" appeared in a mathematics +book written by Englishman Thomas Harriot that was published in 1631. However, Harriot did not invent the +symbols…the editor of the book did! Harriot used triangular symbols to represent less than and greater than; the +editor, for reasons unknown, changed to symbols that are similar to the ones we use today. The symbols used to +represent less than or equal to, and greater than or equal to ( +and +) were first used in 1731 by French hydrologist +and surveyor Pierre Bouguer. Interestingly, English mathematician John Wallis had used similar symbols as early as +1670, but he put the bar above the less than and greater than symbols instead of below them. +Solving Linear Inequalities +A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. A linear +inequality is an inequality in one variable that can be written in one of the forms +or +where +, +, and +are all real numbers. +When we solved linear equations, we were able to use the properties of equality to add, subtract, multiply, or divide both +sides and still keep the equality. Similar properties hold true for inequalities. We can add or subtract the same quantity +from both sides of an inequality and still keep the inequality. For example, we know that 2 is less than 4, i.e., +. If we +add 6 to both sides of this inequality, we still have a true statement: +The same would happen if we subtracted 6 from both sides of the inequality; the statement would stay true: +Notice that the inequality signs stayed the same. This leads us to the Addition and Subtraction Properties of Inequality. +FORMULA +For any numbers +, +, and , if +, then +and +. +For any numbers +, +, and , if +, then +and +. +We can add or subtract the same quantity from both sides of an inequality and still keep the inequality the same. But +what happens to an inequality when we divide or multiply both sides by a number? Let's first multiply and divide both +sides by a positive number, starting with an inequality we know is true, +. We will multiply and divide this +inequality by 5: +5.3 • Linear Inequalities in One Variable with Applications +359 + +The inequality signs stayed the same. Does the inequality stay the same when we divide or multiply by a negative +number? Let's use our inequality +to find out, multiplying it and dividing it by +: +Notice that when we filled in the inequality signs, the inequality signs reversed their direction in order to make it true! To +summarize, when we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we +divide or multiply an inequality by a negative number, the inequality sign reverses. This gives us the Multiplication and +Division Property of Inequality. +FORMULA +For any numbers +, +, and , +multiply or divide by a positive: +if +and +, then +and +if +and +, then +and +multiply or divide by a negative: +if +and +, then +and +if +and +, then +and +To summarize, when we divide or multiply an inequality by: +• +a positive number, the inequality sign stays the same. +• +a negative number, the inequality sign reverses. +Be careful to only reverse the inequality sign when you are multiplying and dividing by a negative. You do NOT +reverse the inequality sign when you add or subtract a negative. For example, +is solved by dividing both +sides of the inequality by 2 to get +. You do NOT reverse the inequality sign because there is a negative 4. As +another example, +is solved by adding +to both sides to get +. This does not reverse the +inequality sign because we were not multiplying or diving by a negative. We then divide both sides by 5 and get +. +EXAMPLE 5.23 +Solving a Linear Inequality Using One Operation +Solve +, graph the solution on the number line, and write the solution in interval notation. +Solution +360 +5 • Algebra +Access for free at openstax.org + +Figure 5.13 +YOUR TURN 5.23 +1. Solve +, graph the solution on the number line, and write the solution in interval notation. +EXAMPLE 5.24 +Solving a Linear Inequality Using Multiple Operations +Solve the inequality +, graph the solution on the number line, and write the solution in interval notation. +Solution +Figure 5.14 +YOUR TURN 5.24 +1. Solve the inequality +, graph the solution on the number line, and write the solution in +interval notation. +Solving Applications with Linear Inequalities +Many real-life situations require us to solve inequalities. The method we will use to solve applications with linear +inequalities is very much like the one we used when we solved applications with equations. We will read the problem +and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to +represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will +solve the inequality. +Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a +whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will +determine whether we round up or down. +EXAMPLE 5.25 +Constructing a Linear Inequality to Solve an Application with Tablet Computers +A teacher won a mini grant of $4,000 to buy tablet computers for their classroom. The tablets they would like to buy cost +$254.12 each, including tax and delivery. What is the maximum number of tablets the teacher can buy? +Solution +Let +5.3 • Linear Inequalities in One Variable with Applications +361 + +times $254.12 has to be less than $4,000, so +. +Solve for : +The teacher can buy 15 tablets and stay under $4,000. +YOUR TURN 5.25 +1. Taleisha’s phone plan costs her $28.80 per month plus $0.20 per text message. How many text messages can she +send/receive and keep her monthly phone bill no more than $50? +EXAMPLE 5.26 +Constructing a Linear Inequality to Solve a Tuition Application +The local community college charges $113 per credit hour. Your budget is $1,500 for tuition this fall semester. What +number of credit hours could you take this fall? +Solution +Let +the number of credit hours you could take. +times $113 has to be less than $1,500, so +. +Solve for : +You can take up to 13 credits and stay under $1,500. +YOUR TURN 5.26 +1. You are awarded a $500 scholarship! In addition to the $1,500 you have saved for tuition, you now have an +additional $500 to spend on credit hours for fall semester. Now, how many credit hours could you take this fall +semester? Assume the cost is still $113 per credit hour. +EXAMPLE 5.27 +Constructing a Linear Inequality to Solve an Application with Travel Costs +Brenda’s best friend is having a destination wedding and the event will last 3 days and 3 nights. Brenda has $500 in +savings and can earn $15 an hour babysitting. She expects to pay $350 for airfare, $375 for food and entertainment, and +$60 a night for her share of a hotel room. How many hours must she babysit to have enough money to pay for the trip? +Solution +Let +number of babysitting hours. +times $15 plus $500 has to be more than +, so +. +Solve for +: +362 +5 • Algebra +Access for free at openstax.org + +Brenda must babysit at least 27 hours. +YOUR TURN 5.27 +1. Malik is planning a 6-day summer vacation trip. He has $840 in savings, and he earns $45 per hour for tutoring. +The trip will cost him $525 for airfare, $780 for food and sightseeing, and $95 per night for the hotel. How many +hours must he tutor to have enough money to pay for the trip? +TECH CHECK +The Desmos activities called "Inequalities on a Number Line" (https://openstax.org/r/Inequalities_on_a) and +"Compound Inequalities on a Number Line" (https://openstax.org/r/Compound_Inequalities) are ways for students to +develop and deepen their understanding of inequalities. Teachers will need a Desmos account to assign the activity +for student use. Once they have assigned the activity to their students, teachers need to share the code for the +activity with their students. Students will input the code (https://openstax.org/r/will_input) to work on the activity. +Check Your Understanding +For the following exercises, choose the correct interval notation for the graph. +22. +a. +b. +c. +d. +e. +23. +a. +b. +c. +d. +e. +24. +a. +b. +c. +d. +e. +25. +a. +b. +c. +5.3 • Linear Inequalities in One Variable with Applications +363 + +d. +e. +26. +is the solution for which inequality? +a. +b. +c. +d. +e. +27. +is the solution for which inequality? +a. +b. +c. +d. +e. +28. +is the solution for which inequality? +a. +b. +c. +d. +e. +29. +is the solution for which inequality? +a. +b. +c. +d. +e. +For the following exercises, choose the equation that best models the situation. +30. Renaldo is hauling boxes of lawn chairs. Each box is the same size, 8 cubic feet. Renaldo’s truck has a capacity of +764 cubic feet. How many boxes of lawn chairs can Renaldo put in his truck? +a. +b. +c. +d. +e. +None of these +31. Bernadette babysits the neighbor’s kids, making on average $50 a night. How many nights will she have to +babysit in order to earn enough money to buy a used car, whose cost is $8,120? +a. +b. +c. +d. +e. +None of these +SECTION 5.3 EXERCISES +For the following exercises, graph the inequality on a number line and write the interval notation. +1. +2. +3. +4. +5. +6. +364 +5 • Algebra +Access for free at openstax.org + +7. +8. +9. +10. +For the following exercises, solve the inequality, graph the solution on the number line, and write the solution in +interval notation. +11. +12. +13. +14. +15. +16. +17. +18. +19. +20. +21. +22. +For the following exercises, construct a linear inequality to solve the application. +23. The elevator in Yehire’s apartment building has a sign that says the maximum weight is 2,100 pounds. If the +average weight of one person is 150 pounds, how many people can safely ride the elevator? +24. Arleen got a $20 gift card for the coffee shop. Her favorite iced drink costs $3.79. What is the maximum number +of drinks she can buy with the gift card? +25. Ryan charges his neighbors $17.50 to wash their car. How many cars must he wash next summer if his goal is to +earn at least $1,500? +26. Kimuyen needs to earn $4,150 per month in order to pay all her expenses. Her sales job pays her $3,475 per +month plus 4 percent of her total sales. What is the minimum Kimuyen’s total sales must be in order for her to +pay all her expenses? +27. Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her +$66,500 plus 15 percent of her total sales. What would her total sales need to be for her salary on the second +offer to be higher than the first? +28. Kiyoshi’s phone plan costs $17.50 per month plus $0.15 per text message. What is the maximum number of text +messages Kiyoshi can use so the phone bill is no more than $56.60? +29. Kellen wants to rent a banquet room in a restaurant for her cousin’s baby shower. The restaurant charges $350 +for the banquet room plus $32.50 per person for lunch. How many people can Kellen have at the shower if she +wants the maximum cost to be $1,500? +30. Noe installs and configures software on home computers. He charges $125 per job. His monthly expenses are +$1,600. How many jobs must he work in order to make a profit of at least $2,400? +5.3 • Linear Inequalities in One Variable with Applications +365 + +5.4 Ratios and Proportions +Figure 5.15 This bar graph shows popular social media app usage. (Source (https://openstax.org /r/media_apps_chart)) +Learning Objectives +After completing this section, you should be able to: +1. +Construct ratios to express comparison of two quantities. +2. +Use and apply proportional relationships to solve problems. +3. +Determine and apply a constant of proportionality. +4. +Use proportions to solve scaling problems. +Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information +from Figure 5.15, we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to +328 M. Note that the "M" stands for million, so 2,006 million is actually 2,006,000,000 and 328 million is 328,000,000. +Similarly, the number of Qzone users compared to the number of Pinterest users is in a ratio of 632 million to 175 +million. These types of comparisons are ratios. +Constructing Ratios to Express Comparison of Two Quantities +Note there are three different ways to write a ratio, which is a comparison of two numbers that can be written as: +to +OR +OR the fraction +. Which method you use often depends upon the situation. For the most part, we will want to +write our ratios using the fraction notation. Note that, while all ratios are fractions, not all fractions are ratios. Ratios +make part to part, part to whole, and whole to part comparisons. Fractions make part to whole comparisons only. +EXAMPLE 5.28 +Expressing the Relationship between Two Currencies as a Ratio +The Euro (€) is the most common currency used in Europe. Twenty-two nations, including Italy, France, Germany, Spain, +Portugal, and the Netherlands use it. On June 9, 2021, 1 U.S. dollar was worth 0.82 Euros. Write this comparison as a +ratio. +Solution +Using the definition of ratio, let +U.S. dollar and let +Euros. Then the ratio can be written as either 1 to 0.82; +or 1:0.82; or +366 +5 • Algebra +Access for free at openstax.org + +YOUR TURN 5.28 +1. On June 9, 2021, 1 U.S. dollar was worth 1.21 Canadian dollars. Write this comparison as a ratio. +EXAMPLE 5.29 +Expressing the Relationship between Two Weights as a Ratio +The gravitational pull on various planetary bodies in our solar system varies. Because weight is the force of gravity acting +upon a mass, the weights of objects is different on various planetary bodies than they are on Earth. For example, a +person who weighs 200 pounds on Earth would weigh only 33 pounds on the moon! Write this comparison as a ratio. +Solution +Using the definition of ratio, let +pounds on Earth and let +pounds on the moon. Then the ratio can be +written as either 200 to 33; or 200:33; or +YOUR TURN 5.29 +1. A person who weighs 170 pounds on Earth would weigh 64 pounds on Mars. Write this comparison as a ratio. +Using and Applying Proportional Relationships to Solve Problems +Using proportions to solve problems is a very useful method. It is usually used when you know three parts of the +proportion, and one part is unknown. Proportions are often solved by setting up like ratios. If +and +are two ratios +such that +then the fractions are said to be proportional. Also, two fractions +and +are proportional +if and only if +. +EXAMPLE 5.30 +Solving a Proportion Involving Two Currencies +You are going to take a trip to France. You have $520 U.S. dollars that you wish to convert to Euros. You know that 1 U.S. +dollar is worth 0.82 Euros. How much money in Euros can you get in exchange for $520? +Solution +Step 1: Set up the two ratios into a proportion; let +be the variable that represents the unknown. Notice that U.S. dollar +amounts are in both numerators and Euro amounts are in both denominators. +Step 2: Cross multiply, since the ratios +and +are proportional, then +. +You should receive +Euros +. +YOUR TURN 5.30 +1. After your trip to France, you have 180 Euros remaining. You wish to convert them back into U.S. dollars. +Assuming the exchange rate is the same +, how many dollars should you receive? Round to the +nearest cent if necessary. +5.4 • Ratios and Proportions +367 + +EXAMPLE 5.31 +Solving a Proportion Involving Weights on Different Planets +A person who weighs 170 pounds on Earth would weigh 64 pounds on Mars. How much would a typical racehorse (1,000 +pounds) weigh on Mars? Round your answer to the nearest tenth. +Solution +Step 1: Set up the two ratios into a proportion. Notice the Earth weights are both in the numerator and the Mars weights +are both in the denominator. +Step 2: Cross multiply, and then divide to solve. +So the 1,000-pound horse would weigh about 376.5 pounds on Mars. +YOUR TURN 5.31 +1. A person who weighs 200 pounds on Earth would weigh only 33 pounds on the moon. A 2021 Toyota Prius +weighs 3,040 pounds on Earth; how much would it weigh on the moon? Round to the nearest tenth if necessary. +EXAMPLE 5.32 +Solving a Proportion Involving Baking +A cookie recipe needs +cups of flour to make 60 cookies. Jackie is baking cookies for a large fundraiser; she is told she +needs to bake 1,020 cookies! How many cups of flour will she need? +Solution +Step 1: Set up the two ratios into a proportion. Notice that the cups of flour are both in the numerator and the amounts +of cookies are both in the denominator. To make the calculations more efficient, the cups of flour +is converted to a +decimal number (2.25). +Step 2: Cross multiply, and then simplify to solve. +Jackie will need 38.25, or +, cups of flour to bake 1,020 cookies. +YOUR TURN 5.32 +1. You are going to bake cookies, using the same recipe as above. You find out that you have 27 cups of flour in +your pantry. Assuming you have all the other ingredients necessary, how many cookies can you make with 27 +368 +5 • Algebra +Access for free at openstax.org + +cups of flour? +Part of the definition of proportion states that two fractions +and +are proportional if +. This is the +"cross multiplication" rule that students often use (and unfortunately, often use incorrectly). The only time cross +multiplication can be used is if you have two ratios (and only two ratios) set up in a proportion. For example, you +cannot use cross multiplication to solve for +in an equation such as +because you do not have just the +two ratios. Of course, you could use the rules of algebra to change it to be just two ratios and then you could use +cross multiplication, but in its present form, cross multiplication cannot be used. +PEOPLE IN MATHEMATICS +Eudoxus was born around 408 BCE in Cnidus (now known as Knidos) in modern-day Turkey. As a young man, he +traveled to Italy to study under Archytas, one of the followers of Pythagoras. He also traveled to Athens to hear +lectures by Plato and to Egypt to study astronomy. He eventually founded a school and had many students. +Eudoxus made many contributions to the field of mathematics. In mathematics, he is probably best known for his +work with the idea of proportions. He created a definition of proportions that allowed for the comparison of any +numbers, even irrational ones. His definition concerning the equality of ratios was similar to the idea of cross +multiplying that is used today. From his work on proportions, he devised what could be described as a method of +integration, roughly 2000 years before calculus (which includes integration) would be fully developed by Isaac Newton +and Gottfried Leibniz. Through this technique, Eudoxus became the first person to rigorously prove various theorems +involving the volumes of certain objects. He also developed a planetary theory, made a sundial still usable today, and +wrote a seven volume book on geography called Tour of the Earth, in which he wrote about all the civilizations on the +Earth, and their political systems, that were known at the time. While this book has been lost to history, over 100 +references to it by different ancient writers attest to its usefulness and popularity. +Determining and Applying a Constant of Proportionality +In the last example, we were given that +cups of flour could make 60 cookies; we then calculated that +cups of +flour would make 1,020 cookies, and 720 cookies could be made from 27 cups of flour. Each of those three ratios is +written as a fraction below (with the fractions converted to decimals). What happens if you divide the numerator by the +denominator in each? +The quotients in each are exactly the same! This number, determined from the ratio of cups of flour to cookies, is called +the constant of proportionality. If the values +and +are related by the equality +then +is the constant of +proportionality between +and +. Note since +then +and +One piece of information that we can derive from the constant of proportionality is a unit rate. In our example (cups of +flour divided by cookies), the constant of proportionality is telling us that it takes 0.0375 cups of flour to make one +cookie. What if we had performed the calculation the other way (cookies divided by cups of flour)? +In this case, the constant of proportionality +is telling us that +cookies can be made with one +cup of flour. Notice in both cases, the "one" unit is associated with the denominator. The constant of proportionality is +also useful in calculations if you only know one part of the ratio and wish to find the other. +EXAMPLE 5.33 +Finding a Constant of Proportionality +Isabelle has a part-time job. She kept track of her pay and the number of hours she worked on four different days, and +recorded it in the table below. What is the constant of proportionality, or pay divided by hours? What does the constant +5.4 • Ratios and Proportions +369 + +of proportionality tell you in this situation? +Pay +$87.50 +$50.00 +$37.50 +$100.00 +Hours +7 +4 +3 +8 +Solution +To find the constant of proportionality, divide the pay by hours using the information from any of the four columns. For +example, +. The constant of proportionality is 12.5, or $12.50. This tells you Isabelle's hourly pay: For every +hour she works, she gets paid $12.50. +YOUR TURN 5.33 +1. The following table contains the lengths of four objects in both inches and centimeters. What is the constant of +proportionality (centimeters divided by inches)? What does the constant of proportionality tell you in this +situation? +Object +floor tile +book +table +pencil +Length (in.) +24 +13 +60 +7.5 +Length (cm) +60.96 +33.02 +152.4 +19.05 +EXAMPLE 5.34 +Applying a Constant of Proportionality: Running mph +Zac runs at a constant speed: 4 miles per hour (mph). One day, Zac left his house at exactly noon (12:00 PM) to begin +running; when he returned, his clock said 4:30 PM. How many miles did he run? +Solution +The constant of proportionality in this problem is 4 miles per hour (or 4 miles in 1 hour). Since +where +is the +constant of proportionality, we have +(30 minutes is +, or +, hours) +, since from the definition we know +Zac ran 18 miles. +YOUR TURN 5.34 +1. One week, Zac ran a total of 122 miles. How much time did he spend running in that week? +370 +5 • Algebra +Access for free at openstax.org + +EXAMPLE 5.35 +Applying a Constant of Proportionality: Filling Buckets +Joe had a job where every time he filled a bucket with dirt, he was paid $2.50. One day Joe was paid $337.50. How many +buckets did he fill that day? +Solution +The constant of proportionality in this situation is $2.50 per bucket (or $2.50 for 1 bucket). Since +where +is the +constant of proportionality, we have +Since we are solving for +, and we know from the definition that +Joe filled 135 buckets. +YOUR TURN 5.35 +1. Suppose one day Joe filled 83 buckets; how much money would he make on that day? +EXAMPLE 5.36 +Applying a Constant of Proportionality: Miles vs. Kilometers +While driving in Canada, Mabel quickly noticed the distances on the road signs were in kilometers, not miles. She knew +the constant of proportionality for converting kilometers to miles was about 0.62—that is, there are about 0.62 miles in 1 +kilometer. If the last road sign she saw stated that Montreal is 104 kilometers away, about how many more miles does +Mabel have to drive? Round your answer to the nearest tenth. +Solution +The constant of proportionality in this situation is 0.62 miles per 1 kilometer. Since +where +is the constant of +proportionality, we have +Rounding the answer to the nearest tenth, Mabel has to drive 64.5 miles. +YOUR TURN 5.36 +1. Later in her trip, Mabel decides to drive to the capital of Canada, Ottawa. As she left Montreal, she saw a road +sign that read that Ottawa is 203 kilometers away. About how many miles is that? Round your answer to the +nearest tenth. +5.4 • Ratios and Proportions +371 + +Using Proportions to Solve Scaling Problems +Figure 5.16 A map of the northeastern United States +Ratio and proportions are used to solve problems involving scale. One common place you see a scale is on a map (as +represented in Figure 5.16). In this image, 1 inch is equal to 200 miles. This is the scale. This means that 1 inch on the +map corresponds to 200 miles on the surface of Earth. Another place where scales are used is with models: model cars, +trucks, airplanes, trains, and so on. A common ratio given for model cars is 1:24—that means that 1 inch in length on the +model car is equal to 24 inches (2 feet) on an actual automobile. Although these are two common places that scale is +used, it is used in a variety of other ways as well. +EXAMPLE 5.37 +Solving a Scaling Problem Involving Maps +Figure 5.17 is an outline map of the state of Colorado and its counties. If the distance of the southern border is 380 +miles, determine the scale (i.e., 1 inch = how many miles). Then use that scale to determine the approximate lengths of +the other borders of the state of Colorado. +Figure 5.17 Outline Map of Colorado (credit: "Map of Colorado Counties" by David Benbennick/Wikimedia Commons, +372 +5 • Algebra +Access for free at openstax.org + +Public Domain) +Solution +When the southern border is measured with a ruler, the length is 4 inches. Since the length of the border in real life is +380 miles, our scale is 1 inch +miles. +The eastern and western borders both measure 3 inches, so their lengths are about 285 miles. The northern border +measures the same as the southern border, so it has a length of 380 miles. +YOUR TURN 5.37 +1. +Outline Map of Wyoming (credit: "Blank map of Wyoming showing counties" by David Benbennick/Wikimedia +Commons, Public Domain) +Consider the outline map of the state of Wyoming and its counties. If the distance of the southern border is 365 +miles, determine the scale (i.e., +how many miles). Then use that scale to determine the approximate +lengths of the other borders of the state of Wyoming. +EXAMPLE 5.38 +Solving a Scaling Problem Involving Model Cars +Die-cast NASCAR model cars are said to be built on a scale of 1:24 when compared to the actual car. If a model car is 9 +inches long, how long is a real NASCAR automobile? Write your answer in feet. +Solution +The scale tells us that 1 inch of the model car is equal to 24 inches (2 feet) on the real automobile. So set up the two +ratios into a proportion. Notice that the model lengths are both in the numerator and the NASCAR automobile lengths +are both in the denominator. +This amount (216) is in inches. To convert to feet, divide by 12, because there are 12 inches in a foot (this conversion from +inches to feet is really another proportion!). The final answer is: +5.4 • Ratios and Proportions +373 + +The NASCAR automobile is 18 feet long. +YOUR TURN 5.38 +1. A toy Jeep is built on a +scale. The website for the toy Jeep says the toy is +inches long. Based on this, +how long is the real Jeep? +Check Your Understanding +32. If +, then +for all non-zero whole numbers +, +, , and +. +a. +True +b. +False +33. If the ratio of wolves to rabbits in a national park is +, then the ratio of rabbits to (wolves and rabbits) is +. +a. +True +b. +False +34. All fractions are ratios but not all ratios are fractions. +a. +True +b. +False +35. In the following equation, +, cross multiplication can be used as the first step towards solving for +. +a. +True +b. +False +36. All fractions are ratios but not all ratios are fractions. +a. +True +b. +False +37. There are 16 math majors and 12 non-math majors in Ms. Kraft’s class. What is not a correct way to express the +ratio of math majors to non-math majors? +38. There are 16 math majors and 12 non-math majors in Ms. Kraft’s class. What shows the ratio of math majors to all +the students in Ms. Kraft’s class? +None of these +39. One U.S. dollar is worth +British pounds. Damon is traveling to Great Britain and wishes to exchange $450 U.S. +dollars for British pounds. How many British pounds should Damon get in return? +625 +6,250 +3,456 +345.6 +None of these +40. The HO scale for model trains is the most common size of model trains. This scale is +. If a real locomotive is +73 feet long, how long should the model locomotive be (in inches)? Round your answer to the nearest inch. +41. Albert’s Honda Civic gets 37 miles per gallon of gasoline. The gas tank on the Civic can hold +gallons of gas. +Albert is driving from Tucson, Arizona to Los Angeles, California, a distance of 485 miles. Albert thinks he can make +it on one full tank of gasoline. Can he? Explain. +374 +5 • Algebra +Access for free at openstax.org + +42. The average price of a gallon of regular gasoline in the California on July 1, 2021 was +per gallon. Albert stops +at a gas station in California and puts 9.5 gallons of gasoline into his Civic. How much did he pay for the gas? +SECTION 5.4 EXERCISES +For the following exercises, use this scenario: Kelly opened a bag of colored chocolate coated candies and counted the +number of each color of candy. She found she had 9 green, 4 yellow, 13 black, 11 orange, 8 blue, and 7 red. What is the +ratio of the following candy colors? +1. Red candies to green candies +2. Green candies to black candies +3. Yellow candies to black candies +4. Black candies to blue candies +5. Orange candies to non-orange candies +6. Yellow candies to non-yellow candies +7. Red candies to all candies +8. Pink candies to all candies +9. Candies with the letter ‘r’ in their name to all candies +10. Candies with the letter ‘r’ in their name to candies without the letter ‘r’ in their name +For the following exercises, solve each proportion for the unknown variable. +11. +12. +13. +14. +15. +16. +17. +18. +19. +(Round answer to the nearest hundredth.) +20. +(Round answer to the nearest hundredth.) +21. Pet Paradise has 20 cats and 16 dogs. Animal Acres has 15 cats. How many dogs must be at Animal Acres so that +Pet Paradise and Animal Acres have the same ratio of cats to dogs? +22. Pet Paradise has 20 cats and 16 dogs. Critter Corral has 28 dogs. How many cats must be at Critter Corral so that +Pet Paradise and Critter Corral have the same ratio of cats to dogs? +23. A high school has 960 students. The ratio of students to high school teachers is +. How many high school +teachers are at the school? +24. A high school has 960 students. The ratio of students to high school teachers is +. How many more teachers +are needed to have a +ratio at the high school of students to teachers? +25. One U.S. dollar is worth $1.23 Canadian dollars. Bernice is traveling to Canada and wants to convert $550 U.S. to +Canadian money. How much in Canadian money should she receive? +26. One U.S. dollar is worth $1.23 Canadian dollars. Rene is traveling from Canada to the United States and wants to +convert $550 of Canadian money to U.S. money. How much in U.S. money should he receive? Round your answer +to the nearest cent. +27. One U.S. dollar is worth $1.23 Canadian dollars. What is one Canadian dollar worth in U.S. funds? Round your +answer to the nearest cent. +28. A salad recipe needs one cup of crushed almonds. It will serve eight people. Rashida needs to make a salad to +serve 20 people. How many cups of crushed almonds does she need? +29. A salad recipe needs one cup of crushed almonds. It will serve eight people. Elmer has 4.75 cups of crushed +almonds. If he uses all of the crushed almonds he has to make this salad, how many people will it serve? +30. Jorge is 6 feet tall and casts a 7-foot shadow. At the same time, a nearby tree has a shadow of 56 feet. How tall is +the tree? +5.4 • Ratios and Proportions +375 + +31. Tony can run 4 kilometers in 30 minutes. At that rate, how far could he run in 1 hour, 45 minutes? +32. Kara’s parent owns a restaurant. When she came in one day, they asked her to figure out how much they were +spending per ounce on steak they were buying from a vendor. They had their last four receipts, but unfortunately +they spilled liquid on them and some parts were unreadable. Find out how much Kara’s parent is spending per +ounce on steak; then use that information to fill in the unreadable parts of the receipts (labeled +, +, and +below). +Receipt +1 +2 +3 +4 +Ounces +128 +460 +541 +Cost +$163.84 +$277.76 +33. The scale for a map reads “ +.” You measure the distance on the map from Fargo, North Dakota +to Winnipeg, Manitoba and get +inches. How far is it from Fargo to Winnipeg? +34. Hot Wheels toy cars are said to be built on a scale of +when compared to the actual car. If a real car is 18 feet +long, how long should the Hot Wheels toy car be (in inches)? +35. The Eiffel Tower in Paris, France, is 1,067 feet tall. The replica Eiffel Tower in Las Vegas, Nevada, is built on the scale +of +. How tall is the replica Eiffel Tower in Las Vegas? Round your answer to the nearest foot. +5.5 Graphing Linear Equations and Inequalities +Figure 5.18 How much would it cost to fill up your gas tank? (credit: "Gas Under 4 Bucks" by Mark Turnauckas, Flickr/CC +BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Graph linear equations and inequalities in two variables. +2. +Solve applications of linear equations and inequalities. +In this section, we will learn how to graph linear equations and inequalities. There are several real-world scenarios that +can be represented by graphs of linear inequalities. Think of filling your car up with gasoline. If gasoline is $3.99 per +gallon and you put 10 gallons in your car, you will pay $39.90. Your friend buys 15 gallons of gasoline and pays $59.85. +You can plot these points on a coordinate system and connect the points with a line to create the graph of a line. You'll +learn to do both in this section. +Plotting Points on a Rectangular Coordinate System +Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between +two variables in a rectangular coordinate system. The rectangular coordinate system is also called the +-plane or the +376 +5 • Algebra +Access for free at openstax.org + +“coordinate plane.” +The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The +horizontal number line is called the +-axis. The vertical number line is called the +-axis. These axes divide a plane into +four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and +proceeding counterclockwise. See Figure 5.19. +Figure 5.19 Quadrants on the Coordinate Plane +In the rectangular coordinate system, every point is represented by an ordered pair (Figure 5.20). The first number in +the ordered pair is the +-coordinate of the point, and the second number is the +-coordinate of the point. The phrase +"ordered pair" means that the order is important. At the point where the axes cross and where both coordinates are +zero, the ordered pair is +. The point +has a special name. It is called the origin. +Figure 5.20 Ordered Pair +We use the coordinates to locate a point on the +-plane. Let's plot the point +as an example. First, locate 1 on the +-axis and lightly sketch a vertical line through +. Then, locate 3 on the +-axis and sketch a horizontal line through +. Now, find the point where these two lines meet—that is the point with coordinates +. See Figure 5.21. +5.5 • Graphing Linear Equations and Inequalities +377 + +Figure 5.21 Point +Plotted on the Coordinate Plane +Notice that the vertical line through +and the horizontal line through +are not part of the graph. The dotted +lines are just used to help us locate the point +. When one of the coordinates is zero, the point lies on one of the +axes. In Figure 5.22, the point +is on the +-axis and the point (−2, 0) is on the +-axis. +Figure 5.22 Points +and +Plotted on the Coordinate Plane +EXAMPLE 5.39 +Plotting Points on a Coordinate System +Plot the following points in the rectangular coordinate system and identify the quadrant in which the point is located: +1. +2. +3. +4. +5. +378 +5 • Algebra +Access for free at openstax.org + +Solution +The first number of the coordinate pair is the +-coordinate, and the second number is the +-coordinate. To plot each +point, sketch a vertical line through the +-coordinate and a horizontal line through the +-coordinate (Figure 5.23). Their +intersection is the point. +1. +Since +, the point is to the left of the +-axis. Also, since +, the point is above the +-axis. The point +is +in quadrant II. +2. +Since +, the point is to the left of the +-axis. Also, since +, the point is below the +-axis. The point +is in quadrant III. +3. +Since +, the point is to the right of the +-axis. Since +, the point is below the +-axis. The point +is in +quadrant IV. +4. +Since +, the point whose coordinates are +is on the +-axis. +5. +Since +, the point is to the right of the +-axis. Since +, which is equal to 2.5, the point is above the +-axis. +The point +is in quadrant I. +Figure 5.23 +YOUR TURN 5.39 +1. Plot the following points in the rectangular coordinate system and identify the quadrant in which the point is +located: +a. +b. +c. +d. +e. +Graphing Linear Equations in Two Variables +Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you +solved the equation, you got exactly one solution. But equations can have more than one variable. Equations with two +variables may be of the form +. An equation of this form, where +and +are both not zero, is called a linear +equation in two variables. Here is an example of a linear equation in two variables, +and +. +5.5 • Graphing Linear Equations and Inequalities +379 + +The equation +is also a linear equation. But it does not appear to be in the form +. We can use +the addition property of equality and rewrite it in +form. +Step 1: Add +to both sides. +Step 2: Simplify. +Step 3: Put it in +form. +By rewriting +as +, we can easily see that it is a linear equation in two variables because it is of the +form +. When an equation is in the form +, we say it is in standard form of a linear equation. +Most people prefer to have +, +, and +be integers and +when writing a linear equation in standard form, +although it is not strictly necessary. +Linear equations have infinitely many solutions. For every number that is substituted for +there is a corresponding +value. This pair of values is a solution to the linear equation and is represented by the ordered pair ( , ). When we +substitute these values of +and +into the equation, the result is a true statement, because the value on the left side is +equal to the value on the right side. +We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We +connect the points with a straight line to get the graph of the linear equation. We put arrows on the ends of each side of +the line to indicate that the line continues in both directions. +A graph is a visual representation of all the solutions of a linear equation. The line shows you all the solutions to that +linear equation. Every point on the line is a solution of that linear equation. And every solution of the linear equation is +on this line. This line is called the graph of the equation. Points not on the line are not solutions! The graph of a linear +equation +is a straight line. +• +Every point on the line is a solution of the equation. +• +Every solution of this equation is a point on this line. +EXAMPLE 5.40 +Determining Points on a Line +Figure 5.24 is the graph of +. +Figure 5.24 Graph of +380 +5 • Algebra +Access for free at openstax.org + +For each ordered pair, decide: +I. +Is the ordered pair a solution to the equation? +II. +Is the point on the line? +Solution +Substitute the +- and +-values into the equation to check if the ordered pair is a solution to the equation. +I. +II. +Plot the points +, +, +, and +. +Figure 5.25 +In Figure 5.25, the points +, +, and +are on the line +, and the point +is not on the line. +The points that are solutions to +are on the line, but the point that is not a solution is not on the line. +YOUR TURN 5.40 +The given figure is the graph of +. +5.5 • Graphing Linear Equations and Inequalities +381 + +Graph of +For each ordered pair below, decide: +I. +Is the ordered pair a solution to the equation? +II. +Is the point on the line? +1. +2. +3. +4. +The steps to take when graphing a linear equation by plotting points are: +Step 1: Find three points whose coordinates are solutions to the equation. Organize them in a table. +Step 2: Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check +your work. +Step 3: Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. +It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two +points and one of them is incorrect, you can still draw a line, but it will not represent the solutions to the equation. It will +be the wrong line. If you use three points, and one is incorrect, the points will not line up. This tells you something is +wrong, and you need to check your work. +EXAMPLE 5.41 +Graphing a Line by Plotting Points +Graph the equation: +. +Solution +Find three points that are solutions to the equation. Since this equation has the fraction +as a coefficient of +, we will +choose values of +carefully. We will use zero as one choice and multiples of 2 for the other choices. Why are multiples of +two a good choice for values of +? By choosing multiples of 2, the multiplication by +simplifies to a whole number. +382 +5 • Algebra +Access for free at openstax.org + +( , +) +0 +3 +2 +4 +4 +5 +Plot the points, check that they line up, and draw the line (Figure 5.26). +Figure 5.26 +YOUR TURN 5.41 +1. Graph the equation +by plotting points. +Solving Applications Using Linear Equations in Two Variables +Many fields use linear equalities to model a problem. While our examples may be about simple situations, they give us +an opportunity to build our skills and to get a feel for how they might be used. +EXAMPLE 5.42 +Pumping Gas +Gasoline costs $3.53 per gallon. You put 10 gallons of gasoline in your car, and pay $35.30. Your friend puts 15 gallons of +5.5 • Graphing Linear Equations and Inequalities +383 + +gasoline in their car and pays $52.95. Your neighbor needs 5 gallons of gasoline, how much will they pay? +Solution +Let +and let +. If gas is $3.53 per gallon, then +. The two +points given are (10, 35.30) and (15, 52.95). Plot the points, check that they line up, and draw the line (Figure 5.27). +Figure 5.27 +We can see the point at +. The +-value is found by multiplying 5 by $3.53 to get $17.65. Your neighbor will pay +$17.65. +YOUR TURN 5.42 +1. If a stamp costs $0.55 and you buy a book of 20 stamps, then you pay $11. If you want to mail 100 letters, you +can buy a roll of stamps for $55. Your friend only needs 3 stamps, how much will they pay? +384 +5 • Algebra +Access for free at openstax.org + +PEOPLE IN MATHEMATICS +René Descartes +Figure 5.28 René Descartes (credit: Flickr, Public Domain) +René Descartes was born in 1596 in La Haye, France. He was sickly as a child, so much so that he was allowed to stay +in bed until 11:00 AM rather than get up at 5:00 AM like the other school children. He kept this habit of rising late for +most of the rest of his life. +After his primary schooling, Descartes attended the University of Poitiers, receiving a law degree in 1616. He then +embarked on a myriad of journeys, joining two different militaries (one in the Netherlands, the other in Bavaria) and +generally travelling around Europe until 1628, when he settled in the Netherlands. It was here that he began to delve +deeply into his ideas of science, mathematics, and philosophy. +In 1637, at the urging of his friends, Descartes published Discourse on the Method for Conducting One's Reason Well +and Seeking the Truth in the Sciences. The book had three appendices: La Dioptrique, a work on optics; Les Météores, +which pertained to meteorology; and La Géométrie, a work on mathematics. It was in this appendix that he proposed +a geometric way of representing many different algebraic expressions and equations. It is this system of +representation that almost all mathematical textbooks use today. +These publications (along with several others) brought much fame to Descartes. So renowned was his reputation that +late in 1649, Queen Christina of Sweden asked Descartes to come to Sweden to tutor her. However, she wished to do +her studies at 5:00 in the morning; Descartes had to break his lifelong habit of sleeping in late. A few months later, in +February 1650, Descartes died of pneumonia. +Graphing Linear Inequalities +Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two +variables that can be written in one of the following forms: +, +, +, and +where +and +are not both zero. We will look at linear inequalities in two variables, which are very +similar to linear equations in two variables. +Like linear equations, linear inequalities in two variables have many solutions. Any ordered pair ( , +) that makes an +inequality true when we substitute in the values is a solution to a linear inequality. +5.5 • Graphing Linear Equations and Inequalities +385 + +EXAMPLE 5.43 +Determining Solutions to an Inequality +Determine whether each ordered pair is a solution to the inequality +: +1. +2. +3. +4. +5. +Solution +1. +2. +3. +4. +5. +YOUR TURN 5.43 +Determine whether each ordered pair is a solution to the inequality +1. +2. +3. +4. +5. +Let us think about +. The point +separated that number line into two parts. On one side of 3 are all the +numbers less than 3. On the other side of 3 all the numbers are greater than 3. See Figure 5.29. +386 +5 • Algebra +Access for free at openstax.org + +Figure 5.29 Solution to +on a Number Line +Similarly, the line +separates the plane into two regions. On one side of the line are points with +. On +the other side of the line are the points with +. We call the line +a boundary line. +For an inequality in one variable, the endpoint is shown with a parenthesis (Figure 5.30) or a bracket (Figure 5.31) +depending on whether or not +is included in the solution: +Figure 5.30 Endpoint with Parenthesis +Figure 5.31 Endpoint with Bracket +Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or +not it the line is included in the solution. +Boundary line is +Boundary line is +Boundary line is not included in solution. +Boundary line is included in solution. +Boundary line is dashed. +Boundary line is solid. +Now, let us take a look at what we found in Example 5.43. We will start by graphing the line +, and then we will +plot the five points we tested, as graphed in Figure 5.32. We found that some of the points were solutions to the +inequality +and some were not. Which of the points we plotted are solutions to the inequality +? The +points +and +are solutions to the inequality +. Notice that they are both on the same side of the +boundary line +. The two points +and +are on the other side of the boundary line +, and +they are not solutions to the inequality +. For those two points, +. What about the point +? Because +, the point is a solution to the equation +, but not a solution to the inequality +. So, the point +is on the boundary line. +5.5 • Graphing Linear Equations and Inequalities +387 + +Figure 5.32 Graph of +Let us take another point above the boundary line and test whether or not it is a solution to the inequality +. The +point +clearly looks to be above the boundary line, doesn’t it? Is it a solution to the inequality? +Yes, +is a solution to +. Any point you choose above the boundary line is a solution to the inequality +. All points above the boundary line are solutions. Similarly, all points below the boundary line, the side with +and +, are not solutions to +, as shown in Figure 5.33. +Figure 5.33 Graph of +, with +Above the Boundary Line and +Below the Boundary Line +The graph of the inequality +is shown in Figure 5.34. The line +divides the plane into two regions. The +388 +5 • Algebra +Access for free at openstax.org + +shaded side shows the solutions to the inequality +. The points on the boundary line, those where +, are +not solutions to the inequality +, so the line itself is not part of the solution. We show that by making the +boundary line dashed, not solid. +Figure 5.34 Graph of +EXAMPLE 5.44 +Writing a Linear Inequality Shown by a Graph +The boundary line shown in this graph is +. Write the inequality shown in Figure 5.35. +5.5 • Graphing Linear Equations and Inequalities +389 + +Figure 5.35 +Solution +The line +is the boundary line. On one side of the line are the points with +and on the other side of +the line are the points with +. Let us test the point +and see which inequality describes its position relative +to the boundary line. At +, which inequality is true: +or +? +True +False +Since +is true, the side of the line with +, is the solution. The shaded region shows the solution of the +inequality +. Since the boundary line is graphed with a dashed line, the inequality does not include the equal +sign. The graph shows the inequality +. +We could use an +point as a test point, provided it is not on the line. Why did we choose +? Because it is the easiest +to evaluate. You may want to pick a point on the other side of the boundary line and check that +. +YOUR TURN 5.44 +1. Write the inequality shown by the graph with the boundary line +. +390 +5 • Algebra +Access for free at openstax.org + +EXAMPLE 5.45 +Graphing a Linear Inequality +Graph the linear inequality +. +5.5 • Graphing Linear Equations and Inequalities +391 + +Solution +Step 1. Identify and graph +the boundary line (Figure +5.36). +• +If the inequality is ≤ or ≥, +the boundary line is solid. +• +If the inequality is < or >, the +boundary line is dashed. +Replace the inequality +sign with an equal sign +to find the boundary +line. +Graph the boundary line +. +The inequality sign is ≥, +so we draw a solid line. +Figure 5.36 +Step 2. Test a point that is +not on the boundary line. Is +it a solution of the +inequality? +We’ll test +. +Is it a solution of the +inequality? +At +, is +? +So, +is a solution. +Step 3. Shade in one side of +the boundary line (Figure +5.37). +• +If the test point is a solution, +shade in the side that +includes the point. +• +If the test point is not a +solution, shade in the +opposite side. +The test point +is a +solution to +. +So we shade in that side. +Figure 5.37 +All points in the shaded region and on the boundary line +represent the solution to +. +392 +5 • Algebra +Access for free at openstax.org + +YOUR TURN 5.45 +1. Graph the linear inequality: +. +VIDEO +Graphing Linear Inequalities in Two Variables (https://openstax.org/r/Graphing_linear) +Solving Applications Using Linear Inequalities in Two Variables +Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us +an opportunity to build our skills and to get a feel for how they might be used. +EXAMPLE 5.46 +Working Multiple Jobs +Hilaria works two part time jobs to earn enough money to meet her obligations of at least $240 a week. Her job in food +service pays $10 an hour and her tutoring job on campus pays $15 an hour. How many hours does Hilaria need to work +at each job to earn at least $240? +1. +Let +be the number of hours she works at the job in food service and let +be the number of hours she works +tutoring. Write an inequality that would model this situation. +2. +Graph the inequality. +3. +Find three ordered pairs ( +) that would be solutions to the inequality. Then, explain what that means for Hilaria. +Solution +1. +Let +be the number of hours she works at the job in food service and let +be the number of hours she works +tutoring. She earns $10 per hour at the job in food service and $15 an hour tutoring. At each job, the number of +hours multiplied by the hourly wage will give the amount earned at that job. +2. +Graph the inequality: +Step 1: Graph the boundary line +Create a table of values +0 +6 +12 +Step 2: Pick a test point. Let us pick +again: +? +is false and not a solution so the shading happens on the other side of the boundary line (Figure 5.38). +5.5 • Graphing Linear Equations and Inequalities +393 + +Figure 5.38 +3. +From the graph, we see that the ordered pairs +, +, +represent three of infinitely many solutions. +Check the values in the inequality. +For Hilaria, it means that to earn at least $240, she can work 15 hours tutoring and 10 hours at her food service job, earn +all her money tutoring for 16 hours, or earn all her money while working 24 hours at the job in food service. +YOUR TURN 5.46 +Harrison works two part time jobs. One at a gas station that pays $11 an hour and the other is as an IT consultant +for $16.50 an hour. Between the two jobs, Harrison wants to earn at least $330 a week. How many hours does +Harrison need to work at each job to earn at least $330? +1. Let +be the number of hours he works at the gas station and let +be the number of hours he works as an IT +consultant. Write an inequality that would model this situation. +2. Graph the inequality. +3. Find three ordered pairs ( , +) that would be solutions to the inequality. Then, explain what that means for +Harrison. +Check Your Understanding +43. Choose the correct solution to the equation +. +a. +b. +c. +d. +394 +5 • Algebra +Access for free at openstax.org + +44. Choose the correct graph for +. +5.5 • Graphing Linear Equations and Inequalities +395 + +45. Choose the correct equation for the graph shown: +a. +b. +c. +d. +46. Choose the correct graph for +. +396 +5 • Algebra +Access for free at openstax.org + +47. Choose the correct inequality for the graph shown. +a. +b. +c. +d. +SECTION 5.5 EXERCISES +1. Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located. +a. +b. +c. +d. +e. +5.5 • Graphing Linear Equations and Inequalities +397 + +For each ordered pair below, decide: +I. +Is the ordered pair a solution to the equation? +II. +Is the point on the line in the given graph? +2. +3. +4. +5. +For each ordered pair below, decide: +I. +Is the ordered pair a solution to the equation? +II. +Is the point on the line in the given graph? +398 +5 • Algebra +Access for free at openstax.org + +6. +7. +8. +9. +For each ordered pair below, decide: +I. +Is the ordered pair a solution to the equation? +II. +Is the point on the line in the given graph? +5.5 • Graphing Linear Equations and Inequalities +399 + +10. +11. +12. +13. +For the following exercises, graph by plotting points. +14. +15. +16. +17. +18. +19. +20. +For the following exercises, determine whether each ordered pair is a solution to the inequality. +21. +22. +23. +24. +25. +26. Write the inequality shown by the graph with the boundary line +. +400 +5 • Algebra +Access for free at openstax.org + +27. Write the inequality shown by the graph with the boundary line +. +28. Write the inequality shown by the graph with the boundary line +. +29. Write the inequality shown by the graph with the boundary line +. +5.5 • Graphing Linear Equations and Inequalities +401 + +30. Write the inequality shown by the shaded region in the graph with the boundary line +. +31. Write the inequality shown by the shaded region in the graph with the boundary line +. +32. Write the inequality shown by the shaded region in the graph with the boundary line +. +402 +5 • Algebra +Access for free at openstax.org + +33. Write the inequality shown by the shaded region in the graph with the boundary line +. +For the following exercises, graph the linear inequality. +34. +35. +36. +37. +38. +39. +40. +41. +42. +43. +44. +5.5 • Graphing Linear Equations and Inequalities +403 + +5.6 Quadratic Equations with Two Variables with Applications +Figure 5.39 The Gateway Arch in St. Louis, Missouri (credit: modification of work "Gateway Arch - St. Louis - Missouri" by +Sam valadi/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Multiply binomials. +2. +Factor trinomials. +3. +Solve quadratic equations by graphing. +4. +Solve quadratic equations by factoring. +5. +Solve quadratic equations using square root method. +6. +Solve quadratic equations using the quadratic formula. +7. +Solve real world applications modeled by quadratic equations. +In this section, we will discuss quadratic equations. There are several real-world scenarios that can be represented by the +graph of a quadratic equation. Think of the Gateway Arch in St. Louis, Missouri. Both ends of the arch are 630 feet apart +and the arch is 630 feet tall. You can plot these points on a coordinate system and create a parabola to graph the +quadratic equation. +Identify Polynomials, Monomials, Binomials and Trinomials +You have learned that a term is a constant, or the product of a constant and one or more variables. When it is of the +form +, where +is a constant and +is a positive whole number, it is called a monomial. Some examples of +monomial are 8, +, +, and +. +A monomial or two or more monomials combined by addition or subtraction is a polynomial. Some examples include: +, +, and +. Some polynomials have special names, based on the number of +terms. A monomial is a polynomial with exactly one term (examples: 14, +, +, and +). A binomial has exactly +two terms (examples: +, +, +, and +), and a trinomial has exactly three terms (examples: +, +, +, and +). +Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” +of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to +these special polynomials and just call all the rest polynomials. +Multiply Binomials +Recall multiplying algebraic expressions from Algebraic Expressions. In this section, we will continue that work and +multiply binomials as well. We can use an area model to do multiplication. +404 +5 • Algebra +Access for free at openstax.org + +EXAMPLE 5.47 +Multiply Binomials +Multiply +. +Solution +Step 1: Use the distributive property: +In the area model (Figure 5.40) multiply each term on the side by each term on the top (think of it as a multiplication +table). +Figure 5.40 +Step 2: After we multiply, we get the following equation: +Step 3: Combine the like terms to arrive at: +YOUR TURN 5.47 +1. Multiply +. +EXAMPLE 5.48 +Multiplying More Complex Binomials +Multiply +. +Solution +Step 1: Use the Distributive Property: +Figure 5.41 +Step 2: After multiplying, get the following equation: +Step 3: Combine the like terms to arrive at: +5.6 • Quadratic Equations with Two Variables with Applications +405 + +YOUR TURN 5.48 +1. Multiply +. +WHO KNEW? +They Are Teaching Multiplication of Binomials in Elementary School +Manipulatives are often used in elementary school for students to experience a hands-on way to experience the +mathematics they are learning. Base Ten Blocks, or Dienes Blocks, are often used to introduce place value and the +operation of whole numbers. When multiplying two-digit numbers, students can make an array to visualize the +Distributive Property. Figure 5.42 shows the value of each Base Ten Block and Figure 5.43 shows how to multiply 17 +and 23 using an area model and Base Ten Blocks. You can see how this helps students visualize the multiplication +using the Distributive Property. Consider how +can easily extend to +in algebra! +Figure 5.42 The Value of Each Base Ten Block +Figure 5.43 How to Multiply 17 and 23 Using an Area Model and Base Ten Blocks +Factoring Trinomials +We’ve just covered how to multiply binomials. Now you will need to “undo” this multiplication—to start with the product +and end up with the factors. Let us review an example of multiplying binomials to refresh your memory. +To factor the trinomial means to start with the product, +, and end with the factors, +. You need +to think about where each of the terms in the trinomial came from. The first term came from multiplying the first term in +each binomial. So, to get +in the product, each binomial must start with an +. +  +  +The last term in the trinomial came from multiplying the last term in each binomial. So, the last terms must multiply to 6. +What two numbers multiply to 6? The factors of 6 could be 1 and 6, or 2 and 3. How do you know which pair to use? +Consider the middle term. It came from adding the outer and inner terms. So the numbers that must have a product of 6 +will need a sum of 5. +406 +5 • Algebra +Access for free at openstax.org + +We’ll test both possibilities and summarize the results in the following table, which will be very helpful when you work +with numbers that can be factored in many different ways. +Factors of 6 +Sum of Factors +1, 6 +2, 3 +We see that 2 and 3 are the numbers that multiply to 6 and add to 5. We have the factors of +. They are +. +You can check if the factors are correct by multiplying. Looking back, we started with +, which is of the form +, where +and +. We factored it into two binomials of the form +and +. +To get the correct factors, we found two number +and +whose product is +and sum is +. With the area model (Figure +5.44), start with an empty box and then put in the +term and . +Figure 5.44 +Continue by putting in two terms that add up to +and +(Figure 5.45): +Figure 5.45 +Then you find the terms of the binomials on the top and side (Figure 5.46): +Figure 5.46 +EXAMPLE 5.49 +Factoring Trinomials +Factor +. +Solution +The numbers that must have a product of 12 will need a sum of 7. We will summarize the results in a table below. +5.6 • Quadratic Equations with Two Variables with Applications +407 + +Factors of 12 +Sum of Factors +1, 12 +2, 6 +3, 4 +We see that 3 and 4 are the numbers that multiply to 12 and add to 7. The factors of +are +. +YOUR TURN 5.49 +1. Factor +. +EXAMPLE 5.50 +Factoring More Complex Trinomials +Factor +. +Solution +The numbers that must have a product of 28 will need a sum of +. We will summarize the results in a table. +Factors of 28 +Sum of Factors +We see that 4 and 7 are the numbers that multiply to 28 and add to 11. But we needed +, so we will need to use +and +because +and +. The factors of +are +. +YOUR TURN 5.50 +1. Factor: +. +VIDEO +Factoring with the Box Method (Area Model) (https://openstax.org/r/Factoring_with_the_Box) +Solving Quadratic Equations by Graphing +We have already solved and graphed linear equations in Graphing Linear Equations and Inequalities, equations of the +form +. In linear equations, the variables have no exponents. Quadratic equations are equations in which +the variable is squared. The following are some examples of quadratic equations: +The last equation does not appear to have the variable squared, but when we simplify the expression on the left, we will +get +. The general form of a quadratic equation is +, where +and +are real numbers, with +. +408 +5 • Algebra +Access for free at openstax.org + +Remember that a solution of an equation is a value of a variable that makes a true statement when substituted into the +equation. The solutions of quadratic equations are the values of the variables that make the quadratic equation +true. +To solve quadratic equations, we need methods different than the ones we used in solving linear equations. We will start +by solving a quadratic equation from its graph. Just like we started graphing linear equations by plotting points, we will +do the same for quadratic equations. Let us look first at graphing the quadratic equation +. We will choose integer +values of +between +and 2 and find their +values, as shown in the table below. +0 +0 +1 +1 +1 +2 +4 +4 +Notice when we let +and +, we got the same value for +. +The same thing happened when we let +and +. Now, we will plot the points to show the graph of +. See +Figure 5.47. +Figure 5.47 +The graph is not a line. This figure is called a parabola. Every quadratic equation has a graph that looks like this. When +the solution to the quadratic +is 0 because +at +. +5.6 • Quadratic Equations with Two Variables with Applications +409 + +EXAMPLE 5.51 +Graphing a Quadratic Equation +Graph +and list the solutions to the quadratic equation. +Solution +We will graph the equation by plotting points. +Step 1: Choose integer values for +, substitute them into the equation, and solve for +. +Step 2: Record the values of the ordered pairs in the chart. +0 +1 +0 +0 +2 +3 +3 +Step 3: Plot the points and then connect them with a smooth curve. The result will be the graph of the equation +(Figure 5.48). The solutions are +and +. +Figure 5.48 +YOUR TURN 5.51 +1. Graph +. +410 +5 • Algebra +Access for free at openstax.org + +EXAMPLE 5.52 +Solving a Quadratic Equation From Its Graph +Find the solutions of +from its graph (Figure 5.49). +Figure 5.49 +Solution +The solutions of a quadratic equations are the values of +that make the equation a true statement when set equal to +zero (i.e. when +). +at +and +. +YOUR TURN 5.52 +1. Find the solutions of +from its graph. +5.6 • Quadratic Equations with Two Variables with Applications +411 + +Solving Quadratic Equations by Factoring +Another way of solving quadratic equations is by factoring. We will use the Zero Product Property that says that if the +product of two quantities is zero, it must be that at least one of the quantities is zero. The only way to get a product +equal to zero is to multiply by zero itself. +EXAMPLE 5.53 +Solving a Quadratic Equation by Factoring +Solve +. +Solution +Step 1. Set each factor equal +to zero. +The product equals zero, so at least one factor must +equal zero. +Step 2. Solve the linear +equations. +Solve each equation. +Step 3. Check. +Substitute each solution separately into the original +equation. +YOUR TURN 5.53 +1. Solve +. +EXAMPLE 5.54 +Solve Another Quadratic Equation by Factoring +Solve +. +Solution +Step 1. Write the quadratic equation in +standard form, +. +The equation is already in standard +form. +Step 2. Factor the quadratic expression. +Factor +412 +5 • Algebra +Access for free at openstax.org + +Step 3. Use the Zero Product Property. +Set each factor equal to zero. +Step 4. Solve the linear equations. +We have two linear equations. +Step 5. Check. +Substitute each solution separately +into the original equation. +YOUR TURN 5.54 +1. Solve +. +VIDEO +Solving Quadratics with the Zero Property (https://openstax.org/r/Zero_Property) +Be careful to write the quadratic equation in standard form first. The equation must be set equal to zero in order for +you to use the Zero Product Property! Often students start in Step 2 resulting in an incorrect solution. For example, +cannot be factored to +and then solved by setting each factor equal to +. +The correct way to solve this quadratic equation is to set it equal to zero FIRST: +which +becomes +, then continue to factor. See the table below for the correct way to apply the Zero Product +Property. +Step 1 +Skipped +Step 2 +Step 3 +5.6 • Quadratic Equations with Two Variables with Applications +413 + +Step 4 +Step 5 +Solving Quadratic Equations Using the Square Root Property +We just solved some quadratic equations by factoring. Let us use factoring to solve the quadratic equation +. +Step 1: Put the equation in standard form. +Step 2: Factor the left side. +Step 3: Use the Zero Product Property. +Step 4: Solve each equation. +Step 5: Combine the two solutions into +The solution is read as “ +is equal to positive or negative 3.” +What happens when we have an equation like +? Since 7 is not a perfect square, we cannot solve the equation by +factoring. These equations are all of the form +. We define the square root of a number in this way: If +, then +is a square root of +. This leads to the Square Root Property. +EXAMPLE 5.55 +Using the Square Root Property to Solve a Quadratic Equation +Solve using the square Root Property: +. +Solution +Step 1: Use the Square Root Property. +Step 2: Simplify the radical. +Step 3: Rewrite to show the two solutions. +YOUR TURN 5.55 +1. Solve using the Square Root Property: +. +EXAMPLE 5.56 +Using the Square Root Property to Solve Another Quadratic Equation +Solve using the Square Root Property: +. +414 +5 • Algebra +Access for free at openstax.org + +Solution +Step 1: Solve for . +Step 2: Use the Square Root Property. +Step 3: Simplify the radical. +Step 4: Rewrite to show the two solutions. +, +YOUR TURN 5.56 +1. Solve using the Square Root Property: +. +Solving Quadratic Equations Using the Quadratic Formula +This last method we will look at for solving quadratic equations is the quadratic formula. This method works for all +quadratic equations, even the quadratic equations we could not factor! To use the quadratic formula, we substitute the +values of +, +, and +into the expression on the right side of the formula. Then, we do all the math to simplify the +expression. The result gives the solution(s) to the quadratic equation. +EXAMPLE 5.57 +Solving a Quadratic Equation Using the Quadratic Formula +Solve using the quadratic formula: +. +Solution +This equation is in standard form. +Step 1: Identify the +, +, and +values. +Step 2: Write the quadratic formula. +Step 3: Substitute in the values of +, +, . +Step 4: Simplify. +5.6 • Quadratic Equations with Two Variables with Applications +415 + +Step 5: Rewrite to show two solutions. +Step 6: Simplify. +Step 7: Check. +YOUR TURN 5.57 +1. Solve using the quadratic formula: +. +EXAMPLE 5.58 +Solving Another Quadratic Equation Using the Quadratic Formula +Solve using the quadratic formula: +. +Solution +Step 1. Write the quadratic equation in standard +form. Identify the +, +, +values. +This equation is in standard +form. +Step 2. Write the quadratic formula. Then +substitute in the values of +, +, . +Substitute in +, +, +Step 3. Simplify the fraction, and solve for +. +416 +5 • Algebra +Access for free at openstax.org + +Step 4. Check the solutions. +Put each answer in the original +equation to check. +Substitute +. +Substitute +. +YOUR TURN 5.58 +1. Solve using the quadratic formula: +. +VIDEO +Solving Quadratics with the Quadratic Formula (https://openstax.org/r/Solving_Quadratics) +Solving Real-World Applications Modeled by Quadratic Equations +There are problem solving strategies that will work well for applications that translate to quadratic equations. Here’s a +problem-solving strategy to solve word problems: +Step 1: Read the problem. Make sure all the words and ideas are understood. +Step 2: Identify what we are looking for. +Step 3: Name what we are looking for. Choose a variable to represent that quantity. +Step 4: Translate into an equation. It may be helpful to restate the problem in one sentence with all the important +information. Then, translate the English sentence into an algebra equation. +Step 5: Solve the equation using good algebra techniques. +Step 6: Check the answer in the problem and make sure it makes sense. +Step 7: Answer the question with a complete sentence. +EXAMPLE 5.59 +Finding Consecutive Integers +The product of two consecutive integers is 132. Find the integers. +Solution +Step 1: Read the problem. +Step 2: Identify what we are looking for. +5.6 • Quadratic Equations with Two Variables with Applications +417 + +We are looking for two consecutive integers. +Step 3: Name what we are looking for. +Let +the first integer +Let +the next consecutive integer. +Step 4: Translate into an equation. Restate the problem in a sentence. +The product of the two consecutive integers is 132. The first integer times the next integer is 132. +Step 5: Solve the equation. +Bring all the terms to one side. +Factor the trinomial. +Use the zero product property. +Solve the equations. +There are two values for +that are solutions to this problem. So, there are two sets of consecutive integers that will +work. +If the first integer is +, then the next integer is 12. If the first integer is +, then the next integer is +. +Step 6: Check the answer. +The consecutive integers are 11, 12 and +, +. The product of 11 and +and the product of +. +Both pairs of consecutive integers are solutions. +Step 7: Answer the question. +The consecutive integers are 11, 12, and +, +. +YOUR TURN 5.59 +1. The product of two consecutive odd integers is 240. Find the integers. +Were you surprised by the pair of negative integers that is one of the solutions? In some applications, negative solutions +will result from the algebra, but will not be realistic for the situation. +EXAMPLE 5.60 +Finding Length and Width of a Garden +A rectangular garden has an area 15 square feet. The length of the garden is 2 feet more than the width. Find the length +and width of the garden. +Solution +Step 1: Read the problem. In problems involving geometric figures, a sketch can help you visualize the situation (Figure +5.50). +418 +5 • Algebra +Access for free at openstax.org + +Figure 5.50 +Step 2: Identify what you are looking for. +We are looking for the length and width. +Step 3: Name what you are looking for. The length is 2 feet more than width. +Let +the width of the garden. +the length of the garden. +Step 4: Translate into an equation. +Restate the important information in a sentence. +The area of the rectangular garden is 15 square feet. +Use the formula for the area of a rectangle. +Substitute in the variables. +Step 5: Solve the equation. Distribute first. +Get zero on one side. +Factor the trinomial. +Use the Zero Product Property. +Solve each equation. +Since +is the width of the garden, it does not make sense for it to be negative. We eliminate that value for +. +Width is 3 feet. +Find the value of the length. +length. +5 +Length is 5 feet. +Step 6: Check the answer (Figure 5.51). +5.6 • Quadratic Equations with Two Variables with Applications +419 + +Does the answer make sense? +Figure 5.51 +Yes, this makes sense. +Step 7: Answer the question. +The width of the garden is 3 feet and the length is 5 feet. +YOUR TURN 5.60 +1. A rectangular sign has an area of 30 square feet. The length of the sign is 1 foot more than the width. Find the +length and width of the sign. +WORK IT OUT +Completing the Square +Recall the two methods used to solve quadratic equations of the form +by factoring and by using the +quadratic formula. There are, however, many different methods for solving quadratic equations that were developed +throughout history. Egyptian, Mesopotamian, Chinese, Indian, and Greek mathematicians all solved various types of +quadratic equations, as did Arab mathematicians of the ninth through the twelfth centuries. It is one of these Arab +mathematicians' methods that we wish to investigate with this activity. +Muhammad ibn Musa al-Khwarizmi was employed as a scholar at the House of Wisdom in Baghdad, located in +present day Iraq. One of the many accomplishments of Al-Khwarizmi was his book on the topic of algebra. In that +book, he asks, “What must be the square which, when increased by ten of its own roots, amounts to 39?” Al- +Khwarizmi, like many Arab mathematicians of his time, was well versed in Euclid's Elements. Like Euclid, he viewed +algebra very geometrically, and thus had a geometric approach to solving a problem like the one above. In his +approach, he used a method which today we refer to as completing the square. +His description of the solution method for the above problem is: halve the number of roots, which in the present +instance yields 5. This you multiply by itself; the product is 25. Add this to 39; the sum is 64. Now take the root of this +which is 8, and subtract from it half the number of the roots, which is 5; the remainder is 3. This is the root of the +square which you sought for. Thus the square is 9. +420 +5 • Algebra +Access for free at openstax.org + +So, what does all of this mean? Al-Khwarizmi would start with a square of unknown length +of side (we will label the side length +). See Figure 5.52 So, this square has area +Figure 5.52 +He would then proceed to halve the number of roots (i.e., there are 10 roots by which the +square is increasing) to get 5; this he would add to the first square. See Figure 5.53 The +area of the two new pieces added into the original square are both +. At this point, we +have +. +Figure 5.53 +Now Al-Khwarizmi needed to “complete the square” by adding into the drawing a small +square. See Figure 5.54 This square has an area of 25. +Figure 5.54 +, or +. +Notice that the completed square has side length +, so the large square has area +. (Notice algebraically +that the left half of the equation +factors to +This means the area of large square +equals 64. If +, then +; so +must be equal to 3 or +to make this true. Note that Al-Kwarimi +would not have considered the possibility of a negative solution, since he approached the solution geometrically, and +negative distances do not exist. +Check Your Understanding +48. Which quadratic equation equals +? +a. +b. +c. +d. +49. Which product is equal to +? +a. +b. +c. +d. +5.6 • Quadratic Equations with Two Variables with Applications +421 + +50. The graph shown is the graph of which quadratic equation? +a. +b. +c. +d. +51. What is the solution to +? +a. +b. +c. +d. +52. +can be factored to +. +a. +True +b. +False +53. +can be solved using the square root method. +a. +True +b. +False +54. +can be solved using the quadratic formula. +a. +True +b. +False +55. +can be graphed as: +422 +5 • Algebra +Access for free at openstax.org + +a. +True +b. +False +56. Using the square root method, find the solutions to +. +SECTION 5.6 EXERCISES +For the following exercises, multiply the binomials. +1. +2. +3. +4. +5. +6. +For the following exercises, factor the trinomials. +7. +8. +9. +10. +11. +12. +For the following exercises, solve the quadratic equations by graphing. +13. Graph and list the solutions to the quadratic equation +. +5.6 • Quadratic Equations with Two Variables with Applications +423 + +14. Graph and list the solutions to the quadratic equation +. +15. Graph and list the solutions to the quadratic equation +. +424 +5 • Algebra +Access for free at openstax.org + +16. Find the solutions of +from its graph. +17. Find the solutions of +from its graph. +5.6 • Quadratic Equations with Two Variables with Applications +425 + +18. Find the solutions of +from its graph. +For the following exercises, solve the quadratic equation by factoring. +19. +20. +21. +22. +23. +24. +For the following exercises, solve the quadratic equation using the square root method. +25. +26. +27. +For the following exercises, solve the quadratic equation using the quadratic formula. +28. +426 +5 • Algebra +Access for free at openstax.org + +29. +30. +31. +32. +33. +34. The product of two consecutive odd integers is 99. Find the integers. +35. The product of two consecutive even integers is 168. Find the integers. +36. A rectangular patio has an area of 180 square feet. The width of the patio is three feet less than the length. Find +the length and width of the patio. +For the following exercises, use the Pythagorean Theorem: In any right triangle, where +and +are the lengths of the +legs and +is the length of the hypotenuse, as shown in the given figure, +. +37. Justine wants to put a deck in the corner of her backyard in the shape of a right triangle, as shown in the given +figure. The hypotenuse will be 17 feet long. The length of one side will be 7 feet less than the length of the +other side. Find the lengths of the sides of the deck. +38. A boat’s sail is a right triangle. The length of one side of the sail is 7 feet more than the other side. The +hypotenuse is 13. Find the lengths of the two sides of the sail. +39. The sun casts a shadow from a flagpole. The height of the flagpole is three times the length of its shadow. The +distance between the end of the shadow and the top of the flagpole is 21 feet. Find the length of the shadow +and the length of the flagpole. Round to the nearest tenth of a foot. +40. Rene is setting up a holiday light display. He wants to make a “tree” in the shape of two right triangles and has +two 10-foot strings of lights to use for the sides. He will attach the lights to the top of a pole and to two stakes +on the ground. He wants the height of the pole to be the same as the distance from the base of the pole to +each stake. How tall should the pole be? Round to the nearest tenth of a foot. +5.6 • Quadratic Equations with Two Variables with Applications +427 + +5.7 Functions +Figure 5.55 A small group of elementary students learning from their teacher. (credit: modification of work "Our school" +by Woodleywonderworks/Flickr, CC BY 2.0 ) +Learning Objectives +After completing this section, you should be able to: +1. +Use function notation. +2. +Determine if a relation is a function with different representations. +3. +Apply the vertical line test. +4. +Determine the domain and range of a function. +In this section, we will learn about relations and functions. As we go about our daily lives, we have many data items or +quantities that are paired to our names. Our social security number, student ID number, email address, phone number, +and our birthday are matched to our name. There is a relationship between our name and each of those items. When +your teacher gets their class roster, the names of all the students in the class are listed in one column and then the +student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where +the first element is a student name and the second element is that student’s ID number, we call this a relation. +(Student name, Student ID #) +The set of all the names of the students in the class is called the domain of the relation and the set of all student ID +numbers paired with these students is the range of the relation. In general terms, a relation is any set of ordered pairs, +( +). All the +-values in the ordered pairs together make up the domain. All the +-values in the ordered pairs together +make up the range. +There are many situations similar to the student's name and student ID # where one variable is paired or matched with +another. The set of ordered pairs that records this matching is a relation. A special type of relation, called a function, +occurs extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one +element in the range. For each ordered pair in the relation, each +-value is matched with only one +-value. +Let us look at the relation between your friends and their birthdays in Figure 5.56. Every friend has a birthday, but no +one has two birthdays. It is okay for two people to share a birthday. It is okay that Danny and Stephen share July 24 as +their birthday and that June and Liz share August 2. Since each person has exactly one birthday, the relation is a function. +428 +5 • Algebra +Access for free at openstax.org + +Figure 5.56 Birthday Mapping +Use Function Notation +It is very convenient to name a function; most often functions are named +, +, +, +, +, or +. In any function, for each +-value from the domain, we get a corresponding +-value in the range. In the function +, we write this range value +as +( ). This notation +( ) is called function notation and is read "f of +" or "the value of f at +." In this case the parentheses +do not indicate multiplication. +We call +the independent variable as it can be any value in the domain. We call +the dependent variable as its value +depends on +. Much like when you first encountered the variable +, function notation may be rather unsettling. But the +more you use the notation, the more familiar you become with the notation, and the more comfortable you will be with +it. +Let’s review the equation +. To find the value of +when +, we know to substitute +into the equation +and then simplify. +Let +. +The value of the function at +is 3. We do the same thing using function notation, the equation +can be +written as +. To find the value when +, we write: +Let +. +The value of the function at +is 3. This process of finding the value of +for a given value of +is called +evaluating the function. +EXAMPLE 5.61 +Evaluating the Function +For the function +, evaluate the function. +1. +2. +3. +5.7 • Functions +429 + +Solution +1. +To evaluate +, substitute 3, for +. +Simplify. +2. +To evaluate +, substitute +for +. +Simplify. +3. +To evaluate +, substitute +for +. +Simplify. +YOUR TURN 5.61 +For the function +, evaluate the function. +1. +2. +3. +EXAMPLE 5.62 +Evaluating the Function in an Application +The number of unread emails in Sylvia’s inbox is 75. This number grows by 10 unread emails a day. The function +represents the relation between the number of emails, +, and the time, , measured in days. Find +(5). +Explain what this result means. +Solution +Find +(5). Explain what this result means. +Substitute in +. +Simplify. +If 5 is the number of days, +is the number of unread emails after 5 days. After 5 days, there are 125 unread emails in +Sylvia’s inbox. +430 +5 • Algebra +Access for free at openstax.org + +YOUR TURN 5.62 +1. The number of unread emails in Bryan’s account is 100. This number grows by 15 unread emails a day. The +function +represents the relation between the number of emails, +, and the time, , measured +in days. Find +(7). Explain what the result means. +Determining If a Relation Is a Function with Different Representations +We can determine whether a relation is a function by identifying the input and the output values. If each input value +leads to only one output value, classify the relation as a function. If any input value leads to two or more outputs, do not +classify the relation as a function. +We will review three different representations of relations and determine if they are functions: ordered pairs, mapping, +and equations. +EXAMPLE 5.63 +Determining If a Relation Is a Function with a Set of Ordered Pairs +Use the set of ordered pairs to determine whether the relation is a function. +1. +2. +Solution +1. +Each +-value is matched with only one +-value. This relation is a function. +2. +The +-value 9 is matched with two +-values, both 3 and +. This relation is not a function. +YOUR TURN 5.63 +Use the set of ordered pairs to determine whether the relation is a function. +1. +2. +A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the +elements of the range. Consider the example of the relation between your friends and their birthdays used in Figure +5.57. In this particular example, the domain is the set of people’s names, and the range is the set of their birthdays. This +mapping was a function because everybody’s name maps to exactly one birthday. +EXAMPLE 5.64 +Determining If a Relation Is a Function with Mapping +Use the mapping in Figure 5.57 to determine whether the relation is a function. +Figure 5.57 +5.7 • Functions +431 + +Solution +Both Lydia and Marty have two phone numbers. Each +-value is not matched with only one +-value. This relation is not a +function. +YOUR TURN 5.64 +1. Use the mapping in the given figure to determine whether the relation is a function. +In algebra functions will usually be represented by an equation. It is easiest to see if the equation is a function when it is +solved for +. If each value of +results in only one value of +, then the equation defines a function. +EXAMPLE 5.65 +Determining If a Relation Is a Function with an Equation +Determine whether each equation is a function. Assume +is the independent variable. +1. +2. +3. +Solution +1. +For each value of +, we multiply it by +and then add 7 to get the +-value. +For example, if +: +We have that when +, then +. It would work similarly for any value of +. Since each value of +, corresponds +to only one value of +the equation defines a function. +2. +For each value of +, we square it and then add 1 to get the +-value. +For example, if +We have that when +, then +. It would work similarly for any value of +. Since each value of +corresponds +to only one value of +, the equation defines a function. +3. +432 +5 • Algebra +Access for free at openstax.org + +Isolate the +term. +Let us substitute +. +This gives us two values for +. +We have shown that when +, then +and +. It would work similarly for any value of +. Since each value +of +does not corresponds to only one value of +the equation does not define a function. +YOUR TURN 5.65 +Determine whether each equation is a function. +1. +2. +3. +VIDEO +Relations and Functions (https://openstax.org/r/Relationsp_and_Functions) +Applying the Vertical Line Test +We reviewed how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered +pairs, a mapping, or an equation. We will now cover how to tell if a graph is that of a function. +An ordered pair +is a solution of a linear equation, if the equation is a true statement when the +-values and +-values of the ordered pair are substituted into the equation. The graph of a linear equation is a straight line where +every point on the line is a solution of the equation, and every solution of this equation is a point on this line. Figure 5.58 +we can see that in the graph of the equation +, for every +-value there is only one +-value, as shown in the +accompanying table. +5.7 • Functions +433 + +Figure 5.58 Graph of the Equation +A relation is a function if every element of the domain has exactly one value in the range. The relation defined by the +equation +is a function. If we look at the graph, each vertical dashed line only intersects the solid line at one +point. This makes sense as in a function, for every +-value there is only one +-value. If the vertical line hit the graph twice, +the +-value would be mapped to two +-values, and so the graph would not represent a function. This leads us a graphical +method of determining functions called the vertical line test, which states that a set of points in a rectangular +coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical +line intersects the graph in more than one point, the graph does not represent a function. +EXAMPLE 5.66 +Applying the Vertical Line Test +Determine whether the graph (Figure 5.59) is the graph of a function applying the vertical line test. +Figure 5.59 +434 +5 • Algebra +Access for free at openstax.org + +Solution +On the graph (Figure 5.60), only three vertical dashed lines are drawn. However, it can be determined that any vertical +dashed line that is drawn will intersect the solid line at exactly one point. It is the graph of a function. +Figure 5.60 +YOUR TURN 5.66 +1. Determine whether the graph is the graph of a function. +EXAMPLE 5.67 +Applying the Vertical Line Test to a Parabola +Determine whether the graph is the graph of a function (Figure 5.61). +5.7 • Functions +435 + +Figure 5.61 +Solution +Figure 5.62 does not represent a function since the vertical dashed lines shown on the graph below intersect the solid +line at two points. +Figure 5.62 +YOUR TURN 5.67 +1. Determine whether the graph is the graph of a function. +436 +5 • Algebra +Access for free at openstax.org + +Determining the Domain and Range of a Function +For the function +is the independent variable as it can be any value in the domain, and +is the dependent +variable since its value depends on +. For the function +, the values of +make up the domain and the values of +make up the range. +EXAMPLE 5.68 +Finding the Domain and Range of Ordered Pairs +For +: +1. +Find the domain of the relation. +2. +Find the range of the relation. +Solution +1. +The domain is the set of all +-values of the relation: +2. +The range is the set of all +-values of the relation: +YOUR TURN 5.68 +For the relation +: +1. Find the domain of the relation. +2. Find the range of the relation. +EXAMPLE 5.69 +Finding the Domain and Range on a Graph +Use Figure 5.63 to: +1. +List the ordered pairs of the relation. +2. +Find the domain of the relation. +3. +Find the range of the relation. +5.7 • Functions +437 + +Figure 5.63 +Solution +1. +The ordered pairs of the relation are: +. +2. +The domain is the set of all +-values of the relation: +. Notice that while +repeats, it is only listed +once. +3. +The range is the set of all +-values of the relation: +Notice that while +repeats, it is only listed +once. +YOUR TURN 5.69 +Use the given figure to: +1. List the ordered pairs of the relation. +2. Find the domain of the relation. +3. Find the range of the relation. +438 +5 • Algebra +Access for free at openstax.org + +VIDEO +Domain and Range on Graphs (https://openstax.org/r/Domain) +WHO KNEW? +Function and Function Notation +In 1673, Gottfried Leibniz, the German mathematician who co-invented calculus, seems to be the first person to use +the word function in a mathematical sense, although his use of it does not exactly fit with the modern use and +definition. The person who is credited with the modern definition of function is Swiss mathematician Johann +Bernoulli, who wrote about it in a letter to Leibniz in 1698. Supposedly, Leibniz wrote Bernoulli back, approving of this +use of the word. In 1734, the use of the notation +for a function was first used by Swiss mathematician Leonhard +Euler (pronounced “Oiler”). Euler had a knack for inventing notation. He also introduced the notation +for the base of +natural logs (1727), +for the square root of +(1777), +for summation (1755), and many others. Euler also +introduced many other ideas associated with functions. Euler defined exponential functions and defined logarithmic +functions as their inverse; he also introduced the beta and gamma functions, and was the first person to consider the +trigonometric identities (sine, cosine, etc.) as functions. +Check Your Understanding +57. If +then +. +a. +True +b. +False +58. +represent the ordered pairs of a function. +a. +True +b. +False +59. The graph shown represents the graph of a function: +a. +True +b. +False +60. The figure shown represents the mapping of a function. +5.7 • Functions +439 + +a. +True +b. +False +61. The domain of the mapping in the figure is +. +a. +True +b. +False +SECTION 5.7 EXERCISES +For the following exercises, evaluate the functions at the values +and +. +1. +2. +3. +4. +5. +6. +For the following exercises, determine whether the ordered pairs represent a function. +7. +8. +9. +10. +11. +For the following exercises, determine whether the mapping represented a function. +12. +440 +5 • Algebra +Access for free at openstax.org + +13. +For the following exercises, determine whether the equations represent +as a function of +. +14. +15. +16. +17. +18. +19. +For the following exercises, use the vertical line test to determine which graph represents a function. +20. +5.7 • Functions +441 + +21. +22. +442 +5 • Algebra +Access for free at openstax.org + +23. +24. +5.7 • Functions +443 + +25. +For the following exercises, use the set of ordered pairs to find the domain and the range. +26. +27. +28. +29. +For the following exercises, use the graph to find the domain and the range. +30. +444 +5 • Algebra +Access for free at openstax.org + +31. +32. +5.7 • Functions +445 + +33. +446 +5 • Algebra +Access for free at openstax.org + +5.8 Graphing Functions +Figure 5.64 The ski lifts and the mountain both have a slope. (credit: modification of work "colorado springs zoo tram" by +woodleywonderworks/Flickr, CC BY 2.0) +Learning Objectives +After completing this module, you should be able to: +1. +Graph functions using intercepts. +2. +Compute slope. +3. +Graph functions using slope and +-intercept. +4. +Graph horizontal and vertical lines. +5. +Interpret graphs of functions. +6. +Model applications using slope and +-intercept. +In this section, we will expand our knowledge of graphing by graphing linear functions. There are many real-world +scenarios that can be represented by graphs of linear functions. Imagine a chairlift going up at a ski resort. The journey a +skier takes travelling up the chairlift could be represented as a linear function with a positive slope. The journey a skier +takes down the slopes could be represented by a linear function with a negative slope. +Graphing Functions Using Intercepts +Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that +when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing +the line might use different sets of three points. At first glance, their two lines might not appear to be the same, since +they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One +way to recognize that they are indeed the same line is to look at where the line crosses the +-axis and the +-axis. These +points are called the intercepts of a line. Let us review the graphs of the lines in Figure 5.65. +5.8 • Graphing Functions +447 + +Figure 5.65 +The table below lists where each of these lines crosses the +- and +-axis. Do you see a pattern? For each line, the +-coordinate of the point where the line crosses the +-axis is zero. The point where the line crosses the +-axis has the +form +and is called the +-intercept of the line. The +-intercept occurs when +is zero. In each line, the +-coordinate of +the point where the line crosses the +-axis is zero. The point where the line crosses the +-axis has the form +and is +called the +-intercept of the line. The +-intercept occurs when +is zero. +Figure +The line crosses the +at: +Ordered Pair for this +Point +The line crosses the +at: +Ordered Pair for This +Point +Figure (a) +3 +6 +Figure (b) +4 +448 +5 • Algebra +Access for free at openstax.org + +Figure +The line crosses the +at: +Ordered Pair for this +Point +The line crosses the +at: +Ordered Pair for This +Point +Figure (c) +5 +Figure (d) +0 +0 +General +Figure +EXAMPLE 5.70 +Finding +- and +-Intercepts +Find the +-intercept and +-intercept on the (a) and (b) graphs in Figure 5.66. +Figure 5.66 +Solution +In Figure 5.66, the graph crosses the +-axis at the point +. The +-intercept is +. The graph crosses the +-axis at +the point +. The +-intercept is +. In Figure 5.66, the graph crosses the +-axis at the point +. The +-intercept is +. The graph crosses the +-axis at the point +. The +-intercept is +. +YOUR TURN 5.70 +1. Find the +-intercept and +-intercept on the given graph. +5.8 • Graphing Functions +449 + +EXAMPLE 5.71 +Graphing a Function Using Intercepts +Find the intercepts of +. Then graph the function using the intercepts. +Solution +Let +to find the +-intercept, and let +to find the +-intercept. +To find the +-intercept, let +. +To find the +-intercept, let +. +Simplify. +Simplify. +The +-intercept is: +The +-intercept is: +Plot the intercepts to get the graph in Figure 5.67. +450 +5 • Algebra +Access for free at openstax.org + +Figure 5.67 +YOUR TURN 5.71 +1. Find the intercepts of +and use them to graph the equation. +Computing Slope +When graphing linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt +down. Some lines are very steep and some lines are flatter. In mathematics, the measure of the steepness of a line is +called the slope of the line. To find the slope of a line, we locate two points on the line whose coordinates are integers. +Then we sketch a right triangle where the two points are vertices of the triangle and one side is horizontal and one side +is vertical. Next, we measure or calculate the distance along the vertical and horizontal sides of the triangle. The vertical +distance is called the rise and the horizontal distance is called the run. +We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the +vertical distance changes while the run measures the horizontal change, as shown in this illustration. Slope (Figure 5.68) +is a rate of change. +Figure 5.68 +FORMULA +To calculate slope +, use the formula +, +where the rise measures the vertical change and the run measures the horizontal change. +5.8 • Graphing Functions +451 + +The concept of slope has many applications in the real world. In construction, the pitch of a roof, the slant of plumbing +pipes, and the steepness of stairs are all applications of slope. As you ski or jog down a hill, you definitely experience +slope. +EXAMPLE 5.72 +Finding the Slope from a Graph +Find the slope of the line shown in Figure 5.69. +Figure 5.69 +Solution +Step 1: Locate two points on the graph whose coordinates are integers, such as +and +. Starting at +, sketch +a right triangle to +as shown in Figure 5.70. +Figure 5.70 +Step 2: Count the rise; since it goes down, it is negative. The rise is −2. +Step 3: Count the run. The run is 3. +Step 4: Use the slope formula +substitute the values of the rise and run. +The slope of the line is +. +The solution is +decreases by 2 units as +increases by 3 units. +452 +5 • Algebra +Access for free at openstax.org + +YOUR TURN 5.72 +1. Find the slope of the line shown in the graph. +Sometimes we will need to find the slope of a line between two points when we don’t have a graph to measure the rise +and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the +slope without graphing. First, we need to introduce some algebraic notation. +We have seen that an ordered pair ( , +) gives the coordinates of a point. But when we work with slopes, we use two +points. How can the same symbol ( , +) be used to represent two different points? Mathematicians use subscripts to +distinguish such points. For example, ( +, +) would be said aloud as “ +sub 1, +sub 1” and ( +, +) read “ +sub 2, +sub +2.” The “sub” is a short way of saying “subscript.” We will use ( +, +) to identify the first point and ( +, +) to identify the +second point in our slope equation. If we had more than two points, (if we were finding more than one slope), we could +use ( +, +), ( +, +), and so on. +Let’s review how the rise and run relate to the coordinates of the two points by taking another look at the slope of the +line between the points +and +, as shown in Figure 5.71. +Figure 5.71 +On the graph, we count the rise of 3 and the run of 5. Notice on the graph that that ( +, +) is the point +and ( +, +) +is the point +. The rise can be found by subtracting the +-coordinates, 6 and 3, and the run can be found by +subtracting the +-coordinates 7 and 2. +We have shown that +is really another version of +. We can use this formula to find the slope of a line. +5.8 • Graphing Functions +453 + +FORMULA +To find the slope of the line between two points ( +, +) and ( +, +), use the formula +EXAMPLE 5.73 +Finding the Slope of the Line Using Points +Use the slope formula to find the slope of the line through the points (−2, −3) and (−7, 4). +Solution +We’ll call (−2, −3) point 1 and (−7, 4) point 2. +Step 1: Use the slope formula: +Step 2: Substitute the values: +Step 3: Simplify: +Step 4: Verify the slope on the graph shown in Figure 5.72. +Figure 5.72 +YOUR TURN 5.73 +1. Use the slope formula to find the slope of the line through the pair of points +and +. +Graphing Functions Using Slope and +-Intercept +We have graphed linear equations by plotting points and using intercepts. Once we see how an equation in slope- +intercept form and its graph are related, we will have one more method we can use to graph lines. Review the graph of +the equation +in Figure 5.73 and find its slope and +-intercept. +454 +5 • Algebra +Access for free at openstax.org + +Figure 5.73 Graph of the equation +. +The vertical and horizontal lines in the graph show us the rise is 1 and the run is 2, respectively. +Substituting into the slope formula: +The +-intercept is +. Look at the equation of this line. +Look at the slope and +-intercept. +When a linear equation is solved for +, the coefficient of the +term is the slope and the constant term is the +-coordinate +of the +-intercept. We say that the equation +is in slope-intercept form. Sometimes the slope-intercept form +is called the +-form. +EXAMPLE 5.74 +Finding the Slope and +-Intercept of a Line +Identify the slope and +-intercept of the line from the equation: +1. +2. +Solution +1. +We compare our equation to the slope-intercept form of the equation. +Step 1: Write the slope-intercept form of the equation of the line. +Step 2: Write the equation of the line. +Step 3: Identify the slope. +5.8 • Graphing Functions +455 + +Step 4: Identify the +-intercept. +2. +When an equation of a line is not given in slope-intercept form, our first step will be to solve the equation for +. +Step 1: Solve for +. +Step 2: Subtract +from each side. +Step 3: Divide both sides by 3. +Step 4: Simplify. +Step 5: Write the slope-intercept form of the equation of the line. +Step 6: Write the equation of the line. +Step 7: Identify the slope. +Step 8: Identify the +-intercept. +. +YOUR TURN 5.74 +Identify the slope and +-intercept from the equation of the line. +1. +2. +EXAMPLE 5.75 +Graphing the Slope and +-Intercept +Graph the line of the equation +using its slope and +-intercept. +Solution +The equation is in slope-intercept form +. +Step 1: Identify the slope and +-intercept. +, +-intercept is +. +456 +5 • Algebra +Access for free at openstax.org + +Step 2: Plot the +-intercept on the coordinate system (Figure 5.74). +1. +Identify the rise over the run. +2. +Count out the rise and run to mark the second point. +rise +, run 1 +Figure 5.74 +YOUR TURN 5.75 +1. Graph the line of the equation +using its slope and +-intercept. +Graphing Horizontal and Vertical Lines +Some linear equations have only one variable. They may have just +without the +, or just +without an +. This changes +how we make a table of values to get the points to plot. Let us consider the equation +. This equation has only one +variable, +. The equation says that +is always equal to +, so its value does not depend on +. No matter what the value +of +is, the value of +is always +. To make a table of values, write +in for all the +-values. Then choose any values for +. Since +does not depend on +, you can choose any numbers you like. But to fit the points on our coordinate graph, we +will use 1, 2, and 3 for the +-coordinates in the table below. +( , +) +−3 +1 +2 +3 +Plot the points from the table and connect them with a straight line (Figure 5.75). Notice that we have graphed a vertical +line. +5.8 • Graphing Functions +457 + +Figure 5.75 Graph of +What is the slope? If we take the two points +and +then the rise is 2 and the run is 0. +Using the slope formula we get: +The slope is undefined since division by zero is undefined. We say that the slope of the vertical line +is undefined. +The slope of any vertical line +(where +is any number) will be undefined. +What if the equation has +but no +? Let’s graph the equation +. This time the +-value is a constant, so in this +equation, +does not depend on +. Fill in 4 for all the +values in the table below and then choose any values for +. We +will use 0, 2, and 4 for the +-coordinates. +( , +) +0 +4 +2 +4 +4 +4 +In Figure 5.76, we have graphed a horizontal line passing through the +-axis at 4. +458 +5 • Algebra +Access for free at openstax.org + +Figure 5.76 Graph of +What is the slope? If we take the two points +and +then the rise is 0 and the run is 2. Using the slope formula, +we get +. The slope of the horizontal line +is 0. The slope of any horizontal line +(where +is +any number) will be 0. When the +-coordinates are the same, the rise is 0. +EXAMPLE 5.76 +Graphing A Vertical Line +Graph: +. +Solution +The equation has only one variable, +, and +is always equal to 2. We create a table where +is always 2 and then put in +any values for +. The graph is a vertical line passing through the +-axis at 2 (Figure 5.77). +( , +) +2 +1 +2 +2 +2 +3 +5.8 • Graphing Functions +459 + +Figure 5.77 +YOUR TURN 5.76 +1. Graph +. +EXAMPLE 5.77 +Graphing A Horizontal Line +Graph: +. +Solution +The equation +has only one variable, +. The value of +is constant. All the ordered pairs in the next table have the +same +-coordinate. The graph is a horizontal line passing through the +-axis at −1 (Figure 5.78). +( , +) +0 +3 +460 +5 • Algebra +Access for free at openstax.org + +Figure 5.78 +YOUR TURN 5.77 +1. Graph the equation +. +The table below summarizes all the methods we have used to graph lines. +Interpreting Graphs of Functions +An important yet often overlooked area in algebra involves interpreting graphs. Oftentimes in math classes, students are +given mathematical functions and can make graphs to represent them. But the interpretation of graphs is a more +applicable skill to the real world. Being able to “read” a graph—understanding its domain and range, what the intercepts +mean, and what the slope (or curve) means— that's a real-world skill. +5.8 • Graphing Functions +461 + +EXAMPLE 5.78 +Interpreting a Graph +In Figure 5.79 the +-axis on the graph represents the 120-minute bike ride Juan went on. The +-axis represents how far +away he was from his home. +Figure 5.79 +1. +Interpret the +- and +-intercept. +2. +For each segment, find the slope. +3. +Create an interpretation of this graph (i.e., make up a story that goes with it). +Solution +1. +is the +- and +-intercept and represents Juan at home before his bike ride. The distance from home is 0 miles +and 0 minutes have passed. +2. +In the first 30 minutes, the slope is +and indicates Juan is traveling 1 mile for every 5 minutes. Between 30 and 60 +minutes, the slope is 0 and indicates that he’s not riding the bike (the distance is not increasing). Then between 60 +and 90 minutes, the slope is +again. Finally, after 90 minutes the slope is +meaning Juan is getting 4 miles +closer to home every 15 minutes. +3. +Answers will vary. Juan left his house for a bike ride. After 30 minutes, he was 6 miles from home and he stopped for +ice cream at his local ice-cream truck. He enjoyed his ice cream for 30 minutes. He then jumped back on his bike and +rode to his friend’s house. He arrived there 30 minutes later. His friend’s house was 12 miles from his home. His +friend was not home so he immediately turned around and quickly rode home in 45 minutes. +YOUR TURN 5.78 +In the given figure the +-axis on the graph represents the years. The y-axis represents the number of teachers at +Jones High School. +Teachers at Jones High +462 +5 • Algebra +Access for free at openstax.org + +1. Interpret the +- and +-intercept. +2. For each segment, find the slope. +3. Create an interpretation of this graph (i.e., make up a story that goes with it). +Modeling Applications Using Slope and +-Intercept +Many real-world applications are modeled by linear equations. We will review a few applications here so you can +understand how equations written in slope-intercept form relate to real-world situations. Usually when a linear equation +model uses real-world data, different letters are used for the variables instead of using only +and +. The variable names +often remind us of what quantities are being measured. Also, we often need to extend the axes in our rectangular +coordinate system to bigger positive and negative numbers to accommodate the data in the application. +EXAMPLE 5.79 +Converting Temperature +The equation +is used to convert temperatures from degrees Celsius ( +) to degrees Fahrenheit ( +). +1. +Find the Fahrenheit temperature for a Celsius temperature of 0°. +2. +Find the Fahrenheit temperature for a Celsius temperature of 20°. +3. +Interpret the slope and +-intercept of the equation. +4. +Graph the equation. +Solution +1. +Find the Fahrenheit temperature for a Celsius temperature of 0°. +Find +when +. +Simplify. +2. +Find the Fahrenheit temperature for a Celsius temperature of 20°. +Find +when +. +Simplify. +Simplify. +3. +Interpret the slope and +-intercept of the equation. +Even though this equation uses +and +, it is still in slope-intercept form. +The slope +means that the temperature Fahrenheit ( +) increases 9 degrees when the temperature Celsius ( +) +increases 5 degrees. +The +-intercept means that when the temperature is 0° on the Celsius scale, it is 32° on the Fahrenheit scale. +4. +Graph the equation. +We will need to use a larger scale than our usual. Start at the +-intercept +, and then count out the rise of 9 +and the run of 5 to get a second point as shown in Figure 5.80. +5.8 • Graphing Functions +463 + +Figure 5.80 +YOUR TURN 5.79 +The equation +is used to estimate a person’s height in inches, +, based on women’s shoe size, . +1. Estimate the height of a child who wears women’s shoe size 0. +2. Estimate the height of a woman with shoe size 8. +3. Interpret the slope and +-intercept of the equation. +4. Graph the equation. +EXAMPLE 5.80 +Calculating Driving Costs +Sam drives a delivery van. The equation +models the relation between his weekly cost, +, in dollars and the +number of miles, +, that he drives. +1. +Find Sam’s cost for a week when he drives 0 miles. +2. +Find the cost for a week when he drives 250 miles. +3. +Interpret the slope and +-intercept of the equation. +4. +Graph the equation. +Solution +1. +Find Sam’s cost for a week when he drives 0 miles. +Find +when += 0. +Simplify. +Sam’s costs are $60 when he drives 0 miles. +2. +Find the cost for a week when he drives 250 miles. +464 +5 • Algebra +Access for free at openstax.org + +Find +when +. +Simplify. +Sam’s costs are $185 when he drives 250 miles. +3. +Interpret the slope and +-intercept of the equation. +The slope, 0.5, means that the weekly cost, +, increases by $0.50 when the number of miles driven, +, increases by +1. The +-intercept means that when the number of miles driven is 0, the weekly cost is $60. +4. +Graph the equation (Figure 5.81). +We’ll need to use a larger scale than usual. Start at the +-intercept (0, 60). To count out the slope +, we rewrite +it as an equivalent fraction that will make our graphing easier. +So to graph the next point go up 50 from the intercept of 60 and then to the right 100. The second point will be (100, +110). +Figure 5.81 +YOUR TURN 5.80 +Stella has a home business selling gourmet pizzas. The equation +models the relation between her +weekly cost, +, in dollars and the number of pizzas, +, that she sells. +1. Find Stella’s cost for a week when she sells no pizzas. +2. Find the cost for a week when she sells 15 pizzas. +3. Interpret the slope and +-intercept of the equation. +4. Graph the equation. +5.8 • Graphing Functions +465 + +Check Your Understanding +62. True or False. The +-intercept of +. +a. +True +b. +False +63. True or False. The slope of the line containing the points (1, 2) and (2, 4) is 1. +a. +True +b. +False +For the following exercises, use the graph shown. +64. True or False. This graph has a slope of 5. +a. +True +b. +False +65. True or False. This is the graph of the equation +. +a. +True +b. +False +66. True or False. All vertical lines have a slope of zero. +a. +True +b. +False +SECTION 5.8 EXERCISES +For the following exercises, find the +- and +-intercepts on the graph. +466 +5 • Algebra +Access for free at openstax.org + +1. +2. +3. +5.8 • Graphing Functions +467 + +4. +For the following exercises, graph using the intercepts. +5. +6. +7. +8. +9. +10. +11. +12. +For the following exercises, find the slope of the line. +13. +468 +5 • Algebra +Access for free at openstax.org + +14. +15. +5.8 • Graphing Functions +469 + +16. +17. +470 +5 • Algebra +Access for free at openstax.org + +18. +19. +5.8 • Graphing Functions +471 + +20. +For the following exercises, use the slope formula to find the slope of the line between each pair of points. +21. (2, 5), (4, 0) +22. (−3, 3), (4, −5) +23. (−1, −2), (2, 5) +24. (4, −5), (1, −2) +For the following exercises, identify the slope and +-intercept of each line. +25. +26. +27. +28. +29. +30. +31. +32. +For the following exercises, graph the line of each equation using its slope and +-intercept. +33. +34. +35. +36. +37. +38. +39. +40. +For the following exercises, find the slope of each line and graph. +41. +42. +43. +44. +For the following exercises, graph and interpret applications of slope-intercept. +The equation +models the relation between the amount of Tuyet’s monthly water bill payment, +, in +dollars, and the number of units of water, +, used. +472 +5 • Algebra +Access for free at openstax.org + +45. Find Tuyet’s payment for a month when 0 units of water are used. +46. Find Tuyet’s payment for a month when 12 units of water are used. +47. Interpret the slope and +-intercept of the equation. +48. Graph the equation. +For the following exercises, graph and interpret applications of slope-intercept. +Bruce drives his car for his job. The equation +models the relation between the amount in dollars, +, +that he is reimbursed and the number of miles, +, he drives in one day. +49. Find the amount Bruce is reimbursed on a day when he drives 0 miles. +50. Find the amount Bruce is reimbursed on a day when he drives 220 miles. +51. Interpret the slope and +-intercept of the equation. +52. Graph the equation. +For the following exercises, graph and interpret applications of slope-intercept. +Cherie works in retail and her weekly salary includes commission for the amount she sells. The equation +models the relation between her weekly salary, +, in dollars and the amount of her sales, , in dollars. +53. Find Cherie’s salary for a week when her sales were $0. +54. Find Cherie’s salary for a week when her sales were $3,600. +55. Interpret the slope and +-intercept of the equation. +56. Graph the equation. +For the following exercises, graph and interpret applications of slope-intercept. +Costa is planning a lunch banquet. The equation +models the relation between the cost in dollars, +, of +the banquet and the number of guests, +. +57. Find the cost if the number of guests is 40. +58. Find the cost if the number of guests is 80. +59. Interpret the slope and +-intercept of the equation. +60. Graph the equation. +5.8 • Graphing Functions +473 + +5.9 Systems of Linear Equations in Two Variables +Figure 5.82 Fruits and vegetables at a farmer’s market. (credit: “California Ave. Farmers’ Market” by Jun Seita/Flickr, CC BY +2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Determine and show whether an ordered pair is a solution to a system of equations. +2. +Solve systems of linear equations using graphical methods. +3. +Solve systems of linear equations using substitution. +4. +Solve systems of linear equations using elimination. +5. +Identify systems with no solution or infinitely many solutions. +6. +Solve applications of systems of linear equations. +In this section, we will learn how to solve systems of linear equations in two variables. There are several real-world +scenarios that can be represented by systems of linear equalities. Suppose two friends, Andrea and Bart, go shopping at +a farmers market to buy some vegetables. Andrea buys 2 tomatoes and 4 cucumbers and spends $2.00. Bart buys 4 +tomatoes and 5 cucumbers and spends $2.95. What is the price of each vegetable? +Determining If an Ordered Pair Is a Solution to a System of Equations +When we solved linear equations in Linear Equations in One Variable with Applications and Linear Inequalities in One +Variable with Applications, we learned how to solve linear equations with one variable. Now we will work with two or +more linear equations grouped together, which is known as a system of linear equations. +In this section, we will focus our work on systems of two linear equations in two unknowns (variables) and applications of +systems of linear equations. An example of a system of two linear equations is shown below. We use a brace to show the +two equations are grouped together to form a system of equations. +A linear equation in two variables, such as +, has an infinite number of solutions. Its graph is a line. Remember, +every point on the line is a solution to the equation and every solution to the equation is a point on the line. To solve a +system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other +words, we are looking for the ordered pairs ( , +) that make both equations true. These are called the solutions of a +system of equations. +To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into +each equation. If the ordered pair makes both equations true, it is a solution to the system. +474 +5 • Algebra +Access for free at openstax.org + +EXAMPLE 5.81 +Determining Whether an Ordered Pair Is a Solution to the System +Determine whether the ordered pair is a solution to the system. +1. +2. +Solution +1. +We substitute +and +into both equations. +does not make both equations true. +is not a solution. +2. +We substitute +and +into both equations. +makes both equations true. +is a solution. +YOUR TURN 5.81 +Determine whether the ordered pair is a solution to the system. +1. +2. +EXAMPLE 5.82 +Determining Whether an Ordered Pair Is a Solution to the System +Determine whether the ordered pair is a solution to the system +1. +2. +Solution +1. +Substitute +for +and +for +into both equations. +5.9 • Systems of Linear Equations in Two Variables +475 + +is a solution. +2. +Substitute +for +and +for +into both equations. +is not a solution. +YOUR TURN 5.82 +Determine whether the ordered pair is a solution to the system. +1. +2. +Solving Systems of Linear Equations Using Graphical Methods +We will use three methods to solve a system of linear equations. The first method we will use is graphing. The graph of a +linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph +two lines. Then we can see all the points that are solutions to each equation. And, by finding what points the lines have +in common, we will find the solution to the system. +Most linear equations in one variable have one solution; but for some equations called contradictions, there are no +solutions, and for other equations called identities, all numbers are solutions. Similarly, when we solve a system of two +linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in +Figure 5.83. +476 +5 • Algebra +Access for free at openstax.org + +Figure 5.83 +Each time we demonstrate a new method, we will use it on the same system of linear equations. At the end you will +decide which method was the most convenient way to solve this system. +The steps to use to solve a system of linear equations by graphing are shown here. +Step 1: Graph the first equation. +Step 2: Graph the second equation on the same rectangular coordinate system. +Step 3: Determine whether the lines intersect, are parallel, or are the same line. +Step 4: Identify the solution to the system. +If the lines intersect, identify the point of intersection. This is the solution to the system. +If the lines are parallel, the system has no solution. +If the lines are the same, the system has an infinite number of solutions. +Step 5: Check the solution in both equations. +EXAMPLE 5.83 +Solving a System of Linear Equations by Graphing +Solve this system of linear equations by graphing. +5.9 • Systems of Linear Equations in Two Variables +477 + +Solution +Step 1: Graph the first +equation. +To graph the first line, +write the equation in +slope-intercept form. +Step 2: Graph the +second equation on the +same rectangular +coordinate system. +To graph the second +line, use intercepts. +478 +5 • Algebra +Access for free at openstax.org + +Step 3: Determine +whether the lines +intersect, are parallel, +or are the same line. +Look at the graph of +the lines. +The lines intersect. +Step 4 and Step 5: +Identify the solution to +the system. +If the lines intersect, +identify the point of +intersection. Check to +make sure it is a +solution to both +equations. This is the +solution to the system. +If the lines are parallel, +the system has no +solution. +If the lines are the +same, the system has +an infinite number of +solutions. +Since the lines +intersect, find the +point of intersection. +Check the point in +both equations. +The lines intersect at +. +The solution is +. +YOUR TURN 5.83 +1. Solve this system of linear equations by graphing. +Solving Systems of Linear Equations Using Substitution +We will now solve systems of linear equations by the substitution method. We will use the same system we used for +graphing. +We will first solve one of the equations for either +or +. We can choose either equation and solve for either +variable—but we’ll try to make a choice that will keep the work easy. Then, we substitute that expression into the other +equation. The result is an equation with just one variable—and we know how to solve those! +After we find the value of one variable, we will substitute that value into one of the original equations and solve for the +other variable. Finally, we check our solution and make sure it makes both equations true. This process is summarized +here: +Step 1: Solve one of the equations for either variable. +Step 2: Substitute the expression from Step 1 into the other equation. +Step 3: Solve the resulting equation. +Step 4: Substitute the solution in Step 3 into either of the original equations to find the other variable. +Step 5: Write the solution as an ordered pair. +Step 6: Check that the ordered pair is a solution to both original equations. +5.9 • Systems of Linear Equations in Two Variables +479 + +EXAMPLE 5.84 +Solving a System of Linear Equations Using Substitution +Solve this system of linear equations by substitution: +Solution +Step 1: Solve one of the equations for +either variable. +We’ll solve the first equation +for +. +Step 2: Substitute the expression from +Step 1 into the other equation. +We replace +in the second +equation with the expression +. +Step 3: Solve the resulting equation. +Now we have an equation with +just 1 variable. We know how +to solve this! +Step 4: Substitute the solution from Step +3 into one of the original equations to +find the other variable. +We’ll use the first equation and +replace +with 4. +Step 5: Write the solution as an ordered +pair. +The ordered pair is ( , +). +Step 6: Check that the ordered pair is a +solution to both original equations. +Substitute +, +into +both equations and make sure +they are both true. +Both equations are true. +is the solution to the system. +YOUR TURN 5.84 +1. Solve this system of linear equations by substitution: +Solving Systems of Linear Equations Using Elimination +We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable +coefficients are small, and the solution has integer values. Substitution works well when we can easily solve one +equation for one of the variables and not have too many fractions in the resulting expression. +The third method of solving systems of linear equations is called the elimination method. When we solved a system by +substitution, we started with two equations and two variables and reduced it to one equation with one variable. This is +480 +5 • Algebra +Access for free at openstax.org + +what we’ll do with the elimination method, too, but we’ll have a different way to get there. +The elimination method is based on the Addition Property of Equality. The Addition Property of Equality says that when +you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of +Equality to say that when you add equal quantities to both sides of an equation, the results are equal. For any +expressions +, +, , and +: +if +and +then +. +To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which +variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so +that we can add the equations together and eliminate that variable. Notice how that works when we add these two +equations together: +The +’s add to zero and we have one equation with one variable. Let us try another one: +This time we do not see a variable that can be immediately eliminated if we add the equations. But if we multiply the first +equation by +, we will make the coefficients of +opposites. We must multiply every term on both sides of the equation +by +. +Then rewrite the system of equations. +Now we see that the coefficients of the +terms are opposites, so +will be eliminated when we add these two equations. +Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original +equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both +of the original equations. Here’s a summary of using the elimination method: +Step 1: Write both equations in standard form. If any coefficients are fractions, clear them. +Step 2: Make the coefficients of one variable opposites. +Decide which variable you will eliminate. +Multiply one or both equations so that the coefficients of that variable are opposites. +Step 3: Add the equations resulting from Step 2 to eliminate one variable. +Step 4: Solve for the remaining variable. +Step 5: Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. +Step 6: Write the solution as an ordered pair. +Step 7: Check that the ordered pair is a solution to both original equations. +5.9 • Systems of Linear Equations in Two Variables +481 + +EXAMPLE 5.85 +Solving a System of Linear Equations Using Elimination +Solve this system of linear equations by elimination: +Solution +Step 1: Write both equations in +standard form. +If any coefficients are fractions, clear +them. +Both equations are in standard +form, +. There are no +fractions. +Step 2: Make the coefficients of one +variable opposites. +Decide which variable you will +eliminate. +Multiply one or both equations so that +the coefficients of that variable are +opposites. +We can eliminate the +’s by +multiplying the first equation by +2. +Multiply both sides of +by 2. +Step 3: Add the equations resulting +from Step 2 to eliminate one variable. +We add the +’s, +’s, and +constants. +Step 4: Solve for the remaining +variable. +Solve for +. +Step 5: Substitute the solution from +Step 4 into one of the original +equations. Then solve for the other +variable. +Substitute +into the second +equation, +. Then solve +for +. +Step 6: Write the solution as an +ordered pair. +Write it as +. +Step 7: Check that the ordered pair is +a solution to both original equations. +Substitute +, +into +and +. Do +they make both equations true? +Yes! +The solution is +. +YOUR TURN 5.85 +1. Solve this system of linear equations by elimination: +482 +5 • Algebra +Access for free at openstax.org + +Identifying Systems with No Solution or Infinitely Many Solutions +In all the systems of linear equations so far, the lines intersected, and the solution was one point. In Example 5.86 and +Example 5.87, we will look at a system of equations that has no solution and at a system of equations that has an infinite +number of solutions. +EXAMPLE 5.86 +Solving a System of Linear Equations with No Solution +Solve the system by a method of your choice: +Solution +Let us solve the system of linear equations by graphing. +To graph the first equation, we will use its slope and +-intercept. +To graph the second equation, we will use the intercepts. +0 +−2 +4 +0 +Graph the lines (Figure 5.84). +5.9 • Systems of Linear Equations in Two Variables +483 + +Figure 5.84 +Determine the points of intersection. The lines are parallel. Since no point is on both lines, there is no ordered pair that +makes both equations true. There is no solution to this system. +YOUR TURN 5.86 +1. Solve the system by a method of your choice: +EXAMPLE 5.87 +Solving a System of Linear Equations with Infinite Solutions +Solve the system by a method of your choice: +Solution +Let us solve the system of linear equations by graphing. +Find the slope and +-intercept of the first equation. +Find the intercepts of the second equation. +484 +5 • Algebra +Access for free at openstax.org + +0 +−3 +0 +Graph the lines (Figure 5.85). +Figure 5.85 +The lines are the same! Since every point on the line makes both equations true, there are infinitely many ordered pairs +that make both equations true. +There are infinitely many solutions to this system. +YOUR TURN 5.87 +1. Solve the system by a method of your choice: +In the previous example, if you write the second equation in slope-intercept form, you may recognize that the equations +have the same slope and same +-intercept. Since every point on the line makes both equations true, there are infinitely +many ordered pairs that make both equations true. There are infinitely many solutions to the system. We say the two +lines are coincident. Coincident lines have the same slope and same +-intercept. A system of equations that has at least +one solution is called a consistent system. A system with parallel lines has no solution. We call a system of equations +like this an inconsistent system. It has no solution. +We also categorize the equations in a system of equations by calling the equations independent or dependent. If two +equations are independent, they each have their own set of solutions. Intersecting lines and parallel lines are +independent. If two equations are dependent, all the solutions of one equation are also solutions of the other equation. +When we graph two dependent equations, we get coincident lines. Let us sum this up by looking at the graphs of the +three types of systems. See Figure 5.86 and the table that follows +5.9 • Systems of Linear Equations in Two Variables +485 + +Figure 5.86 +Lines +Intersecting +Parallel +Coincident +Number of Solutions +1 point +No solution +Infinitely many +Consistent/Inconsistent +Consistent +Inconsistent +Consistent +Dependent/Independent +Independent +Independent +Dependent +WORK IT OUT +Using Matrices and Cramer’s Rule to Solve Systems of Linear Equations +An +by +matrix is an array with +rows and +columns, where each item in the matrix is a number. Matrices are used +for many things, but one thing they can be used for is to represent systems of linear equations. For example, the +system of linear equations +can be represented by the following matrix: +To use Cramer’s Rule, you need to be able to take the determinant of a matrix. The determinant of a 2 by 2 matrix +, +denoted +, is +For example, the determinant of the matrix +Cramer’s Rule involves taking three determinants: +1. +The determinant of the first two columns, denoted +; +2. +The determinant of the first column and the third column, denoted +; +3. +The determinant of the third column and the first column, denoted +. +Going back to the original matrix +486 +5 • Algebra +Access for free at openstax.org + +Now Cramer’s Rule for the solution of the system will be: +Putting in the values for these determinants, we have +The solution to the system is the +ordered pair +. +Solving Applications of Systems of Linear Equations +Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two +variables easier than setting them up with just one variable. To solve an application, we will first translate the words into +a system of linear equations. Then we will decide the most convenient method to use, and then solve the system. +Step 1: Read the problem. Make sure all the words and ideas are understood. +Step 2: Identify what we are looking for. +Step 3: Name what we are looking for. Choose variables to represent those quantities. +Step 4: Translate into a system of equations. +Step 5: Solve the system of equations using good algebra techniques. +Step 6: Check the answer in the problem and make sure it makes sense. +Step 7: Answer the question with a complete sentence. +EXAMPLE 5.88 +Applying System to a Real-World Application +Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 a year plus +$15 for each training session. Option B would pay her $10,000 a year plus $40 for each training session. How many +training sessions would make the salary options equal? +Solution +Step 1: Read the problem. +Step 2: Identify what we are looking for. +We are looking for the number of training sessions that would make the pay equal. +Step 3: Name what we are looking for. +Let +, and +Step 4: Translate into a system of equations. +Option A would pay her $25,000 plus $15 for each training session. +Option B would pay her $10,000 + $40 for each training session. +The system is shown. +5.9 • Systems of Linear Equations in Two Variables +487 + +Step 5: Solve the system of equations. +We will use substitution. +Substitute +for +in the second equation +Solve for +. +Step 6: Check the answer. +Are 600 training sessions a year reasonable? +Are the two options equal when +? +Substitute into each equation. +Step 7: Answer the question. +The salary options would be equal for 600 training sessions. +YOUR TURN 5.88 +1. Translate to a system of equations and then solve. +When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for 20 minutes, her fitness app +says she burned 278 calories. When she spent 20 minutes on the elliptical trainer and 30 minutes circuit training, +she burned 473 calories. How many calories does she burn for each minute on the elliptical trainer? How many +calories for each minute of circuit training? +VIDEO +Practice with Solving Applications of Systems of Equations (https://openstax.org/r/Practice_with_Solving) +Applications of Systems of Linear Equations (https://openstax.org/r/Applications_of_Systems) +Check Your Understanding +Decide whether it would be more convenient to solve the system of equations by substitution or elimination. +67. +68. +69. +70. +71. +488 +5 • Algebra +Access for free at openstax.org + +72. +73. +74. +SECTION 5.9 EXERCISES +For the following exercises, determine if the points are solutions to the given system of equations. +1. +2. +3. +4. +For the following exercises, solve the following systems of equations by graphing. +5. +6. +7. +8. +9. +10. +11. +12. +For the following exercises, solve the systems of equations by substitution. +13. +14. +15. +16. +17. +18. +19. +5.9 • Systems of Linear Equations in Two Variables +489 + +20. +For the following exercises, solve the systems of equations by elimination. +21. +22. +23. +24. +25. +26. +27. +28. +For the following exercises, solve the system of equations by graphing, substitution, or elimination. +29. +30. +31. +32. +33. +34. +35. +36. +For the following exercises, translate to a system of equations and solve. +37. Jackie has been offered positions by two cable companies. The first company pays a salary of $14,000 plus a +commission of $100 for each cable package sold. The second pays a salary of $20,000 plus a commission of $25 +for each cable package sold. How many cable packages would need to be sold to make the total pay the same? +38. Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. Saturday she spent 2 +hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did she burn per +hour when playing basketball? How many calories did she burn per hour when canoeing? +39. Mitchell currently sells stoves for company A at a salary of $12,000 plus a $150 commission for each stove he +sells. Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. +How many stoves would Mitchell need to sell for the options to equal? +40. The total number of calories in 2 hot dogs and 3 cups of cottage cheese is 960 calories. The total number of +calories in 5 hot dogs and 2 cups of cottage cheese is 1,190 calories. How many calories are in a hot dog? How +many calories are in a cup of cottage cheese? +41. Andrea and Bart go to the local farmers market to purchase some fruit. Andrea buys 4 apples and 5 oranges, +which cost $3.10; Bart buys 4 apples and 6 oranges, which cost $3.40. What is the cost of an orange? What is the +cost of an apple? +490 +5 • Algebra +Access for free at openstax.org + +42. Jack and Jill go to a local farmers market to purchase some fruit. Jack buys 3 peaches and 2 limes, which cost +$1.50; Jill buys 6 peaches and 5 limes, which cost $3.45. What is the cost of a peach? What is the cost of a lime? +5.10 Systems of Linear Inequalities in Two Variables +Figure 5.87 Many college students find part-time jobs at places such as coffee shops to help pay for college. (credit: +modification of work “TULLY’s COFFEE” by MIKI Yoshihito/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Demonstrate whether an ordered pair is a solution to a system of linear inequalities. +2. +Solve systems of linear inequalities using graphical methods. +3. +Graph systems of linear inequalities. +4. +Interpret and solve applications of linear inequalities. +In this section, we will learn how to solve systems of linear inequalities in two variables. In Systems of Linear Equations in +Two Variables, we learned how to solve for systems of linear equations in two variables and found a solution that would +work in both equations. We can solve systems of inequalities by graphing each inequality (as discussed in Graphing +Linear Equations and Inequalities) and putting these on the same coordinate system. The double-shaded part will be our +solution to the system. There are many real-life examples for solving systems of linear inequalities. +Consider Ming who has two jobs to help her pay for college. She works at a local coffee shop for $7.50 per hour and at a +research lab on campus for $12 per hour. Due to her busy class schedule, she cannot work more than 15 hours per week. +If she needs to make at least $150 per week, can she work seven hours at the coffee shop and eight hours in the lab? +Determining If an Ordered Pair Is a Solution of a System of Linear Inequalities +The definition of a system of linear inequalities is similar to the definition of a system of linear equations. A system of +linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two +linear inequalities is shown here. +To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We +solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the +plane that contains all ordered pairs +that make both inequalities true. The solution of a system of linear +inequalities is shown as a shaded region in the +-coordinate system that includes all the points whose ordered pairs +make the inequalities true. +To determine if an ordered pair is a solution to a system of two inequalities, substitute the values of the variables into +each inequality. If the ordered pair makes both inequalities true, it is a solution to the system. +5.10 • Systems of Linear Inequalities in Two Variables +491 + +EXAMPLE 5.89 +Determining Whether an Ordered Pair Is a Solution to a System +Determine whether the ordered pair is a solution to the system: +1. +2. +Solution +1. +Is the ordered pair +a solution? +We substitute +and +into both inequalities. +The ordered pair +made both inequalities true. Therefore +is a solution to this system. +2. +Is the ordered pair +a solution? +We substitute +and +into both inequalities. +The ordered pair +made one inequality true, but the other one false. Therefore +is not a solution to this +system. +YOUR TURN 5.89 +Determine whether the ordered pair is a solution to the system: +1. +2. +Solving Systems of Linear Inequalities Using Graphical Methods +The solution to a single linear inequality was the region on one side of the boundary line that contains all the points that +make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to +both inequalities. We will review graphs of linear inequalities and solve the linear inequality from its graph. +EXAMPLE 5.90 +Solving a System of Linear Inequalities by Graphing +Use Figure 5.88 to solve the system of linear inequalities: +492 +5 • Algebra +Access for free at openstax.org + +Figure 5.88 +Solution +To solve the system of linear inequalities we look at the graph and find the region that satisfies BOTH inequalities. To do +this we pick a test point and check. Let's us pick +. +Is +a solution to +Is +a solution to +The region containing +is the solution to the system of linear inequalities. Notice that the solution is all the +points in the area shaded twice, which appears as the darkest shaded region. +YOUR TURN 5.90 +1. Use the graph shown to solve the system of linear inequalities: +5.10 • Systems of Linear Inequalities in Two Variables +493 + +Graphing Systems of Linear Inequalities +We learned that the solution to a system of two linear inequalities is a region that contains the solutions to both +inequalities. To find this region by graphing, we will graph each inequality separately and then locate the region where +they are both true. The solution is always shown as a graph. +Step 1: Graph the first inequality. +Graph the boundary line. +Shade in the side of the boundary line where the inequality is true. +Step 2: On the same grid, graph the second inequality. +Graph the boundary line. +Shade in the side of that boundary line where the inequality is true. +Step 3: The solution is the region where the shading overlaps. +Step 4: Check by choosing a test point. +EXAMPLE 5.91 +Solving a System of Linear Inequalities by Graphing +Solve the system by graphing: +Solution +Graph +by graphing +and testing a point (Figure 5.89). The intercepts are +and +and the +boundary line will be dashed. Test +which makes the inequality false so shade the side that does not contain +. +494 +5 • Algebra +Access for free at openstax.org + +Figure 5.89 +Graph +by graphing +using the slope +and +-intercept +(Figure 5.90). The +boundary line will be dashed. Test +which makes the inequality true, so shade the side that contains +. +Figure 5.90 +Choose a test point in the solution and verify that it is a solution to both inequalities. The point of intersection of the two +lines is not included as both boundary lines were dashed. The solution is the area shaded twice—which appears as the +darkest shaded region. +YOUR TURN 5.91 +1. Solve the system by graphing: +5.10 • Systems of Linear Inequalities in Two Variables +495 + +EXAMPLE 5.92 +Graphing a System of Linear Inequalities +Solve the system by graphing: +Solution +Graph +by graphing +(Figure 5.91) and testing a point. The intercepts are +and +and +the boundary line will be dashed. Test +, which makes the inequality true, so shade the side that contains +. +Figure 5.91 +Graph +by graphing +and recognizing that it is a horizontal line through +(Figure 5.92). The boundary +line will be dashed. Test +, which makes the inequality true so shade the side that contains +. +Figure 5.92 +The point +is in the solution, and we have already found it to be a solution of each inequality. The point of +496 +5 • Algebra +Access for free at openstax.org + +intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice, +which appears as the darkest shaded region. +YOUR TURN 5.92 +1. Solve the system by graphing: +Systems of linear inequalities where the boundary lines are parallel might have no solution. We will see this in the next +example. +EXAMPLE 5.93 +Graphing Parallel Boundary Lines with No Solution +Solve the system by graphing: +Solution +Graph +, by graphing +(Figure 5.93) and testing a point. The intercepts are +and +and +the boundary line will be solid. Test +, which makes the inequality false, so shade the side that does not contain +. +Figure 5.93 +Graph +by graphing +using the slope +and +-intercept +(Figure 5.94). The +boundary line will be dashed. Test +, which makes the inequality true, so shade the side that contains +. +5.10 • Systems of Linear Inequalities in Two Variables +497 + +Figure 5.94 +No shared point exists in both shaded regions, so the system has no solution. +YOUR TURN 5.93 +1. Solve the system by graphing: +Some systems of linear inequalities where the boundary lines are parallel will have a solution. We will see this in the next +example. +EXAMPLE 5.94 +Graphing Parallel Boundary Lines with a Solution +Solve the system by graphing: +Solution +Graph +by graphing +using the slope +and the +-intercept +(Figure 5.95). The +boundary line will be dashed. Test +, which makes the inequality true, so shade the side that contains +. +498 +5 • Algebra +Access for free at openstax.org + +Figure 5.95 +Graph +by graphing +(Figure 5.96) and testing a point. The intercepts are +and +and +the boundary line will be dashed. Choose a test point in the solution and verify that it is a solution to both inequalities. +Test +, which makes the inequality false, so shade the side that does not contain +. +Figure 5.96 +No point on the boundary lines is included in the solution as both lines are dashed. The solution is the region that is +shaded twice which is also the solution to +. +YOUR TURN 5.94 +1. Solve the system by graphing: +5.10 • Systems of Linear Inequalities in Two Variables +499 + +Interpreting and Solving Applications of Linear Inequalities +When solving applications of systems of inequalities, first translate each condition into an inequality. Then graph the +system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both +variables are positive, so add inequalities to the system as additional requirements. +EXAMPLE 5.95 +Applying Linear Inequalities to Calculating Photo Costs +A photographer sells their prints at a booth at a street fair. At the start of the day, they want to have at least 25 photos to +display at their booth. Each small photo they display costs $4 and each large photo costs $10. They do not want to spend +more than $200 on photos to display. +1. +Write a system of inequalities to model this situation. +2. +Graph the system. +3. +Could they display 10 small and 20 large photos? +4. +Could they display 20 large and 10 small photos? +Solution +1. +Let +the number of small photos and +. To find the system of equations translate +the information. They want to have at least 25 photos. +The number of small plus the number of large should be at least 25. +$4 for each small and $10 for each large must be no more than $200 +The number of small photos must be greater than or equal to 0. +The number of large photos must be greater than or equal to 0. +We have our system of equations. +2. +Since +and +(both are greater than or equal to) all solutions will be in the first quadrant. As a result, our +graph shows only Quadrant I. To graph +, graph +as a solid line. Choose +as a test point. +Since it does not make the inequality true, shade the side that does not include the point +. +To graph +, graph +as a solid line. Choose +as a test point. Since it does make the +inequality true, shade (bottom left) the side that include the point +. +Figure 5.97 +The solution of the system is the region of Figure 5.97 that is shaded the darkest. The boundary line sections that +border the darkly shaded section are included in the solution as are the points on the +-axis from +to +. +500 +5 • Algebra +Access for free at openstax.org + +3. +To determine if 10 small and 20 large photos would work, we look at the graph to see if the point +is in the +solution region. We could also test the point to see if it is a solution of both equations. It is not, so the photographer +would not display 10 small and 20 large photos. +4. +To determine if 20 small and 10 large photos would work, we look at the graph to see if the point +is in the +solution region. We could also test the point to see if it is a solution of both equations. It is, so the photographer +could choose to display 20 small and 10 large photos. Notice that we could also test the possible solutions by +substituting the values into each inequality. +YOUR TURN 5.95 +Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers and cookies, +and he doesn’t want to spend more than $5. At the hamburger restaurant near his college, each hamburger has 240 +calories and costs $1.40. Each cookie has 160 calories and costs $0.50. +1. Write a system of inequalities to model this situation. +2. Graph the system. +3. Could he eat 3 hamburgers and 2 cookies? +4. Could he eat 2 hamburgers and 4 cookies? +VIDEO +Solving Systems of Linear Inequalities by Graphing (https://openstax.org/r/Solving_Systems) +Systems of Linear Inequalities (https://openstax.org/r/Systems_of_Linear) +Check Your Understanding +Match the correct graph to its system of inequalities. +75. +a. +5.10 • Systems of Linear Inequalities in Two Variables +501 + +b. +c. +502 +5 • Algebra +Access for free at openstax.org + +d. +e. +76. +5.10 • Systems of Linear Inequalities in Two Variables +503 + +a. +b. +504 +5 • Algebra +Access for free at openstax.org + +c. +d. +5.10 • Systems of Linear Inequalities in Two Variables +505 + +e. +77. +a. +506 +5 • Algebra +Access for free at openstax.org + +b. +c. +5.10 • Systems of Linear Inequalities in Two Variables +507 + +d. +e. +78. +508 +5 • Algebra +Access for free at openstax.org + +a. +b. +5.10 • Systems of Linear Inequalities in Two Variables +509 + +c. +d. +510 +5 • Algebra +Access for free at openstax.org + +e. +79. +a. +5.10 • Systems of Linear Inequalities in Two Variables +511 + +b. +c. +512 +5 • Algebra +Access for free at openstax.org + +d. +e. +SECTION 5.10 EXERCISES +For the following exercises, determine whether each ordered pair is a solution to the system. +1. +A: +B: +2. +A: +B: +5.10 • Systems of Linear Inequalities in Two Variables +513 + +3. +A: +B: +4. +A: +B: +5. +A: +B: +6. +A: +B: +For the following exercises, determine whether each ordered pair is a solution to the darkest shaded region of the +graph. +7. +A: +B: +514 +5 • Algebra +Access for free at openstax.org + +8. +A: +B: +9. +A: +B: +10. +A: +5.10 • Systems of Linear Inequalities in Two Variables +515 + +B: +11. +A: +B: +12. +A: +B: +13. +516 +5 • Algebra +Access for free at openstax.org + +A: +B: +For the following exercises, solve the systems of linear equations by graphing. +14. +15. +16. +17. +18. +19. +20. +21. +22. +23. +24. +25. +26. +27. +28. +29. +30. +For the following exercises, translate to a system of inequalities and solve. +A gardener does not want to spend more than $50 on bags of fertilizer and peat moss for their garden. Fertilizer costs +$2 a bag and peat moss costs $5 a bag. The gardener’s van can hold at most 20 bags. +31. Write a system of inequalities to model this situation. +32. Graph the system. +33. Can they buy 15 bags of fertilizer and 4 bags of peat moss? +34. Can they buy 10 bags of fertilizer and 10 bags of peat moss? +For the following exercises, translate to a system of inequalities and solve. +A student is studying for their final exams in chemistry and algebra. They only have 24 hours to study, and it will take +them at least 3 times as long to study for algebra than chemistry. +35. Write a system of inequalities to model this situation. +36. Graph the system. +37. Can they spend 4 hours on chemistry and 20 hours on algebra? +5.10 • Systems of Linear Inequalities in Two Variables +517 + +38. Can they spend 6 hours on chemistry and 18 hours on algebra? +For the following exercises, translate to a system of inequalities and solve. +Mara is attempting to build muscle mass. To do this, she needs to eat an additional 80 grams of protein or more in a +day. A bottle of protein water costs $3.20 and a protein bar costs $1.75. The protein water supplies 27 grams of protein +and the bar supplies 16 grams. Let +be the number of water bottles Mara can buy, and let +be the number of protein +bars she can buy. If Mara has $10 dollars to spend: +39. Write a system of inequalities to model this situation. +40. Graph the system. +41. Could she buy 3 bottles of protein water and 1 protein bar? +42. Could she buy no bottles of protein water and 5 protein bars? +For the following exercises, translate to a system of inequalities and solve. +Mark is increasing his exercise routine by running and walking at least 4 miles each day. His goal is to burn a minimum +of 1,500 calories from this exercise. Walking burns 270 calories/mile and running burns 650 calories/mile. +43. Write a system of inequalities to model this situation. +44. Graph the system. +45. Could he meet his goal by walking 3 miles and running 1 mile? +46. Could he meet his goal by walking 2 miles and running 2 miles? +For the following exercises, translate to a system of inequalities and solve. +Tension needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only $25 to spend +on the extra food he needs and will spend it on $0.75 donuts, which have 360 calories each, and $2 energy drinks, +which have 110 calories. +47. Write a system of inequalities that models this situation. +48. Graph the system. +49. Can he buy 8 donuts and 4 energy drinks and satisfy his caloric needs? +50. Can he buy 1 donut and 3 energy drinks and satisfy his caloric needs? +5.11 Linear Programming +Figure 5.98 The aftermath of an earthquake and tsunami. (credit: modification of work "Earthquake and Tsunami Japan" +by Climate and Ecosystems Change Adaptation Research University Network/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Compose an objective function to be minimized or maximized. +2. +Compose inequalities representing a system application. +3. +Apply linear programming to solve application problems. +518 +5 • Algebra +Access for free at openstax.org + +Imagine you hear about some natural disaster striking a far-away country; it could be an earthquake, a fire, a tsunami, a +tornado, a hurricane, or any other type of natural disaster. The survivors of this disaster need help—they especially need +food, water, and medical supplies. You work for a company that has these supplies, and your company has decided to +help by flying the needed supplies into the disaster area. They want to maximize the number of people they can help. +However, there are practical constraints that need to be taken into consideration; the size of the airplanes, how much +weight each airplane can carry, and so on. How do you solve this dilemma? This is where linear programming comes into +play. Linear programming is a mathematical technique to solve problems involving finding maximums or minimums +where a linear function is limited by various constraints. +As a field, linear programming began in the late 1930s and early 1940s. It was used by many countries during World War +II; countries used linear programming to solve problems such as maximizing troop effectiveness, minimizing their own +casualties, and maximizing the damage they could inflict upon the enemy. Later, businesses began to realize they could +use the concept of linear programming to maximize output, minimize expenses, and so on. In short, linear programming +is a method to solve problems that involve finding a maximum or minimum where a linear function is constrained by +various factors. +WHO KNEW? +A Mathematician Invents a “Tsunami Cannon” +On December 26, 2004, a massive earthquake occurred in the Indian Ocean. This earthquake, which scientists +estimate had a magnitude of 9.0 or 9.1 on the Richter Scale, set off a wave of tsunamis across the Indian Ocean. The +waves of the tsunami averaged over 30 feet (10 meters) high, and caused massive damage and loss of life across the +coastal regions bordering the Indian Ocean. +Usama Kadri works as an applied mathematician at Cardiff University in Wales. His areas of research include fluid +dynamics and non-linear phenomena. Lately, he has been focusing his research on the early detection and easing of +the effects of tsunamis. One of his theories involves deploying a series of devices along coastlines which would fire +acoustic-gravity waves (AGWs) into an oncoming tsunami, which in theory would lessen the force of the tsunami. Of +course, this is all in theory, but Kadri believes it will work. There are issues with creating such a device: they would +take a tremendous amount of electricity to generate an AGW, for instance, but if it would save lives, it may well be +worth it. +Compose an Objective Function to Be Minimized or Maximized +An objective function is a linear function in two or more variables that describes the quantity that needs to be +maximized or minimized. +EXAMPLE 5.96 +Composing an Objective Function for Selling Two Products +Miriam starts her own business, where she knits and sells scarves and sweaters out of high-quality wool. She can make a +profit of $8 per scarf and $10 per sweater. Write an objective function that describes her profit. +Solution +Let +represent the number of scarves sold, and let +represent the number of sweaters sold. Let +represent profit. +Since each scarf has a profit of $8 and each sweater has a profit of $10, the objective function is +. +YOUR TURN 5.96 +1. For a fundraiser at school, the Robotics Club is selling bags of apples and bunches of bananas during lunch. They +will make a profit of $4 per bag of apples and $6 per bunch of bananas. Write an objective function that +describes the profit the Robotics Club will make. +5.11 • Linear Programming +519 + +EXAMPLE 5.97 +Composing an Objective Function for Production +William’s factory produces two products, widgets and wadgets. It takes 24 minutes for his factory to make 1 widget, and +32 minutes for his factory to make 1 wadget. Write an objective function that describes the time it takes to make the +products. +Solution +Let +equal the number of widgets made; let +equal the number of wadgets made; let +represent total time. The +objective function is +. +YOUR TURN 5.97 +1. Suppose William has a second factory that can make widgets in 20 minutes and wadgets in 28 minutes. Write an +objective function that describes the time it takes to make the products. +Composing Inequalities Representing a System Application +For our two examples of profit and production, in an ideal world the profit a person makes and/or the number of +products a company produces would have no restrictions. After all, who wouldn’t want to have an unrestricted profit? +However in reality this is not the case; there are usually several variables that can restrict how much profit a person can +make or how many products a company can produce. These restrictions are called constraints. +Many different variables can be constraints. When making or selling a product, the time available, the cost of +manufacturing and the amount of raw materials are all constraints. In the opening scenario with the tsunami, the +maximum weight on an airplane and the volume of cargo it can carry would be constraints. Constraints are expressed as +linear inequalities; the list of constraints defined by the problem forms a system of linear inequalities that, along with the +objective function, represent a system application. +EXAMPLE 5.98 +Representing the Constraints for Selling Two Products +Two friends start their own business, where they knit and sell scarves and sweaters out of high-quality wool. They can +make a profit of $8 per scarf and $10 per sweater. To make a scarf, 3 bags of knitting wool are needed; to make a +sweater, 4 bags of knitting wool are needed. The friends can only make 8 items per day, and can use not more than 27 +bags of knitting wool per day. Write the inequalities that represent the constraints. Then summarize what has been +described thus far by writing the objective function for profit and the two constraints. +Solution +Let +represent the number of scarves sold, and let +represent the number of sweaters sold. There are two constraints: +the number of items the business can make in a day (a maximum of 8) and the number of bags of knitting wool they can +use per day (a maximum of 27). The first constraint (total number of items in a day) is written as: +Since each scarf takes 3 bags of knitting wool and each sweater takes 4 bags of knitting wool, the second constraint, +total bags of knitting wool per day, is written as: +In summary, here are the equations that represent the new business: +; This is the profit equation: The business makes $8 per scarf and $10 per sweater. +520 +5 • Algebra +Access for free at openstax.org + +YOUR TURN 5.98 +1. For a fundraiser at school, the Robotics Club is selling bags of apples and bunches of bananas during lunch. They +will make a profit of $4 per bag of apples and $6 per bunch of bananas. Due to school health regulations, the +club is allowed to have only 20 bags and bunches of fruit on school grounds each day to sell. Another regulation: +the container where the Robotics Club keeps the fruit has a maximum weight capacity of 70 pounds. Each bag of +apples weighs 3 pounds, while each bunch of bananas weighs 5 pounds. Write the inequalities that represent +these constraints. Then summarize the equations that represent this system. +EXAMPLE 5.99 +Representing Constraints for Production +A factory produces two products, widgets and wadgets. It takes 24 minutes for the factory to make 1 widget, and 32 +minutes for the factory to make 1 wadget. Research indicates that long-term demand for products from the factory will +result in average sales of 12 widgets per day and 10 wadgets per day. Because of limitations on storage at the factory, no +more than 20 widgets or 17 wadgets can be made each day. Write the inequalities that represent the constraints. Then +summarize what has been described thus far by writing the objective function for time and the two constraints. +Solution +Let +equal the number of widgets made; let +equal the number of wadgets made. Based on the long-term demand, we +know the factory must produce a minimum of 12 widgets and 10 wadgets per day. We also know because of storage +limitations, the factory cannot produce more than 20 widgets per day or 17 wadgets per day. Writing those as +inequalities, we have: +The number of widgets made per day must be between 12 and 20, and the number of wadgets made per day must be +between 10 and 17. Therefore, we have: +The system is: +is the variable for time; it takes 24 minutes to make a widget and 32 minutes to make a wadget. +YOUR TURN 5.99 +1. Suppose a second factory can make widgets in 20 minutes and wadgets in 28 minutes. Research for this factory +indicates that long-term demand for products from this second factory will result in average sales of 15 widgets +per day and 13 wadgets per day. Because of limitations on storage at his factory, no more than 22 widgets or 19 +wadgets can be made each day. Write the inequalities that represent the constraints. Then summarize what has +been described thus far by writing the objective function for time and the two constraints. +Applying Linear Programming to Solve Application Problems +There are four steps that need to be completed when solving a problem using linear programming. They are as follows: +5.11 • Linear Programming +521 + +Step 1: Compose an objective function to be minimized or maximized. +Step 2: Compose inequalities representing the constraints of the system. +Step 3: Graph the system of inequalities representing the constraints. +Step 4: Find the value of the objective function at each corner point of the graphed region. +The first two steps you have already learned. Let’s continue to use the same examples to illustrate Steps 3 and 4. +EXAMPLE 5.100 +Solving a Linear Programming Problem for Two Products +Three friends start their own business, where they knit and sell scarves and sweaters out of high-quality wool. They can +make a profit of $8 per scarf and $10 per sweater. To make a scarf, 3 bags of knitting wool are needed; to make a +sweater, 4 bags of knitting wool are needed. The friends can only make 8 items per day, and can use not more than 27 +bags of knitting wool per day. Determine the number of scarves and sweaters they should make each day to maximize +their profit. +Solution +Step 1: Compose an objective function to be minimized or maximized. From Example 5.98, the objective function is +. +Step 2: Compose inequalities representing the constraints of the system. From Example 5.98, the constraints are +and +. +Step 3: Graph the system of inequalities representing the constraints. Using methods discussed in Graphing Linear +Equations and Inequalities, the graphs of the constraints are shown below. Because the number of scarves ( ) and the +number of sweaters ( ) both must be non-negative numbers (i.e., +and +), we need to graph the system of +inequalities in Quadrant I only. Figure 5.99 shows each constraint graphed on its own axes, while Figure 5.100 shows the +graph of the system of inequalities (the two constraints graphed together). In Figure 5.100, the large shaded region +represents the area where the two constraints intersect. If you are unsure how to graph these regions, refer back to +Graphing Linear Equations and Inequalities. +Figure 5.99 Graphs of each constraint +522 +5 • Algebra +Access for free at openstax.org + +Figure 5.100 Graph of the System of Inequalities +Step 4: Find the value of the objective function at each corner point of the graphed region. The “graphed region” is the +area where both of the regions intersect; in Figure 5.101, it is the large shaded area. The “corner points” refer to each +vertex of the shaded area. Why the corner points? Because the maximum and minimum of every objective function will +occur at one (or more) of the corner points. Figure 5.101 shows the location and coordinates of each corner point. +Figure 5.101 Graph of Region with Corner Points +Three of the four points are readily found, as we used them to graph the regions; the fourth point, the intersection point +of the two constraint lines, will have to be found using methods discussed in Systems of Linear Equations in Two +Variables, either using substitution or elimination. As a reminder, set up the two equations of the constraint lines: +For this example, substitution will be used. +Substituting +into the first equation for +, we have +5.11 • Linear Programming +523 + +Now, substituting the 5 in for +in either equation to solve for +. Choosing the second equation, we have: +Therefore, +, and +. +To find the value of the objective function, +, put the coordinates for each corner point into the equation +and solve. The largest solution found when doing this will be the maximum value, and thus will be the answer to the +question originally posed: determining the number of scarves and sweaters the new business should make each day to +maximize their profit. +Corner ( , +) +Objective Function +The maximum value for the profit +occurs when +and +. This means that to maximize their profit, the new +business should make 5 scarves and 3 sweaters every day. +YOUR TURN 5.100 +1. For a fundraiser at school, the Robotics Club is selling bags of apples and bunches of bananas during lunch. They +will make a profit of $4 per bag of apples and $6 per bunch of bananas. Due to school health regulations, the +club is allowed to have only 20 bags and bunches of fruit on school grounds each day to sell. Another regulation: +the container where the Robotics Club keeps the fruit has a maximum weight capacity of 70 pounds. Each bag of +apples weighs 3 pounds, while each bunch of bananas weighs 5 pounds. Determine the number of bags of +apples and the number of bags of bananas the Robotics Club should sell each day to maximize their profit. +PEOPLE IN MATHEMATICS +Leonid Kantorovich +Leonid Vitalyevich Kantorovich was born January 19, 1912, in St. Petersburg, Russia. Two major events affected young +Leonid’s life: when he was five, the Russian Revolution began, making life in St. Petersburg very difficult; so much so +that Leonid’s family fled to Belarus for a year. When Leonid was 10, his father died, leaving his mother to raise five +children on her own. +Despite the hardships, Leonid showed incredible mathematical ability at a young age. When he was only 14, he +enrolled in Leningrad State University to study mathematics. Four years later, at age 18, he graduated with what +524 +5 • Algebra +Access for free at openstax.org + +would be equivalent to a Ph.D. in mathematics. +Although his primary interests were in pure mathematics, in 1938 he began working on problems in economics. +Supposedly, he was approached by a local plywood manufacturer with the following question: how to come up with a +work schedule for eight lathes to maximize output, given the five different kinds of plywood they had at the factory. +By July 1939, Leonid had come up with a solution, not only to the lathe scheduling problem but to other areas as well, +such as an optimal crop rotation schedule for farmers, minimizing waste material in manufacturing, and finding +optimal routes for transporting goods. The technique he discovered to solve these problems eventually became +known as linear programming. He continued to use this technique for solving many other problems involving +optimization, which resulted in the book The Best Use of Economic Resources, which was published in 1959. His +continued work in linear programming would ultimately result in him winning the Nobel Prize of Economics in 1975. +Check Your Understanding +80. Kellie makes tables and chairs. Kellie profits $20 from a table ( ), and $10 from a chair ( ). The objective function for +profit in this situation is: +a. +b. +c. +d. +81. Dave grows wheat ( +) and barley ( ) on a farm. Dave expects to profit $150 per acre for wheat and $180 per acre +for barley. The objective function for profit in this situation is: +a. +b. +c. +d. +82. An antique music store sells two types of vinyl records; 45 rpm records ( ) and 33 rpm records ( ). It makes a profit +of $2.50 for each 45 rpm record and $6.75 for each 33 rpm record. The objective function for profit in this situation +is: +a. +b. +c. +d. +None of these +83. Kellie makes tables and chairs. Kellie profits $20 from a table ( ), and $10 from a chair ( ). A table requires 15 board +feet of wood, while a chair requires 4 board feet of wood. Kellie has 70 board feet available. What is the constraint +inequality in this situation? +a. +b. +c. +d. +84. Kellie makes tables and chairs. Kellie profits $20 from a table ( ), and $10 from a chair ( ). The maximum number of +tables and chairs Kellie can make in any one day is 12. What is the constraint inequality in this situation? +a. +b. +c. +d. +85. Dave grows wheat ( +) and barley ( ) on a farm. Dave expects to profit $150 per acre for wheat and $180 per acre +for barley. The cost of seed is $10 per acre for wheat and $15 per acre for barley. Dave can only afford to spend +$945 on seed. What is the constraint inequality in this situation? +a. +b. +5.11 • Linear Programming +525 + +c. +d. +86. Dave grows wheat ( +) and barley ( ) on a farm. Dave expects to profit $150 per acre for wheat and $180 per acre +for barley. The cost of raising each crop is $30 per acre for wheat and $25 per acre for barley. Dave budgets $1,635 +for the raising of both crops. What is the constraint inequality in this situation? +a. +b. +c. +d. +87. Kellie makes tables and chairs. Kellie profits $20 from a table ( ), and $10 from a chair ( ). A table requires 15 board feet +of wood, while a chair requires 4 board feet of wood. Kellie has 70 board feet available. The maximum number of tables +and chairs Kellie can make in any one day is 12. The graph of the system of inequalities representing the constraints is: +88. Kellie makes tables and chairs. Kellie profits $20 from a table ( ), and $10 from a chair ( ). A table requires 15 board +feet of wood, while a chair requires 4 board feet of wood. Kellie has 70 board feet available. The maximum number +of tables and chairs Kellie can make in any one day is 12. The four corner points of the system are: +a. +b. +c. +89. Kellie makes tables and chairs. Kellie profits $20 from a table ( ), and $10 from a chair ( ). A table requires 15 board +feet of wood, while a chair requires 4 board feet of wood. Kellie has 70 board feet available. The maximum number +of tables and chairs Kellie can make in any one day is 12. The maximum profit Kellie can make in one day is: +SECTION 5.11 EXERCISES +For the following exercises, find the value of the objective function at each corner of the graphed region. +1. Objective Function +. +526 +5 • Algebra +Access for free at openstax.org + +2. Objective Function +3. Objective Function +4. Objective Function +5.11 • Linear Programming +527 + +5. Objective Function +For the following exercises, write the constraint inequalities. The variables to use are given in parentheses. +6. Fernando builds birdbaths +and birdhouses +. Fernando can make a total of 7 birdbaths and birdhouses per +day. A birdbath costs $8 to make, while a birdhouse costs $6 to make. Fernando has $48 to spend on building +materials for the day. When he sells them, Fernando makes $12 in profit on a birdbath and $9 in profit on a +birdhouse. +7. A fruit pie +requires 12 ounces of fruit and 15 ounces of dough; a fruit tart +requires 4 ounces of fruit and 3 +ounces of dough. There are 72 ounces of fruit and 60 ounces of dough. +8. One recipe for chocolate cake +calls for 9 ounces of chocolate chips and 4 eggs; a recipe for dark chocolate +cake +requires 12 ounces of chocolate chips but only 3 eggs. There are 90 ounces of chocolate chips and 36 +eggs. +9. To build an outdoor bench +, a carpenter needs 10 pieces of wood and 26 nails; to build an outdoor chair +, +the carpenter need 8 pieces of wood and 33 nails. There are 92 pieces of wood and 286 nails. +For the following exercises, graph each of the system of inequalities from Exercises 6–9. Assume all graphs are in the +first quadrant. +10. Graph of Exercise 6 +11. Graph of Exercise 7 +12. Graph of Exercise 8 +13. Graph of Exercise 9 +For the following exercises, use the four steps for solving linear programming problems to solve. +14. A restaurant sells both regular milk and chocolate milk. To make a glass of regular milk ( ), it takes 16 ounces +of, well, milk. To make a glass of chocolate milk ( ), it takes 15 ounces of milk and 1 ounce of chocolate +flavoring. The restaurant makes a profit of $1.50 per glass on regular milk and $1.00 per glass on chocolate +528 +5 • Algebra +Access for free at openstax.org + +milk. At the beginning of the day, the restaurant has 600 ounces of milk and 24 ounces of chocolate flavoring. +To maximize profits, how much of each should they sell that day? +15. To make a package of all-beef hot dogs ( ), a factory uses one pound of beef; to make their regular all-meat hot +dogs ( ), they use ½ pound of beef and ½ pound of pork. The profit on the package of all-beef hot dogs is $2.40 +per pack; the profit on the all-meat hot dogs is $3.20 per pack. If there are 400 pounds of beef and 250 pounds +of pork available, how many of each product should the factory make to maximize their profit? +16. A toy maker makes two plastic toys, the Ring ( ) and the Stick ( ). The toy maker makes $5 per Ring and $4 per +Stick. The Ring uses 4 feet of plastic, while the Stick uses 3 feet of plastic. Today the toy maker has 36 feet of +plastic available. The toy maker also only makes 10 plastic toys per day. To maximize profit, how many of each +toy should the toy maker make? +17. The toy maker also makes exactly two toys out of wood, the Box ( ) and the Bat ( ). The toy maker makes $6 per +Box and $7 per Bat. Each Box requires 25 ounces of wood, and each Bat requires 40 ounces of wood. Today the +toy maker has 260 ounces of wood available. The toy maker also only makes 8 wooden toys per day. To +maximize profit, how many of each wooden toy should the toy maker make? +18. Sara makes two kinds of kites out of fabric and popsicle sticks. Her Famous Flyer ( ) needs 2 yards of fabric and +9 popsicle sticks; her Gallant Glider ( ) needs 3 yards of fabric and 18 popsicle sticks. She makes a profit of $4 +on the Famous Flyer and $6 on the Gallant Glider. Today she has 30 yards of fabric and 153 popsicle sticks. How +many of each kite should she make to maximize her profit? +19. Randy’s RV Storage stores two types of Recreational Vehicles (RVs), The Xtra RV ( ) takes up 400 square feet of +space, while the Yosemite RV ( ) takes up 600 square feet of space. Randy has 55,000 square feet of storage +space. By local law, he is only allowed to have a maximum of 100 RVs on his property at any one time. He +charges $60 a month to store an Xtra RV, and $80 a month to store a Yosemite RV. How many of each should he +store in order to maximize his profit? +20. A Belgian chocolatier wants to introduce two new chocolate bar creations. The first chocolate bar is called Super +Dark ( ), and it consists of 90 grams of chocolate and 10 grams of sugar. The second chocolate bar is called +Special Dark ( ), containing 80 grams of chocolate and 20 grams of sugar. She calculates that her company will +make 1 Euro per bar of Super Dark, and 2 Euros per bar on Special Dark. She first will create some samples to +sell out of 1,260 grams of chocolate and 240 grams of sugar. How many of each bar should the chocolatier +create to maximize profit? +21. A juice bottler makes two kinds of specialty juices using different mixtures of pineapple ( ) and orange ( ) +juices. A 16-ounce bottle of Island Delight has 10 ounces of pineapple juice and 6 ounces of orange juice. A +16-ounce bottle of Sun Fun has 4 ounces of pineapple juice and 12 ounces of orange juice. The bottler makes +$1.60 per bottle on Island Delight and $1.20 per bottle on Sun Fun. The amounts of juice available today are 640 +ounces of pineapple juice and 768 ounces of orange juice. To maximize profit, how many of each bottle of juice +should the juice bottler make? +22. Fernando builds birdbaths ( ) and birdhouses ( ). Fernando can make a total of 7 birdbaths and birdhouses per +day. A birdbath costs $8 to make, while a birdhouse costs $6 to make. Fernando has $48 to spend on building +materials for the day. When he sells them, Fernando makes $12 in profit on a birdbath and $9 in profit on a +birdhouse. Determine how many of each Fernando should make to maximize his profit for the day. +23. A farmer grows wheat ( ) and barley ( ) on his 500 acres of cropland. He expects to profit $150 per acre for +wheat and $180 per acre for barley. The cost of raising each crop (seed, pesticide, etc.) is $60 per acre for wheat +and $90 per acre for barley. The farmer can budget $36,000 for the growing of the crops. To maximize his profit, +how many acres of each crop should be grown? +24. A company is going to ship food ( ) and water ( ) to the victims of a tsunami. Each container of food will feed 8 +people for a day, and each container of water will give 12 people their daily water. The food containers each +weigh 30 pounds and take up 8 cubic feet of space; each container of water weighs 120 pounds, but takes up +only 2 cubic feet of space. The airplanes lined up to carry the supplies to the victims cannot have its cargo +exceed 24,000 pounds; also, the total cargo area in the airplanes is 4,000 cubic feet. How many containers of +food and water can be sent with each plane shipment that maximizes the shipment? +25. Another company will send clothing ( ) and medical supplies ( ) to the victims of the tsunami. Each container of +clothing contains enough clothing for 12 people; each container of medical supplies can aid 8 people. The +clothing containers each weigh 50 pounds and take up 6 cubic feet of space; each container of medical supplies +weighs 20 pounds, and takes up 4 cubic feet of space. The airplanes lined up to carry the supplies to the victims +cannot have its cargo exceed 24,000 pounds; also, the total cargo area in the airplanes is 3,000 cubic feet. How +many containers of clothing and medical supplies can be sent with each plane shipment that maximizes the +shipment? +5.11 • Linear Programming +529 + +Chapter Summary +Key Terms +5.1 Algebraic Expressions +• +variable +• +constant +• +expression +• +equation +• +equal sign +• +term +• +coefficient +• +like terms +• +Distributive Property +5.2 Linear Equations in One Variable with Applications +• +linear equation +5.3 Linear Inequalities in One Variable with Applications +• +linear inequality +• +Addition and Subtraction Property of Linear Inequalities +• +Multiplication and Division Property of Linear Inequalities +5.4 Ratios and Proportions +• +ratio +• +proportion +• +constant of proportionality +• +scale +• +construct ratios +• +solve proportions +• +use proportions to solve scaling problems +5.5 Graphing Linear Equations and Inequalities +• +ordered pair +• +origin +• +points on the axes +• +linear equation in two variables +• +standards form of a linear equation +• +solution +• +linear inequality in two variables +• +solution to a linear inequality +• +boundary line +5.6 Quadratic Equations with Two Variables with Applications +• +monomial +• +polynomial +• +binomial +• +trinomial +• +quadratic equation +• +Zero Product Property +5.7 Functions +• +relation +• +domain +• +function +• +mapping +• +vertical line test +530 +5 • Chapter Summary +Access for free at openstax.org + +5.8 Graphing Functions +• +intercepts of a line +• +slope +• +slope-intercept form +5.9 Systems of Linear Equations in Two Variables +• +system of linear equations +• +solutions of a system of equations +• +contradictions +• +identities +• +coincident lines +• +consistent system of linear equations +• +inconsistent system of linear equations +5.10 Systems of Linear Inequalities in Two Variables +• +system of linear inequalities +5.11 Linear Programming +• +linear programming +• +objective function +• +constraint +Key Concepts +5.1 Algebraic Expressions +• +Algebra is useful because it allows us to understand many situations in real life by modeling them with expressions. +• +Algebraic expressions are the building blocks of algebra. From algebraic expressions we can create algebraic +equations. +• +Algebraic expressions are the building blocks of algebra. From algebraic expressions we can create algebraic +equations. +• +Algebraic expressions are often simplified and evaluated using the four arithmetic operations. +5.2 Linear Equations in One Variable with Applications +• +Solving linear equations means discovering what the value of the variable in a linear equation represents in the +given conditions. +• +When solving a linear equation, most often you will have one solution; however, a linear equation may have no +solutions or infinitely many solutions. +5.3 Linear Inequalities in One Variable with Applications +• +Inequalities can be used when the possible values (answers) in a certain situation are numerous, or when the exact +value (answer) is not known, but it is known to be within a range of possible values. +• +Linear inequalities can be represented using a number line or using interval notation. +5.4 Ratios and Proportions +• +A ratio is a comparison of two numbers. The ratio of two numbers +and +can be written as: +to +OR +: +OR the +fraction +/ . +• +All fractions are ratios, but not all ratios are fractions. Ratios make part to part, part to whole, and whole to part +comparisons. Fractions make part to whole comparisons only. +• +When two ratios are equal, we say they are in proportion or are proportional. +• +Setting up proportions allows us to solve many various situations where three of the four values of the proportion +are known. +5.5 Graphing Linear Equations and Inequalities +• +Linear equations can be represented graphically on a rectangular coordinate system. +• +Solving linear equations in two variables means finding the point where two lines intersect. There are three +possibilities: The lines intersect at exactly one point; the lines do not intersect (they are parallel); or the lines +intersect everywhere (they are the same line). +5 • Chapter Summary +531 + +• +Solving linear inequalities in two variables means finding a region of possible answers. Every point in this region will +make both inequalities true statements. +• +Plotting points is a standard way to help graph linear equations and linear inequalities. +5.6 Quadratic Equations with Two Variables with Applications +• +A quadratic equation is an algebraic equation where the highest power (degree) of the equation is two. +• +To solve a quadratic equation is to find the value(s) that when substituted in for the variables, will make the +equation equal to zero. +• +There can be two, one, or no solutions to any quadratic equation. +• +There are several methods to solve a quadratic equation. These methods include factoring quadratic equations, +graphic quadratic equations, using the square root method, and using the quadratic formula. +5.7 Functions +• +A relation is any set of ordered pairs +. All of the +-values of the set are the domain, and all of the +-values of the +set are the range. +• +A relation is a function if each +-value in the domain is assigned to exactly one element in the range. A +-value in the +range can have more than one +-value assigned to it; but each +-value can only be assigned to one +-value. +• +For the function +is the name of the function, +is the domain value variable, and +is the range +value variable. +• +The vertical line test is a test that can be done on the graph of a relation to determine if it is a function. +5.8 Graphing Functions +• +Every linear function can be graphically represented by a unique line that shows all the solutions of the equation. +• +The points where the graph of a line intersects the +-axis and +-axis are called the intercepts of the line. +• +Most lines will have one +-intercept and one +-intercept. Only if the line is straight vertical (no +-intercept) or +straight horizontal (no +-intercept) will it not have both intercepts. Note that a line that is straight vertical is not a +function, but a line that is straight horizontal is a function. +• +Since any two points determine a straight line, any linear function can be graphed if both intercepts are known. +• +The slope of a linear function is the ratio of the vertical change divided by the horizontal change. It is often referred +to as +. +• +A formula for finding the slope of linear functions is +for any two points of the linear function +and +. +5.9 Systems of Linear Equations in Two Variables +• +To solve a system of linear equations means finding the point or points where the two linear equations intersect. +• +Two lines can intersect at one point, no points if they are parallel, or every point if they are the same equation. +• +Systems of linear equations can be solved by graphing, by using substitution, or by using the elimination method. +5.10 Systems of Linear Inequalities in Two Variables +• +To solve a system of linear inequalities means to find the area(s) where the points in that area make all the linear +inequalities true. +• +Systems of linear inequalities can be solved by graphing the linear equations associated with the inequalities, then +'testing' points to see whether the values of the point make the equation true or not. +5.11 Linear Programming +• +Linear programming is a mathematical technique to solve problems involving finding maximums or minimums +where a linear function is limited by various constraints. +• +An objective function is a linear function in two or more variables that describes the quantity that needs to be +maximized or minimized. +• +In linear programming, a constraint is a restriction that affects the maximum or minimum values of an objective +function. +• +Through the creation of objective functions and restraints, a linear system can be developed and solved through +linear programming. +532 +5 • Chapter Summary +Access for free at openstax.org + +Videos +5.1 Algebraic Expressions +• +Q&A: Why We Teach Algebra (https://openstax.org/r/Teach_Algebra) +5.2 Linear Equations in One Variable with Applications +• +Solving for a Variable in an Equation (https://openstax.org/r/Solving_for_a_variable) +5.5 Graphing Linear Equations and Inequalities +• +Graphing Linear Inequalities in Two Variables (https://openstax.org/r/Graphing_linear) +5.6 Quadratic Equations with Two Variables with Applications +• +Factoring with the Box Method (Area Model) (https://openstax.org/r/Factoring_with_the_Box) +• +Solving Quadratics with the Zero Property (https://openstax.org/r/Zero_Property) +• +Solving Quadratics with the Quadratic Formula (https://openstax.org/r/Solving_Quadratics) +5.7 Functions +• +Relations and Functions (https://openstax.org/r/Relationsp_and_Functions) +• +Domain and Range on Graphs (https://openstax.org/r/Domain) +5.9 Systems of Linear Equations in Two Variables +• +Practice with Solving Applications of Systems of Equations (https://openstax.org/r/Practice_with_Solving) +• +Applications of Systems of Linear Equations (https://openstax.org/r/Applications_of_Systems) +5.10 Systems of Linear Inequalities in Two Variables +• +Solving Systems of Linear Inequalities by Graphing (https://openstax.org/r/Solving_Systems) +• +Systems of Linear Inequalities (https://openstax.org/r/Systems_of_Linear) +Formula Review +5.1 Algebraic Expressions +• +Distributive Property: +5.3 Linear Inequalities in One Variable with Applications +• +For any numbers +, +, and +if +, then +and +. +• +For any numbers +, +, and , if +, then +and +. +• +For any numbers +, +, and , +multiply or divide by a positive: +if +and +, then +and +if +and +, then +and +multiply or divide by a negative: +if +and +, then +and +if +and +, then +and +5.8 Graphing Functions +• +To calculate slope +, use the formula +, +where the rise measures the vertical change and the run measures the horizontal change. +• +To find the slope of the line between two points +and +, use the formula +Projects +Ratio and Proportion—Comparing Prices, Part 1 +Go to your favorite coffee shops and find out what a same sized drink costs at each. You can do something similar for +pizza as well. Find the unit rate (i.e., price per ounce or price per square inch). For example, go to your favorite coffee +place and find the price per units on all their large coffee drinks. Or go to your favorite pizza place and compare prices of +5 • Chapter Summary +533 + +all their extra-large pizzas (by price per square inch). Write a report on the best deals. +Ratio and Proportion—Comparing Prices, Part 2 +Rather than comparing prices of different, but same sized drinks (or pizzas), compare unit prices of the same drinks but +of different sizes. Find out what the best bargain is based on price per ounce, price per square inch, etc. For example, +compare the prices of your favorite soft drink sold at a local store, but in various sizes (i.e., 12-ounce can, 16-ounce +bottle, 20-ounce bottle, 1-liter bottle, and multipacks). Or go to a pizza place and find out what the best bargain is on +their menu, based on price per square inch of pizza. Write a report on the best deals. +Systems of Linear Inequalities—Comparing Cell Phone Plans +Go to the websites of different cell phone companies and compare their plans. Write a report on “the best deals. "Best +Deals” doesn’t necessarily mean “cheapest.” You will need to look at what each company provides concerning restrictions +(constraints) on minutes to talk. What are the constraints on the cell phone coverage for each company? Do they cover +your area of the country well? Do they cover the entire United States well, or at least areas where you will be travelling? +Is this coverage 5G, or is it less? Can you add a phone easily? Can you bring your previous phone number to this plan? +The possibilities of constraints affecting each plan are several. So your task is to determine which plan is best, based on +not only cost but also all constraints you deem important. +534 +5 • Chapter Summary +Access for free at openstax.org + +Chapter Review +Algebraic Expressions +1. Translate from algebra to words: +2. Translate from words to algebra: the quotient of +and 7. +3. Translate from an English phrase to an expression: A gym charges $5.00 per class +and a $20 membership fee. +4. Use parentheses to make the following statement true: +5. Evaluate and simplify +when +. +6. Perform the indicated operation for the expression: +Linear Equations in One Variable with Applications +7. Solve the linear equations using a general strategy: +8. Solve the linear equations using properties of equations: +9. It costs 30 cents for an ear of corn. Construct a linear equation and solve how much it costs to buy 23 ears of corn. +10. State whether the following equation has exactly one solution, no solution, or infinitely many solutions +11. Solve the formula +for +Linear Inequalities in One Variable with Applications +12. Graph the inequality +on a number line and write the interval notation. +13. Solve the inequality +, graph the solution on the number line, and write the solution in interval notation. +14. Construct a linear inequality to solve the application: Daniel wants to surprise his girlfriend with a birthday party at +her favorite restaurant. It will cost $42.75 per person for dinner, including tip and tax. His budget for the party is +$500. What is the maximum number of people Daniel can have at the party? +Ratio and Proportions +Christer opened a bag of marbles and counted the number of each color. They found they had 9 green, 4 yellow, 13 +black, 11 orange, 8 blue, and 7 red. +15. What is the ratio of green marbles to orange marbles? +16. What is the ratio of red marbles to blue marbles? +17. What is the ratio of sum of the marbles with an odd number of marbles to the sum of the marbles with an even +number of marbles? +18. What is the ratio of black marbles to all marbles? +19. Solve: +20. Basil the cat is 17 pounds and 24 inches long from head to tail. In his new movie Claws, he is supersized to 50 +pounds. How many inches long will he be? Round your answer to the nearest tenth. +Graphing Linear Equations and Inequalities +21. For each ordered pair, decide: +I. +Is the ordered pair a solution to the equation? +II. +Is the point on the line in the graph shown? +5 • Chapter Summary +535 + +22. Graph +by plotting points. +23. Determine whether each ordered pair is a solution to the inequality +24. Write the inequality shown by the graph with the boundary line +. +25. Graph the linear inequality: +. +Quadratic Equations with Two Variables with Applications +26. Multiply +. +27. Factor +. +28. Graph and list the solutions to the quadratic equation +. +29. Solve +by factoring. +536 +5 • Chapter Summary +Access for free at openstax.org + +30. Solve +using the square root method. +31. Solve +using the quadratic formula. +32. The height in feet, +, of an object shot upwards into the air with initial velocity, +, after +seconds is given by the +formula +. A firework is shot upwards with initial velocity 130 feet per second. How many seconds +will it take to reach a height of 260 feet? Round to the nearest tenth of a second. +Functions +33. Evaluate the function +at the values +, +, +, +, and +. +34. Determine whether +represents a function. +35. Determine whether the mappings represent a function: +36. Determine whether +represent +as a function of +. +37. Use the vertical line test to determine if the graph represents a function. +38. Use the set of ordered pairs to find the domain and the range. +39. Use the graph shown to find the domain and the range. +5 • Chapter Summary +537 + +Graphing Functions +40. Find the +- and +-intercepts on the graph. +41. Graph +using the intercepts. +42. Find the slope of the line in the graph shown. +538 +5 • Chapter Summary +Access for free at openstax.org + +43. Use the slope formula to find the slope of the line between (2, 4) and (5, 7). +44. Identify the slope and +-intercept of +. +45. Graph the line of +using its slope and +-intercept. +The equation +models the relation between the monthly water bill payment, +, in dollars, and the +number of units of water, +, used. +46. Find the payment for a month when 0 units of water were used. +47. Find the payment for a month when 15 units of water were used. +48. Interpret the slope and +-intercept of the equation. +49. Graph the equation. +System of Linear Equations in Two Variables +50. Determine if (0, 1) and (2, 3) are solutions to the given system of equations. +51. Solve the system of equations by graphing. +52. Solve the system of equations by substitution. +53. Solve the systems of equations by elimination. +54. Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. +Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits +would Kenneth need to sell for the options to be equal? +Systems of Linear Inequalities in Two Variables +55. Determine whether (0, 0) and (2, 3) are solutions to the system. +56. Determine whether (0, 0) and (6, –8) are solutions to the darkest shaded region of the graph. +5 • Chapter Summary +539 + +57. Solve the systems of linear equations by graphing. +Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least 35 more grams +of protein each day and no more than an additional 200 calories daily. An ounce of cheddar cheese has 7 grams of +protein and 110 calories. An ounce of parmesan cheese has 11 grams of protein and 22 calories. +58. Write a system of inequalities to model this situation. +59. Graph the system. +60. Could she eat 1 ounce of cheddar cheese and 3 ounces of parmesan cheese? +61. Could she eat 2 ounces of cheddar cheese and 1 ounce of parmesan cheese? +Linear Programming +A toy maker makes two plastic toys, the Ring ( ) and the Stick ( ). The toy maker makes $4 per Ring and $6 per Stick. +The Ring uses 3 feet of plastic, while the Stick uses 5 feet of plastic. Today the toy maker has 40 feet of plastic available. +The toy maker also only makes 10 plastic toys per day. To maximize profit, how many of each toy should the toy maker +make? +62. Find the objective function. +63. Write the constraints as a system of inequalities. +64. Graph of the system of inequalities. +65. Find the value of the objective function at each corner point of the graphed region. +66. To maximize his profit, how many of each toy should the toymaker make? +Chapter Test +1. Perform the indicated operation for the expression: +2. Solve the linear equations using properties of equations: +3. It costs 55 cents for a stamp. Construct a linear equation and solve how much it cost to buy 50 stamps. +4. Solve the formula +for +. +5. Solve the inequality +, graph the solution on the number line, and write the solution in interval notation. +6. Construct a linear inequality to solve the application: Bella wants to buy a round of shakes for her friends. It will +cost $4.75 per shake, including tip and tax. Her budget is $50. What is the maximum number of friends Bella can +buy shakes for? +7. Manneken Pis is a famous statue in Brussels, Belgium. It is 24 inches tall and weighs 37.5 pounds. The average +540 +5 • Chapter Summary +Access for free at openstax.org + +man is 69 inches tall and weighs 198 pounds. Is Manneken Pis proportional to the average male? +8. Graph +by plotting points. +9. Graph the linear inequality: +10. Graph and list the solutions to the quadratic equation +. +11. Solve +by factoring. +12. Solve +using the quadratic formula. +13. Evaluate the function +at the values +, +, +, +, and +. +14. Use the vertical line test to determine if the given graph represents a function. +15. Use the graph shown to find the domain and the range. +16. Graph +using the intercepts. +5 • Chapter Summary +541 + +17. Use the slope formula to find the slope of the line between (1, 4) and (3, 5). +18. Identify the slope and +-intercept of +. +19. Graph the line of +using its slope and +-intercept. +The equation +, models the cost of visiting the Cat Café in San Diego for one hour. +, in dollars, is the +total cost and the cost per person, +, is $15 plus a $4.50 reservation fee. +20. Find the payment for two people. +21. Find the payment for five people. +22. Interpret the slope and +-intercept of the equation. +23. Graph the equation. +24. Solve the system of equations by graphing. +25. Solve the system of equations by substitution. +26. Solve the systems of equations by elimination. +27. Anna goes to the concession stand at a movie theater. She buys 5 popcorns and 4 large sodas and pays a total of +$60. During intermission, Isabelle goes to the concession stand. She buys 1 popcorn and 2 large sodas and pays a +total of $18. What is the cost of one popcorn, and the cost of one large soda? +28. Solve the systems of linear equations by graphing. +Juliette is selling fresh lemonade and cupcakes. She sells a cup of lemonade for $2 and a cupcake for $3. She needs to +make at least $100 to donate to the local cat sanctuary. She needs to sell at least 20 cups of lemonade. +29. Write a system of inequalities to model this situation. +30. Graph the system. +31. Could she sell 30 cups of lemonade and 10 cupcakes and make $100? +32. Could she sell 20 cups of lemonade and 30 cupcakes and make $100? +A toy maker makes exactly two toys out of wood; the Box ( ) and the Bat ( ). He makes $5 per Box and $6 per Bat. Each +Box requires 30 ounces of wood, and each Bat requires 45 ounces of wood. Today the toy maker has 270 ounces of +wood available. The toy maker also only makes 8 wooden toys per day. To maximize profit, how many of each wooden +toy should the toy maker make? +33. Find the objective function. +34. Write the constraints as a system of inequalities. +35. Graph of the system of inequalities. +36. Find the value of the objective function at each corner point of the graphed region. +37. To maximize profit, how many of each toy should the toymaker make? +542 +5 • Chapter Summary +Access for free at openstax.org + +Figure 6.1 Financial health helps you realize your goals. (credit: modification of work "Budget and Bills" by Alabama +Extension/Flickr, Public Domain) +Chapter Outline +6.1 Understanding Percent +6.2 Discounts, Markups, and Sales Tax +6.3 Simple Interest +6.4 Compound Interest +6.5 Making a Personal Budget +6.6 Methods of Savings +6.7 Investments +6.8 The Basics of Loans +6.9 Understanding Student Loans +6.10 Credit Cards +6.11 Buying or Leasing a Car +6.12 Renting and Homeownership +6.13 Income Tax +Introduction +The topic of money management is a broad and sometimes complex one. Ultimately, personal money management +involves managing both our debt and also our savings and investments. +In 2021, the average American had consumer debt balance of $96,371. Nearly $100,000 per person. And less than 25% of +Americans are debt free. Consumer debt can include mortgages, credit cards, as well as student loans. A key question all +consumers should consider is how to manage debt and not become overburdened by it. The first step is to create a +budget, which puts earnings into perspective, indicating what we can, and cannot, afford. A budget also entails setting +aside certain funds for savings and investment, which help us achieve our short- and long-term goals. +Creating a budget requires an understanding of how money—debt and savings—works. Initially, percentages and +interest need to be understood. They drive most of what happens with debt and savings. With that understanding, +discussions of buying a house, a car, or incurring credit card debt can be addressed from a financial perspective. All the +while, retirement is waiting. Preparing for retirement involves saving and saving earlier rather than later. The power of +compound interest is on full display when saving early. +This chapter covers some of the basics of money management: percentages, interest, budgeting, debt (student loans, +mortgage, car, credit cards), savings, investments, and taxes. +6 +MONEY MANAGEMENT +6 • Introduction +543 + +6.1 Understanding Percent +Figure 6.2 The federal budget describes how money is spent and how money is earned. (credit: "Breakdown of revenues +and outlays in 2021 US Federal budget" Wikimedia Commons, Public Domain) +Learning Objectives +After completing this section, you should be able to: +1. +Define and calculate percent. +2. +Convert between percent, decimal, and fractional values. +3. +Calculate the total, percent, or part. +4. +Solve application problems involving percents. +In 2020, the U.S. federal government budgeted $3.5 billion for the National Park Service (https://openstax.org/r/ +National_Park_Service), which appears to be a very large number (and is!) and a large portion of the total federal budget. +However, the total outlays from the U.S. federal government in 2020 was $6.6 trillion (https://openstax.org/r/ +U.S._federal_government). So, the amount budgeted for the National Park Service was less than one-tenth of 1 percent, +or 1/10%, of the total outlays. This percent describes a specific number. Understanding that ratio puts the $3.5 billion +budgeted to the National Park Service in perspective. +This chapter focuses on percent as a primary tool for understanding money management. The interest paid on debt, the +interest earned through investments, and even taxes are entirely determined using percent. This section introduces the +basics of working with this invaluable tool. +Define and Calculate Percent +The word percent comes from the Latin phrase per centum, which means “by the hundred.” So any percent is a number +divided by 100. Changing a percent to a fraction is to write the percent in its fractional form. To write +% in its fractional +form is to write the percent as the fraction +. +A percent need not be an integer and does not have to be less than 100. +EXAMPLE 6.1 +Rewriting a Percent as a Fraction +Rewrite the following as fractions: +1. +18% +544 +6 • Money Management +Access for free at openstax.org + +2. +84% +3. +38.7% +4. +213% +Solution +1. +Using the definition and += 18, 18% in fractional form is +. +2. +Using the definition and += 84, 84% in fractional form is +. +3. +Using the definition and += 38.7, 38.7% in fractional form is +. +4. +Using the definition and += 213, 213% in fractional form is +. +YOUR TURN 6.1 +Rewrite the following as fractions: +1. 3% +2. 94% +3. 67.2% +4. 670% +Convert Between Percent, Decimal, and Fractional Values +When any calculation with a percent is to be performed, the form of the percent must be changed, either to its fractional +form or its decimal form. We can change a percent into decimal form by dividing the percent by 100 and representing +the result as a decimal. +FORMULA +The decimal form of +% is found by calculating the decimal value of +. +EXAMPLE 6.2 +Converting a Percent to Decimal Form +Convert the following percents to decimal form: +1. +17% +2. +7% +3. +18.45% +Solution +1. +To convert 17% to its decimal form, divide 17 by 100. This moves the decimal two places to the left, resulting in 0.17. +The decimal form of 17% is 0.17. +2. +To convert 7% to its decimal form, divide 7 by 100. This moves the decimal two places to the left, resulting in 0.07. +The decimal form of 7% is 0.07. +3. +To convert 18.45% to its decimal form, divide 18.45 by 100. This moves the decimal two places to the left, resulting in +0.1845. The decimal form of 18.45% is 0.1845. +YOUR TURN 6.2 +Convert the following percents to decimal form: +1. 9% +2. 24% +3. 2.18% +6.1 • Understanding Percent +545 + +You should notice that, to convert from percent to decimal form, you can simply move the decimal two places to the left +without performing the division. +FORMULA +To convert the number +from decimal form to percent, multiply +by 100 and place a percent sign, %, after the +number, +. +EXAMPLE 6.3 +Converting the Decimal Form of a Percent to Percent +Convert each of the following to percent: +1. +0.34 +2. +4.15 +3. +0.0391 +Solution +1. +Using the formula and += 0.34, we calculate +, which gives us 34%. +2. +Using the formula and += 4.15, we calculate +, which gives us 415%. +3. +Using the formula and += 0.0391, we calculate +, which gives us 3.91%. +YOUR TURN 6.3 +Convert the following to percent: +1. 0.41 +2. 0.02 +3. 9.2481 +You should notice that, to convert from decimal form to percent form, you can simply move the decimal two places to +the right without performing the multiplication. +Calculate the Total, Percent, or Part +The word “of” is used to indicate multiplication using fractions, as in “one-fourth of 56.” To find “one-fourth of 56” we +would multiply 56 by one-fourth. We can think of percents as fractions with a specific denominator—100. So, to calculate +“25% of 52,” we multiply 52 by 25%. But, first we need to convert the percent to either fractional form (25/100) or decimal +form. Using the decimal form of 25% we have 0.25 × 52, which equals 13. +In this problem, 52 is the total or base, 25 is the percentage, and 13 is the percentage of 52, or the part of 52. This is +sometimes referred to as the amount. +FORMULA +The mathematical formula relating the total (base), the percent in decimal form, and the part (amount) is +, or, +. +In all calculations, the percent is expressed in decimal form. +Knowing any two of the values in our formula allows us to calculate the third value. In the following example, we know +the total and the percent, and are asked to find the percentage of the total. +546 +6 • Money Management +Access for free at openstax.org + +EXAMPLE 6.4 +Finding the Percent of a Total +1. +Determine 70% of 3,500 +2. +Determine 156% of 720 +Solution +1. +The total is += 3,500, and the percent is += 70. The decimal form of 70% is 0.70. To find the part, or percent of the +total, substitute those values into the formula and calculate. +From this, we say that 70% of 3,500 is 2,450. +2. +The total is += 720, and the percent is += 156. The decimal form of 156% is 1.56. To find the part, or percent of the +total, substitute those values into the formula and calculate. +From this, we say that 156% of 720 is 1,123.2. +YOUR TURN 6.4 +1. Determine 26% of 1,300. +2. Determine 225% of 915. +VIDEO +Finding Percent of a Total (https://openstax.org/r/solve_percent_problem1) +In the previous example, we knew the total and the percent and found the part using our formula. We may instead know +the percent and the part, but not the total. We can use our formula again to solve for the total. +EXAMPLE 6.5 +Finding the Total from the Percent and the Part +1. +What is the total if 35% of the total is 70? +2. +What is the total if 10% of the total is 4,000? +Solution +1. +Step 1: The percent is 35, which in decimal form is 0.35. We were given that 35% of the total is 70, so the part is 70. +We are to find the total. Substituting into the formula, we have +Step 2: To find the total, we solve the equation for the total. +From this we see that 200 is the total, or, that 35% of 200 is 70. +6.1 • Understanding Percent +547 + +2. +Step 1: The percent is 10, which in decimal form is 0.1. We were given that 10% of the total is 4,000, so the part is +4,000. Substituting into the formula, we have +Step 2: To find the total, we solve the equation for the total. +From this we see that 40,000 is the total, or that 10% of 40,000 is 4,000. +YOUR TURN 6.5 +1. What is the total if 18% of the total is 45? +2. What is the total if 15% of the total is 900? +VIDEO +Finding the Total from the Percent and the Part (https://openstax.org/r/solve_percent_problem2) +Similarly, the percent can be found if the total and the percent of the total (the part) are known. This will result in the +decimal form of the percent, so it must be converted to percent form. +EXAMPLE 6.6 +Finding the Percent from the Total and the Part +1. +What percent of 500 is 175? +2. +What percent of 228 is 155? +Solution +1. +Step 1: The total is 500, the percent of the total is 175. Substituting into the formula, we have +Step 2: To find the percent, we solve the equation for the percent. +We see the percent in decimal form is 0.35. Converting from the decimal form yields 35%. We say that 175 is 35% of +500. +2. +Step 1: The total is 228, the percent of the total is 155. Substituting into the formula, we have +548 +6 • Money Management +Access for free at openstax.org + +Step 2: To find the percent, we solve the equation for the percent. +We see the percent is 0.6798 (rounded to four decimal places). Converting from the decimal form yields 67.98%. We +say that 155 is 67.98% of 228. +YOUR TURN 6.6 +Find the percent in the following: +1. Total is 40, percent of the total is 25 +2. Total is 730, percent of the total is 292 +VIDEO +Finding the Percent When the Total and the Part Are Known (https://openstax.org/r/solve_percent_problem3) +Solve Application Problems Involving Percents +Percents are frequently used in finance, research, science experiments, and even casual conversation. Understanding +these types of values helps when consuming media or discussing finances, for instance. Effectively working with and +interpreting numbers and percents will help you become an informed consumer of this information. +In most cases, working through what is presented requires you to identify that you are indeed working with a question +of percents, which two of the three values that are related through percents are known, and which of the three values +you need to find. +EXAMPLE 6.7 +Retention Rate at College +Justine applies to a medium size university outside her hometown and finds out that the retention rate (percent of +students who return for their sophomore year) for the 2021 academic year at the university was 84%. During a visit to +the registrar’s office, she finds out that 1,350 people had enrolled in academic year 2021. How many students from the +academic year 2021 are returning for the 2022 academic year? +Solution +The percent of students who will return for the 2022 academic year (the retention rate) is 84%. The total number of +students who enrolled in the 2021 academic year was 1,350. This means the percent is known and the total is known. +From this, we can determine the number of students who will return (percent of the total) for the 2022 academic year +using the formula +. Substituting into the formula and calculating, we find that the number of +students that are returning is +So 1,134 students will return for the 2022 academic year. +YOUR TURN 6.7 +1. Harris works the bookstore in their hometown. During one particular day, the store had total sales of $1,765, of +6.1 • Understanding Percent +549 + +which Harris sold 30%. What were Harris’s total sales that day? +EXAMPLE 6.8 +Percent of Chemistry Majors +Cameron enrolls in a calculus class. In this class of 45 students, there are 18 chemistry majors. What percent of the class +are chemistry majors? +Solution +In this situation, the percent is to be determined. We know the total number of students, 45, and the part of the students +that are chemistry majors, 18. Using that information and the formula +, the percent can be found. +Substituting and solving, we have +Converting the 0.4 from decimal form, we find that 40% of the students in the calculus class are chemistry majors. +YOUR TURN 6.8 +1. At the Fremont County fair, there were 2,532 adult visitors. Of these, 1,679 purchased the Adult Mega Pass. What +percent of the adult visitors purchased the Adult Mega Pass? +EXAMPLE 6.9 +Total Sales and Commission +Mariel makes a 20% commission on every sale she makes. One week, her commission check is for $153.00. What were +her total sales that week? +Solution +In this problem, Mariel’s total sales is to be determined. We know the percent she earns is 20%. We also know that her +sales commission was $153.00, which is the percent of the total. Using this information and the formula +we can find Mariel’s total sales. The decimal form of 20% is 0.2. The part, or percent of the total, is +153. Substituting and solving, we obtain +Mariel’s total sales were $765.00. +YOUR TURN 6.9 +1. Mina’s family has replaced 65% of their home’s older light bulbs with LED bulbs. If they now have 52 LED bulbs, +how many total lightbulbs are in Mina’s house? +550 +6 • Money Management +Access for free at openstax.org + +WHO KNEW? +LED Lightbulbs +According to the energy website from the U.S. government (https://openstax.org/r/U.S._government), LED lightbulbs +use at least 75% less energy than incandescent bulbs. They also last up to 25 times as long as an incandescent bulb. If +lighting is a significant percent of your electrical use, replacing all incandescent bulbs with LED bulbs will significantly +reduce your electric bill. +Check Your Understanding +1. What is the denominator for any percent? +2. Convert 38.7% to decimal form. +3. What is 68% of 280? +4. Find the total if 41% of the total is 342. If necessary, round to two decimal places. +5. TikTok has an estimated 80,000,000 (80 million) registered users in the United States. The population of the United +States is 332,403,650. What percent of the U.S. population are registered TikTok users? If necessary, round to two +decimal places. +6. An Amazon fulfillment center needs to hire 20% more drivers. If there are currently 80 drivers, how many more +drivers will be hired? +SECTION 6.1 EXERCISES +For any answer, round to two decimal places, if necessary. +In the following exercises, rewrite the percent as a fraction +1. 45% +2. 9.1% +3. 8% +4. 673% +In the following exercises, rewrite the percent in decimal form. +5. 18% +6. 9% +7. 71.2% +8. 934% +9. Find 35% of 250 +10. Calculate 83.1% of 390 +11. Calculate 3.1% of 500 +12. Calculate 750% of 620 +13. If 40% of the total is 32, how much is the total? +14. If 3% of the total is 6.32, how much is the total? +15. If 150% of the total is 61.9, how much is the total? +16. If 18.1% of the total is 18.5, how much is the total? +17. 13 is what percent of 40? +18. 89 is what percent of 500? +19. 31 is what percent of 73? +20. 593.2 is what percent of 184.5? +21. 36 people in a village of 150 want to install a new splashpad at the local playground. What percent of the village +6.1 • Understanding Percent +551 + +wants to install the new splashpad? +22. Mitena is enrolled in a movie appreciation course. There are 84 students (including Mitena) in the course. After +having the students fill out a survey, the professor informs the students that 45.2% chose horror as their favorite +movie genre. How many students in Mitena’s class chose horror as their favorite movie genre? Round off to the +nearest integer. +23. Jadyn’s dorm has a “Rick and Morty night” every Wednesday during the semester. One Wednesday, 27 students +from the dorm come to watch the TV show Rick and Morty. Jadyn knows this is 30% of the dorm’s residents. How +many students reside in the dorm? +24. Percent Error. When performing a scientific experiment that results in quantities of some sort, such as mass in +chemistry or momentum in physics, the percent error is often computed. Percent error, % +, is the percent by +which the value obtained in an experiment, the observed value +, is different than the value that was expected, +the expected value +, in the experiment. The formula is below. +Jim and Kelly are working on a chemistry experiment and expect the result to be 50 grams. However, their result +was 48.7 grams. Find Jim and Kelly’s percent error. +25. Percent Error. See Exercise 24 for the definition of percent error. +Hailey and Elsbeth are using an experiment to determine Earth’s gravity. The expected value is +. Their +experiment gives them a value of +. Find the percent error for Hailey and Elsbeth’s experiment. +6.2 Discounts, Markups, and Sales Tax +Figure 6.3 Sale prices are often described as percent discounts. (credit: "Close-up of a discount sign" by Ivan Radic/Flickr, +CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Calculate discounts. +2. +Solve application problems involving discounts. +3. +Calculate markups. +4. +Solve application problems involving markups. +5. +Compute sales tax. +6. +Solve application problems involving sales tax. +Many people first encounter percentages during a retail transaction such as a percent discount (SALE! 25% off!!), or +through sales tax ("Wait, I thought this was $1.99?"), a report that something has increased by some percentage of the +previous value (NOW! 20% more!!). These are examples of percent decreases and percent increases. In this section, we +552 +6 • Money Management +Access for free at openstax.org + +discuss decrease, increase, and then the case of sales tax. +Calculating Discounts +Retailers frequently hold sales to help move merchandise. The sale price is almost always expressed as some amount off +the original price. These are discounts, a reduction in the price of something. The price after the discount is sometimes +referred to as the reduced price or the sale price. +When a reduction is a percent discount, it is an application of percent, which was introduced in Understanding Percent. +The formula used was +. In a discount application, the discount plays the role of the part, the +percent discount is the percentage, and the original price plays the role of the total. +FORMULA +The formula for a discount based on a percentage is +, with the percent +discount expressed as a decimal. The price of the item after the discount is +. +These are often combined into the following formula +When the original price and the percent discount are known, the discount and the sale price can be directly computed. +EXAMPLE 6.10 +Calculating Discount for a Percent Discount +Calculate the discount for the given price and discount percentage. Then calculate the sale price. +1. +Original price = $75.80; percent discount is 25% +2. +Original price = $168.90; percent discount is 30% +Solution +1. +Substituting the values into the formula +, we find that the discount is +. The discount is $18.95. +The sale price of the item is then +, or $56.85. +2. +Substituting the values into the formula +, we find that the discount is +. The discount is $50.67. +The sale price of the item is then +, or $118.23. +YOUR TURN 6.10 +Calculate the discount for the given original price and discount percentage. Then calculate the sale price. +1. Original price = $1,550.00; percent discount is 32% +2. Original price = $27.50; percent discount is 10% +Sometimes the original price and the sale price of an item is known. From this, the percent discount can be computed +using the formula +, by solving for the percent discount. +EXAMPLE 6.11 +Calculating the Percent Discount from the Original and Sale Prices +Determine the percent discount based on the given original and sale prices. +1. +Original price = $1,200.00; sale price = $900.00 +2. +Original price = $36.70; sale price = $29.52 +6.2 • Discounts, Markups, and Sales Tax +553 + +Solution +1. +Step 1. Find the discount. Using the original price and the sale price, we can find the discount with the formula +. Substituting and calculating, we find the discount to be +. Solving for the discount gives $300.00. +Step 2. Find the percent discount. Substituting the discount of $300.00 and the original price of $1,200.00, into the +formula +, we can find the percent discount. +Converting to percent form, the percent discount is 25%. +2. +Step 1. Find the discount. Using the original price and the sale price, we can find the discount with the formula +. Substituting and calculating, we find the discount to be +. Solving for the discount gives $7.38. +Step 2. Find the percent discount. Substituting the discount of $7.38 and the original price of $36.70, into the +formula +, we can find the percent discount. +Converting to percent form, the percent discount is 20%. +YOUR TURN 6.11 +Determine the percent discount based on the given original and sale prices. +1. Original price = $250.00; sale price = $162.50 +2. Original price = $19.50; sale price = $17.16 +Sometimes the sale price and the percent discount of an item are known. From this, the original price can be found. To +avoid multiple steps, though, the formula that we will use is +. The +original price can be found by solving this equation for the original price. +EXAMPLE 6.12 +Calculating the Original Price from the Percent Discount and Sale Price +Determine the original price based on the percent discount and sale price. +1. +Percent discount 10%; sale price = $450.00 +2. +Percent discount 75%, sale price = $90.00 +Solution +1. +Using the percent discount and the sale price, we can find the original price with the formula +. Substituting and solving for the original price, we find +The original price of the item was $500.00. +2. +Using the percent discount and the sale price, we can find the original price with the formula +554 +6 • Money Management +Access for free at openstax.org + +. Substituting and solving for the original price, we find +The original price of the item was $360.00. +YOUR TURN 6.12 +Determine the original price based on the percent discount and sale price. +1. Percent discount 15%; sale price = $11.05 +2. Percent discount 9%; sale price = $200.20 +Solve Application Problems Involving Discounts +In application problems, identify what is given and what is to be found, using the terms that have been learned, such as +discount, original price, percent discount, and sale price. Once you have identified those, use the appropriate formula (or +formulas) to find the solution(s). +EXAMPLE 6.13 +Determine Discount and New Price a Sale Rack Item +The sale rack at a clothing store is marked “All Items 30% off.” Ian finds a shirt that had an original price of $80.00. What +is the discount on the shirt? What is the sale price of the shirt? +Solution +We are asked to find the discount, and the sale price. We know the percent discount is 30%, or 0.30 in decimal form. The +original price was $80. +Substituting into the percent discount formula, we find that the discount is +. +The discount is $24 on that shirt. The sale price is the original price minus the discount, so the sale price is $80 – $24 = +$56. +YOUR TURN 6.13 +1. A bed originally priced at $550, but is on sale, with a 60% discount. What is the discount on the bed? What is the +sale price of the bed? +EXAMPLE 6.14 +Determine the Percent Discount of a Bus Pass +An annual pass on the city bus is priced at $240. The student price, though, is $168. What is the percent discount for +students for the bus pass? +Solution +We know the original price of the item, $240. We also know the sale price of the item, $168. From this we know the +discount is +. Substituting these values into the formula +, +we can find the percent discount. +6.2 • Discounts, Markups, and Sales Tax +555 + +The student percent discount on the bus pass is 30%. +YOUR TURN 6.14 +1. A pharmacy offers students at a nearby college a discount. Jerry purchases ibuprofen, which had an original +price of $15.80. The cost to Jerry after the student discount was $13.43. What is the percent discount for students +at the pharmacy? +EXAMPLE 6.15 +Finding the Original Price of a New Pair of Tires +Kendra’s car developed a flat, and the tire store told her that two tires had to be replaced. She got a 10% discount on the +pair of tires, and the sale price came to $189.00. What was the original price of the tires? +Solution +Using the percent discount and the sale price, we can find the original price with the +formula +. Substituting and solving for the original price, we find +The original price of the two tires Kendra bought was $210.00. +YOUR TURN 6.15 +1. Marisol needed to buy a new microwave. She got a 26% discount. The sale price Marisol paid was $43.66. What +was the original price of the microwave? +VIDEO +Computing Price Based on a Percent Off Coupon (https://openstax.org/r/Computing_Price_Based) +WORK IT OUT +There are cases where retailers allow multiple discounts to be applied. However, it is rare that the discount +percentages are added together. For example, if you have a 15% coupon and qualify for a 20% price reduction, the +retailer typically does not add those two percentages together to determine the new price. The retailer instead +applies one discount, then applies the second discount to the price obtained after the first discount was deducted. +Research the original prices of two different laptops offered by one retail outlet. Assume you will receive a student +discount of 12% and your outlet of choice is having a 15% off sale on all laptops. +For each laptop: +1. +List the original price and calculate the price after applying the student discount (12%) only. +556 +6 • Money Management +Access for free at openstax.org + +2. +Then find the price after applying the sale discount (15% off) to the price found in Step 1. +3. +Determine the total saved on the laptop and what percent discount the total savings represents. +4. +Now, apply the discounts in reverse order (first the sale discount, then the student discount). +5. +Note anything interesting about your findings. +Calculate Markups +When retailers purchase goods to sell, they pay a certain price, called the cost. The retailer then charges more than that +amount for the goods. This increase is called the markup. This selling price, or retail price, is what the retailer charges +the consumer in order to pay their own costs and make a profit. Markup, then is very similar to discount, except we add +the markup, while we subtract the discount. +FORMULA +The formula for a markup based on a percentage is +, with the percent markup +expressed as a decimal. The price of the item after the markup is +. +These are often combined into the following formula +It should be noted that the formulas used for a markup are very similar to those for a discount, with addition replacing +the subtraction. +EXAMPLE 6.16 +Determining the Retail Price Based on the Cost and the Percent Markup +Calculate the markup for the given cost and markup percentage. Then calculate the retail price. +1. +Cost = $62.00; percent markup is 15% +2. +Cost = $750.00; percent markup is 45% +Solution +1. +Substituting the values into the formula +, we find that the markup is +. The markup is $9.30. +The retail price of the item is then +, or $62.00 + $9.30 = $71.30. +2. +Substituting the values into the formula +, we find that the markup is +. The markup is $337.50. +The retail price of the item is then +, or $750.00 + $337.50 = $1,087.50. +YOUR TURN 6.16 +Calculate the markup for the given cost and markup percentage. Then calculate the retail price. +1. Cost = $1,800.00; percent markup is 22% +2. Cost = $10.50; percent markup is 10% +Sometimes the cost and the retail price of an item are known. From this, the percent markup can be computed using the +formula +, by solving for the percent markup. +EXAMPLE 6.17 +Calculating the Percent Markup from the Cost and Retail Price +Determine the percent markup based on the given cost and retail price. Round percentages to two decimal places. +1. +Cost = $90.00; retail price = $103.50 +6.2 • Discounts, Markups, and Sales Tax +557 + +2. +Cost = $5.20; retail price = $9.90 +Solution +1. +Step 1: Using the cost and the retail price, we can find the markup with the formula +. +Substituting and calculating, we find the markup to be +. Solving for the markup gives +$13.50. +Step 2: After substituting the markup, $13.50, and the original price, $90.00, into the formula +, we can find the percent markup. +Converting to percent form, the percent markup is 15%. +2. +Step 1: Using the cost and the retail price, we can find the markup with the formula +. +Substituting and calculating, we find the markup to be +. Solving for the markup gives $4.70. +Step 2: After substituting the markup, $4.70, and the original price, $5.20, into the formula +, we can find the percent markup. +Converting to percent form, the percent markup is 90.38%. +YOUR TURN 6.17 +Determine the percent markup based on the given cost and retail price. Round percentages to two decimal places. +1. Cost = $120.00; retail price = $190.00 +2. Cost = $0.38; retail price = $1.14 +Sometimes the retail price and the percent markup of an item are known. From this, the cost can be found. To avoid +multiple steps, though, the formula that we will use is +. The cost can be found +by solving this equation for the cost. +EXAMPLE 6.18 +Calculating the Cost from the Percent Markup and Retail Price +Determine the cost based on the percent markup and retail price. +1. +Percent markup 20%; retail price = $10.62 +2. +Percent markup 125%; retail price = $26.55 +Solution +1. +Using the percent markup and the retail price, we can find the cost with the formula +. Substituting and solving for the cost, we find +The cost of the item was $8.85. +558 +6 • Money Management +Access for free at openstax.org + +2. +Using the percent markup and the retail price, we can find the cost with the formula +. Substituting and solving for the original price, we find +The cost of the item was $11.80. +YOUR TURN 6.18 +Determine the cost based on the percent markup and retail price. +1. Percent markup 15%; retail price = $40.25 +2. Percent markup 300%; retail price = $35.96 +Solve Application Problems Involving Markups +As before when working with application problems, be sure to look for what is given and identify what you are to find. +Once you have evaluated the problem, use the appropriate formula to find the solution(s). These application problems +address markups. +EXAMPLE 6.19 +Determine Retail Price of a Power Bar +Janice works at a convenience store near campus. It sells protein bars at a 60% markup. If a bar costs the store $1.30, +how much is the retail price at the convenience store? +Solution +We are asked to find the retail price. We know the percent markup is 60%. The cost of the bar was $1.30. Substituting +into the percent markup formula, we find that the markup is +. The +markup is $0.78 on that protein bar. The retail price is the cost plus the markup, so the retail price is +. The retail price is $2.08. +YOUR TURN 6.19 +1. A furniture outlet spends $360.00 to buy a bed. The store marks up the bed by 250%. What is the retail price of +the bed? +EXAMPLE 6.20 +Determine the Percent Markup of a Phone +Javi began working at a phone outlet. In a recent shipment, he noticed that the cost of the phone to the store was +$480.00. The phone sells for $840.00 in the store. What is the percent markup on the phone? +Solution +We know the cost of the phone, $480. We also know the retail price of the phone, $840.00. From this we know the +markup is +. Substituting these values into the formula +, +we can find the percent markup. +6.2 • Discounts, Markups, and Sales Tax +559 + +The markup on the phone is 75%. +YOUR TURN 6.20 +1. Maggie does some research into textbook costs. The Sociology of the Family text she finds sells for $234.36 but +costs the store only $189.00. What is the percent markup on the sociology book? +EXAMPLE 6.21 +Finding the Cost of a T-Shirt +Bob decided to order a t-shirt for his gaming friend online for $29.50. He knows the markup on such t-shirts is 18%. What +was the t-shirt’s cost before the markup? +Solution +Using the percent markup and the retail price, $29.50, we can find the cost with the formula +. Substituting and solving for cost we find +The cost of the t-shirt was $25.00. +YOUR TURN 6.21 +1. Tina has opened a retail shop and purchased a unique hat for resale. Tina uses a 50% markup and sells the hat +for $57.00. How much did the hat cost Tina? +Compute Sales Tax +Sales tax is applied to the sale or lease of some goods and services in the United States but is not determined by the +federal government. It is most often set, collected, and spent by individual states, counties, parishes, and municipalities. +None of these sales tax revenues go to the federal government. +For example, North Carolina has a state sales tax of 4.75% while New Mexico has a state sales tax of 5%. Additionally, +many counties in North Carolina charge an additional 2% sales tax, bringing the total sales tax for most (72 of the 100) +counties in North Carolinians to 6.75%. However, in Durham, the county sales tax is 2.25% plus an additional 0.5% tax +used to fund public transportation, bringing Durham County’s sales tax to 7%. To find the sales tax in a particular place, +then, add other locality sales taxes to the base state sales tax rate. +How much we pay in sales tax depends on where we are, and what we are buying. +To determine the amount of sales tax on taxable purchase, we need to find the product of the purchase price, or marked +price, and the sales tax rate for that locality. +560 +6 • Money Management +Access for free at openstax.org + +FORMULA +To calculate the amount of sales tax paid on the purchase price in a locality with sales tax given in decimal form, +calculate +The total price is then +When the sales tax calculation results in a fraction of a penny, then normal rounding rules apply, round up for half a +penny or more, but round down for less than half a penny. +You should notice that this the same as markup, except using a different term. Sales tax plays the role of markup, the +purchase price plays the role of cost, and the tax rate plays the role of percent markup. This means all the strategies +developed for markups apply to this situation, with the changes indicated. +EXAMPLE 6.22 +Sales Tax in Kankakee Illinois +The sales tax in Kankakee, Illinois, is 8.25%. Find the sales tax and total price of items based on the purchase price listed. +1. +Purchase price = $428.99 +2. +Purchase price = $34.88 +Solution +1. +The sales tax is found using +. The purchase price is $428.99 and the tax rate is +8.25%. Substituting and calculating, the sales tax is +. The sales tax needs +to be rounded off. Since the third decimal place (fraction of a penny) is 1, we round down and the sales tax is $35.39. +The total price is the sales tax plus the purchase price, so is +. +2. +The sales tax on the item is found using +. The purchase price is $34.88 and the +tax rate is 8.25%. Substituting and calculating, the sales tax is +. The sales tax +needs to be rounded off. Since the third decimal place (fraction of a penny) is 7, we round up and the sales tax is +$2.88. The total price of the item is the sales tax plus the purchase price, so is +. +YOUR TURN 6.22 +The sales tax in Union County, Oregon, is 7%. Find the sales tax and total price of items based on the purchase price +listed. +1. Purchase price = $1,499.00 +2. Purchase price = $26.89 +As before, the information available might be different than only the purchase price and the sales tax rate. In these +cases, use either +or +and solve for the +indicated tax, price, or rate. These problems mirror those for percent markup. +Be aware, almost all sales tax rates are structured as full percentages, or half percent, or one-quarter percent, or three- +quarter percent. This means the decimal value of the sales tax rate, written as a percent, will be either 0, as in 5.0%, 5 as +in 7.5%, 25 as in 3.25%, or 75 as in 4.75%. When rounding for the sales tax percentage, be sure to use this guideline. +EXAMPLE 6.23 +Calculating the Sales Tax from the Purchase Price and the Total Price +Find the sales tax rate for the indicated purchase price and total price. Round using the guideline for sales tax +percentages. +1. +Purchase price = $329.50; total price = $354.21 +2. +Purchase Price = $13.77; total price = $14.39 +6.2 • Discounts, Markups, and Sales Tax +561 + +Solution +1. +Step 1. Find the sales tax paid. First, the amount of sales tax must be found. Subtracting the purchase price from +the total price, the amount of sales tax is $24.71. +Step 2. Find the sales tax rate. Using the purchase price, the sales tax, and the formula +, the sales tax rate can be found. Substituting and solving yields +Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 7.5%. +2. +Step 1. Find the sales tax paid. First, the amount of sales tax must be found. Subtracting the purchase price from +the total price, the amount of sales tax is $0.62. +Step 2. Find the sales tax rate. Using the purchase price, the sales tax, and the formula +, the sales tax rate can be found. Substituting and solving yields +Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 4.5%. +YOUR TURN 6.23 +Find the sales tax rate for the indicated purchase price and total price. Round using the guideline for sales tax +percentages. +1. Purchase price = $83.90; total price = $88.30 +2. Purchase price = $477.00; total price = $509.20 +EXAMPLE 6.24 +Calculating the Purchase Price from the Sales Tax and Total Price +Find the purchase price for the indicated sales tax rate and total price. +1. +Sales tax rate = 5.75%; total price = $36.56 +2. +Sales tax rate = 4.25%; total price = $97.17 +Solution +1. +When the sales tax rate and the total price are known, the formula +can +be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find +The purchase price, the price before tax, was $34.57. +2. +When the sales tax rate and the total price are known, the formula +can +be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find +562 +6 • Money Management +Access for free at openstax.org + +The purchase price, the price before tax, was $93.21. +YOUR TURN 6.24 +Find the purchase price for the indicated sales tax rate and total price. +1. Sales tax rate = 8.25%; total price = $157.81 +2. Sales tax rate = 6.75%; total price = $522.01 +Solve Application Problems Involving Sales Tax +Solving problems involving sales tax follows the same ideas and steps as solving problems for markups. But here we will +use the following formula: +We can also use the formula: +. +This can be seen in the following examples. +EXAMPLE 6.25 +Compute Sales Tax for Denver, Colorado +The sales tax rate in Denver Colorado is 8.81%. Keven buys a TV in Denver, and the purchase price (before taxes) is +$499.00. How much will Keven pay in sales tax and what will be the total amount he spends when he buys the TV? +Solution +The sales tax rate in Denver is 8.81%. To find the sales tax Keven will pay, find 8.81% of the purchase price. In decimal +form, that sales tax rate is 0.0881. Using the formula and substituting 499.00 for purchase price, we find that Keven will +pay +in sales tax for the TV. +The total price that Keven will pay is the purchase price plus the sales tax, or +. +YOUR TURN 6.25 +1. Daryl decides to buy a new scooter in St. Louis, Missouri, where the sales tax is 9.68%. The scooter he chooses +has a purchase price of $1,149. How much will Daryl pay in sales tax and what is the total price he spends on the +scooter? +EXAMPLE 6.26 +Compute Sales Tax for Austin, Texas +Jillian visits Austin, Texas, and purchases a new set of weights for her home. She spends, including sales tax, $467.64. The +sales tax rate in Austin Texas is 8.25%. How much of the total price is sales tax? +Solution +The sales tax paid for this purchase is the difference in the total price and the purchase price. We know the total price is +$467.64. We also know the sales tax rate, which is 8.25%. In decimal form, this is 0.0825. Using these values and the +6.2 • Discounts, Markups, and Sales Tax +563 + +formula +to find the purchase price. +Knowing both the total price and the now the purchase price, we can find the difference, which is the sales tax. +The total price was $467.64. The purchase price was $432. The difference of the total price and the purchase price, or the +sales tax, is then $467.64 − $432.00, which is $35.64. Jillian pays $35.64 in sales tax. +YOUR TURN 6.26 +1. Elizabeth decides to buy new running shoes in her hometown of Springfield, Illinois, where the sales tax rate is +6.25%. If her total bill comes to $153, how much of the total price is sales tax? +VIDEO +Finding Sales Tax Percentage (https://openstax.org/r/Finding_Sales_Tax) +WHO KNEW? +West Virginia was the first state to impose a sales tax. This happened on May 3, 1921. +Look up your locality on this website that lists standard state-level sales tax rates (https://openstax.org/r/ +resources_rates) and compare the sales tax structure in your state to two nearby states (for the lower 48) and for any two +states (Alaska and Hawaii). +Check Your Understanding +7. What is a discount? +8. What is a markup? +9. An item has a retail price of $45.00. What is the sale price after a 32% discount? +10. A retailer buys an item for $311.00. What is the retail price if their markup is 60%? +11. Does sales tax have the same formula as markup? +12. If the sales tax is 6.8%, what is the total price for an item that has a purchase price of $39.95? +SECTION 6.2 EXERCISES +For the following exercises, use the given values to find the indicated value. Round percent results to 2 decimal places. +Round money results to the penny (2 decimal places). +1. Retail price = $399.00, percent discount = 30%, find the sale price. +2. Retail Price = $75.00, percent discount = 65%, find the sale price. +3. Retail price = $125.00, sale price = $90.00, find the percent discount. +4. Retail price = $47.00, sale price = $41.50, find the percent discount. +5. Sale price = $145.70, percent discount = 20%, find the retail price. +6. Sale price = $1,208.43, percent discount = 13%, find the retail price. +7. Retail price = $26,790.00, percent discount = 8%, find the sale price. +8. Sale price = $314.06, percent discount = 33%, find the retail price. +9. Retail price = $145.50, sale price = $117.90, find the percent discount. +10. Retail price = $28.90, percent discount = 18%, find the sale price. +11. Sale price = $17.59, percent discount = 12%, find the retail price. +564 +6 • Money Management +Access for free at openstax.org + +12. Retail price = $57.50, sale price = $46.00, find the percent discount. +13. Cost = $130.00, percent markup = 34%, find the retail price. +14. Cost = $2.27, percent markup = 42%, find the retail price. +15. Cost = $68.45, retail price = $109.90, find the percent markup. +16. Cost = $466.16, retail price = $699.00, find the percent markup. +17. Retail price = $98.99, percent markup = 25%, find the cost. +18. Retail price = $799.00, percent markup = 55%, find the cost. +In the following exercises, find the sales tax and total paid. +19. Retail price = $17.99; sales tax = 7.5% +20. Retail price = $799.00; sales tax = 8.5% +21. Retail price = $176.83; sales tax = 6.25% +22. Retail price = $223.93; sales tax = 4.5% +In the following exercises, find the sales tax rate. +23. Purchase price = $257.45; total price = $273.54 +24. Purchase price = $14.99; total price = $15.74 +25. Purchase price = $26.83; total price = $28.84 +26. Purchase price = $2,399.90; total price = $2,609.89 +In the following exercises, find the purchase price. +27. Sales tax rate = 4.75%; total price = $50.15 +28. Sales tax rate = 8%; total price = $1,069.20 +29. Sales tax rate = 9.5%; total price = $51.45 +30. Sales tax rate = 5.75%; total price = $3,065.69 +31. Harris has a coupon for 20% off for any purchase. She finds a new tennis racket for $278.00. How much is the price +after the coupon is applied? +32. After the employee discount, Mariam will pay $46.55. What is her employee discount rate if the retail price was +$53.50? Round to nearest full percent. +33. Resa purchased a new game for her cousin. After sales tax, she paid $41.13. Find the sales tax rate she paid if the +purchase price of the game was $38.99. +34. Larissa opens a new secondhand bookstore. She buys a book for $2.75. What is her percent markup if the sells the +book for $8.50. Round to the nearest percent. +35. Doug opens a used auto parts store. He pays $30 for a car door. How much will he charge if his percent markup is +60%? +36. Gaia and Seth live in Osceola County in Florida, where the sales tax rate is 7.5%. They purchased some new +camping gear. The price before taxes came to $784.62. How much do they pay after the sales tax is applied? +37. Theresa decides to purchase a new phone, which has a retail price of $799.00. Her discount is 20% through a +friends and family plan. The sales tax in her county is 6.75%. How much will she pay after the discount? How much +will she pay after the tax is applied? +38. Sakari manages a retail outlet. They receive a shipment of shirts. She sees on the shipping list that each shirt cost +24.50 to the store. The store marks up the shirts by 45%. The county in which she lives charges sales tax of 6.5%. +What is the retail price of one of the shirts? After sales tax, how much will a customer pay for the shirt? +6.2 • Discounts, Markups, and Sales Tax +565 + +6.3 Simple Interest +Figure 6.4 Interest is how savings earns money. (credit: “Interest Rates” by Mike Mozart/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Compute simple interest. +2. +Understand and compute future value. +3. +Compute simple interest loans with partial payments. +4. +Understand and compute present value. +There is truth in the phrase “You need to have money to make money.” In essence, if you have money to lend, you can +lend it at a cost to a borrower and make money on that transaction. +When money is borrowed, the person borrowing the money (borrower) typically has to pay the person or entity that lent +the money (the lender) more than the amount of money that was borrowed. This extra money is the interest that is to +be paid. Interest is sometimes referred to as the cost to borrow, the cost of the loan, or the finance cost. +This idea also applies when someone deposits money in a bank account or some other form of investment. That person +is essentially lending the money to the bank or company. The money earned by the depositor is also called interest. The +interest is typically based on the amount borrowed, or the principal. +The pairing of borrower and lender can take various forms. The borrower may be a consumer using a credit card or +taking out a loan from a bank, the lender. Companies also borrow from lending banks. Someone who invests in a +company’s stock is the lender in this case; the company is essentially the borrower. +In this section, we examine the basic building block of interest paid on loans and borrowed credit and also the returns on +investments like bank accounts, simple interest. +Compute Simple Interest +Let’s get some terminology understood. Interest to be paid by a borrower is often expressed as an annual percentage +rate, which is the percent of the principal that is paid as interest for each year the money is borrowed. This means that +the more that is borrowed, the more that must be paid back. Sometimes, the interest to be paid back is simple interest, +which means that the interest is calculated on the amount borrowed only. +The length of time until the loan must be paid off is the term of the loan. The date when the loan must be paid off is +when the loan is due. The day that the loan is issued is the origination date. We’ll put this terminology to use in the +following examples. Note that in this section we will use letters, called variables, to represent the different parts of the +formulas we’ll be using. This will help keep our formulas and calculations manageable. +566 +6 • Money Management +Access for free at openstax.org + +Simple Interest Loans with Integer Year Terms +Calculating simple interest is similar to the percent calculations we made in Understanding Percent and Discounts, +Markups, and Sales Tax, but must be multiplied by the term of the loan (in years, if dealing with an annual percentage +rate). +FORMULA +The simple interest, +, to be paid on a loan with annual interest rate +for a number of years (term of the loan) , with +principal +, is found using +, where the decimal form of the interest rate, , is used. The total repaid, then +is +or, more directly, +. This total is often referred to as the loan payoff amount, or more +simply just the payoff. +When the annual interest rate, the principal, and the number of years that the money is borrowed is known, the interest +to be paid can be found and from there the total to be repaid can be calculated. +Be aware, interest paid to a lender is almost uniformly rounded up to the next cent. +EXAMPLE 6.27 +Simple Interest on Loans with Integer Year Terms +Calculate the simple interest to be paid on a loan with the given principal, annual percentage rate, and number of years. +Then, calculate the loan payoff amount. +1. +Principal += $4,000, annual interest rate += 5.5%, and number of years += 4 +2. +Principal += $14,800, annual interest rate += 7.9%, and number of years += 7 +Solution +1. +Substitute the principal += $4,000, the decimal form of the annual interest rate += 0.055, and number of years += 4 +into the formula for simple interest, and calculate. +. +The simple interest, or cost of the loan, to be paid on the loan is $880. +The loan payoff amount, or the total to be repaid, is +, or $4,880.00. +2. +Substitute the principal += $14,800, the decimal form of the annual interest rate += 0.079, and number of years += +7 into the formula for simple interest, and calculate. +. +The simple interest, or cost to borrow, to be paid on the loan is $8,184.40. +The loan payoff amount, or the total to be repaid, is +, or $22,984.40. +YOUR TURN 6.27 +Calculate the simple interest to be paid on a loan with the given principal, annual percentage rate, and number of +years. Then calculate the loan payoff, or total to be repaid. +1. Principal += $6,700, annual interest rate += 11.99%, and number of years += 3 +2. Principal += $25,800, annual interest rate += 6.9%, and number of years += 5 +EXAMPLE 6.28 +Simple Interest Equipment Loan +Riley runs an auto repair shop, and needs to purchase a new brake lathe, which costs $11,995. She takes out a two-year, +simple interest loan at an annual interest rate of 14.9%. How much interest will she pay and how much total will she +repay on the loan? +Solution +Step 1. Determine the variables, or parts of the formula. The principal +is the cost of the brake lathe, so += $11,995. +6.3 • Simple Interest +567 + +The interest rate Riley pays is 14.9%, or += 0.149 in decimal form. The length of the loan is two years, so += 2. We are first +asked to find +, the interest Riley will pay. +Step 2. Substitute the known variables into the formula for simple interest +and solve for +. +From Step 1 we have +. +This tells us that the simple interest, or cost to borrow, to be paid on the loan is $3,574.51. +Step 3. Use the formula +to determine the total amount Riley will repay, +. +The total to be repaid is +, or $15,569.51. +YOUR TURN 6.28 +1. Beth is the owner of a small retail store in downtown St. Louis. The windows in the storefront need replacing, so +she needs to take out a $9,500 loan to get the repairs done. The rate she secures is 9.25% and the term of the +loan is one year. How much interest will she pay and how much total will she repay on the loan? +Simple Interest Loans with Other Lengths of Terms +In the previous example and Your Turn exercise, the loans were paid back in one payment after an integer number of +years. However, there are also loans lasting a length of time not equal to an integer number of years (like 1, 2, or 3 years +or more), but in a number of months (like 4 months, 18 months, and so on). What model would apply to these situations? +When the loan is paid back after a term that is not an integer number of years but is instead a number of months, the +term of the loan, or time, , is expressed as a fraction of the year. So for a 2-month loan, the time, in years, is 2/12 = 1/6. +For a 5-month loan, the time in years is 5/12. For an 18-month term, the term in years is 18/12 = 1.5. +EXAMPLE 6.29 +Loan to Purchase Equipment +Abeje needs a loan to purchase equipment for the gym she is going to open. She visits the bank and secures a 4-month +loan of $20,000. Her annual percentage rate is 6.75%. How much interest will Abeje pay and what is her loan payoff +amount? +Solution +Abeje’s loan is for $20,000, so her principal is += 20,000. The interest rate Abeje will pay is 6.75%, or += 0.0675 in +decimal form. The length of the loan is 4 months, so +. Substituting these in the formula for simple interest, we find +her interest to be +The simple interest, or cost to borrow, to be paid on the loan is $450.00. +The payoff is +, or $20,450.00. +YOUR TURN 6.29 +1. Samuel needs to borrow $8,400 to pay for repairs to his small manufacturing facility. He manages to get a simple +interest loan at 17.33%, to be paid after 6 months. How much interest will Samuel pay and what is Samuel’s loan +payoff amount? +Those examples dealt in months. However, some loans are for days only (45 days, 60 days, 120 days). In such cases, we +find the daily interest rate. The fraction we will use for the daily interest rate is the interest rate (as a decimal) divided by +365. This may be referred to as Actual/365. In order to find the term of the loan, divide the number of days in the term of +the loan by 365. +568 +6 • Money Management +Access for free at openstax.org + +FORMULA +To determine the interest, +, on a loan with term +expressed in days, with principal of +, and interest rate in decimal +form of , calculate +. Here, +represents the daily interest rate. +Alternately, the above formula is equivalent to +, where the interest rate remains an annual rate, but +the time is expressed as a fraction of the year. +WHO KNEW? +It seems reasonable to use 365 as the number of days in the year, since there are 365 days in most years. However, +sometimes, banks have used (and continue to use) 360 as the number of days in a year. They may also treat all +months as if they have 30 days. These differences lead to (sometimes small) differences in how much interest is paid. +Since the number of days is in the denominator, a smaller denominator (360) will result in larger numbers (interest) +that is 365 is used for the denominator. See this page from ACRE (https://openstax.org/r/ACRE) for a comparison. +EXAMPLE 6.30 +Loan for Moving Costs +David plans to move his family from Raleigh, North Carolina to Tempe, Arizona. His company will reimburse (pay after +the move) David for the move. David does research and determines that movers will cost $5,600 to move his family’s +belongings to Tempe. He takes out a simple interest, 45-day loan at 11.75% interest to pay this cost. How much interest +will be paid on this 45-day loan, and what is David’s loan payoff amount? +Solution +This loan is in terms of days, so we will use the formula +, where t is the number of days and +is the +annual interest rate. +The principal for the loan is the moving cost, or += 5,600. The annual interest rate that David will pay is 11.75%, which in +decimal is 0.1175. The length of time for the loan is 45 days, so += 45. +Substituting these values into the formula and calculating, we find that the interest to be paid is +, or $81.13 (remember, interest is almost always rounded up to the next cent). +The payoff for the loan is $5,681.13. +YOUR TURN 6.30 +1. Heather runs a silk screen t-shirt shop, and seeks a short-term loan to pay for new inventory (paints, blades, +shirts). They secure a $3,700 simple interest loan for 60 days, at an annual rate of 18.99%. How much will +borrowing the money cost Heather, and what is her loan payoff amount? +Understand and Compute Future Value +Money can be invested for a specific amount of time and earn simple interest while invested. The terminology and +calculations are the same as we’ve already seen. However, instead of the total to be paid back, the investor is interested +in the total value of the investment after the interest is added. This is called the future value of the investment. +FORMULA +The future value, +, of an investment that yields simple interest is +, where +is the +principal (amount invested at the start), +is the annual interest rate in decimal form, and +is the length of time the +6.3 • Simple Interest +569 + +money is invested. The time +will be an integer if the term of the deposit is an integer number of years, will be +number of months/12 if the term is in months, will be actual/365 if the deposit is for a number of days. +EXAMPLE 6.31 +Simple Interest on a Deposit +In the following, determine how much interest was earned on the investment and the future value of the investment, if +the investment yields simple interest. +1. +Principal is $1,000, annual interest rate is 2.01%, and time is 5 years +2. +Principal is $5,000, annual interest rate is 1.85%, and time is 30 years +3. +Principal is $10,000, annual interest rate is 1.25%, and time is 18 months +4. +Principal is $7,000, annual interest rate is 3.26%, and time is 100 days +Solution +1. +The principal is += $1,000, the annual interest rate, in decimal form, is 0.0201, and the term is 5 years, or += 5. +Since the term is an integer number of years, the interest earned on the investment is +, or the interest earned was $100.50. +To find the future value, we use the formula +. Substituting the values and calculating, we find the future +value of the investment to be +. The future value of the investment at the end +of 5 years is $1,100.50. +Notice that the future value could have been calculated directly with +2. +The principal is += $5,000, the annual interest rate, in decimal form, is 0.0185, and the term is 30 years, or += 30. +Since the term is an integer number of years, the interest earned on the investment is +, or the interest earned was $2,775.00. To find the future value, we use +the formula +. Substituting the values and calculating, we find the future value of the investment to be +. The future value of the investment at the end of 30 years is $7,775.00. +3. +The principal is += $10,000, the annual interest rate, in decimal form, is 0.0125, and the term is 18 months. Since +the term is in months, we have to write the months in terms of years. For 18 months, we use 18/12 as . The interest +earned on the investment is +, or the interest earned was $187.50. To +find the future value, we use the formula +. Substituting the values and calculating, we find the future +value of the investment to be +. The future value of the investment at the +end of 18 months is $10,187.50. +4. +Principal is $7,000, annual interest rate is 3.26%, and time is 100 days. The principal is += $7,000, the annual +interest rate, in decimal form, is 0.0326, and the term is 100 days. Since the term is in days, we have to write the +time using actual/365, or += 100/365. The interest earned on the investment is +, or the interest earned was $62.52. To find the future value, we use +the formula +. Substituting the values and calculating, we find the future value of the investment to be +. The future value of the investment at the end of 100 days is $7,062.52. +YOUR TURN 6.31 +In the following, determine how much interest was earned on the investment and the future value of the investment +if the investment yields simple interest. +1. Principal is $4,500, annual interest rate is 1.88%, and time is 3 years +2. Principal is $2,000, annual interest rate is 2.03%, and time is 10 years +3. Principal is $120,000, annual interest rate is 3.1%, and time is 100 days +4. Principal is $4,680, annual interest rate is 1.55%, and time is 42 months +You may have noticed that for these problems, the future value was rounded down. When the future value is paid, the +amount is typically rounded down. +A certificate of deposit (CD) is a savings account that holds a single deposit (the principal) for a fixed term at a fixed +interest rate. Once the term of the CD is over, the CD may be redeemed (cashed in or withdrawn) and the owner of the +570 +6 • Money Management +Access for free at openstax.org + +CD receives the original principal plus the interest earned. The deposit often cannot be withdrawn until the term is up; if +it can be withdrawn early, there is often a penalty imposed to do so. +EXAMPLE 6.32 +Certificate of Deposit +Jonas deposits $2,500 in a CD bearing 3.25% simple interest for a term of 3 years. When he redeems his CD at the end of +the 3 years, how much will he receive? +Solution +This is a future value example. We know that += $2,500 is the amount deposited. The annual simple interest rate in +decimal form is += 0.0325. The term of the investment is += 3 years. +Substituting those values into the future value formula, we have +. +When the CD is redeemed, Jonas will receive $2,743.75. +YOUR TURN 6.32 +1. Mia deposits $4,900 in a CD bearing 3.95%. The CD term is 7 years. When she redeems the CD, how much will +Mia receive? +WORK IT OUT +The reason CD (certificate of deposit) rates look so small is because they are extremely safe investments. Though +overall interest rates for CDs change over time and individual returns vary with the terms of the CD, investors are +offered predictable interest income for their investments. +To investigate this yourself, search online to determine the strengths and weaknesses of CDs (investopedia.com +offers good, basic information on investing). Then, online, identify five national banks and two local banks who offer +CDs. +• +Track the interest rates for the CDs at various terms (1 year, 3 years, 5 years) for each of the banks you found that +offer CDs. +• +Calculate the amount of interest earned for a $10,000 deposit for each CD at each of the terms. +• +Compare the results from the various banks, CDs, and terms and decide which is the best investment. You may +want to consider both the length of time that the money is locked up, and the return. +Paying Simple Interest Loans with Partial Payments +In every example above, there was one payment for the loan, or one withdrawal for the investment. However, for many +loans (house, car, in-ground swimming pool), the loan will be paid back in two or more payments. Such a payment is +called a partial payment, because they only pay off part of the loan. +When a partial payment is made, some of the payment pays for the principal, but the rest of the payment pays for +interest on the principal. When making the first partial payment, the interest is calculated on the principal for the time +between the origination date of the loan and the date of the payment. If another partial payment is made, the interest is +calculated based on the remaining principal and the time between the previous partial payment and the current partial +payment date. +EXAMPLE 6.33 +Interest Paid in a Partial Payment on a Loan +1. +A simple interest loan for $6,500 is taken out at 12.6% annual percentage rate. A partial payment is made 45 days +into the loan period. How much of the partial payment will be for interest? +6.3 • Simple Interest +571 + +2. +A simple interest loan for $13,700 is taken out at 6.55% annual interest rate. A partial payment is to be made after +60 days. How much of the partial payment will be for interest? +Solution +1. +To find the interest paid in this partial payment, we calculate the interest on the principal for the time between the +origination of the loan and the payment day, or 45 days. +The principal is $6,500. The annual interest rate, in decimal form, is 0.126. +The interest paid for 45 days is found by substituting the values for principal +, rate , and time +into the formula +. +Calculating, we have +. Rounding up, the portion of the partial +payment that will be paid for interest is $100.98. +2. +To find the interest paid in this partial payment, we calculate the interest on the principal for the time between the +origination of the loan and the payment day, or 60 days. +The principal is $13,700. The annual interest rate, in decimal form, is 0.0655. +The interest paid for those 60 days is found by substituting those values into the formula +. +Calculating, we have +. Rounding up, the portion of the partial +payment that will be paid for interest is $147.51. +YOUR TURN 6.33 +1. A simple interest loan for $50,000 is taken out at 5.15% annual percentage rate. A partial payment is made +120 days into the loan period. How much of the partial payment will be for interest? +2. A simple interest loan for $8,500 is taken out at 9.9% annual interest rate. A partial payment is to be made +after 75 days. How much of the partial payment will be for interest? +Remaining Balance +The previous examples demonstrated how to determine the interest paid in a partial payment. Using this, we can +determine the remaining balance after a partial payment. +Step 1: determine the amount of the payment, +, that is applied to interest, +. +Step 2: subtract the amount paid in interest from the payment, +. This is the amount applied to the balance. +Step 3: subtract the amount applied to the balance (the value obtained in Step 2) from the balance of the loan, +. This is the remaining balance after the partial payment. +EXAMPLE 6.34 +Determining the Remaining Balance on a Loan After a Partial Payment +1. +A simple interest loan for $45,500 is taken out at 11.8% annual percentage rate. A partial payment of $20,000 is +made 50 days into the loan period. After this payment, what will the remaining balance of the loan be? +2. +A simple interest loan for $150,000 is taken out at 5.85% annual percentage rate. A partial payment of $50,000 is +made 70 days into the loan period. After this payment, what will the remaining balance of the loan be? +Solution +1. +The principal is $45,500, which will be treated as the balance, +, of the loan. The annual simple interest rate, in +decimal form, is 0.118. The time is += 50 days. +Step 1: Determine the amount of the partial payment that is applied to interest. To find this, substitute the values +above into the formula +and calculate. Calculating, the amount of the payment that is applied to +interest is +. Rounding up, we have += $735.48. +Step 2: The amount of the payment that is to be applied to the balance of the loan is partial payment minus the +amount of the partial payment that is applied to the interest. The payment is $2,000. The amount that is applied to +the balance is +. +Step 3: The remaining balance is found by subtracting the amount applied to the balance from the previous +balance, or +. +572 +6 • Money Management +Access for free at openstax.org + +The remining balance after the partial payment is $26,235.48. +2. +The principal is $150,000, which will be treated as the balance, +, of the loan. The annual simple interest rate, in +decimal form, is 0.0585. The time is += 70 days. +Step 1: Determine the amount of the partial payment that is applied to interest. To find this, substitute the values +above into the formula +and calculate. Calculating, the amount of the payment that is applied to +interest is +. Rounding up, we have += $1,682.88. +Step 2: The amount of the payment that is to be applied to the balance of the loan is partial payment minus the +amount of the partial payment that is applied to the interest. The payment is $50,000. The amount that is applied to +the balance is +. +Step 3: The remaining balance is found by subtracting the amount applied to the balance from the previous +balance, or +. +The remining balance after the partial payment is $101,682.88. +YOUR TURN 6.34 +1. A simple interest loan for $1,400 is taken out at 12.5% annual percentage rate. A partial payment of $700 is +made 20 days into the loan period. After this payment, what will the remaining balance of the loan be? +2. A simple interest loan for $23,000 is taken out at 7.25% annual percentage rate. A partial payment of $10,000 +is made 40 days into the loan period. After this payment, what will the remaining balance of the loan be? +Loan Payoff +Finally, we will determine the amount to be paid at the end of the loan. To do so, we apply the formula for the loan +payoff to the remaining balance. However, the length of time for that remaining balance is the time between the partial +payment and the day the loan is paid off. +Step 1: Determine the remaining balance after the partial payment. +Step 2: Calculate the number of days between the partial payment and the date the loan is paid off. This will be the time +in the payment formula. +Step 3: Calculate the amount to be paid at the end of the loan, or the payoff amount, using +, +where +is the remaining balance and +is the time found in Step 2. +EXAMPLE 6.35 +Finding Loan Pay Off After a Partial Payment +Laura takes out an $18,400 loan for 120 days at 17.9% simple interest. She makes a partial payment of $7,500 after 45 +days. What is her payoff amount at the end of the loan? +Solution +The initial balance, or principal, of her loan is $18,400. The interest rate in decimal form is 0.179. Her partial payment of +$7,500 is made after 45 days. Using these values, we can determine how much of the partial payment is applied to the +balance. From there, we can determine her final loan payoff after 120 days. +Step 1: Determine the remaining balance after the partial payment. Using the partial payment process outlined in the +previous example, we first find that the amount of the partial payment that is applied to the balance. Their interest paid +in the partial payment is +, or $406.07 (remember to round up!). Using +this and that the loan amount was for $18,400, the remaining balance on the loan after the partial payment is +. +Step 2: The number of days between the partial payment and the date that the loan is to be paid off is 120 – 45 = 75. This +means that the time between the partial payment and the final payment is 75 days. +Step 3: To calculate the payoff amount, use +, with += $11,306.07 (the remaining balance), += +75 (from Step 2) and += 0.179. The payoff amount, then, is +. +Rounding up, the payoff amount is $11,721.92. +6.3 • Simple Interest +573 + +YOUR TURN 6.35 +1. Paola takes out a 75-day loan for $3,500.00. Her interest rate is 11.2%. If she makes a partial payment of +$1,250.00 after 30 days, what will her payoff be at the end of the loan? +Repeated Partial Payments +Car loans and mortgages (loans for homes) are paid off through repeated partial payments, most often monthly +payments. Since car loans are often 3 to 6 years, and mortgages 15 to 30 years, calculating each individual monthly +payment one at a time is time consuming and tedious. Even a 3-year loan would involve applying the above steps 36 +times! Fortunately, there is a formula for determining the amount of each partial payment for monthly payments on a +simple interest loan. +FORMULA +The amount of monthly payments, +, for a loan with principal +, monthly simple interest rate +(in decimal form), for +number of months is found using the formula +. The monthly interest rate is the annual rate +divided by 12. The number of months is the number of years times 12. +EXAMPLE 6.36 +Calculating Car Payments +Desiree buys a new car, by taking a loan out from her credit union. The balance of her loan is $27,845.00. The annual +interest rate that Desiree will pay is 7.3%. She plans to pay this off over 4 years. How much will Desiree’s monthly +payment be? +Solution +To use the formula for monthly payments, we need the principal, the interest rate, and the number of years. The +principal is $27,845. The annual rate, in decimal form, is 0.073. Dividing 0.073 by 12 gives the monthly interest rate +. She takes the loan out for 4 years, which is +months. Substituting these values into +the formula, +, we calculate: +Using the formula and rounding up to the next cent, we see that Desiree’s monthly payment will be $670.67. +YOUR TURN 6.36 +1. Russell buys a new car, by taking a loan out from the dealership. After all of the discussions are over, he finances +(gets a loan for) $23,660. The annual interest rate that Russell will pay is 4.76%. He plans to pay this off over 5 +years. How much will Russell’s monthly payment be? +574 +6 • Money Management +Access for free at openstax.org + +TECH CHECK +The calculation of payments is long, and involves many steps. However, most spreadsheet programs, including +Google Sheets, have a payment function. In Google Sheets, that function is PMT. To find the payment for an +installment loan (like for a car), you need to enter the interest rate per period, the number of payments, and the loan +amount. From Your Turn 6.36, the rate was 0.0476/12, the number of payments was 60, and the loan amount was +$23,660. In Google Sheets, select any cell and enter the following: +=PMT(0.0476/12,60,23660) +And click the enter key. Immediately, in the cell you selected, the payment of $443.90 appears, though with a negative +sign. The negative sign indicates it is a payment out of an account. Since we want to know the payment amount, we +ignore the negative sign. The result, with the formula in the formula bar, is shown in Figure 6.5. +Figure 6.5 Google Sheets payment function +In general, to use the PMT function in Google Sheets, enter +=PMT( /12, *12, +) +where +is the annual interest rate, +is the number of years, and +is the principal of the loan. +Understand and Compute Present Value for Simple Interest Investments +When finding the future value of an investment, we know how much is deposited, but we have no idea how much that +money will be worth in the future. If we set a goal for the future, it would be useful to know how much to deposit now so +an account reaches the goal. The amount that needs to be deposited now to hit a goal in the future is called the present +value. +FORMULA +The present value, +, of money deposited at an annual, simple interest rate of +(in decimal form) for time +(in +years) with a specified future value of +, is calculated with the formula +. +Note: Present value, in this calculation, is always rounded up. Otherwise, future value may fall short of the target +future value. +Understanding what this tells you is important. When you find the present value, that is how much you need to invest +now to reach the goal +, under the conditions (time and rate) at which the money will be invested. +EXAMPLE 6.37 +Compute the present value of the investment described. Interpret the result. +1. += $10,000, += 15 years, annual simple interest rate of 5.5% +2. += $150,000, += 20 years, annual simple interest rate of 6.25% +3. += $250,000, += 486 months, annual simple interest rate of 4.75% +6.3 • Simple Interest +575 + +Solution +1. +The future value is += $10,000. The time of the investment is in years, so += 15. The annual, simple interest rate is +5.5%, which in decimal form is 0.055. We substitute those values into the formula and calculate. +. Rounding up, we see that the present value +of $10,000 invested at a simple annual interest rate of 5.5% for 15 years is $5,479.46. This means that $5,479.46 +needs to be invested so that, after 15 years at 5.5% interest, the investment will be worth $10,000. +2. +The future value is += $150,000. The time of the investment is in years, so += 20. The annual, simple interest rate +is 6.25%, which in decimal form is 0.0625. We substitute those values into the formula and calculate. +. Rounding up, we see that the present value +of $150,000 invested at a simple annual interest rate of 6.25% for 20 years is $66,666.67. This means that $66,666.67 +needs to be invested so that, after 20 years at 6.25% interest, the investment will be worth $150,000. +3. +The future value is += $250,000. The time of the investment is 486 months. This needs to be converted to years. To +do so, divide the number of months by 12, giving +, so += 40.5 years. The annual, simple interest +rate is 4.75%, which in decimal form is 0.0475. We substitute those values into the formula and calculate. +. Rounding up, we see that the +present value of $250,000 invested at a simple annual interest rate of 4.75% for 486 months is $129,954.52. This +means that $129,954.52 needs to be invested so that, after 486 months at 4.75% interest, the investment will be +worth $150,000. +YOUR TURN 6.37 +Compute the present value of the investment described. Interpret the result. +1. += $25,000, += 10 years, annual simple interest rate of 7.5% +2. += $320,000, += 35 years, annual simple interest rate of 6.5% +3. += $90,000, += 270 months, annual simple interest rate of 3.75% +EXAMPLE 6.38 +Present Value of a CD +Beatriz will invest some money in a CD that yields 3.99% simple interest when invested for 30 years. How much must +Beatriz invest so that after those 30 years, her CD is worth $300,000? +Solution +Beatriz needs to know how much to deposit now so that her CD is worth $300,000 after 30 years. This means she needs +to know the present value of that $300,000. The time is 30 years and the annual simple interest rate, in decimal form, is +0.0399. Using that information and the formula for present value, we calculate the present value of that $300,000. +. Rounding up, Beatriz needs to invest +$136,549.85 so that she has $300,000 in 30 years. +YOUR TURN 6.38 +1. Kentaro will invest some money in a CD that yields 2.75% simple interest when invested for 5 years. How much +must Kentaro invest so that after those 5 years his CD is worth $12,000? +Check Your Understanding +13. What is interest? +14. What is the principal in a loan? +15. Calculate the simple interest to be paid for a 6-year loan with principal $1,500.00 and annual interest rate of +12.99% +576 +6 • Money Management +Access for free at openstax.org + +16. A simple interest loan for $24,200 is taken out at 10.55% annual percentage rate. A partial payment of $13,000 is +made 25 days into the loan period. After this payment, what will the remaining balance of the loan be? +17. Find the monthly payment for a $9,800.00 loan at a 13.8% interest for 4 years. +18. Find the present value of an investment with future value $30,000 with a simple interest rate of 3.75% invested for +10 years. +SECTION 6.3 EXERCISES +1. If $1,500.00 is invested in an account bearing 3.5% interest, what is the principal? +2. If $1,500 is invested in an account bearing 3.5% interest, what is the interest rate? +3. What is simple interest? +4. What is the present value of an investment? +5. What is the future value of an investment? +6. What is a partial payment on a loan? +In the following exercises, calculate the simple interest and payoff for the loan with the given principal, simple interest +rate, and time. +7. Principal += $5,000, annual interest rate += 6.5%, and number of years += 6 +8. Principal += $3,500, annual interest rate += 12%, and number of years += 7 +9. Principal += $7,800, annual interest rate += 11.5%, and number of years += 10 +10. Principal += $62,500, annual interest rate += 4.88%, and number of years += 4 +11. Principal += $4,600, annual interest rate += 9.9%, for 18 months +12. Principal += $19,000, annual interest rate += 16.9%, for 14 months +13. Principal += $8,500, annual interest rate += 10.66%, for 6 months +14. Principal += $17,600, annual interest rate += 17.9%, for 20 months +15. Principal += $4,000, annual interest rate += 8.5%, for 130 days +16. Principal += $9,900, annual interest rate += 15.9%, for 90 days +17. Principal += $600, annual interest rate += 16.8%, for 25 days +18. Principal += $890, annual interest rate += 9.75%, for 200 days +In the following exercises, find the future value of the investment with the given principal, simple interest rate, and +time. +19. Principal is $5,300, annual interest rate is 2.07%, and time is 18 years. +20. Principal is $14,700, annual interest rate is 3.11%, and time is 10 years. +21. Principal is $5,600, annual interest rate is 2.55%, for 30 months. +22. Principal is $10,000, annual interest rate is 1.99%, for 15 months. +23. Principal is $2,000, annual interest rate is 3.22%, for 100 days. +24. Principal is $900, annual interest rate is 3.75%, for 175 days. +In the following exercises, determine the amount applied to principal for the indicated partial payment on the loan with +the given principal, interest rate, and time when the partial payment was made. +25. A simple interest loan for $2,700 is taken out at 11.6% annual percentage rate. A partial payment of $1,500 is +made 28 days into the loan period. +26. A simple interest loan for $900 is taken out at 18.9% annual percentage rate. A partial payment of $400 is made +30 days into the loan period. +27. A simple interest loan for $13,500 is taken out at 14.8% annual percentage rate. A partial payment of $8,000 is +made 75 days into the loan period. +28. A simple interest loan for $9,900 is taken out at 9.875% annual percentage rate. A partial payment of $4,000 is +made 65 days into the loan period. +In the following exercises, determine the remaining principal for the indicated partial payment on the loan with the +given principal, interest rate, and time when the partial payment was made. +29. A simple interest loan for $2,700 is taken out at 11.6% annual percentage rate. A partial payment of $1,500 is +made 28 days into the loan period. +30. A simple interest loan for $900 is taken out at 18.9% annual percentage rate. A partial payment of $400 is made +30 days into the loan period +6.3 • Simple Interest +577 + +31. A simple interest loan for $13,500 is taken out at 14.8% annual percentage rate. A partial payment of $8,000 is +made 75 days into the loan period. +32. A simple interest loan for $9,900 is taken out at 9.875% annual percentage rate. A partial payment of $4,000 is +made 65 days into the loan period. +In the following exercises, find the payoff value of the loan with the given principal, annual simple interest rate, term, +partial payment, and time at which the partial payment was made. +33. Principal = $1,500, rate = 6.99%, term is 5 years, partial payment of $900 made 2 years into the loan. +34. Principal = $21,500, rate = 7.44%, term is 10 years, partial payment of 15,000 made after 6 years. +35. Principal = $6,800, rate = 11.9%, term is 200 days, partial payment of $4,000 made after 100 days. +36. Principal = $800, rate = 13.99%, term is 150 days, partial payment of $525 made after 50 days. +In the following exercises, find the monthly payment for a loan with the given principal, annual simple interest rate and +number of years. +37. Principal = $4,500, rate = 8.75%, years = 3 +38. Principal = $2,700, rate = 15.9%, years = 5 +39. Principal = $13,980, rate = 10.5%, years = 4 +40. Principal = $8,750, rate = 9.9%, years = 10 +In the following exercises, find the present value for the given future value, +, annual simple interest rate , and +number of years . +41. += $25,000, += 15 years, annual simple interest rate of 6.5% +42. += $12,000, += 10 years, annual simple interest rate of 4.5% +43. += $15,000, += 16 years, annual simple interest rate of 3.5% +44. += $100,000, += 30 years, annual simple interest rate of 5.5% +45. Rita takes out a simple interest loan for $4,000 for 5 years. Her interest rate is 7.88%. How much will Rita pay when +the loan is due? +46. Humberto runs a private computer networking company, and needs a loan of $31,500 for new equipment. He +shops around for the lowest interest rate he can find. He finds a rate of 8.9% interest for a 10-year term. How much +will Humberto’s payoff be at the end of the 10 years? +47. Jaye needs a short-term loan of $3,500. They find a 75-day loan that charges 14.9% interest. What is Jaye’s payoff? +48. Theethat’s car needs new struts, which cost $1,189.50 installed, but he doesn’t have the money to do so. He asks +the repair shop if they offer any sort of financing. It offers him a short-term loan at 18.9% interest for 60 days. +What is Theethat’s payoff for the struts? +49. Michelle opens a gaming shop in her small town. She takes out an $8,500 loan to get started. The loan is at 9.5% +interest and has a term of 5 years. Michelle decides to make a partial payment of $4,700 after 3 years. What will +Michelle pay when the loan is due? +50. A small retailer borrows $3,750 for a repair. The loan has a term of 100 days at 13.55% interest. If the retailer pays +a partial payment of $2,000 after 30 days, what will the loan payoff be when the loan is due? +51. Sharon invests $2,500 in a CD for her granddaughter. The CD has a term of 5 years and has a simple interest rate +of 3.11%. After that 5-year period, how much will the CD be worth? +52. Jen and Fred have a baby, and deposit $1,500 in a savings account bearing 1.76% simple interest. How much will +the account be worth in 18 years? +53. Yasmin decides to buy a used car. Her credit union offers 7.9% interest for 5-year loans on used cars. The cost of +the car, including taxes and fees, is $11,209.50. How much will Yasmin’s monthly payment be? +54. Cleo runs her own silk-screening company. She needs new silk-screening printing machines, and finds two that will +cost her, in total, $5,489.00. She takes out a 3-year loan at 8.9% interest. What will her monthly payments be for the +loan? +55. Kylie wants to invest some money in an account that yields 4.66% simple interest. Her goal is to have $20,000 in 15 +years. How much should Kylie invest to reach that goal? +56. Ishraq wants to deposit money in an account that yields 3.5% simple interest for 10 years, to help with a down +payment for a home. Her goal is to have $25,000 for the down payment. How much does Ishraq need to deposit to +reach that goal? +578 +6 • Money Management +Access for free at openstax.org + +In the following exercises, use the Cost of Financing. The difference between the total paid for a loan, along with all +other charges paid to obtain the loan, and the original principal of the loan is the cost of financing. It measures how +much more you paid for an item than the original price. In order to find the cost of financing, find the total paid over +the life of the loan. Add to that any fees paid for the loan. Then subtract the principal. +57. Yasmin decides to buy a used car. Her credit union offers 7.9% interest for 5-year loans on used cars. The cost of +the car, including taxes and fees, is $11,209.50. How much did she pay the credit union over the 5 years? What +was the cost of financing for Yasmin? +58. Cleo runs her own silk-screening company. She needs new silk-screening printing machines, and finds two that +will cost her, in total, $5,489.00. She takes out a 3-year loan at 8.9% interest. What was the cost of financing for +Cleo? +6.4 Compound Interest +Figure 6.6 The impact of compound interest (credit: "English Money" by Images Money/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Compute compound interest. +2. +Determine the difference in interest between simple and compound calculations. +3. +Understand and compute future value. +4. +Compute present value. +5. +Compute and interpret effective annual yield. +For a very long time in certain parts of the world, interest was not charged due to religious dictates. Once this restriction +was relaxed, loans that earned interest became possible. Initially, such loans had short terms, so only simple interest was +applied to the loan. However, when loans began to stretch out for years, it was natural to add the interest at the end of +each year, and add the interest to the principal of the loan. After another year, the interest was calculated on the initial +principal plus the interest from year 1, or, the interest earned interest. Each year, more interest was added to the money +owed, and that interest continued to earn interest. +Since the amount in the account grows each year, more money earns interest, increasing the account faster. This growth +follows a geometric series (Geometric Sequences). It is this feature that gives compound interest its power. This module +covers the mathematics of compound interest. +Understand and Compute Compound Interest +As we saw in Simple Interest, an account that pays simple interest only pays based on the original principal and the term +of the loan. Accounts offering compound interest pay interest at regular intervals. After each interval, the interest is +added to the original principal. Later, interest is calculated on the original principal plus the interest that has been added +previously. +After each period, the interest on the account is computed, then added to the account. Then, after the next period, when +interest is computed, it is computed based on the original principal AND the interest that was added in the previous +periods. +6.4 • Compound Interest +579 + +The following example illustrates how compounded interest works. +EXAMPLE 6.39 +Interest Compounded Annually +Abena invests $1,000 in a CD (certificate of deposit) earning 4% compounded annually. How much will Abena’s CD be +worth after 3 years? +Solution +Since the interest is compounded annually, the interest will be computed at the end of each year and added to the CD’s +value. The interest at the end of the following year will be based on the value found form the previous year. +Step 1: After the first year, the interest in Abena’s CD is computed using the interest formula +. The principal +is += 1,000, the rate, as a decimal, is 0.04, and the time is one year, so += 1. Using that, the interest earned in the first +year is +, so the interest earned in the first year was $40.00. This is added to the +value of the CD, making the CD worth +. +Step 2: At the end of the second year, interest is again computed, but is computed based on the CD’s new value, $1,040. +Using this new value and the interest formula ( and +are still 0.04 and 1, respectively), we see that the CD earned +, or $41.60. This is added to the value of the CD, making the CD now worth +. +Step 3: At the end of the third year, interest is again computed, but is computed based on Abena’s CD’s new value, +$1,081.60. Using this value and the interest formula ( and +are still 0.04 and 1, respectively), we see that the CD earned +, or $43.26 (remember to round down). This is added to the value of the +CD, making the CD now worth +. +After 3 years, Abena’s CD is worth $1,124.86. +YOUR TURN 6.39 +1. Oksana deposits $5,000 in a CD that earns 3% compounded annually. How much is the CD worth after 4 years? +Determine the Difference in Interest Between Simple and Compound Calculations +It is natural to ask, does compound interest make much of a difference? To find out, we revisit Abena’s CD. +EXAMPLE 6.40 +Comparing Simple to Compound Interest on a 3-Year CD +Abena invested $1,000 in a CD that earned 4% compounded annually, and the CD was worth $1,124.86 after 3 years. Had +Abena invested in a CD with simple interest, how much would the CD have been worth after 3 years? How much more +did Abena earn using compound interest? +Solution +Had Abena invested $1,000 in a 4% simple interest CD for 3 years, her CD would have been worth +, or $1,120.00. With interest compounded annually, Abena’s CD was +worth $1,124.86. The difference between compound and simple interest is +. So +compound interest earned Abena $4.86 more than the simple interest did. +YOUR TURN 6.40 +1. Oksana deposits $5,000 in a CD that earned 3% compounded annually and was worth $5,627.54 after 4 years. +Had Oksana invested in a CD with simple interest, how much would the CD have been worth after 4 years? How +much more did Oksana earn using compound interest? +580 +6 • Money Management +Access for free at openstax.org + +VIDEO +Compound Interest (https://openstax.org/r/compound_interest_beginners) +Understand and Compute Future Value +Imagine investing for 30 years and compounding the interest every month. Using the method above, there would be 360 +periods to calculate interest for. This is not a reasonable approach. Fortunately, there is a formula for finding the future +value of an investment that earns compound interest. +FORMULA +The future value of an investment, +, when the principal +is invested at an annual interest rate of +(in decimal form), +compounded +times per year, for +years, is found using the formula +. This is also referred to as the +future value of the investment. +Note, sometimes the formula is presented with the total number of periods, +, and the interest rate per period, . In +that case the formula becomes +. +EXAMPLE 6.41 +Computing Future Value for Compound Interest +In the following, compute the future value of the investment with the given conditions. +1. +Principal is $5,000, annual interest rate is 3.8%, compounded monthly, for 5 years. +2. +Principal is $18,500, annual interest rate is 6.25%, compounded quarterly, for 17 years. +Solution +1. +The principal is += $5,000, interest rate, in decimal form, += 0.038, compounded monthly so += 12, and for += 5 +years. Substituting these values into the formula, we find +The future value of the investment is $6,044.43. +2. +The principal is += $18,500, interest rate, in decimal form, += 0.0625, compounded quarterly so += 4, and for += 17 +years. Substituting these values into the formula, we +The future value of the investment is $53,093.54. +YOUR TURN 6.41 +In the following, compute the future value of the investment with the given conditions. +1. Principal is $7,600, annual interest rate is 4.1%, compounded monthly, for 10 years. +6.4 • Compound Interest +581 + +2. Principal is $13,250, annual interest rate is 2.79%, compounded quarterly, for 25 years. +EXAMPLE 6.42 +Interest Compounded Quarterly +Cody invests $7,500 in an account that earns 4.5% interest compounded quarterly (4 times per year). Determine the +value of Cody’s investment after 10 years. +Solution +Cody’s initial investment is $7,500, so += $7,500. The annual interest rate is 4.5%, which is 0.045 in decimal form. +Compounding quarterly means there are four periods in a year, so += 4. He invests the money for 10 years. Substituting +those values into the formula, we calculate +After 10 years, Cody’s initial investment of $7,500 is worth $11,732.82. +YOUR TURN 6.42 +1. Maggie invests $3,000 in an account that earns 5.1% interest compounded monthly. How much is the account +worth after 13 years? +EXAMPLE 6.43 +Interest Compounded Daily +Kathy invests $10,000 in an account that yields 5.6% compounded daily. How much money will be in her account after 20 +years? +Solution +Kathy’s initial investment is $10,000, so += $10,000. The annual interest rate is 5.6%, which is 0.056 in decimal form. +Compounding daily means there are 364 periods in a year, so += 365. She invests the money for 20 years, so += 20. +Substituting those values into the formula, we calculate +After 20 years, Kathy’s initial investment of $10,000 is worth $30,645.90. +YOUR TURN 6.43 +1. Jacob invests $3,000 in a CD that yields 3.4% compounded daily for 5 years. How much is his CD worth after 5 +years? +582 +6 • Money Management +Access for free at openstax.org + +VIDEO +Compare Simple Interest to Interest Compounded Annually (https://openstax.org/r/ +compare_simple_compound_interest1) +Compare Simple Interest and Compound Interest for Different Number of Periods Per Year +(https://openstax.org/r/compare_simple_compound_interest2) +WORK IT OUT +To truly grasp how compound interest works over a long period of time, create a table comparing simple interest to +compound interest, with different numbers of periods per year, for many years would be useful. In this situation, the +principal is $10,000, and the annual interest rate is 6%. +1. +Create a table with five columns. Label the first column YEARS, the second column SIMPLE INTEREST, the third +column COMPOUND ANNUALLY, the fourth column COMPOUND MONTHLY and the last column COMPOUND +DAILY, as shown below. +YEARS +SIMPLE INTEREST +COMPOUND ANNUALLY +COMPOUND MONTHLY +COMPOUND DAILY +2. +In the years column, enter 1, 2, 3, 5, 10, 20, and 30 for the rows. +3. +Calculate the account value for each column and each year. +4. +Compare the results from each of the values you find. How do the number of periods per year (compoundings +per year) impact the account value? How does the number of years impact the account value? +5. +Redo the chart, with an interest rate you choose and a principal you choose. Are the patterns identified earlier +still present? +Understand and Compute Present Value +When investing, there is often a goal to reach, such as “after 20 years, I’d like the account to be worth $100,000.” The +question to be answered in this case is “How much money must be invested now to reach the goal?” As with simple +interest, this is referred to as the present value. +FORMULA +The money invested in an account bearing an annual interest rate of +(in decimal form), compounded +times per +year for +years, is called the present value, +, of the account (or of the money) and found using the formula +, where +is the value of the account at the investment’s end. Always round this value up to the +nearest penny. +EXAMPLE 6.44 +Computing Present Value +Find the present value of the accounts under the following conditions. +1. += $250,000, invested at 6.75 interest, compounded monthly, for 30 years. +2. += $500,000, invested at 7.1% interest, compounded quarterly, for 40 years. +Solution +1. +To reach a final account value of += $250,000, invested at 6.75% interest, in decimal form += 0.0675 (decimal +form!), compounded monthly, so += 12, for 30 years, substitute those values into the formula for present value. +6.4 • Compound Interest +583 + +Calculating, we find the present value of the $250,000. +In order for this account to reach $250,000 after 30 years, $33,186.23 needs to be invested. +2. +To reach a final account value of += $500,000, invested at 7.1% interest, in decimal form += 0.071, compounded +quarterly, so += 4, for 40 years, substitute those values into the formula for present value. Calculating, we find the +present value of the $500,000. +In order for this account to reach $500,000 after 40 years, $29,949.69 needs to be invested. +YOUR TURN 6.44 +Find the present value of the accounts under the following conditions. +1. += $1,000,000, invested at 5.75% interest, compounded monthly, for 40 years. +2. += $175,000, invested at 3.8% interest, compounded quarterly, for 20 years. +EXAMPLE 6.45 +Investment Goal with Compound Interest +Pilar plans early for retirement, believing she will need $1,500,000 to live comfortably after the age of 67. How much will +she need to deposit at age 23 in an account bearing 6.35% annual interest compounded monthly? +Solution +Knowing how much to deposit at age 23 to reach a certain value later is a present value question. The target value for +Pilar is $1,500,000. The interest rate is 6.35%, which in decimal form is 0.0635. Compounded monthly means += 12. She’s +23 and will leave the money in the account until the age of 67, which is 44 years, making += 44. Using this information +and substituting in the formula for present value, we calculate +584 +6 • Money Management +Access for free at openstax.org + +Pilar will need to invest $92,442,51 in this account to have $1,500,000 at age 67. +YOUR TURN 6.45 +1. Hajun turns 30 this year and begins to think about retirement. He calculates that he will need $1,200,000 to retire +comfortably. He finds a fund to invest in that yields 7.23% and is compounded monthly. How much will Hajun +need to invest in the fund when he turns 30 so that he can reach his goal when he retires at age 65? +Compute and Interpret Effective Annual Yield +As we’ve seen, quarterly compounding pays interest 4 times a year or every 3 months; monthly compounding pays 12 +times a year; daily compounding pays interest every day, and so on. Effective annual yield allows direct comparisons +between simple interest and compound interest by converting compound interest to its equivalent simple interest rate. +We can even directly compare different compound interest situations. This gives information that can be used to identify +the best investment from a yield perspective. +Using a formula, we can interpret compound interest as simple interest. The effective annual yield formula stems from +the compound interest formula and is based on an investment of $1 for 1 year. +Effective annual yield is +where += effective annual yield, += interest rate in decimal form, and += +number of times the interest is compounded in a year. +is interpreted as the equivalent annual simple interest rate. +EXAMPLE 6.46 +Determine and Interpret Effective Annual Yield for 6% Compounded Quarterly +Suppose you have an investment paying a rate of 6% compounded quarterly. Determine and interpret that effective +annual yield of the investment. +Solution +Here, += 4 (quarterly) and += 0.06 (decimal form). Substituting into the formula we find that the effective annual yield is +Therefore, a rate of 6% compounded quarterly is equivalent to a simple interest rate of 6.14%. +YOUR TURN 6.46 +1. Calculate and interpret the effective annual yield for an investment that pays at a 7% interest compounded +quarterly. +6.4 • Compound Interest +585 + +EXAMPLE 6.47 +Determine and Interpret Effective Annual Yield for 5% Compounded Daily +Calculate and interpret the effective annual yield on a deposit earning interest at a rate of 5% compounded daily. +Solution +In this case, the rate is += 0.05 and += 365 (daily). Using the formula +, we have +This tells us that an account earning 5% compounded daily is equivalent to earning 5.13% as simple interest. +YOUR TURN 6.47 +1. Calculate and interpret the effective annual yield on a deposit earning 2.5% compounded daily. +EXAMPLE 6.48 +Choosing a Bank +Minh has a choice of banks in which he will open a savings account. He will deposit $3,200 and he wants to get the best +interest he can. The banks advertise as follows: +Bank +Interest Rate +ABC Bank +2.08% compounded monthly +123 Bank +2.09% compounded annually +XYZ Bank +2.05% compounded daily +Which bank offers the best interest? +Solution +To compare these directly, Minh could change each interest rate to its effective annual yield, which would allow direct +comparison between the rates. Computing the effective annual yield for all three choices gives: +ABC Bank: +123 Bank: +XYZ Bank: +ABC Bank has the highest effective annual yield, so Minh should choose ABC bank. +YOUR TURN 6.48 +1. Isabella decides to deposit $5,500 in a CD but needs to choose between banks that offer CDs. She identifies four +banks and finds out the terms of their CDs. Her findings are in the table below. +586 +6 • Money Management +Access for free at openstax.org + +Bank +Interest Rate +Smith Bank +3.08% compounded quarterly +Park Bank +3.11% compounded annually +Town Bank +3.09% compounded daily +Community Bank +3.10% compounded monthly +Which bank has the best yield? +Check Your Understanding +19. What is compound interest? +20. Which yields more money, simple interest or compound interest? +21. Find the future value after 15 years of $8,560.00 deposited in an account bearing 4.05% interest compounded +monthly. +22. $10,000 is deposited in an account bearing 5.6% interest for 5 years. Find the difference between the future value +when the interest is simple interest and when the interest is compounded quarterly. +23. Find the present value of $75,000 after 28 years if money is invested in an account bearing 3.25% interest +compounded monthly. +24. What can be done to compare accounts if the rates and number of compound periods per year are different? +25. Find the effective annual yield of an account with 4.89% interest compounded quarterly. +SECTION 6.4 EXERCISES +1. What is the difference between simple interest and compound interest? +2. What is a direct way to compare accounts with different interest rates and number of compounding periods? +3. Which type of account grows in value faster, one with simple interest or one with compound interest? +How many periods are there if interest is compounded? +4. Daily +5. Weekly +6. Monthly +7. Quarterly +8. Semi-annually +In the following exercises, compute the future value of the investment with the given conditions. +9. Principal = $15,000, annual interest rate = 4.25%, compounded annually, for 5 years +10. Principal = $27,500, annual interest rate = 3.75%, compounded annually, for 10 years +11. Principal = $13,800, annual interest rate = 2.55%, compounded quarterly, for 18 years +12. Principal = $150,000, annual interest rate = 2.95%, compounded quarterly, for 30 years +13. Principal = $3,500, annual interest rate = 2.9%, compounded monthly, for 7 years +14. Principal = $1,500, annual interest rate = 3.23%, compounded monthly, for 30 years +15. Principal = $16,000, annual interest rate = 3.64%, compounded daily, for 13 years +16. Principal = $9,450, annual interest rate = 3.99%, compounded daily, for 25 years +In the following exercises, compute the present value of the accounts with the given conditions. +17. Future value = $250,000, annual interest rate = 3.45%, compounded annually, for 25 years +18. Future value = $300,000, annual interest rate = 3.99%, compounded annually, for 15 years +19. Future value = $1,500,000, annual interest rate = 4.81%, compounded quarterly, for 35 years +6.4 • Compound Interest +587 + +20. Future value = $750,000, annual interest rate = 3.95%, compounded quarterly, for 10 years +21. Future value = $600,000, annual interest rate = 3.79%, compounded monthly, for 17 years +22. Future value = $800,000, annual interest rate = 4.23%, compounded monthly, for 35 years +23. Future value = $890,000, annual interest rate = 2.77%, compounded daily, for 25 years +24. Future value = $345,000, annual interest rate = 2.99%, compounded daily, for 19 years +In the following exercises, compute the effective annual yield for accounts with the given interest rate and number of +compounding periods. Round to three decimal places. +25. Annual interest rate = 2.75%, compounded monthly +26. Annual interest rate = 3.44%, compounded monthly +27. Annual interest rate = 5.18%, compounded quarterly +28. Annual interest rate = 2.56%, compounded quarterly +29. Annual interest rate = 4.11%, compounded daily +30. Annual interest rate = 6.5%, compounded daily +The following exercises explore what happens when a person deposits money in an account earning compound +interest. +31. Find the present value of $500,000 in an account that earns 3.85% compounded quarterly for the indicated +number of years. +a. +40 years +b. +35 years +c. +30 years +d. +25 years +e. +20 years +f. +15 years +32. Find the present value of $1,000,000 in an account that earns 6.15% compounded monthly for the indicated +number of years. +a. +40 years +b. +35 years +c. +30 years +d. +25 years +e. +20 years +f. +15 years +33. In the following exercises, the number of years can reflect delaying depositing money. 40 years would be +depositing money at the start of a 40-year career. 35 years would be waiting 5 years before depositing the +money. Thirty years would be waiting 10 years before depositing the money, and so on. What do you notice +happens if you delay depositing money? +34. For each 5-year gap for exercise 32, compute the difference between the present values. Do these differences +remain the same for each of the 5-year gaps, or do they differ? How do they differ? What conclusion can you +draw? +35. Daria invests $2,500 in a CD that yields 3.5% compounded quarterly for 5 years. How much is the CD worth after +those 5 years? +36. Maurice deposits $4,200 in a CD that yields 3.8% compounded annually for 3 years. How much is the CD worth +after those 3 years? +37. Georgita is shopping for an account to invest her money in. She wants the account to grow to $400,000 in 30 years. +She finds an account that earns 4.75% compounded monthly. How much does she need to deposit to reach her +goal? +38. Zak wants to create a nest egg for himself. He wants the account to be valued at $600,000 in 25 years. He finds an +account that earns 4.05% interest compounded quarterly. How much does Zak need to deposit in the account to +reach his goal of $600,000? +39. Eli wants to compare two accounts for their money. They find one account that earns 4.26% interest compounded +monthly. They find another account that earns 4.31% interest compounded quarterly. Which account will grow to +Eli’s goal the fastest? +40. Heath is planning to retire in 40 years. He’d like his account to be worth $250,000 when he does retire. He wants to +deposit money now. How much does he need to deposit in an account yielding 5.71% interest compounded semi- +588 +6 • Money Management +Access for free at openstax.org + +annually to reach his goal? +41. Jo and Kim want to set aside some money for a down payment on a new car. They have 6 years to let the money +grow. If they want to make a $15,000 down payment on the car, how much should they deposit now in an account +that earns 4.36% interest compounded monthly? +42. A newspaper’s business section runs an article about savings at various banks in the city. They find six that offer +accounts that offer compound interest. +Bank A offers 3.76% compounded daily. +Bank B offers 3.85% compounded annually. +Bank C offers 3.77% compounded weekly. +Bank D offers 3.74% compounded daily. +Bank E offers 3.81% compounded semi-annually. +To earn the most interest on a deposit, which bank should a person choose? +43. Paola reads the newspaper article from exercise 32. She really wants to know how different they are in terms of +dollars, not effective annual yield. She decides to compute the future value for accounts at each bank based on a +principal of $100,000 that are allowed to grow for 20 years. What is the difference in the future values of the +account with the highest effective annual yield, and the account with the second highest effective annual yield? +44. Paola reads the newspaper article from exercise 32. She really wants to know how different they are in terms of +dollars, not effective annual yield. She decides to compute the future value for accounts at each bank based on a +principal of $100,000 that are allowed to grow for 20 years. What is the difference in the future values of the +account with the highest effective annual yield, and the account with the lowest effective annual yield? +45. Jesse and Lila need to decide if they want to deposit money this year. If they do, they can deposit $17,400 and allow +the money to grow for 35 years. However, they could wait 12 years before making the deposit. At that time, they’d +be able to collect $31,700 but the money would only grow for 23 years. Their account earns 4.63% interest +compounded monthly. Which plan will result in the most money, depositing $17,400 now or depositing $31,700 in +12 years? +46. Veronica and Jose are debating if they should deposit $15,000 now in an account or if they should wait 10 years +and deposit $25,000. If they deposit money now, the money will grow for 35 years. If they wait 10 years, it will grow +for 25 years. Their account earns 5.25% interest compounded weekly. Which plan will result in the most money, +depositing $15,000 now or depositing $25,000 in 10 years? +6.5 Making a Personal Budget +Figure 6.7 Calculating a budget is important to your financial health. (credit: “Budget planning concept on white desk” by +Marco Verch Professional Photographer/Flickr, CC BY 2.0) +6.5 • Making a Personal Budget +589 + +Learning Objectives +After completing this section, you should be able to: +1. +Create a personal budget with the categories of expenses and income. +2. +Apply general guidelines for a budget. +“That doesn’t fit in the budget.” +“We didn’t budget for that.” +“We need to figure out our budget and stick to it.” +A budget is an outline of how money and resources should be spent. Companies have them, individuals have them, your +college has one. But do you have one? +Creating a realistic budget is an important step in careful stewardship of your financial health. Designing your budget +will help understand the financial priorities you have, and the constraints on your life choices. You want to have enough +income to pay not only for the necessities, but also for things that represent your wants, like trips or dinner out. You also +may want to save money for large purchases or retirement. You do not want to just get by, and you do not want the +problems associated with overdue balances, rising debt, and possibly losing something you have worked hard to obtain. +While creating a budget may seem intimidating at first, coming up with your basic budget outline is the hardest part. +Over time, you will adjust not only the numbers, but the categories. +Creating a Budget +You should view creating a budget as a financial tool that will help you achieve your long-term goals. A budget is an +estimation of income and expenses over some period of time. You will be able to track your progress, which will help you +to prepare for the future by making smart investment decisions. +There are several budget-creating tools available, such as the apps Good budget and Mint, and Google Sheets. Getting +started, though, begins well before you find an app. The following are steps that can be used to create your monthly +budget. +1. +Track your income and expenses Review your income and expenses for the past 6 months to a year. This will give +you an idea of your current habits. +2. +Set your income baseline Determine all the sources of income you will have. This income may from paychecks, +investments, or freelance work. It even includes child support and gifts. Be sure to use income after taxes. This +allows you to determine your maximum expenditures per month. +For income that is not steady, such as gig work or freelance work, use the previous 6 to 12 months of income to find +an average income from that gig or freelance work. Use this average in the budgeting process. +3. +Determine your expenses Review your bills from the past 6 months. You should include mortgage payments or +rent, insurance, car payments, utilities, groceries, transportation expenses, personal care, entertainment, and +savings. Using your credit card statements and bank statements will help you determine these amounts. Be aware +that some of the expenses will not change over time. These are referred to as fixed expenses, like rent, car +payments, insurance, internet service, and the like. Other expenses may vary widely from month to month and are +appropriately called variable expenses, and include such expenses as gasoline, groceries. +Some expenses are yearly, such as insurance or property taxes. Other expanses may be quarterly (four times per +year) or semiannual (twice per year). To budget for such bills by month, divide the bill total by the number of months +the bill covers. +4. +Categorize your expenses These categories may be housing, transportation, or food, for broad categories, or may +get more specific, where you categorize car payments, car insurance, and gasoline separately. The categories are +your choices. Be sure to account for the cost of maintaining a vehicle or home. The more specific you are, the better +you’ll understand your spending needs and habits. +5. +Total your monthly income and monthly expenses and compare These values should be compared. If your +expenses are higher than your income, then adjustments have to be made. Decisions of what to do with any extra +income is part of the planning process also. +6. +Make plans for unplanned expenses Ask anyone, an unexpected car repair can ruin a carefully crafted budget. +Have a plan for how you can be ready for these random expenses. This often means creating a cushion in your +590 +6 • Money Management +Access for free at openstax.org + +budget. +7. +Use your budget to make decisions and adjust for any changes Your budget is a changeable document. Add to it +when you wish, refer to it when special purchases are to be made. Keeping your budget up to date helps +accommodate changes in income and expenses. +VIDEO +Creating a Budget (https://openstax.org/r/creating_budget) +In this section, we will focus on income and expenses. One of the easiest ways to manage a budget is to create a table, +with one column containing income sources, another with income values, a third with expense categories, and a last +containing expenses. An example is shown in Table 6.1. +Income Source +Amount +Expense +Amount +Full-time job +$3,565 +Rent +$975 +Uber +$185 +Car Payment +$355 +TOTAL +$3,750 +Student Loan +$418 +Electric +$76 +Food +$400 +Gasoline +$250 +Car Insurance +$165 +Clothing +$100 +Entertainment +$100 +TOTAL +$2,839 +Table 6.1 Table with Budget +WHO KNEW? +Gross Pay and Take Home Pay +If you’ve ever had a paycheck, you know that taxes are taken out of your pay before you get your check. This amount +of money varies from state to state, and sometimes even city to city. For a person making $50,000 per year gross +salary in Salt Lake City, Utah, take home pay is about 75.6% of gross salary. In Detroit, Michigan, take home pay is +about 74.5% of gross salary. Lakeland, Florida, take home pay is about 80.5% of gross salary. These also change based +on how much a person earns! Before choosing a place to live, it makes sense to determine how much deductions +from pay will impact your income. +EXAMPLE 6.49 +Creating a Budget +Heather has graduated college and currently works as a nurse for a rural medical group. Her net monthly income from +that job is $3,765.40. She also works part-time on the weekends, earning another $672.00 per month. Her monthly +expenses are rent at $1,050, car payments at $489, student loan payments at $728, car insurance at $139, utilities at +$130, clothing at $150, entertainment (going out with friends, Netflix, Amazon Prime, movies) at $300, credit card debt at +6.5 • Making a Personal Budget +591 + +$200, food at $360, and gasoline at $275. Create her budget in a table, compare the total income to total expenses, and +determine how much excess income per month she has or how much she falls short by each month. +Solution +Step 1: To begin, we create the table with appropriate headings. +Income Source +Amount +Expense +Amount +Step 2: Her income categories are her nursing job, with $3,765.40 per month, and her part-time job, with $672.00 per +month. Entering these into the table, we have the following. +Income Source +Amount +Expense +Amount +Nursing +$3,765.40 +Part-time +$672.00 +Step 3: Her monthly expenses are listed above. Entering the categories and the amount for each of those expenses, the +table is now +Income Source +Amount +Expense +Amount +Nursing +$3,765.40 +Rent +$1,050 +Part-time +$672.00 +Car Payment +$489 +Student Loan +$728 +Car Insurance +$139 +Utilities +$130 +Clothing +$150 +Entertainment +$300 +Credit Card +$200 +Food +$400 +Gasoline +$250 +Step 4: Totaling the income and expenses, we see that her total income is $4,437.40 per month, and her total expenses +are $3,836 per month. Comparing these, we see that Heather has $601.40 in excess income per month. This provides a +cushion in her budget. +592 +6 ��� Money Management +Access for free at openstax.org + +YOUR TURN 6.49 +1. Mateo works as a union electrician in a suburban area. Monthly, his take home pay is $3,375. He sometimes does +small side jobs for family or friends, and averages about $300 per month from these little jobs. His monthly +expenses are his mortgage at $986.78, truck payments at $589.00, truck insurance at $312, utilities at $167, +clothing at $150, entertainment at $400, credit card debt at $325, food at $470, and gasoline at $375. Create +Mateo’s budget in a table, compare the total income to total expenses, and determine how much excess income +per month he has or how much he falls short by each month. +EXAMPLE 6.50 +Creating a Budget +Carol is working in a dental lab, creating dentures and bridges. Monthly her take home pay is $2,816 (based on $22 per +hour minus payroll taxes). She also receives $320 per month in child support for her one daughter. Her monthly +expenses are rent at $700, car payments at $229, student loan payments at $250, car insurance at $119, health insurance +at $225, utilities at $80, clothing at $75, entertainment at $200, food at $275, and gasoline at $275. Create Carol’s budget +in a table, compare the total income to total expenses, and determine how much excess income per month she has or +how much she falls short by each month. +Solution +Step 1: To begin, we create the table with appropriate headings. +Income Source +Amount +Expense +Amount +Step 2: Her income categories are from work, $2,816, and child support, $320, per month. Entering these into the table, +we have the following. +Income Source +Amount +Expense +Amount +Job +$2,816.00 +Child support +$320.00 +Step 3: Her monthly expenses are listed above. Entering the categories and the amount for each of those expenses, the +table is now +Income Source +Amount +Expense +Amount +Job +$2,816.00 +Rent +$700 +Child support +$320.00 +Car Payment +$229 +Student Loan +$250 +Car Insurance +$119 +Utilities +$80 +6.5 • Making a Personal Budget +593 + +Income Source +Amount +Expense +Amount +Health insurance +$225 +Clothing +$75 +Entertainment +$200 +Food +$275 +Gasoline +$275 +Step 4: Totaling the income and expenses, we see that her total income is $3,136.00 per month, and her total expenses +are $2,428.00 per month. Comparing these, we see that Carol has $708.00 in excess income per month. This is the +cushion in her budget. +YOUR TURN 6.50 +1. Maddy works as a mechanical engineer, making $6,093.75 monthly after payroll taxes. Her monthly expenses are +her mortgage at $1,452.89, car payments at $627.38, car insurance at $179.00, health insurance at $265.00, +utilities at $320, clothing at $150, entertainment at $400, credit card debt at $450, food at $370, and gasoline at +$175. Create Maddy’s budget in a table, compare the total income to total expenses, and determine how much +excess income per month she has or how much she falls short by each month. +Using the budget process, we can make decisions on adding expenses to the budget. To do so, check the cushion of the +budget to see if there is room in the budget for the new expense. +EXAMPLE 6.51 +Adding to an Existing Budget +In the example above, Carol had excess income of $708.00. She looks up the cost of before-school care for her daughter. +She finds that, monthly, the cost would be $252.00 per month. Is this an affordable program for Carol? Add this expense +to her budget table. +Solution +She can afford this, as the cost for the before school program is $252.00 and she had extra income of $708.00. Adding +this to her budget, her budget table is now +Income Source +Amount +Expense +Amount +Job +$2,816.00 +Rent +$700 +Child support +$320.00 +Car Payment +$229 +Student Loan +$250 +Car Insurance +$119 +Utilities +$80 +594 +6 • Money Management +Access for free at openstax.org + +Income Source +Amount +Expense +Amount +Health insurance +$225 +Clothing +$75 +Entertainment +$200 +Food +$275 +Gasoline +$275 +Before-school care +$252 +Now, she has $456.00 in excess income per month. +YOUR TURN 6.51 +1. Recall Heather’s budget from Example 6.49. She decides she wants to buy her own home, which would increase +her expenses. Instead of $1,050.00 in rent, she would pay $1,240.00 for her mortgage. Her utilities costs would +increase to $295.00 per month. Add these to Heather’s budget to determine if the changes are affordable. +The 50-30-20 Budget Philosophy +It isn’t clear, obvious, or easy to decide how much of your income to allocate to various categories of expenses. Many +people pay their bills and then consider all the leftover money to be spending money. However, when developing your +own budget, you may want to follow the 50-30-20 budget philosophy, which provides a basic guideline for how your +income could be allocated. Fifty percent of your budget is allotted to your needs, 30% of your budget is allotted to pay +for your wants, and 20% of your budget is allotted for savings and debt service (paying off your debts). +Knowing what expenses are necessary and what expenses are wants is important, since wants and needs are often +confused. The following are necessary expenses that represent basic living requirements and debt services. This list +isn’t complete: mortgage/rent, utilities, car, car insurance, health care, groceries, gasoline, child care (for working +parents), and minimum debt payments. The 50-30-20 budget philosophy suggests that 50%, or half, your income go to +these necessities. +Wants, though, are things you could live without but still wish to have, such as Amazon Prime, restaurant dinners, coffee +from Starbucks, vacation trips, and hobby costs. Even a gym membership or that new laptop are wants. Creating the +room to afford these wants is important to our mental health. Not budgeting for things we want will negatively impact +our quality of life. +The remaining 20% should be set aside, either in retirement funds, stocks, other investments, an emergency fund +(recommendations are that an emergency fund have 3 months of income), and perhaps extra spent to pay down debt. +This 20% is very useful for addressing those unexpected costs, such as repairs or replacement of items that no longer +work. Without budgeting this cushion, any expense that is a surprise can cause us to miss necessary payments. +The list of necessary expenses was not complete. There are other expenses that could be included. +Necessary Expenses and Expenses that are Wants +For some people, an expense will be necessary while the same expense for someone else will be a want. A good example +of this is internet service. Many people consider internet service as a need, especially those who work from home or who +are not able to leave their homes. One could also call internet service a need if they have children in school. For others, +internet service is a want. If a person’s job doesn’t require them to be online, if they are not in school, if they do not have +kids, then internet service can be dropped. There are public options for internet service. One could even use their phone +as a hot spot. +Cars often fall into the category of need, but could also fall into the want category, depending on where and how you +6.5 • Making a Personal Budget +595 + +live. Bikes, public transportation, and walking are all options that could replace a car. This would then remove the cost of +gasoline and car insurance. +Another consideration when deciding if an item on your budget is a need or a want is about your choices and priorities. +A car is a need for many. But the need for a car is not the same as the need for a specific car. If you choose to buy a car +with payments that exceed your budgeted amount for the car, then that car is a want. The amount you exceed the +budget now belongs in the want category. +The same can be said for housing. If you want an apartment that costs $1,250 per month, but your budget only allows +for an apartment that costs $900, then $350 of the rent is a want. +The point of that is to carefully consider if an expense is a need as opposed to a want. +When your expenses exceed your income, you may want to change how you budget your income to line up with these +guidelines. This may mean cutting back, finding less-expensive living arrangements, finding a less-expensive (and more +fuel-efficient) car, or sacrificing some specialty groceries. Using these guidelines keeps your financial life manageable. +Better still, they can guide you as you begin your life after graduation. +EXAMPLE 6.52 +Evaluate a Budget Using 50-30-20 model +In the example above, after Carol added before school care for her daughter to the budget, her budget was as shown +below. Evaluate Carol’s budget using the 50-30-20 budget philosophy. +Income Source +Amount +Expense +Amount +Job +$2,816.00 +Rent +$700 +Child support +$320.00 +Car Payment +$229 +Student Loan +$250 +Car Insurance +$119 +Utilities +$80 +Health insurance +$225 +Clothing +$75 +Entertainment +$200 +Food +$275 +Gasoline +$275 +Before-school care +$252 +Solution +Carol’s total income is $3,136.00. Applying the 50-30-20 budget philosophy to this income requires the calculation of +each of those percentages. +For the necessities, Carol should budget 50% of her income, or +. +For her wants, she should budget 30% of her income, or +. +For savings and extra debt service, she should budget 20% of her income, or +. +596 +6 • Money Management +Access for free at openstax.org + +In her budget, her necessities include all expenses except for entertainment. These expenses total $2,480, which exceeds +the suggested budget amount of $1568.00. To follow the guidelines, Carol would have to cut back on these necessities. +For her wants, she spends $200.00 on entertainment, which is well below the suggested budget amount of $940.80. If +she modifies how much she spends on needs, she may be able to increase the spending on her wants. +Her excess income is $456.00, which is below what she should be saving and using to pay down extra debt. If she does +adjust how much she spends on needs, she could increase the amount for savings. +YOUR TURN 6.52 +1. Recall Heather’s budget from Example 6.49, before she thought of moving. That budget is below. Evaluate +Heather’s budget using the 50-30-20 budget philosophy. +Income Source +Amount +Expense +Amount +Nursing +$3,765.40 +Rent +$1,050 +Part-time +$672.00 +Car Payment +$489 +Student Loan +$728 +Car Insurance +$139 +Utilities +$130 +Clothing +$150 +Entertainment +$300 +Credit Card +$200 +Food +$400 +Gasoline +$250 +EXAMPLE 6.53 +Creating a Budget Based on the 50-30-20 Budget Philosophy +Carmen is about to graduate and has been offered a job at a bank as a data scientist. She estimates her monthly take +home pay to be $5,662.50. Apply the 50-30-20 philosophy to that monthly income. How should Carmen use this +information? +Solution +Step 1. To apply the 50-30-20 budget philosophy to Carmen’s income, she needs to calculate 50%, 30%, and 20% of her +income. Fifty percent of her income is +. Thirty percent of her income is +. Twenty percent of her income is +. +Step 2. She would then budget $2,831.25 for her needs, $1,698.75 for her wants, and $1,132.50 for savings and debt +service. +Step 3. When choosing where to live, what to eat, and what to drive, she should make choices that keep those costs, +combined with her debt service costs, gasoline, and utilities, below $2,831.25. This means she will have to make +6.5 • Making a Personal Budget +597 + +decisions about what her priorities are. +Step 4. She should then figure out what she wants to do with her money, and stay within the limits, that is, keep those +costs below $1,698.75. +Step 5. Finally, she can begin building her savings with the remaining $1,132.50. +YOUR TURN 6.53 +1. Elijah has finished an apprenticeship and is about to start his first job as an HVAC (heating, ventilation, and air +conditioning) tech. He estimates that his net monthly income will be $3,263.44. Apply the 50-30-20 budget +philosophy to his income to set guidelines for Elijah’s budget. How should Elijah use this information? +EXAMPLE 6.54 +Using the 50-30-20 Budget Philosophy to Analyze Affordability +Steve is thinking of moving out of his family’s home. He currently works at a full-time job making $18 per hour, which will +give him, approximately, a net annual income of $29,180 (working 40 hours per week for 52 weeks per year). He has +student debt that he pays off at $218.00 per month, and already owns a car that he pays $162.00 per month for. +1. +Apply the 50-30-20 budget philosophy to Steve’s income. +2. +If he follows the budget, how much does he have, after paying his car payment and student loan, to spend on +necessities. +3. +If he follows the budget, how much will he set aside for wants? For savings? +4. +Discuss the affordability of moving out, based on Steve’s budget. +Solution +Before the 50-30-20 philosophy can be applied, Steve’s monthly income needs to be determined. This is found by +dividing his annual income by 12. This gives +. This will be used for his monthly budget. +1. +To apply the 50-30-20 philosophy to Steve’s income, find 50%, 30%, and 20% of his monthly income. +Needs (50%): 50% of his income is +. +Wants (30%): +Savings (20%): +2. +The total for Steve’s needs is $1,215.83. From this, he already pays $218.00 for his student loans, and $162.00 for his +car payment. Together that is $380.00. Subtracting from the amount he should budget for his needs, he can spend +$835.83 on other needs. +3. +Steve budgeted $729.50 for wants, and $486.33 for savings and other debt servicing. +4. +Steve will have other needs to pay for, including rent, utilities, food, heath care, gasoline, and car insurance. It is +difficult to imagine Steve being able to afford to move out, unless he reallocates money that he would want to save, +or use for entertainment and other wants, or takes on another job. Even if Steve uses all the money that the +50-30-20 budget sets aside for savings, he still only has $1,322.16 to spend on those necessities. It does not appear +he can afford to move out. +YOUR TURN 6.54 +Fran wants to take a new job but will have to move to an area with a higher cost of living. With her current income, +she can use the 50-30-20 budget philosophy. The new job will have a net pay of $43,700 annually. She will still have +to pay her car payment of $295.00, her student loans that cost $264.00 per month, and her outstanding credit card +debt, on which she pays $200 per month. +1. Apply the 50-30-20 budget philosophy to Fran’s new income. +2. If she follows the budget, how much does she have, after paying her credit card debt, car payment and +student loan, to spend on necessities. +3. If she follows the budget, how much will she set aside for wants? For savings? +598 +6 • Money Management +Access for free at openstax.org + +4. Discuss the affordability of changing jobs and moving, based on Fran’s budget. +VIDEO +50-30–20 Budget Philosophy (https://openstax.org/r/50-30-20_budgeting_rule) +Check Your Understanding +26. What is a budget? +27. What are necessary expenses? +28. David gathers his paystubs and bills from the past 6 months. His income, after taxes, is $3,450 per month. His rent, +utilities included, is $925. His car payments are $178.54 per month, his car insurance is $129.49 per month, his +credit cards cost him $117.00 per month, he spends $195 per month on gas, his food costs are $290 per month. He +also spends $21.99 on Amazon prime, $49.99 on his internet bill, and $400 per month going out. Create David’s +monthly budget, including totals, based on that information. +29. Using David’s Budget from Exercise 28, how much income does he have per month after accounting for his +expenses? +30. Apply the 50-30-20 budget philosophy to David’s budget. +31. Evaluate David’s budget with respect to the 50-30-20 budget philosophy. +SECTION 6.5 EXERCISES +In the following exercises, categorize each expense as a necessary expense or an expense that is a want. +1. Rent +2. Dinner at a restaurant. +3. Car payment +4. New game system +5. Gym membership +6. Electric bill +7. Heating bill +8. Phone bill +9. Netflix +10. Student Loan Payment +11. Explain how a necessary expense for one person could be a want expense for another person. +12. Explain how a necessary expense may be partly a necessary expense and partially a want expense. +In the following exercises, create the budget, including totals and how much the income exceeds or falls short of the +expenses, based on the information given. +13. Per month: paychecks = $3,680, consulting = $900, Mortgage = $1,198.00, Utilities = $376, Cell phone = $67.50, +Car payments = $627.85, Car insurance = $183.50, Student loans = $833, Food = $450, Gasoline = $275, Internet += $69, Dining out = $250, Credit cards = $375, entertainment = $300 +14. Per month: paychecks = $2,750, child support = $500, Mortgage = $945.50, Utilities = $195, Cell phone = $37.50, +Car payments = $298.23, Car insurance = $163.50, Student loans = $438, Food = $250, Gasoline = $175, Internet += $49, Netflix = $15, After school care = $711, Credit cards = $150, entertainment = $150 +15. Per month: paychecks = $4,385, Rent = $1095, Utilities = $165, Cell phone = $67.50, Car payments = $467.35, Car +insurance = $243.75, Student loans = $1,150, Food = $325, Gasoline = $260, Internet = $99, Netflix = $15, +Amazon = $23, Gym membership = $49, entertainment = $650 +16. Per month: paychecks = $3,460, Gig job = $173, Rent = $895, Utilities = $165, Car payments = $195.80, Car +insurance = $123.30, Food = $265, Gasoline = $185, Internet = $39, Hulu = $15, Amazon = $23, Credit cards +$97.60, Entertainment = $600 +In the following exercises, determine the amount of money that should be allocated to each of the three categories of +the 50-30-20 budget philosophy guidelines. +17. Referring to Exercise 13: Monthly income = $4,580.00 +18. Referring to Exercise 14: Monthly income = $3,250.00 +6.5 • Making a Personal Budget +599 + +19. Referring to Exercise 15: Monthly income = $4,385.00 +20. Referring to Exercise 16: Monthly income = $3,633.00 +In the following exercises, evaluate the given budget with respect to the 50-30-20 budget philosophy guidelines. +21. The budget and 50-30-20 rule from exercises 13 and 17. +22. The budget and 50-30-20 rule from exercises 14 and 18. +23. The budget and 50-30-20 rule from exercises 15 and 19. +24. The budget and 50-30-20 rule from exercises 16 and 20. +For the following exercises, Kiera and Logan sit down to make their budget. Kiera works full time as a mental health +counselor and sells kids toys on her own. Logan works as a branch manager at a local bank and works part-time at the +nearby bar. They collect their financial document to work out their budget. Kiera’s paychecks from her job as a mental +health counselor, after taxes and per month, total $3,021. Logan’s paychecks from the bank, after taxes and per month, +total $3,827. Kiera’s income from toy sales for the last 3 months were $140, $87, and $475. Logan’s take-home pay from +the bartending job for the last 3 months were $540, $310, and $449. +25. Determine how much income Kiera and Logan have per month. +26. Apply the 50-30-20 budget philosophy to their income. +For the following exercises, Kiera and Logan gather their bills from the last 6 months. Their fixed expenses, with costs, +are rent for $1,350, Kiera’s car payment for $275, Logan’s car payment of $380, student loans (they each have students +loans) for $934, car insurance for $289, internet service for $39, Netflix for $15, Amazon Prime for $24, gym +membership for $99, and cell phones for $250. The variable cost expenses, and their average costs for the last 6 +months, are utilities for $370, gasoline for $500, food for $475, clothing for $225, and miscellaneous entertainment +expenses for $535. They always pay off their credit card bill and carry no balance. +27. Create their budget, using the income from Exercise 25. +28. Categorize each expense as a need or a want. Find the total for each, along with remaining income. +29. Compare their budget to the guidelines from the 50-30-20 budget from Exercise 27. +30. Determine if Kiera and Logan can afford to buy a new computer, which would cost $330 per month for the next +6 months. +In the following exercises, the Federal Paycheck Calculator (https://openstax.org/r/smartasset) was used to estimate +monthly take-home pay. The annual salary, before taxes and deductions, is provided. Then, the monthly take-home pay +after taxes and deductions is given (which means the monthly take-home pay is not just the annual salary divided by +12!). In each case, apply the 50-30-20 budget philosophy to the monthly take-home income. Note: These are based on +living in Indianapolis, Indianapolis, unmarried and with no dependents. +31. Annual salary: $30,000. Monthly take home: $1,938 +32. Annual salary: $40,000.00. Monthly take home: $2,564 +33. Annual salary: $50,000. Monthly take home: $3,144 +34. Annual salary: $70,000. Monthly take home: $4,229 +35. Annual salary: $100,000. Monthly take home: $5,840 +36. Annual salary: 150,000. Monthly take home: $8,506 +In the following exercises, the Federal Paycheck Calculator (https://openstax.org/r/smartasset) was used to estimate +monthly take-home pay. The hourly pay, before taxes and deductions, is provided. Then, the monthly take-home pay +after taxes and deductions is given (which means the monthly take-home pay is not just the hourly pay times 174 +hours!). In each case, apply the 50-30-20 budget philosophy to the monthly take-home income. Note: These are based +on living in Tempe, Arizona, unmarried and with no dependents. +37. Hourly pay: $12.15 (minimum wage in Tempe, Arizona as of September 2022). Monthly take home: $1,698 +38. Hourly pay: $15.00. Monthly take home: $2,083 +39. Hourly pay: $17.50. Monthly take home: $2,421 +40. Hourly pay: $19.75. Monthly take home: $2,725 +41. Hourly pay: $25.00. Monthly take home: $3,369 +42. Hourly pay: $35.00. Monthly take home: $4,547 +600 +6 • Money Management +Access for free at openstax.org + +6.6 Methods of Savings +Figure 6.8 Money wisely invested grows over time. (credit: “Stack of Cash” by Janak Raja/Flickr, Public Domain Mark 1.0) +Learning Objectives +After completing this section, you should be able to: +1. +Distinguish various basic forms of savings plans. +2. +Compute return on investment for basic forms of savings plans. +3. +Compute payment to reach a financial goal. +The stock market crash of 1929 led to the Great Depression, a decade-long global downturn in productivity and +employment. A state of shock swept through the United States; the damage to people’s lives was immeasurable. +Americans no longer trusted established financial institutions. By October 1931, the banking industry’s biggest challenge +was restoring confidence to the American public. In the next 10 years, the federal government would impose strict +regulations and guidelines on the financial industry. The Emergency Banking Act of 1933 created the Federal Deposit +Insurance Corporation (FDIC), which insures bank deposits. The new federal guidelines helped ease suspicions among +the general public about the banking industry. Gradually, things returned to normal, and today we have more +investment instruments, many insured through the FDIC, than ever before. +In this section, we will first look at the different types of savings accounts and proceed to discuss the various types of +investments. There is some overlap, but we will try to differentiate among these financial instruments. Saving money +should be a goal of every adult, but it can also be a difficult goal to attain. +Distinguish Various Basic Forms of Savings Plans +There are at least three types of savings accounts. Traditional savings accounts, certificates of deposit (CDs), and money +market accounts are three main savings account vehicles. +Savings Account +A savings account is probably the most well-known type of investment, and for many people it is their first experience +with a bank. A savings account is a deposit account, held at a bank or other financial institution, which bears some +interest on the deposited money. Savings accounts are intended as a place to save money for emergencies or to achieve +short-term goals. They typically pay a low interest rate, but there is virtually no risk involved, and they are insured by the +FDIC for up to $250,000. +Savings accounts have some strengths. They are highly flexible. Generally, there are no limitations on the number of +withdrawals allowed and no limit on how much you can deposit. It is not unusual, however, that a savings account will +have a minimum balance in order for the bank to pay maintenance costs. If your account should dip below the +minimum, there are usually fees attached. +6.6 • Methods of Savings +601 + +WHO KNEW? +Many banks are covered by FDIC insurance. The FDIC is the Federal Deposit Insurance Corporation and is an +independent agency created by the U.S. Congress. One of its purposes is to provide insurance for deposits in banks, +including savings accounts. Be aware, not all banks are FDIC insured. The FDIC insures up to $250,000 for a savings +account, so you do not want your balance to exceed that federally insured limit. +Having your savings account at the same bank as your checking account does offer a real advantage. For example, if +your checking account is approaching its lower limit, you can transfer funds from your savings account and avoid any +bank fees. Similarly, if you have an excess of funds in your checking account, you can transfer funds to your savings +account and earn some interest. Checking accounts rarely pay interest. +PEOPLE IN MATHEMATICS +J.P. Morgan +J.P. Morgan was a wealthy banker around the turn of the 20th century. His business interests included railroads and +the steel industry. However, it was in 1907 that a financial crisis, caused by poor banking decisions and followed by +such great distrust in the banking system that a frenzy of withdrawals from banks occurred, that J.P. Morgan and +other wealthy bankers lent from their own funds to help stabilize and save the system. +There are some weaknesses to savings accounts. Primarily, it is because savings accounts earn very low interest rates. +This means they are not the best way to grow your money. Experts, though, recommend keeping a savings account +balance to cover 3 to 6 months of living expenses in case you should lose your job, have a sudden medical expense, or +other emergency. +Around tax time, you will receive a 1099-INT form stating the amount of interest earned on your savings, which is the +amount that must be reported when you file your tax return. A 1099 form is a tax form that reports earnings that do not +come from your employer, including interest earned on savings accounts. These 1099 forms have the suffix INT to +indicate that the income is interest income. +Savings accounts earn interest, and those earnings can be found using the interest formulas from previous sections. The +final value of these accounts is sometimes called the future value of the account. +EXAMPLE 6.55 +Single Deposit in a Savings Account +Violet deposits $4,520.00 in a savings account bearing 1.45% interest compounded annually. If she does not add to or +withdraw any of that money, how much will be in the account after 3 years? +Solution +To find the compound interest, use the formula from Compound Interest, +, where +represents the +amount in the account after +years, with initial deposit (or principal) of +, at an annual interest rate, in decimal form, of +, compounded +times per year. Violet has a principal of $4,520.00, which will earn an interest of += 0.0145, +compounded yearly (so += 1), for += 3 years. Substituting and calculating, we find that Violet’s account will be worth +Or, Violet will have $4,719.48 after 3 years. +602 +6 • Money Management +Access for free at openstax.org + +YOUR TURN 6.55 +1. Brian deposits $5,600 in a savings account that yields 1.23% interest compounded annually. If he leaves that +deposit in the account and adds nothing new to the account, what will the account be worth in 5 years? +WHO KNEW? +Banks have not always offered interest on savings accounts. An 1836 publication from Indiana noted that banks in +other states allow small interest on deposits. It specifically says that in these other states, these deposits are what +business transactions are based upon. And that giving interest would encourage deposits, and thus increase the +business that banks can do. +Journal of the House of Representatives of the Sate of Indiana (https://openstax.org/r/onepage) +Certificates of Deposit, or CDs +We discussed certificates of deposit (CDs) in earlier sections. CDs differ from savings accounts in a few ways. First, the +investment lasts for a fixed period of time, agreed to when the money is invested in the CD. These time periods often +range from 6 months to 5 years. Money from the CD cannot be withdrawn (without penalty) until the investment period +is up. Also, money cannot be added to an existing CD. +Certificates of deposit have features similar to savings accounts. They are insured by the FDIC. They are entirely safe. +They do, though, offer a better interest rate. The trade-off is that once the money is invested in a CD, that money is +unavailable until the investment period ends. +EXAMPLE 6.56 +5-Year CD +Silvio deposits $10,000 in a CD that yields 2.17% compounded semiannually for 5 years. How much is the CD worth after +5 years? +Solution +This also uses the compound interest formula from Compound Interest, +, Substituting the values += +$10,000, += 0.0217, += 2 (semiannually means twice per year), and += 5, we find the account will be worth +The CD will be worth $11,219.53 after 5 years. +YOUR TURN 6.56 +1. Denise deposits $3,500 in a CD bearing 2.23% interest compounded quarterly for 3 years. How much will Denise’s +CD be worth after those 3 years? +Money Market Account +A money market account is similar to a savings account, except the number of transactions (withdrawals and transfers) +is generally limited to six each month. Money market accounts typically have a minimum balance that must be +maintained. If the balance in the account drops below the minimum, there is likely to be a penalty. Money market +accounts offer the flexibility of checks and ATM cards. Finally, the interest rate on a money market account is typically +6.6 �� Methods of Savings +603 + +higher than the interest rate on a savings account. +EXAMPLE 6.57 +Single Deposit to a Money Market Account +Marietta opens a money market account, and deposits $2,500.00 in the account. It bears 1.76% interest compounded +monthly. If she makes no other transactions on the account, how much will be in the account after 4 years? +Solution +This, once again, uses the compound interest formula from Compound Interest: +, Substituting the +values += $25,000, += 0.0176, += 12, and += 4, we find the account will be worth +The money market account will be worth $2,682.20 after 4 years. +YOUR TURN 6.57 +1. Chuck opens a money market account, and deposits $8,500.00 in the account. It bears 1.83% interest +compounded quarterly. If he leaves makes no other transactions on the account, how much will be in the +account after 3 years? +Return on Investment +If we want to compare the profitability of different investments, like savings accounts versus other investment tools, we +need a measure that evens the playing field. Such a measure is return on investment. +FORMULA +The return on investment, often denoted ROI, is the percent difference between the initial investment, +, and the +final value of the investment, +, or +, expressed as a percentage. +The length of time of the investment is not considered in ROI. +EXAMPLE 6.58 +Calculating Return on Investment +1. +Determine the return on investment for the 5-year CD from Example 6.56. Round the percentage to two decimal +places. +2. +Determine the return on investment for the money market account from Example 6.57. Round the percentage to +two decimal places. +Solution +1. +The initial deposit in the CD was $10,000, so += $10,000. The value at the end of 5 years was $11,239.53. so += +$11,239.53. Substituting and computing we find the return on investment. +604 +6 • Money Management +Access for free at openstax.org + +The ROI is 12.40%. +2. +The initial deposit in the money market was $2,500, so += $2,500. The value at the end of 4 years was $2,682.20. so += $2,682.20. Substituting and computing we find the return on investment. +The ROI is 7.29%. +YOUR TURN 6.58 +1. The amount of $13,000 is invested in a savings account. After 10 years the account has $15,250.00. Find the +return on investment for this account. +2. The amount of $6,500 is deposited in a money market account. After 7 years, the account has $7,358.00. Find +the return on investment for this account. +VIDEO +Return on Investment, ROI (https://openstax.org/r/This_video) +Annuities as Savings +In Compound Interest, we talked about the future value of a single deposit. In reality, people often open accounts that +allow them to add deposits, or payments, to the account at regular intervals. This agrees with the 50-30-20 budget +philosophy, where some income is saved every month. When a deposit is made at the end of each compounding period, +such a savings account is called an ordinary annuity. +The formula for the future value of an ordinary annuity is +, where +is the future value +of the annuity, +is the payment, +is the annual interest rate (in decimal form), +is the number of compounding +periods per year, and +is the number of years. +It is important to note that the number of deposits per year and the number of periods per year are the same. +Another form of annuity if the annuity due, which has deposits at the start of each compounding period. This other +annuity type has different formulas and is not addressed in this text. +EXAMPLE 6.59 +Future Value of an Ordinary Annuity +Jill has an account that bears 3.75% interest compounded monthly. She decides to deposit $250.00 each month, at the +6.6 • Methods of Savings +605 + +end of the compounding period, into this account. What is the future value of this account, after 8 years? +Solution +These are regular payments into an account bearing compound interest. She is depositing them at the end of each +compounding period. This makes this an ordinary annuity. Substituting the values += 250, += 0.0375, += 12, and += 8 +into the formula, we find the future value of the account. +The account, after 8 years, will contain $27,938.20. +YOUR TURN 6.59 +1. Kelly invests $525 every third month, at the end of the compounding period, into an account bearing 3.89% +interest compounded quarterly. How much will be in the account after 15 years? +WHO KNEW? +Setting Savings Account Interest Rates +There are a number of factors that contribute to the amount a bank gives for savings accounts. The interest rate +reflects how much the bank values deposits. It also reflects the money that the bank will earn when they lend out +money. Finally, interest rates are impacted by the Federal Reserve Bank. When the Fed raises interest rates, so do +banks. +PEOPLE IN MATHEMATICS +The Federal Reserve Chairperson +The Federal Reserve Board monitors the risks in the financial system to help ensure a healthy economy for +individuals, companies, and communities. The Board oversees the 12 regional reserve banks. The Chairperson of the +Federal Reserve Board testifies to Congress twice per year, meets with the secretary of the Treasury, chairs the Federal +Open Market Committee, and is the face of federal monetary policy. Currently, the Fed Chair is Jerome Powell, who +has served since 2018. +EXAMPLE 6.60 +Saving for College +When Yusef was born, Rita and George began to save for Yusef’s college years by investing $2,500 each year in a savings +account bearing 3.4% interest compounded annually. How much will they have saved after 18 years? +606 +6 • Money Management +Access for free at openstax.org + +Solution +To find the future value of the account, we use the ordinary annuity formula +. The +payment is $2,500, rate is 0.034, the number of compounding periods is 1, and the number of years is 18. Substituting +these values and computing, we have +After saving for 18 years, Rita and George will have $60,694.77 for Yusef’s college. +YOUR TURN 6.60 +1. Bemnet saves $280 per month in a savings account bearing 3.11% interest compounded monthly. After 20 years, +how much does Bemnet have in the account? +TECH CHECK +Google Sheets offers a function to calculate the future value of an ordinary annuity. To get Google Sheets to calculate +the future value, you use the following: +=fv(rate,number_of_periods, payment, present_value, end_or_beginning). +To explain, the rate is the rate per compounding period. From our formula, that is +. Also, the number of periods +must be entered. From our formula, that is +. The payment is the amount deposited each period. Present value is +0 if we begin with no money and rely only on the payments to be made. However, if some money is available to put in +the account before the payments start, that amount, an initial deposit, would be the value of +. Finally, for an +ordinary annuity, enter 0 for end or beginning. Using the values for Jill, the payment amount is $250, += 0.0375, += +12, += 8, and that there is no initial deposit, += 0, the Google Sheets formula is +=fv(0.0375/12,12*8,250,0,0). +Figure 6.9 shows the formula in Google Sheets. +Figure 6.9 Google Sheets formula +Hitting the enter key shows the payment value (Figure 6.10). +6.6 • Methods of Savings +607 + +Figure 6.10 Payment value +Notice that the future value is negative, since it is a payment leaving an account. +VIDEO +Future Value Using Google Sheets (https://openstax.org/r/cell_references) +Compute Payment to Reach a Financial Goal +The formula used to get the future value of an ordinary annuity is useful, finding out what the final amount in the +account will be. However, that isn’t how planning works. To plan, we need to know how much to put into the ordinary +annuity each compounding period in order to reach a goal. Fortunately, that formula exists. +FORMULA +The formula for the amount that needs to be deposited per period, +, of an ordinary annuity to reach a specified +goal, +, is +, where +is the annual interest rate (in decimal form), +is the number of periods +per year, and +is the number of years. +With this formula, it is possible to plan the amount to be saved. +EXAMPLE 6.61 +Saving for a Car +Yaroslava wants to save in order to buy a car, in 3 years, without taking out a loan. She determines that she’ll need +$35,500 for the purchase. If she deposits money into an ordinary annuity that yields 4.25% interest compounded +monthly, how much will she need to deposit each month? +Solution +Yaroslava has a goal and needs to know the payments to make to reach the goal. Her goal is += $35,500, with an +interest rate += 0.0425, compounded per month so += 12, and for 3 years, making += 3. Substituting into the formula, +Yaroslava finds the necessary payment. +To reach her goal, Yaroslava would need to deposit $926.33 in her account each month. +This has been rounded up, so that the deposits don’t fall short of the goal. However, some round off using the +608 +6 • Money Management +Access for free at openstax.org + +standard rounding rules: if the last digit is 1, 2, 3, or 4, the number is rounded down; if the last digit is 5, 6, 7, 8, or 9 +the number is rounded up. +YOUR TURN 6.61 +1. Chione decides to put new siding on her house. She finds that it will cost about $27,800. She decides to begin +saving for the purchase so that she doesn’t take on debt to side the house. How much would Chione need to +deposit each quarter in an ordinary annuity that yields 5.16% compounded quarterly for 5 years? +TECH CHECK +Google Sheets offers a function to calculate the payment necessary to reach a goal using ordinary annuities. To get +Google Sheets to calculate the payment, you use the following: +=pmt(rate,number_of_periods, present_value, future_value, end_or_beginning). +To explain, the rate is per compounding period. From our formula, that is +. Also, the number of periods must be +entered. From our formula, that is +. The present value is the amount of money that the account begins with. If we +begin with no money and rely only on the payments to be made, then this number is 0. However, if some money is +available to put in the account before the payments start, that amount, an initial deposit, would be the value of +. +Next, enter the future value, +. Finally, for an ordinary annuity, enter 0 for end or beginning. Using the values for +Yaroslava, += 0.0425, += 12, += 3, and that there is no initial deposit, += 0, the Google Sheets formula is +=pmt(0.0425/12,12*3,0,35500,0). +Figure 6.11 shows the formula in Google Sheets. +Figure 6.11 Google Sheets formula +Hitting the enter key shows the payment value (Figure 6.12). +Figure 6.12 Payment value +Notice that the payment is negative, since it is a payment leaving an account. Additionally, the payment is $926.32. +We rounded that up, but Google Sheets rounded off. +Check Your Understanding +32. What is a savings account? +33. How does a CD differ from a savings account? +34. Which is more flexible, a CD or a money market account? Why? +35. If $7,500 is deposited in a 4-year CD earning 3.28% interest compounded monthly, how much is in the account +after 4 years? +6.6 • Methods of Savings +609 + +36. If the initial deposit in an account is $10,000 and the account is worth $12,560 after 7 years, what is the return on +investment? +37. Find the future value of an account if $450.00 per quarter is invested in a savings account bearing 3.5% interest +compounded quarterly for 10 years. +38. How much must be deposited per quarter in an account bearing 2.98% interest compounded quarterly if the +account is to be worth $300,000 after 25 years? +SECTION 6.6 EXERCISES +1. Which account has the greatest flexibility, savings, certificate of deposit, or money market? +2. Why are interest rates on savings accounts, CDs, and money market accounts low? +3. Which of savings accounts, certificates of deposit, and money market accounts, allow for transactions? +4. How does number of years impact the return on investment? +In the following exercises, find the future value of the account based on the information given. +5. The amount of $3,000 deposited in a CD bearing 2.6% compounded semi-annually for 3 years. +6. The amount of $1,500 deposited in a money market account bearing 3.11% interest compounded monthly for +10 years. +7. The amount of $8,450 deposited in a savings account bearing 1.75% interest compounded monthly for 2 years. +8. The amount of $10,500 deposited in a savings account bearing 1.35% interest compounded quarterly for 20 +years. +9. The amount of $24,800 deposited in a money market account bearing 2.53% interest compounded semi- +annually for 13 years. +10. The amount of $16,400 deposited in a CD bearing 2.55% interest compounded quarterly for 18 years. +In the following exercises, find the return on investment based on the specified exercise. Round to two decimal places. +11. Account from Exercise 5. +12. Account from Exercise 6. +13. Account from Exercise 7. +14. Account from Exercise 8. +In the following exercises, find the future value of the ordinary annuities based on the payment, interest rate, +compounding periods and length of time given. +15. The amount of $150 deposited monthly in an account bearing 4.22% interest compounded monthly for 20 +years. +16. The amount of $500 deposited semi-annually in an account bearing 3.62% interest compounded semi-annually +for 30 years. +17. The amount of $250 deposited quarterly in an account bearing 3.61% interest compounded quarterly for 25 +years. +18. The amount of $250 deposited monthly in an account bearing 3.09% interest compounded monthly for 40 +years. +19. The amount of $1,500 deposited annually in an account bearing 3.34% interest compounded annually for 10 +years. +20. The amount of $1400 deposited semi-annually in an account bearing 2.78% interest compounded semi- +annually for 30 years. +In the following exercises, find the payment per period necessary to reach a specified future value based on the given +interest rate, compounding periods per year, and number of years. Recall, the number of payments per year and the +number of compounding periods per year are the same. +21. Future value of $1,000,000 from an account bearing 3.94% interest compounded monthly for 40 years. +22. Future value of $500,000 from an account bearing 2.11% interest compounded quarterly for 30 years. +23. Future value of $750,000 from an account bearing 3.27% interest compounded monthly for 25 years. +24. Future value of $300,000 from an account bearing 3.59% interest compounded semiannually for 35 years. +25. Future value of $1,000,000 from an account bearing 3.62% interest compounded annually for 25 years. +26. Future value of $600,000 from an account bearing 4.02% interest compounded quarterly for 30 years. +27. Dina deposits $3,000 in a 5-year CD that bears 3.25% interest compounded quarterly. What is the CD worth after +those 5 years? +610 +6 • Money Management +Access for free at openstax.org + +28. Timothy deposits $1,200 in a savings account that bears 1.85% interest compounded monthly. If Timothy does not +deposit or withdraw money from the account how much is in Timothy’s account after 3 years? +29. Leslie deposits $13,000 in a money market account that bears 2.55% interest compounded semi-annually. If Leslie +does not withdraw or deposit money into the account, how much is in Leslie’s account after 6 years? +30. Jennifer deposits $8,500 in a 3-year CD bearing 2.71% interest compounded annually. How much is Jennifer’s CD +worth after those 3 years? +31. Yasmin has analyzed her budget and decides to deposit $425 per month in an account bearing 3.99% interest +compounded monthly. How much will be in the account after 20 years? After 30 years? After 40 years? +32. Brad applied the 50-30-20 budget philosophy to his income and decides that he can afford $380 per month for +savings. He finds an account bearing 3.47% interest compounded monthly. How much will he have in the account +after 25 years? 30 years? 35 years? +33. Ashliegh wants to save for an early retirement. She thinks she needs $1,250,000 to retire at the age of 55, which is +30 years from now. How much must she deposit per month in an account bearing 3.48% interest compounded +monthly to reach her goal? +34. Colin plans out the next 38 years of his life. In order to retire in 38 years (age 65) with $1,450,000, how much +should he deposit quarterly in an account bearing 4.21% interest compounded quarterly to reach his goal? +In the following exercises, different savings strategies will be compared. +35. Sam is 23 years old and has just landed her first post-college job. She creates a budget, and using the 50-30-20 +budget philosophy, she sees she should save or pay down debt with $650. She decides to apply $300 per month +to long-term savings. She finds an account bearing 3.75% interest compounded monthly. Sam begins investing +$300 per month in that account on her 24th birthday. How much will be in the account at age 65 (41 years)? +36. Sam decides instead to delay investing in the account until her 35th birthday. How much will be in the account +at age 65 (30 years)? +37. Sam decides to deposit the $300 per month until she turns 35 years old (11 years). She will then stop investing +the $300 monthly, and just allow the money to earn interest until her 65th birthday (30 more years). How much +will be in her account on her 65th birthday? Hint: First, compute the FV of the deposits. Then use that FV as the +principal for a single deposit into an account bearing 3.75% interest compounded monthly. +38. Compare the results of the three investment strategies. +In the following exercises, different savings strategies will be compared. +39. Dahlia is 22 years old and has just landed a banking job. She creates a budget, and using the 50-30-20 budget +philosophy, she sees she should save or pay down debt with $400. She decides to apply $250 per month to +long-term savings. She finds an account bearing 6.2% interest compounded monthly. Dahlia begins investing +$250 per month in that account on her 23rd birthday. How much will be in the account on her 68th birthday (45 +years)? +40. Dahlia decides instead to delay investing in the account until her 34th birthday. How much will be in the +account on her 68th birthday (34 years)? +41. Dahlia decides to deposit the $250 per month until her 34th birthday (11 years). She will then stop investing the +$250 monthly, and just allow the money to earn interest until her 68th birthday (34 more years). How much will +be in her account on her 65th birthday? Hint: First compute the +of the deposits. Then use that +as the +principal for a single deposit into an account bearing 6.2% interest compounded monthly. +42. Compare the results of the three investment strategies. +6.6 • Methods of Savings +611 + +6.7 Investments +Figure 6.13 Stocks are bought and sold to improve investment values. (credit: modification of work "FT ringing the +Closing Bell at the NYSE" by Financial Times/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Distinguish between basic forms of investments including stocks, bonds, and mutual funds. +2. +Understand what bonds are and how bond investments work. +3. +Understand how stocks are purchased and gain or lose value. +4. +Read and derive information from a stock table. +5. +Define a mutual fund and how to invest. +6. +Compute return on investment for basic forms of investments. +7. +Compute future value of investments. +8. +Compute payment to reach a financial goal. +9. +Identify and distinguish between retirement savings accounts. +You can save your money in a safe or a vault (or worse, under the mattress!), but that money does not grow. It would be +hard to save enough for retirement that way. What can be done to increase the value of the money you already have? +The answer is to invest it. Use the money that you have to earn more money back. For instance, as we saw in Methods of +Savings, you can save it in a bank. Or, to reach loftier goals, invest in something more likely to grow, such as stocks. +A great example of this is Apple stock. Anyone who bought stock in Apple Inc. (formerly Apple Computer, Inc.) in 1997 +and held onto the shares earned a lot of money. To be more specific, $100 worth of Apple shares bought in 1980, when it +was first sold to the public, was valued at $67,564 in 2019, or 676 times more! (https://openstax.org/r/much_aspx) +Perhaps you have heard a story like that, of an investment opportunity taken that paid off, or the story of an investment +opportunity missed. But such stories are the exceptions. +In this section, we’ll investigate bonds, stocks, and mutual funds and their comparative strengths and weaknesses. We +close the section with a discussion of retirement savings accounts. +Distinguish Between Basic Forms of Investments +Bonds, stocks, and mutual funds tend to offer higher returns, but to varying degrees, come with higher risks. Stocks and +mutual funds also vary in how much they earn. Their predicted rates of return on investment are not guaranteed, but +educated guesses based on market trends and historical performance. +We will use the methods and formulas we learned earlier to evaluate these forms of investment. +Bonds +Bonds are issued from big companies and from governments. Selling bonds is an alternative to an institution taking a +loan from a bank. The funds from the selling of bonds are often used for large projects, like funding the building of a +new highway or hospital. +Bonds are considered a conservative investment. They are bought for what is known as the issue price. The interest is +fixed (does not change) at the time of purchase and is based on the issue price of the bond. The interest rate is often +referred to as the coupon rate; the interest paid is often called the coupon yield. The interest paid is often higher than +savings accounts and the risk is exceptionally low. The bond is for a fixed length of time. The end of this time is the +612 +6 • Money Management +Access for free at openstax.org + +maturity date of the bond. +There are several types of bonds: +• +Treasury bonds are issued by the federal government. +• +Municipal bonds are issued by state and local governments. +• +Corporate bonds are issued by major corporations. +There are other types of bonds available, but they are beyond the scope of this section. +WHO KNEW? +Trading Bonds +Bonds are often part of larger investment portfolios. These bonds may be traded. However, the interest paid is based +on the price when the bond was bought (the issue price). These bonds can be bought and sold for more or less +money than the issue price. If the bond is bought for more than the issue price, the interest is still paid on the issue +price, not on the purchase price when the trade was made. This means the actual return on the bond decreases. If the +bond is bought for less than the issue price, the return on the bond goes up. +VIDEO +Bonds (https://openstax.org/r/investing_basics_bonds) +EXAMPLE 6.62 +Bond Investment +Muriel purchases a $3,000 bond with a maturity of 4 years at a fixed coupon rate of 5.5% paid annually. How much is +Muriel paid each year, and how much does she receive on the maturity date? +Solution +The coupon rate is 5.5%. 5.5% of her bond value is +. After year 1, Muriel receives $165. She +receives $165 after years 2 and 3 also. In year 4, when the bond matures, Muriel receives $3,165, or the interest and the +initial investment, or principal. +YOUR TURN 6.62 +1. Maureen invests $5,000 in a bond with a maturity date in 5 years at a fixed coupon rate of 4.75%. How much is +Maureen paid each year and how much does she receive on the maturity date? +Stocks +Stocks are part ownership in a company. They come in units called shares. The performance and earnings of stocks is +not guaranteed, which makes them riskier than any other investment discussed earlier. However, they can offer higher +return on investment than the other investments. Their value grows in two ways. They offer dividends, which is a +portion of the profit made by the company. And the price per share can increase based on how others see that value of +the company changing. If the value of the company drops, or the company folds, the money invested in the stock also +drops. +Most stock transactions are executed through a broker. Brokers’ commissions can be a percentage of value of the trades +made or a flat fee. There are full-service brokers who charge higher commission rates, but they also offer financial advice +and perform the research that you may not have the time or the expertise to do on your own. A discount broker only +executes the stock transactions, buying or selling, so they charge lower rates than full-service brokers. There are also +brokers that offer commission-free trading. +An important thing to remember is that stocks might provide a very large return on investment, but the trade-off is the +risk associated with owning stocks. +6.7 • Investments +613 + +WHO KNEW? +Chapter 11 Bankruptcy and Stocks +In the fall of 2022, the parent company of Regal Theaters, named Cineworld, filed for Chapter 11 bankruptcy. +According to news articles, the bankruptcy was necessitated due to its heavy debt load. Generally, a company can file +for Chapter 11 bankruptcy to allow them time to reorganize and restructure debts. When this happens, the company, +after the Chapter 11 process is over, offers new stock. This makes the previous stock worthless. However, the company +may allow an exchange of old stock for a discounted amount of the new stock. This in effect reduces (maybe vastly) +the wealth held by those who owned the original stock. +EXAMPLE 6.63 +Buying Stock in Company ABC +Haniah buys stock in the ABC company, investing a total of $13,000. She expects the stock to grow, through stock price +increase and reinvestment of dividends, by 12.3% per year and compounded annually. If she leaves that money invested, +how much will the stocks be worth in 20 years? +Solution +Calculating this is a compound interest calculation, if Haniah’s assumption about the stock’s performance is correct. If so, +then the principal is $13,000, the rate is 0.123, the number of compounding periods per year is 1, and the time is 20 +years. Substituting into the compound interest formula from Methods of Savings, and computing, we have +. After 20 years, her stock is +now worth $132,293.49. +YOUR TURN 6.63 +1. Rixie deposits $23,000 in the stock of DEF company. She assumes the stock value to grow though stock price +increase and reinvestment of dividends by 13.8% compounded annually. How much will her stock be worth in 12 +years? +WHO KNEW? +Risk and Volkswagen +The question of risk hovers over every investment. How risky can it get? Volkswagen seems to be a rather safe +investment. But in 2015, Volkswagen’s stock tumbled 30% over a few days when it was revealed that the company had +installed software that altered the emission performance of some of their diesel engines. Volkswagen’s hope was that +lower emissions would bolster US sales of some of their diesel models. This was a drastic drop, and many investors +lost a lot of money. However, the stock has come back since then. This was mild compared to the 65% drop in the +Martha Stewart Living Omnimedia stocks. +PEOPLE IN MATHEMATICS +Warren Buffett +Warren Buffett is an investment legend. He began his career as an investment salesman in the 1950s. He formed +Buffett associates in 1956. In 1965, he was in control of Berkshire Hathaway, which began as a merger between two +textile companies. In his role there, he began to invest in a variety of companies. It is now a conglomerate holding +company, and fully owns GEICO, Duracell, Diary Queen, and other large companies. +614 +6 • Money Management +Access for free at openstax.org + +His investment philosophy involves finding stocks and bonds from companies that have high intrinsic worth +compared to their stock or bond prices. This means he focuses not on the supply and demand side of stock investing, +but instead on the company’s worth in total. Using this philosophy, he has become one of the world’s most successful +investors. +Reading Stock Tables +Information about particular stocks is contained in stock tables. This information includes how much the stock is selling +for, and its high and low values form the past year (52 weeks). In a newspaper, the stock table may look like this: +52-Week High +Low +Stock +SYM +Div +Yld +% +P/E +Vol +100s +High +Low +Close +Net +Chg +41.66 +18.90 +McDonald’s +MCD +.72 +2.9 +12 +7588 +25.73 +23.87 +25.42 ++0.31 +22.60 +13.20 +Monsanto +MON +.52 +2.4 +55 +15474 +21.86 +21.48 +21.64 +-0.29 +17.05 +8.30 +Motorola +MOT +.16 +1.7 +dd +16149 +10.57 +8.88 +10.43 ++0.14 +31.75 +22.99 +Mueller +MLI +- +- +16 +1564 +29.32 +27.03 +27.11 +-0.02 +Table 6.2 excerpt from a stock table, 2008 +The symbols and abbreviations are defined here: +52-week +High +52-week +Low +The highest and lowest price of the stock over the past 52 weeks +Stock +SYM +The name of the company and the symbol used for trading +Annual DIV +The current annual dividend per share +Yld % +Percent yield is +P/E +Price to earnings ratio, share price divided by earnings per share over past year (dd +indicates loss) +Vol 100s +The number of shares traded yesterday in 100s +High +Low +The highest and lowest prices at which stocks traded yesterday +Close +The price at which the stock traded at the close of the market yesterday +Net Chg +Net change; change in price from market close 2 days ago to yesterday’s close +The formulas for yield and price to earnings is a good way to measure how much the stock returns per share. Their +values are calculated in the stock table, but deserve attention here. +FORMULA +The price to earnings ratio of a stock, P/E, is +. The percent yield for a stock, Yld%, is +. +6.7 • Investments +615 + +It should be noted that the price of a stock increases and decreases every moment, and so these value change as the +share price changes. +EXAMPLE 6.64 +Computing Percent Yield +1. +Find the percent yield for a stock with a price of $30.69 and an annual dividend of $1.48. +2. +Find the percent yield for a stock with a price of $62.25 and an annual dividend of $1.76. +Solution +1. +Substituting the values for price, $30.69, and annual dividend, $1.48, we find the percent yield for the stock to be +2. +Substituting the values for price, $62.25, and annual dividend, $1.76, we find the percent yield for the stock to be +YOUR TURN 6.64 +1. Find the percent yield for a stock with a price of $37.40 and an annual dividend of $1.60. +2. Find the percent yield for a stock with a price of $73.22 and an annual dividend of $2.41. +The stock table information is now, and has been, available online, from websites such as cnn.com/markets, +markets.businessinsider.com/stocks, and marketwatch.com. The same information is available from these sites as from +the newspaper listings, but are often accessed one stock at a time. Figure 6.14 shows the stock table for Lowe’s on +September 7, 2022. +Figure 6.14 Key data for Lowe's stock 9/7/2022 (data source: marketwatch.com) +Other key data is further down on the website, and is shown in Figure 6.15, below. +616 +6 • Money Management +Access for free at openstax.org + +Figure 6.15 Key data for Lowe's stock 9/7/2022 (data source: marketwatch.com) +Notice that the 52-week high and low are now shown as the 52-week range. However, you get additional information, +including the stock performance over the past 5 days, past month, past 3 months, the year to date (YTD), and over the +past year. You can also read the number of shares outstanding, the expected date for the dividend (EX-DIVIDEND DATE), +and importantly for the P/E ratio, the earning per share (EPS). +EXAMPLE 6.65 +Reading an Online Stock Table +Consider the stock table (Figure 6.16), and answer the questions based on the table. +6.7 • Investments +617 + +Figure 6.16 Key data for McDonald's stock 9/7/2022 (data source: marketwatch.com) +1. +What is the current price for McDonald’s Corp on this date? +2. +What is the 52-wk high? 52-wk low? +3. +When is the dividend expected? +4. +What is its yield? +5. +What is the earnings per share? +Solution +1. +Looking at the table, the current price of a share is $258.87. +2. +The high was $271.15, and the low was $217.68. +3. +August 31, 2022 +4. +2.13% +5. +The EPS value is $8.12. +YOUR TURN 6.65 +Consider the stock table below, and answer the questions based on the table. +618 +6 • Money Management +Access for free at openstax.org + +Key data for Intel stock 9/7/2022 (data source: marketwatch.com) +1. What is the current price for Intel Corp on this date? +2. What is the 52-wk high? 52-wk low? +3. When is the dividend expected? +4. What is its yield? +5. What is the earnings per share? +As mentioned, stocks earn money in two ways, through dividends and increase in share price. +EXAMPLE 6.66 +Dividends Paid +Darma owns 150 shares of stock in the GDW company. This quarter, GDW is paying $0.87 per share in dividends. How +much will Darma earn in dividends this quarter? +Solution +Each share pays $0.87, so Darma earns +. +6.7 • Investments +619 + +YOUR TURN 6.66 +1. Yulia owns 300 shares of stock in YYZ company. It pays $1.12 per share this quarter. How much did Yulia earn this +quarter on stock in YYZ? +EXAMPLE 6.67 +Stock Price Increases +Vincent buys 100 stocks in the REM company for $21.87 per share. One year later, he sells those 100 shares for $29.15 +per share. +1. +How much money did Vincent make? +2. +What was his return on investment for that one year? +Solution +1. +Vincent spent $21.87 per share to buy the stock. The total he spent on the stock was +. +When he sold the stock, the price was $29.15, so he received +. He made +. +2. +His return on investment was +. +YOUR TURN 6.67 +Ginny buys 200 shares of stock in UUK company for $9.76 per share. At the end of the year, she sells those stocks for +$10.02 per share. +1. How much money did Ginny make? +2. What was her return on investment for that one year? +VIDEO +Reading Stock Summary Online (https://openstax.org/r/Reading_Stock) +Mutual Funds +A mutual fund is a collection of investments that are all bundled together. When you buy shares of a mutual fund, your +money is pooled with the assets of other investors. This pooled money is invested in stocks, bonds, money market +instruments, and other assets. Mutual funds are typically operated by professional money managers who allocate the +fund's assets and attempt to produce capital gains or income for the fund's investors. +A key benefit of mutual funds is that they allow small or individual investors to invest in professionally managed +portfolios of equities, bonds, and other securities. This means each shareholder participates proportionally in the gains +or losses of the fund. The performance of a mutual fund is usually stated as how much the mutual fund’s total value has +increased or decreased. Since there are many different investments inside the mutual fund, the risk is reduced +significantly, compared to direct ownership of stocks. Even so, mutual funds historically perform well and can earn more +than 10% annually. +The investments that make up a mutual fund are structured and maintained to match stated investment objectives, +which are specified in its prospectus. A prospectus is a pamphlet or brochure that provides information about the +mutual fund. Before buying shares of a mutual fund, consult its prospectus, consider its goals and strategies to see if +they match your goals and values and also research any associated fees. +VIDEO +Mutual Funds (https://openstax.org/r/investing_basics_mutual_funds) +620 +6 • Money Management +Access for free at openstax.org + +EXAMPLE 6.68 +Investing in a Mutual Fund +Kaitlyn has analyzed her $12,862.50 quarterly budget using the 50-30-20 budget philosophy, and sees she should be +saving or paying down debt with $2,572.50 per quarter. She decides to invest $1,300 quarterly a mutual fund that reports +an average return of 11.62% over the 18-year life of the mutual fund. Assuming that this interest rate continues, and is +compounded quarterly, how much will her mutual fund account be worth after 5 years? +Solution +Kaitlyn’s plan is an ordinary annuity, and so the future value of her account can be found using the formula +, with a payment of $1,300, a rate of 0.1162, number of compounding periods 4, after 5 +years. Substituting these values into the formula and calculating, we find +Kaitlyn’s mutual fund will be worth $34,595.88 after 5 years. +YOUR TURN 6.68 +1. Aidan decides to invest $3,200 annually in a mutual fund. He expects the fund to have a 10.8% interest rate +compounded annually. How much will Aidan’s mutual fund account have after 15 years? +EXAMPLE 6.69 +Investing in a Mutual Fund to Reach a Goal +Kaitlyn wants to retire with $1,500,000 in her mutual fund account. She will invest for 35 years. The mutual fund reports +an average return of 11.62% over the 18-year-long life of the mutual fund. Assuming that this interest rate continues, +and is compounded quarterly, how much will she need to pay annually into her mutual fund to reach her goal? +Solution +Kaitlyn’s plan is an ordinary annuity, and so the payment to reach her goal can be found using the formula +, with a +, or goal, of $1,500,000, a rate of 0.1162, for 35 years. Substituting these values into the +formula and calculating, we find +Kaitlyn needs to invest $6,689.49 per year (or $557.46 per month) into the mutual fund to reach $1,500,000 in 35 years. +YOUR TURN 6.69 +1. How much does Aidan need to invest annually in his mutual fund to reach a goal of $1,000,000 in 40 years. He +expects the fund to have a 10.8% interest rate compounded annually. +6.7 • Investments +621 + +Return on Investment +As in Methods of Savings, the formula for return on investment is +. As indicated before, this formula does +not take into account how long the investment took to reach its current value. It depends only on the initial value, +, and +the value at the end of the investment, +. +EXAMPLE 6.70 +Return on Investment for a Bond +Recall Example 6.62, in which Muriel purchased a $3,000 bond with a maturity of 4 years at a fixed coupon rate of 5.5% +paid annually. What was Muriel’s return on investment? +Solution +Each year, Muriel received $165. She received this money four times, so earned a total of $660. This represents +– +, +or just the earnings. Using that we find that the ROI is +, or 22%. +YOUR TURN 6.70 +1. Maureen invests $5,000 in a bond with a maturity date in 5 years at a fixed coupon rate of 4.75%. What is +Maureen’s return on investment? +As mentioned, the ROI does not address the length of time of the investment. A good way to do that is to equate the ROI +to an account bearing interest that is compounded annually. +The annual return is the average annual rate, or the annual percentage yield (APY) that would result in the same amount +were the interest paid once a year. +FORMULA +The formula for annual return is +, where += the number of years, += new value, and += starting principal. +We apply this to the previous example. +EXAMPLE 6.71 +Annual Return on Investment for a Bond +Recall Example 6.70, in which Muriel purchased a $3,000 bond with a maturity of 4 years at a fixed coupon rate of 5.5% +paid annually. What was Muriel’s annual return on investment? Interpret this as compound interest. +Solution +Muriel earned a total of $660. This represents +– +, or just the earnings. The starting principal was $3,000. The value +at the end of 4 years was $3,000 + $660 = $3,660. The time of the investment was 4 years. Using that we find that the +annual return is +, or 5.10%. +The 5.5% bond earned the equivalent of 5.10% compounded annually. +YOUR TURN 6.71 +1. Maureen invests $5,000 in a bond with a maturity date in 5 years at a fixed coupon rate of 4.75%. What is +Maureen’s annual return on investment? +In Example 6.71 and Your Turn, the annual return was lower than the interest rate of the investment. This is because the +622 +6 • Money Management +Access for free at openstax.org + +interest from a bond is simple interest, but annual yield equates to compounded annually. +EXAMPLE 6.72 +Return on Investment for Stock in Company ABC +Haniah buys stock in the ABC company, investing a total of $13,000. After 20 years, the stock is worth $132,293.49, +including reinvestment of dividends. +1. +What is Haniah’s return on investment? +2. +What is Haniah’s annual return? +Solution +1. +To calculate Hanniah’s return on investment, substitute $13,000 for +and $132,293.49 for +in the formula +and calculate. Doing so we find Haniah’s return on investment to be +, or 917.64% +2. +To calculate Haniah’s annual return, substitute $13,000 for +and $132,293.49 for +in the formula +and calculate. Doing so we find her annual return to be +, or 12.3% +YOUR TURN 6.72 +Rixie deposits $23,000 in the stock of DEF company. After 12 years, her stock is worth $108,501.30. +1. What is Rixie’s return on investment? +2. What is Rixie’s annual return? +You should see that the annual return is equal to the annual compounded interest that was assumed for the stocks. +Compute Payment to Reach a Financial Goal +As in Methods of Savings, determining the payment necessary to reach a financial goal uses the payment formula for an +ordinary annuity, +. If dealing with mutual funds or stocks, an assumed annual interest rate, +compounded, will be used. This value is often determined through research and informed speculation. +EXAMPLE 6.73 +Richard is saving for new siding for his home. He and his partner believe they will need $37,500 in 10 years to pay for the +siding. How much should they invest yearly in a mutual fund they believe will have an annual interest rate of 12%, +compounded annually, in order to reach their goal? +Solution +The necessary annual payment is found using the function +with += 37,500, += 0.12, and += 1. +Substituting and calculating, we find the annual payment should be +6.7 • Investments +623 + +YOUR TURN 6.73 +1. Pete and Erin want to save for their child’s college. They think they will need $90,000 in 14 years. How much +should they invest annually in a mutual fund they believe will yield 9.5% compounded annually? +Retirement Savings Plans +We close this section by investigating the three main forms of retirement savings accounts: traditional individual +retirement accounts (IRAs), Roth IRAs, and 401(k) accounts. Each has distinct characteristics that are suited to different +investors’ needs. +Individual Retirement Accounts +A traditional IRA lets you contribute up to an amount set by the government, which may change from year to year. For +example, the maximum contribution for 2022 is $6,000; $7,000 over age 50. Anyone is eligible to contribute to a +traditional IRA, regardless of your income level. Your money grows tax-deferred, but withdrawals after age 59½ are taxed +at current rates. Traditional IRAs also allow you to use the contribution itself as a deduction on a current year tax return. +Roth IRAs allow contributions at the same levels as traditional IRAs, with a maximum $6,000 for 2022; $7,000 over age +50. However, to be eligible to make contributions, your earned income must be below a certain level. A Roth IRA allows +after-tax contributions. In other words, the contribution itself is not tax-deductible, as it is with the traditional IRA. +However, your money grows tax-free. If you make no withdrawals until you are age 59½, there are no penalties. IRAs pay +a modest interest rate. +In either case, IRA deposits have to be from earned income, which in effect means if your earned income is over $6,000 +($7,000) then you can deposit the maximum. +EXAMPLE 6.74 +Comparing Roth IRAs to Traditional IRAs +Which type of IRA, Roth or traditional, has an income limit for its use? +Solution +Roth IRAs require income to be below a certain limit. +YOUR TURN 6.74 +1. Which type of IRA, Roth or traditional, allow deposits before tax, but have earnings that are taxed after the age of +59 ½? +WHO KNEW? +In 2022, the maximum that can be added to a Roth IRA was $6,000 for those under 50 years of age. For those over 50 +years of age, the maximum that can be added to a Roth IRA is $7,000. However, to qualify for a Roth IRA in fall of +2022, a single person’s modified adjusted gross income (MAGI) must be below $129,000. Then, if a single person’s +income is between $129,000 and $144,000, the maximum contribution is reduced from the limit for incomes below +$129,000. For a married couples filing a joint tax return those values are $204,000 to $214,000. +401(k) Accounts +Your employer may offer a retirement account to you. These are often in the form of a 401(k) account. There are +traditional and Roth 401(k) accounts, which differ in how they are taxed, much as with other IRAs. In the traditional +401(k) plans, the money is deposited before tax is assessed, which means you do not pay taxes on this money. However, +that means when money is withdrawn, it is taxed. These accounts are similar to mutual funds, in that the money is +invested in a wide range of assets, spreading the risk. +One of the perks some employers offer is to match some amount of your contributions to the 401(k) plan. For instance, +624 +6 • Money Management +Access for free at openstax.org + +they may match your deposits up to 5% of your income. This is an instant 100% return on the money that was matched. +VIDEO +401(k) Accounts (https://openstax.org/r/401(k)s) +EXAMPLE 6.75 +Matching 401(k) Deposit +Alice signs up for her employer-based 401(k). The employer matches any 401(k) contribution up to 6% of the employee +salary. Alice’s annual salary is $51,600. +1. +What is the most money that Alice can deposit that will be fully matched by the company? +2. +How much total will be deposited into Alice’s account if she deposits the full 6%? +3. +How much return does Alice earn if she deposits exactly 6% in her 401(k)? +Solution +1. +The employer will match up to 6% of any employee’s salary. 6% of Alice’s salary is +. So Alice +can deposit up to $3,096 and receive that amount in matching funds in her account. +2. +Alice’s contribution plus the company’s contribution is +, which is the total that is deposited +into Alice’s account. +3. +She earns a 100% return on the day she deposits her $3,096. +YOUR TURN 6.75 +Jameis signs up for his employer-based 401(k). The employer matches any 401(k) contribution up to 7.5% of the +employee salary. Jameis’ annual salary is $72,800. +1. What is the most money that Jamie can deposit that will be fully matched by the company? +2. How much total will be deposited into Jameis’ account if he deposits the full 7.5%? +3. How much return does Jameis earn if he deposits exactly 7.5% in her 401(k)? +401(k) plans with matching funds provide great value, as their rates of return are high compared to savings accounts, +and are less risky that stocks since such funds invest across many investment vehicles. The next example demonstrates +the power of constant deposits into a 401(k) plan that has some employer match. +EXAMPLE 6.76 +Constant Deposits into a 401(k) Plan +DeJean begins depositing $300 per month from his paycheck each month in his employer-based 401(k) account. The +employer matches this deposit as it falls below their matching threshold. DeJean expects the return to average 10% per +year, compounded annually. +1. +How much will DeJean’s account be worth if he keeps making those payments for 30 years? +2. +What will his account be worth without the matching funds? +Solution +1. +This is a form of an ordinary annuity, so the formula +will be used. The company +matches DeJean’s full deposit, so each month $600 will be deposited. He is assuming the money will compound +annually, so the amount deposited each year is needed as the value of pmt. For the year, he will deposit +. The rate is 0.1, the number of compounding periods is 1, and the number of years is 30. +Substituting and calculating, the value of DeJean’s account after 30 years will be +6.7 • Investments +625 + +2. +This is a form of an ordinary annuity, so the formula +will be used but the deposit is now +only $300 per month without the matching funds. For the year, he will deposit +. The rate is 0.1, +the number of compounding periods is 1, and the number of years is 30. Substituting and calculating, the value of +DeJean’s account after 30 years will be +YOUR TURN 6.76 +Crystal begins depositing $450 per month from her paycheck each month in her employer-based 401(k) account. +The employer matches $350 of each deposit. Crystal expects the return to average 9% per year, compounded +annually. +1. How much will Crystal’s account be worth if she keeps making those payments for 25 years? +2. What will her account be worth without the matching funds? +Check Your Understanding +39. Which investment has the highest risk? +40. What is different about who can use a Roth IRA versus a traditional IRA?? +41. How do mutual funds reduce risk? +42. How much in earnings does a 10-year bond with issue price $5,000 that pays 4% interest annually is generated? +43. If an individual makes $115,000 annually, is the person eligible for a Roth IRA? +44. An employer 401(k) matches up to 4% of income by employees. Merisol’s annual salary is $87,500. If Merisol wants +to deposit $4,400 annually in the 401(k), how much will be matched? +45. A person owns 300 shares of stock. It pays $.38 per share this quarter. How much does the share owner earn from +the stock that quarter? +46. David purchased stocks for $3,500. The stocks paid dividends by reinvesting them in the stock. After 3 years, he +sold the socks for $4,650. What was David’s annual return on that investment? +47. How much must be deposited annually in a mutual fund that is expected to bear 12.5% interest compounded +annually if the account is to be worth $750,000 after 25 years? +SECTION 6.7 EXERCISES +1. What is the maturity date for a bond? +2. What is the issue price for a bond? +3. Stock investments increase in value in what two ways? +626 +6 • Money Management +Access for free at openstax.org + +4. Which is the least risky of stocks, bonds, mutual funds, CDs? +5. Which type of individual retirement account allows pre-tax deposits? +6. What are the limits on contributions to individual retirement accounts in 2022? +7. Of bonds, stocks, mutual funds, CDs, and money market accounts, which do not allow for withdrawal until a +certain time period has passed? +8. Which type of IRA allows the account to grow tax free, provided no withdrawals are made until after the age of +? +9. Which of bonds, mutual funds, and CDs are professionally managed? +10. Why do mutual funds and IRAs have relatively low risk? +For the bonds with the given properties, find a. the amount paid each year and b. the total amount earned with the +bond. +11. Issue price of $10,000 pays 3.5% annually, matures in 5 years. +12. Issue price of $3,400 pays 2.75% annually, matures in 10 years. +13. Issue price of $1,000 pays 2.8% annually, matures in 5 years. +14. Issue price of $5,000 pays 3.75% annually, matures in 15 years. +In the following exercises, find: a. the return on investment and b. the annual return for the bond described. Round to +two decimal places. +15. Issue price of $10,000 pays 3.5% annually, matures in 5 years. +16. Issue price of $3,400 pays 2.75% annually, matures in 10 years. +17. Issue price of $1,000 pays 2.8% annually, matures in 5 years. +18. Issue price of $5,000 pays 3.75% annually, matures in 15 years. +19. 40 shares of stock are owned. The dividend per share is $0.38 for a quarter. How much was earned in dividends on +this stock this quarter? +20. 150 shares of stock are owned. The dividend per share is $0.78 for a quarter. How much was earned in dividends +on this stock this quarter? +21. 100 shares of stock are owned. The dividend per share is $0.18 for a quarter. How much was earned in dividends +on this stock this quarter? +22. 250 shares of stock are owned. The dividend per share is $0.41 for a quarter. How much was earned in dividends +on this stock this quarter? +For the following exercises, 70 shares of stock were purchased for $31.50 per share 5 years ago. Over the 5 years, the +total of all dividends earned from this stock was $6.34 per share. The stock is sold for $34.83. +23. How much was earned with dividends and share price increase combined? +24. What was the return on investment for these stocks? +25. What was the annual return for these stocks? +For the following exercises, 10 shares of stock were purchased for $18.91 per share 3 years ago. Over the 3 years, the +total of all dividends earned from this stock was $3.18 per share. The stock is sold for $22.01. +26. How much was earned with dividends and share price increase combined? +27. What was the return on investment for these stocks? +28. What was the annual return for these stocks? +For the following exercises, use the given stock table to answer the following: +a. +What was the 52-week low? +b. +What was the dividend? +c. +What is its year-to-date performance? +d. +What is its yield? +6.7 • Investments +627 + +29. +Stock table (data source: marketwatch.com) +30. +Stock table (data source: marketwatch.com) +In the following exercises, find the future value of the mutual fund or IRA with the given annual deposit, the duration of +the investment, and the assumed annual, compounded, percentage rate. +31. IRA, annual deposit = $6,000, 25 years, assumed percentage rate of 11.3% +32. IRA, annual deposit = $4,800, 15 years, assumed percentage rate of 9.7% +628 +6 • Money Management +Access for free at openstax.org + +33. Mutual fund, annual deposit = $7,500, 35 years, assumed percentage rate of 10% +34. Mutual find, annual deposit = $12,000, 40 years, assumed percentage rate of 9% +In the following exercises, find the annual deposit necessary, into an IRA or mutual fund, to reach the stated financial +goal, given the assumed annual, compounded, interest rate, and the duration of the deposits. Convert that annual +deposit to a monthly amount. +35. Mutual fund, goal = $1,000,000, 25 years, assumed percentage rate of 11.3% +36. Mutual fund, goal = $100,000, 15 years, assumed percentage rate of 9.7% +37. IRA, goal = $750,000, 35 years, assumed percentage rate of 10% +38. IRA, goal = $1,750,000, 40 years, assumed percentage rate of 9% +39. Francis’s employer matches 401(k) contributions up to 7% of annual salary. She makes $98,500. What is the +maximum amount that the company will match for Francis? +40. Miles’ employer matches 401(k) contributions up to 4% of annual salary. He makes $38,500. What is the maximum +amount that the company will match for Miles? +41. Ila’s employer matches 401(k) contributions, up to 4% of salary. She makes $49,000. She wants to deposit $3,000 of +her own earnings per year ($250 per month). How much, including her employer’s matching funds, will be +deposited into Ila’s account each year? +42. Georgia works at a company that matches 401(k) contributions up to 5.5% of salary. Georgia wants to deposit +$6,000 of her own earnings per year ($500 per month) in her 401(k). If she makes $65,000 annually, how much, +including employer contributions, will be deposited in Georgia’s account annually? +In the following exercises, Cheryl’s company offers a 401(k) account to all employees. The company will match +employee contributions up to 6% of the employee’s salary. She earns $81,000 per year. Cheryl decides to deposit, or +contribute, $10,000 annually in the 401(k). She expects a return of 9.5% per year. +43. How much will the company match? +44. What is her total contribution per year? +45. If she deposits into the account for 25 years, how much will her 401(k) be worth? +In the following exercises, Lavanya’s company offers a 401(k) account to all employees. Her annual salary is $58,000. +The company will match employee contributions up to 7% of the employee’s salary. Lavanya decides to deposit $3,200 +annually in the 401(k). She expects a return of 8.5% per year. +46. How much will the company match? +47. What is her total contribution per year? +48. If she deposits into the account for 20 years, how much will her 401(k) be worth? +In the following exercises, Ruslana wants to use her 401(k) to save $1,400,000 when she retires in 36 years. She +assumes the plan will yield 11% compounded annually. Her company will match contributions up to 5% of annual +salary. Her salary is $48,000. +49. How much total will need to be added to her account each year to reach her goal? +50. What is 5% of Ruslana’s salary? +51. The company will match up to 5% of Ruslana’s salary, which means half the payment (up to 5% of salary) will be +contributed by the company. How much is half the necessary payment? +52. Does the answer to Exercise 51 exceed the result from Exercise 50? +53. If the answer to Exercise 52 is no, then the company contributes half the deposit in the 401(k). How much will +the company contribute annually to the 401(k) if this is the case? +54. If the answer to Exercise 52 is yes, then the company contributes only the 5% match of Ruslana's salary. In this +case, how much does the company contribute? +55. How much will Ruslana need to contribute per year, after the employer contribution? +56. Divide the answer to Exercise 55 by 12 to find the monthly contribution Ruslana will make. +In the following exercises, Remy wants to use his 401(k) to save $1,750,000 when he retires in 30 years. He assumes the +plan will yield 10.4% compounded annually. His company will match contributions up to 6% of annual salary. His salary +is $78,000. +57. How much total will need to be added to his account each year to reach his goal? +58. What is 6% of Remy’s salary? +59. The company will match up to 6% of Remy’s salary, which means half the payment (up to 6% of salary) will be +contributed by the company. How much is half the necessary payment? +60. Does the answer to Exercise 59 exceed the result from Exercise 58? +61. If the answer to Exercise 60 is no, then the company contributes half the deposit in the 401(k). How much will +6.7 • Investments +629 + +the company contribute annually to the 401(k) if this is the case? +62. If the answer to Exercise 60 is yes, then the company contributes only the 6% of Remy’s salary. In this case, how +much does the company contribute? +63. How much will Remy need to contribute per year, after the employer contribution? +64. Divide the answer to Exercise 63 by 12 to find the monthly contribution Remy will make. +6.8 The Basics of Loans +Figure 6.17 Loans are contracts that allow people to buy now but require them to pay more. (credit: "Closing" by Tim +Pierce/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Describe various reasons for loans. +2. +Describe the terminology associated with loans. +3. +Understand how credit scoring works. +4. +Calculate the payment necessary to pay off a loan. +5. +Read an amortization table. +6. +Determine the cost to finance for a loan. +New car envy is real. Some people look at a new car and feel that they too should have a new car. The search begins. +They find the model they want, in the color they want, with the features they want, and then they look at the price. That’s +often the point where the new car fever breaks and the reality of borrowing money to purchase the car enters the +picture. This borrowing takes the form of a loan. +In this section, we look at the basics of loans, including terminology, credit scores, payments, and the cost of borrowing +money. +Reasons for Loans +Even if you want a new car because you need one, or if you need a new computer since your current one no longer runs +as fast or smoothly as you would like, or you need a new chimney because the one on your house is crumbling, it’s likely +you do not have that cost in cash. Those are very large purchases. How do you buy that if you don ’t have the cash? +You borrow the money. +And for helping you with your purchase, the company or bank charges you interest. +Loans are taken out to pay for goods or services when a person does not have the cash to pay for the goods or services. +We are most familiar with loans for the big purchases in our lives, such as cars, homes, and a college education. Loans +are also taken out to pay for repairs, smaller purchases, and home goods like furniture and computers. +Loans can come from a bank, or from the company selling the goods or providing the service. The borrower agrees to +630 +6 • Money Management +Access for free at openstax.org + +pay back more than the amount borrowed. So there is a cost to borrowing that should be considered when deciding on a +purchase bought with credit or a borrowed money. +Even using a credit card is a form of a loan. +Essentially, a loan can be obtained for just about any purchase, large or small, that has a cost beyond a person’s cash on +hand. +The Terminology of Loans +There are many words and acronyms that get used in relation to loans. A few are below. +APR is the annual percentage rate. It is the annual interest paid on the money that was borrowed. The principal is the +total amount of the loan, or that has been financed. A fixed interest rate loan has an interest rate that does not change +during the life of the loan. A variable interest rate loan has an interest rate that may change during the life of the loan. +The term of the loan is how long the borrower has to pay the loan back. An installment loan is a loan with a fixed +period, and the borrower pays a fixed amount per period until the loan is paid off. The periods are almost uniformly +monthly. Loan amortization is the process used to calculate how much of each payment will be applied to principal and +how much is applied to interest. Revolving credit, also known as open-end credit, is how most credit cards work but is +also a kind of loan account. (We will learn about credit cards in Credit Cards) You can use up to some specified value, +called the limit, any way you want, and as long as you pay the issuer of the credit according to their terms, you can keep +borrowing from this account. +These and other terminologies can be researched further at Forbes (https://openstax.org/r/terminologies). +WHO KNEW? +Credit Scores +Not everyone pays the same interest for the same loan. One person might get an APR of 2.9% while another pays +6.9%. These rates are based on your credit score. +Data about you and your credit is collected by three credit bureaus—Experian, Equifax, and TransUnion. They +calculate your score using one of two main models: FICO and Vantage Score. The score they develop is based on the +following categories: +• +Payment History: Making your payments on time and not missing payments is by far the most important factor. +All three credit types—revolving, installment, and open—contribute to this factor. +• +Credit Utilization or Amount Owed: How much do you owe on your credit card accounts? This category is +concerned with the ratio of how much you owe on revolving credit accounts relative to your available credit, also +known as your credit utilization ratio. +This is the only category that depends solely on your revolving credit accounts. +• +Length of Credit History: This is the average age of your credit history, including the age of the oldest and +newest accounts. All three types of credit accounts play a role in this category. +• +Credit Mix: This number represents the different types of credit accounts you have, such as credit cards, car +loan, mortgages, and whether you are successful managing both revolving and installment accounts. +• +New Credit: Have you recently opened a new account or applied for new credit? Lenders want to know how +much new credit you are taking on. So, if you are planning to buy a car and make another large purchase with a +credit card, you may want to space these purchases out. +If you have done well in these categories, your credit score will be high, and you will qualify for lower interest rates +because you are not perceived as being a risky investment. However, if you do poorly in these categories, your score +will be low and you will pay higher interest rates since you present a greater risk. +Check out this nerdwallet article on credit scores (https://openstax.org/r/how_to_improve) to learn more! +VIDEO +Credit Scores Explained (https://openstax.org/r/credit_scores_explained) +6.8 • The Basics of Loans +631 + +WHO KNEW? +Where Do Interest Rates Come From? +There are many factors that impact your interest rate beyond your credit score. Banks have the authority to set their +own rates, so competition between banks impacts interest rates. Bank A doesn’t want to charge interest rates that are +too high, since borrowers will find banks with better rates. Banks also don’t want to charge too little interest. +The too little interest is more involved than the too high. The bank needs to make a profit on its loans. Deposits at the +bank are used by the bank to generate loans. The bank has to pay those depositors interest. The bank must charge +more for loans they give than they pay to people with deposits in the bank. +Banks also borrow money from each other. These loans have an interest rate, and once more, the bank making a loan +must make a profit. More directly, they must charge more for loans they give than they pay for loans they take. In the +United States, banks may also borrow from the Federal Reserve, which also charges an interest rate, which is also +called the discount rate. +This is where a bank’s prime rate comes from. A bank’s prime rate is the interest rate it will give to its very best +customers, which means most customers will pay more than the bank’s prime rate. To confuse the issue, there is also +the Wall Street Journal’s prime rate. It is the average of the prime rates charged by individual banks. The Wall Street +Journal surveys several banks to generate this value. +Banks also increase the interest rate charged to customers based on both the credit risk presented by the customer, +and the risk associate with what the loan will be used for. +Calculating Loan Payments +Loan payments are made up of two components. One component is the interest that accrued during the payment +period. The other component is part of the principal. This should remind you of partial payments from Simple Interest. +Over the course of the loan, the amount of principal remaining to be paid decreases. The interest you pay in a month is +based on the remaining principal, just as in the partial payments of Simple Interest. +FORMULA +The amount of interest, +, to be paid for one period of a loan with remaining principal +is +, where +is the +interest rate in decimal form and +is he number of payments in a year (most often += 12). Since the interest is for the +one period, the time is 1 and does not impact the calculation. Note, interest paid to lenders is always rounded up to +the next penny. +EXAMPLE 6.77 +Interest for a Monthly Payment of a Loan +Find the interest to be paid for the period on loans with the following remaining principal and given annual interest rate. +Each period is a month. +1. +Remaining principal is $13,450, interest rate is 6.75% +2. +Remaining principal is $8,460, interest rate is 5.99% +Solution +1. +Substituting $13,450 for the remaining principal +, 0.0675 for , and += 12 since the period is a month into the +formula, we find that the interest to be paid this period is +. +2. +Substituting $8,460 for the remaining principal +, 0.0599 for , and += 12 since the period is a month into the +formula, we find that the interest to be paid this period is +. +632 +6 • Money Management +Access for free at openstax.org + +YOUR TURN 6.77 +Find the interest to be paid for the period on loans with the following remaining principal and given annual interest +rate. Each period is a month. +1. Remaining principal is $56,945, interest rate is 7.5% +2. Remaining principal is $25,850, interest rate is 2.9% +The payment of the loan has to be such that the principal of the loan is paid off with the last payment. In any period, the +amount of interest is defined by the formula above, but changes from period to period since the principal is decreasing +with each payment. The trick is knowing how much principal should be paid each payment so that the loan is paid off at +the stated time. Fortunately, that is found using the following formula. +FORMULA +The payment, +, per period to pay down a loan with beginning principal +is +, where +is the annual interest rate in decimal form, +is the number of years of the loan, and +is the number of payments per +year (typically, loans are paid monthly making += 12). +Note, payment to lenders is always rounded up to the next penny. +Often, the formula takes the form +, where +is the interest rate per period (annual rate +divided by the number of periods per year), and +is the total number of payments to be made. +EXAMPLE 6.78 +Calculating the Payment for a Loan +In the following, calculate the payment necessary to pay off the loan with the given details. The payments are monthly. +1. +A car loan taken out for $28,500 at an annual interest rate of 3.99% for 5 years. +2. +A home loan taken out for $136,700 and an annual interest rate of 5.75% for 15 years. +Solution +1. +The loan is for $28,500, which is the principal. The rate is 3.99%, so += 0.0399. The term of the loan is 5 years, so +=5. Monthly payments means += 12. Substituting these values for +, , +, and +into the formula +and calculating, we find the payment for the loan. +The monthly payment needed is $524.75. +2. +The loan is for $136,000, which is the principal +. The rate is 5.75% so += 0.0575. The term of the loan is 15 years, so += 15. Monthly payments mean +=12. Substituting these values for +, , +, and +into the formula +and calculating, we find the payment for the loan. +6.8 • The Basics of Loans +633 + +The monthly payment needed is $1,135.18. +YOUR TURN 6.78 +In the following, calculate the payment necessary to pay off the loan with the given details. The payments are +monthly. +1. A home improvement loan taken out for $17,950 at an annual interest rate of 7.5% for 10 years. +2. A solar panel installation loan taken out for $33,760 and an annual interest rate of 4.3% for 20 years. +TECH CHECK +Using Google Sheets to Calculate Loan Payments +Google Sheets has a formula to calculate the monthly payment necessary to pay off a loan with a specified interest +rate and term. The formula is the PMT formula. It uses the principal +, interest rate , and +years. Follow these steps +Open a Google worksheet and click on any cell. +Type =-PMT(r/12,12*t,P). +Hit the enter key. +The cell displays the payment. +Please note the negative sign. Since Google Sheets is a spreadsheet program, is sees the payment as funds leaving +the account, and so they are, by default, negative. The negative sign in front of the formula makes the result positive. +For example, for a $50,000 loan at 10.9% interest for 7 years, you would type =-PMT(0.109/12,12*7,50000). The +formula and the result are shown in Figure 6.18. +Figure 6.18 Google Sheets formula +Alternatively, you can also use an online calculator to find monthly payments for a loan, such as the one at +Caluculator.net (https://openstax.org/r/payment_calculator) +634 +6 • Money Management +Access for free at openstax.org + +Reading Amortization Tables +An amortization table or amortization schedule is a table that provides the details of the periodic payments for a loan +where the payments are applied to both the principal and the interest. The principal of the loan is paid down over the life +of the loan. Typically, the payments each period are equal. Importantly, one of the columns will show how much of each +payment is used for interest, another column shows how much is applied to the outstanding principal, and another +column shows the remaining principal or balance Figure 6.19. +Figure 6.19 Amortization table +EXAMPLE 6.79 +Reading from an Amortization Table +Using the partial amortization table (Figure 6.23), answer the following questions. +6.8 • The Basics of Loans +635 + +Figure 6.20 Amortization table +1. +What is the loan amount (principal), the interest rate, and the term of the loan? +2. +How much is the monthly payment? +3. +How much remaining balance is there after the payment in month 15? +4. +How much was the interest in payment 10? +5. +What is the total of the interest paid after payment 18? +6. +What happens to the amount paid in interest each month? +Solution +1. +Reading the values at the top of the table, we see the principal is $10,000, the interest rate is 4.75%, and has a term +of 20 years. +2. +The monthly payment is listed below the term of the loan, and is $64.62. +636 +6 • Money Management +Access for free at openstax.org + +3. +$9,613.83 +4. +$38.68 +5. +$697.01 +6. +The amount paid to interest decreases each month. +YOUR TURN 6.79 +Using the partial amortization table below, answer the following questions. +Amortization table +1. What is the loan amount (principal), the interest rate, and the term of the loan? +2. How much is the monthly payment? +6.8 • The Basics of Loans +637 + +3. How much remaining balance is there after the payment in month 20? +4. How much was the interest in payment 5? +5. What is the total of the interest paid after payment 24? +6. What happens to the amount paid to the principal each month? +VIDEO +Reading an Amortization Table (https://openstax.org/r/Amortization_Table) +Cost of Finance +There are often costs associated with a loan beyond the interest being paid. The cost of finance of a loan is the sum of +all costs, fees, interest, and other charges paid over the life of the loan. +EXAMPLE 6.80 +Cost of Financing a Personal Loan +Irena signed for a loan of $15,000 at 6.33% for 5 years. When she took out the loan, Irena paid a $750 origination fee. +Over the course of the loan, she pays $2,537.96 in interest. What was her cost to finance the loan? +Solution +The cost of finance is the sum is all interest and any fees paid for the loan. The fees paid were $750.00 and the interest +was $2,537.96. Her cost of finance for this loan was $3,287.96. +YOUR TURN 6.80 +1. Samantha takes out a $28,000 loan for 6 years at 5.5% interest. When she took out the loan, she paid $4,937.18 +in interest, an origination fee of $1,400 and an application fee of $100. What was her cost of finance for this loan? +Check Your Understanding +48. Give three reasons for taking out a loan. +49. What is an installment loan? +50. In an installment loan what happens to the amount of the payment that is applied to interest in later payments +compared to earlier payments? +51. An installment loan is paid monthly. The remaining principal is $5,498. If the interest rate is 12.9%, how much +interest is paid that month? +52. A business loan is taken out for $250,000 for 15 years at 6.9% interest. What are the payments for the loan? +53. Consider the amortization table below. What is the balance after the eleventh payment? +54. The total interest paid on a loan is $38,519.50. The origination fee was $300 while the processing fee was $250. An +additional fee was charged to the borrowers, amounting to another $475. How much was the cost of financing for +this loan? +638 +6 • Money Management +Access for free at openstax.org + +SECTION 6.8 EXERCISES +1. How do fixed interest rates and variable interest rates differ? +2. What is revolving credit? +3. What is loan amortization? +4. What are the two components of a loan payment? +5. Name three details that are presented on an amortization schedule. +For the following exercises, calculate the interest due for the monthly installment payment given the remaining +principal and interest rate. +6. Remaining principal = $21,872.99, interest rate = 13.9% +6.8 • The Basics of Loans +639 + +7. Remaining principal = $2,845.43, interest rate = 4.99% +8. Remaining principal = $78,913.76, interest rate = 2.9% +9. Remaining principal = $6,445.22, interest rate = 5.65% +In the following exercises, calculate the monthly payment for the loan. +10. Principal = $8,600, annual interest rate = 6.75%, term is 5 years +11. Principal = $19,400, annual interest rate = 2.25%, term is 6 years +12. Principal = $11,870, annual interest rate = 3.59%, term is 3 years +13. Principal = $41,900, annual interest rate = 8.99%, term is 15 years +14. Principal = $26,150, annual interest rate = 11.1%, term is 7 years +15. Principal = $46,350, annual interest rate = 2.9%, term is 6 years +16. Principal = $175,800, annual interest rate = 4.73%, term is 25 years +17. Principal = $225,000, annual interest rate = 5.06%, term is 30 years +In the following exercises, use the amortization table below to answer the questions. +640 +6 • Money Management +Access for free at openstax.org + +18. What was the loan amount, or starting principal? +19. What was the interest rate? +20. What is the term of the loan? +21. What are the monthly payments? +22. What is the remaining principal after payment 17? +23. How much of payment 27 was for interest? +24. How much of payment 19 was for principal? +25. How much total interest was paid after payment 22? +In the following exercises, use the amortization table for payments 131–152 of a mortgage below to answer the +questions. +6.8 • The Basics of Loans +641 + +26. What was the loan amount, or starting principal? +27. What was the interest rate? +28. What is the term of the loan? +29. What are the monthly payments? +30. What is the remaining principal after payment 140? +31. How much of the principal has been paid off after payment 151? +32. How much of payment 138 was for principal? +33. How much total interest was paid after payment 135? +In the following exercises find the cost to finance for the loan. +34. Total interest paid is $94,598.36, origination fee was $450, processing fee was $300, commission fee was +$1,457.50. +35. Total interest paid was $3,209.34, origination fee was $100, processing fee was $200. +36. Total interest paid was $8,295.50, fees were $875. +37. Total interest paid was $56,114.90, origination fee was $1,230, and filing fee was $250. +38. Kylie takes out a $16,780 loan from her credit union for a new car. The loan’s term is 4 years with an interest rate of +6.77%. What are Kylie’s monthly payments? +39. Crissy and Jonesy take out a $13,200 loan for repairs to their roof. The loan is for 10 years at 7.15% interest. What +are their payments for this loan? +642 +6 • Money Management +Access for free at openstax.org + +Below are two amortization tables. They are for loans with the same principal and interest rate, but different terms, the +first a 20-year term and the second a 30-year term. The tables show the last payments of each loan. Use those two +tables for the following exercises. +6.8 • The Basics of Loans +643 + +40. How much total interest was paid for the 20-year loan? The 30-year loan? Calculate the difference. +41. What was the payment for the 20-year loan? The 30-year loan? Calculate the difference. +42. Compare the two loans. Why would the 30-year be preferable over the 20-year loan? Why would the 20-year loan +be preferable to the 30-year loan? +In the following exercises, compare payments per month and interest rates. +43. A loan for $20,000 is taken out for 5 years. Find the payment is the interest rate is: +a. +2.9% +b. +3.9% +c. +4.9% +d. +5.9% +e. +Did the payments increase by the same amount for each 1% jump in interest rate? Describe the pattern. +44. A loan for $50,000 is taken out for 7 years. Find the payment is the interest rate is: +a. +3.5% +b. +4.5% +c. +5.5% +d. +6.5% +e. +Did the payments increase by the same amount for each 1% jump in interest rate? +644 +6 • Money Management +Access for free at openstax.org + +6.9 Understanding Student Loans +Figure 6.21 College is often paid for through loans. (credit: "Sidewalk Scene in front of Science Building" by IMCBerea +College Wikimedia Commons, CC- BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Describe how to obtain a student loan. +2. +Distinguish between federal and private student loans and state distinctions. +3. +Understand the limits on student loans. +4. +Summarize the standard prepayment plan. +5. +Understand student loan consolidation. +6. +Summarize and describe benefits or drawbacks of other repayment plans. +7. +Summarize possible courses of action if a student loan defaults. +Obtaining a Student Loan +All college students are eligible to apply for a loan regardless of their financial situation or credit rating. Federal student +loans do not require a co-signer or a credit check. Most students do not have a credit history when they begin college, +and the federal government is aware of this. However, private loans will generally require a co-signer as well as a credit +check. The co-signer will assume responsibility for paying off the loan if the student cannot make the payments. +The first step in applying for student loans is to fill out the FAFSA (Free Application for Student Aid). FAFSA determines +financial need and what type of loan the student is qualified to obtain. For students who are still dependents on their +parent’s taxes, the parents also fill out the FAFSA, as their wealth and income impacts what the dependent student is +eligible for. Students who cannot demonstrate financial need will also be helped by applying with FAFSA, as it will help +guide them to the type of loan most appropriate. The FAFSA must be submitted each year. +The FAFSA deadline is the spring of the student’s next academic year. The deadline is often in March. Do not allow +this deadline to pass. +As soon as an offer letter from the college is received, the student should start the application process. The college will +determine the loan amount needed. Also, there are limits on the amount a student can borrow. There are both yearly +limits and aggregate limits. See the table later in this section that outlines the loan limits per school year and in the +aggregate. +If the student receives a direct subsidized loan, there is a limit on the eligibility period. The time limit on eligibility +depends on the college program into which the student enrolls. The school publishes how long a program is expected to +take. The eligibility period is 150% of that published time. For example, if a student is enrolled in a 4-year program, such +as a bachelor’s degree program, their eligibility period is 6 years, as 1.50(4) = 6. Therefore, the student may receive direct +6.9 • Understanding Student Loans +645 + +subsidized loans for a period of 6 years. +Types and Features of Student Loans +Once a tuition statement is received, and all the non-loan awards are analyzed that are applicable to the costs of college +(such as scholarships and grants), there still may be quite of bit of an expense to attend college. This difference between +what college will cost (including tuition, room and board, books, computers) and the non-loan awards received is the +college funding gap. +EXAMPLE 6.81 +College Funding Gap +Ishraq receives her award and tuition letter from the college she wants to attend. Her tuition, fees, books, and room and +board all come to $24,845 for the year. Her non-loan awards include an instant scholarship from the school for $7,500, a +scholarship she earned for enrolling in a STEM program for $3,750, and a $1,000 scholarship from her church. What is +Ishraq’s college funding gap? +Solution +Her awards total to $12,250. Her cost to attend is $24,845. Her college funding gap is then +. She will need to find $12,595 in funding. +YOUR TURN 6.81 +1. Yuan-Teng receives his award and tuition letter from the college he wants to attend. His tuition, fees, books, and +room and board all come to $34,845 for the year. His non-loan awards include an instant scholarship from the +school for $17,500, three scholarships he earned from sources he found online, which total $5,600, and a $2,000 +scholarship from his employer. What is Yuan-Teng’s college funding gap? +There are several loan types, which basically break down into four broad categories: subsidized loans, unsubsidized +loans, PLUS loans, and private loans. These loans are meant to fill the college funding gap. +Federal subsidized loans are backed by the U.S. Department of Education. These loans are intended for undergraduate +students who can demonstrate financial need. Subsidized federal loans, including Stafford loans, defer payments until +the student has graduated. During the deferment, the government pays the interest while the student is enrolled at +least half-time. These loans are generally made directly to students. However, there are restrictions on how the money +can be used. It can only be used for tuition, room and board, computers, books, fees, and college-related expenses. +Interest rates are not based on the financial markets but determined by Congress. Federal loans are backed by the +Department of Education. +Federal unsubsidized loans, including unsubsidized Stafford loans, are available for undergraduate and graduate +students who cannot demonstrate financial need. If the student meets the program requirements, they are +automatically approved. The student is not required to pay these loans during their time in college (enrolled at least half- +time). However, the interest rate is generally higher and there is no deferment period, as with subsidized loans. Interest +begins accruing as soon as the money is disbursed. +The immediate accrual of interest means the balance of the loan grows as the student attends school. A loan that +was for $10,000 can grow past $13,000 over five years of college. Some advisors tell students to pay the interest +portion of the loan while it is deferred to prevent this growth of debt. +Parent Loans for Undergraduate Students (PLUS) are federal loans made directly to parents. They are available even if +parents are not deemed financially needy. A credit check is performed and approval is not automatic. The limit to what +parents can borrow from a PLUS loan each year is still the college funding gap, but the aggregate of the PLUS loans does +not have a limit. This means the PLUS loan can cover whatever is left in the funding gap once all other aid and loans are +applied. Payments do not begin until the student is out of school, but interest begins to accrue the moment funds are +disbursed. Because the parents take out the loan, the parents are responsible for paying back the loan. +Private student loans are backed by a bank or credit institution and require a credit check, and interest rates are +variable. As private loans are not subsidized by the government, no one pays the interest but the borrower. The student +646 +6 • Money Management +Access for free at openstax.org + +does not have to start repaying the loan until after graduation, but interest starts to accrue immediately. This loan has +fewer repayment options, more fees and penalties, and the loan cannot be discharged through bankruptcy. Many +students need a co-signer to acquire a private loan. Like PLUS loans, private student loans can cover whatever is left in +the funding gap once all other aid and loans are applied +Student loans, in general, have a term of 10 years, that is, the loans are paid back over 10 years. This can vary, but 10 +years is the standard. +WHO KNEW? +School-Channel Loans and Direct-to-Consumer Loans +Private loans can fall into one of two categories: school-channel loans and direct to consumer loans. School-channel +loans are disbursed directly to the school. The school verifies the loan does not exceed the cost to attend school. +Direct–to-consumer loans do not have the verification process. Those proceeds are sent directly to the borrower. +They are processed more quickly, but often have higher interest rates. +VIDEO +Types of Student Loans (https://openstax.org/r/Student_Loans) +Limits on Student Loans +As mentioned earlier, there are limits to how much a student can borrow, per year and in total. The following table shows +a general breakdown of the amounts the federal government and private lenders will lend. Amounts are based on level +of need and whether the student is a dependent or an independent student. Independent students include those who +are at least 24 years old, married, a professional, a graduate student, a veteran, a member of the armed forces, an +emancipated minor, or an orphan. The amounts shown are as of this writing in 2022. +Year +Dependent Students +Maximum Amounts +Independent Students Maximum Amounts +First-Year +Undergraduate +$5,500 but no more than +$3,500 may be in +subsidized loans +$9,500 but no more than $3,500 may be in subsidized loans +Second-Year +Undergraduate +$6,500 but no more than +$4,500 may be +subsidized loans +$10,500 but no more than $4,500 in subsidized loans +Third Year and +Additional +Years +$7,500 but no more than +$5,500 may be in +subsidized loans +$12,500 but no more than $5,500 may be in subsidized loans +Graduate and +Professional +Not applicable +$20,500 in unsubsidized loans +Limits +$31,000 but no more +than $23,000 in +subsidized loans +$57,500 for undergraduates but no more than $23,000 in subsidized. +$138,500 for graduate or professional but no more than $65,500 may +be subsidized loans. +Check out this Edvisors page on the limits of student borrowing (https://openstax.org/r/loan_limits) to learn more! +6.9 • Understanding Student Loans +647 + +EXAMPLE 6.82 +Loan for Year 5 of College +Efraim is a dependent undergraduate student enrolled in a biology program. He’s about to attend for the fifth year. In +year 1 he took out $5,000 in federal subsidized and unsubsidized loans, in year 2 he took out $6,400 in federal subsidized +and unsubsidized loans, in years 3 and 4, he took out the maximum federal subsidized and unsubsidized loans amounts. +He needs federal subsidized and unsubsidized loans for his fifth year of school. How much can he obtain in federal +subsidized and unsubsidized student loans? +Solution +The sum of his previous loans is +. The limit for federal subsidized and +unsubsidized loans is $31,000, so in year 5 he can get student loans in the amount of +. +YOUR TURN 6.82 +1. Tiana in an independent college student entering the sixth and final year of her engineering program. In the +previous years, she took out student loans of $5,500, $8,470, $10,000, $7,890, and $11,900. She needs a loan to +finish her final year. How much in federal subsidized and unsubsidized student loans can she obtain? +Putting this all together, we have a way to determine the student loans needed for a student to attend college. +• +First, determine the funding gap. If the student or family can cover the gap, then no loans are necessary. +• +Second, determine how much in federal subsidized and unsubsidized student loans can be taken out. If the total +federal loans available is more than the funding gap, no other loans are needed. +• +Third, if the federal subsidized and unsubsidized loans do not cover the gap, PLUS and private student loans can be +taken out to cover the remainder of the gap. +At each step, if the student and family can cover some or all of the gap, they can do so without taking out a loan. +EXAMPLE 6.83 +College Funding Gap and PLUS and Private Student Loans +Olivia receives her award and tuition letter from the college she wants to attend. Her tuition, fees, books, and room and +board all come to $44,845 for her second year. Her non-loan awards include an instant scholarship from the school for +$13,500, a scholarship she earned for enrolling in an engineering program for $5,750, and a $2,000 scholarship from her +parent’s workplace. For her first year, what is Olivia’s college funding gap? How much can Olivia borrow in federal +subsidized and unsubsidized student loans? Once Olivia takes out her maximum subsidized and unsubsidized federal +student loans, how much will have to be paid for using PLUS and private student loans? +Solution +Her awards total to $21,250. Her cost to attend is $44,845. Her college funding gap is then +. The maximum in federal student loans that Olivia can borrow is $6,500 in year 2. The +remaining funding gap is +. Private student loans, PLUS loans, or other sources must be used +to cover this gap. +YOUR TURN 6.83 +1. Makenzy receives her award and tuition letter from the college she wants to attend. Her tuition, fees, books, and +room and board all come to $39,200 for her third year. Her non-loan awards include an instant scholarship from +the school for $19,500, three scholarships she earned from sources she found online which total $3,850, and a +$5,000 scholarship from her employer. What is Makenzy’s college funding gap? How much can Makenzy borrow +in federal subsidized and unsubsidized student loans? Once she takes out her maximum subsidized and +unsubsidized federal student loans, how much will Makenzy and her family have to pay using PLUS and private +student loans? +648 +6 • Money Management +Access for free at openstax.org + +Student Loan Interest Rates +Student loans are first and foremost loans. Students will pay them back and will pay interest. In the fall of 2022, the +federal student loan interest rate was 4.99%. Private student loans rates ranged between 3.22% and 13.95%. Finding the +lowest interest rate you can helps with the payments, and especially helps if the loan is not federally subsidized. +Remember, if the loan is not federally subsidized, the student is on the hook for the interest that is accumulating with the +loan. +Interest Accrual +The interest on student loans begins as soon as the loan is disbursed (paid to the borrower). When the loan is federally +subsidized, the government pays that interest for the student. This means the loan for a subsidized loan of $3,000 is still +a loan for $3,000 when the student graduates. However, if the loan is not federally subsidized, the student is responsible +for the interest that accrues on the loan. The $3,000 loan from year 1 of college is now a loan for more due to that added +interest. The interest on that loan grew while the student was in college. The formula for growth of the loan’s balance is +the same as compound interest formula from Compound Interest, +. +EXAMPLE 6.84 +Denise takes out unsubsidized student loan, in August, in her first year of college for $2,000. She manages an interest +rate of 8%. She graduates after her fifth year of college, in May. She does not pay the interest on the loan during her +time in college. What is the balance of her first year loan in May of her graduation year? +Solution +The principal of the loan is $2,000. Her interest rate is 8%. Since student loans are typically paid monthly, there are 12 +periods per year. Since the time she has had the loan is not in years, we will use the number of months for the value of +in the formula. She has had the loan for 4 years and 9 months, meaning 57 period have passed. Substituting those +values into the formula and calculating, we find her balance in May of her graduating year is +. +YOUR TURN 6.84 +1. Priya takes out unsubsidized student loan, in August, in her first year of college for $2,000. She manages an +interest rate of 7%. She graduates after her sixth year of college, in May. She does not pay the interest on the +loan during her time in college. What is the balance of her first year loan in May of her graduation year? +The Standard Repayment Plan +There are various repayment plans available. The one most likely to apply to a student loan is the standard repayment +plan, which is available to everyone. Borrowers pay a fixed amount monthly so the loan is paid in full within 10 years. +Consolidated loans, discussed later in this section, also qualify for the standard repayment plan, and may allow the +payoff period to range from 10 to 30 years. Direct subsidized and unsubsidized loans, PLUS loans, and federal Stafford +loans are eligible. +Since these are loans, they are paid back with interest. As with most installment loans, their payments are due monthly. +The formula for paying back these loans is the same as the formula used for paying loans in The Basics of Loans: +. +Using that formula, we can calculate how much the payment is for a student loan. Remember that all loan payments are +rounded up to the next penny. +EXAMPLE 6.85 +Standard Repayment Plan +Find the payment for the following student loans using the standard repayment plan: +6.9 • Understanding Student Loans +649 + +1. +Loan is $3,500, interest is 4.99% +2. +Loan is for $6,200, interest is 6.75% +Solution +1. +The principal is += $3,500 and the rate is += 0.0499. Since this is the standard repayment plan, there are += 12 +payments per year for 10 years. Substituting those values into +and calculating gives a +monthly payment of +2. +The principal is += $6,200 and the rate is += 0.0675. Since this is the standard repayment plan, there are += 12 +payments per year for 10 years. Substituting those values into +and calculating gives a +monthly payment of +YOUR TURN 6.85 +Find the payment for the following student loans using the standard repayment plan: +1. Loan is $7,800, interest is 4.99% +2. Loan is for $11,450, interest is 7.75% +EXAMPLE 6.86 +Standard Repayment Plan for an Unsubsidized Loan +Erson has a balance of $8,132.55 when he starts paying off the 8.6% unsubsidized student loan he took out in his third +year. How much are his payments if the term for his loan is the standard 10 years? +Solution +Using the payment formula, +, with += $8,132.55, += 0.086, += 10 and += 12, we calculate +that his monthly payment will be +650 +6 • Money Management +Access for free at openstax.org + +YOUR TURN 6.86 +1. Kathleen has an unsubsidized student loan from graduate school. She begins to pay it back in the September +after she earned her doctorate. The balance at that point was $22,666.21. The interest rate on the loan is 6.7%. +How much are her payments if she pays it off in the standard of 10 years? +Student Loan Consolidation +When a student graduates, they may have multiple different student loans. Keeping track of them and paying them off +separately can be a burden. Instead, these loans can be consolidated into a single loan. If they are federal loans the +combination is called federal consolidation. Combining private loans is often referred to as refinancing. Refinancing, +or private consolidation, can be used to combine both private and federal student loans. Be aware that consolidated +federal loans may still be subject to the rules and protections that govern subsidized loans. Refinancing loans, private or +federal, are no longer subject to those rules and guidelines. Check out this Experian article about consolidation and +refinancing (https://openstax.org/r/my_student_loans) for more deatil. +In consolidation of federal direct student loans, the interest rate is the weighted average of the interest rates on the +subsidized loans. This means the interest rate remains the same. However, if the term is extended, then the student will +pay back more over time than if they did not extend the loan term. +In refinancing, it is possible to obtain a lower interest rate on the student loans, which may lower how much is paid per +month and lower the total paid back over time. These monthly payments are calculated using the same formula as for +any other loan payment, +. The term of the refinanced loan may also be changed, which +would also impact the payment per month. +In either case, consolidating or refinancing, the monthly financial burden on the student can decrease. However, if the +term is extended, the total amount repaid may increase. +EXAMPLE 6.87 +Federal Loan Consolidation and Interest Rates +Ernest has four federal student loans that he wants to consolidate. He combines them into one loan. What is the +maximum Ernest can reduce the interest rate by? +Solution +Consolidating subsidized loans has no impact on the interest rate of the loans, so the maximum that the interest rate +can be reduced is 0%. +YOUR TURN 6.87 +1. Ryann has five federal student loans that he wishes to consolidate. He wants to extend the term to 15 years. How +does this impact Ryann’s interest rate? The total Ryann pays back? +6.9 • Understanding Student Loans +651 + +EXAMPLE 6.88 +Payments for Consolidated Student Loans +Brianna consolidates her student loans, some federal and some private, into a single refinanced student loan with a +principal of $27,800. The interest rate that Brianna received was 8.375%. If Brianna’s new term is 15 years, how much are +her payments per month? +Solution +The principal is += $27,800 and the rate is += 0.08375. Since the payments are monthly, +=12. The loan term is for 15 +years, so += 15. years. Substituting those values into +and calculating gives a monthly +payment of +YOUR TURN 6.88 +1. Brian consolidates his federal student loans into one loan at 4.99%. The consolidated balance is $20,340, and he +extends the term to 12 years. What are Brian’s monthly payments for the consolidated loan? +Other Repayment Plans +There are various other repayment plans available to students. Plans other than the standard repayment plan typically +require the student to meet certain criteria. The following plans are independent of student income, but may make early +payments easier. +• +Graduated repayment plans are plans where the amount of payments gradually increases so that the loan is paid +off in 10 years, or within 10 to 30 years for consolidated loans. Payments start off small and increase approximately +every 2 years. Almost all loan types are eligible, including direct subsidized and unsubsidized loans, Stafford loans, +PLUS loans, and consolidated loans. +• +Extended repayment plans are available to the direct loan borrower if the outstanding direct loans are over +$30,000. The payments, fixed or graduated, are designed so that the loans are satisfied within 25 years. Eligible +loans include both direct subsidized or unsubsidized loans, Stafford loans, PLUS loans, and consolidated loans. +If student earnings are such that the standard, graduated, or extended repayment plans are unaffordable, then one can +make payments that are based on their discretionary income. Discretionary income is federally defined to be the +difference between (adjusted) gross income and 150% of the poverty guideline for location and family size. This +discretionary income then depends on where one lives (contiguous United States or Hawaii or Alaska) and how many +dependents one has. If married, a spouse’s income will be included in the adjusted gross income. Understanding +discretionary income is necessary to understand how income driven payments plans work. +EXAMPLE 6.89 +Discretionary Income +1. +The poverty guideline for a single person living in Arkansas, is $12,000. If Harriet is a single person in Arkansas with +a (adjusted) gross income of $23,500, what is her discretionary income? +2. +For California, the poverty guideline for a person with four people in the household is $27,750. If such a person has +a (adjusted) gross income of $48,600, what is their discretionary income? +652 +6 • Money Management +Access for free at openstax.org + +Solution +1. +The poverty guideline for a single person in Arkansas is $12,000. 150% of that guideline value is +. The gross income that Harriet makes over that $18,000 is her discretionary income. That +gross income is $23,500, so her discretionary income is +. +2. +The poverty guideline for a household of four in California is $27,750. 150% of that guideline value is +. The gross income that Harriet makes over that $41,625 is her discretionary income. That +gross income is $48,600, so her discretionary income is +. +YOUR TURN 6.89 +1. The poverty guideline for two-person household in Hawaii is $21,060. If Jamie lives in a two-person household +and has a gross income of $36,900, what is his discretionary income? +The following plans all depend on discretionary income. +• +Pay As You Earn (PAYE) repayment plans have monthly payments that are 10% of discretionary income based on a +student’s updated income and family size. If a borrower files a joint tax return, their spouse’s income and debt may +also be considered. Eligible loans include direct subsidized and unsubsidized loans, PLUS loans made to students, +and some consolidated loans. These loans are forgiven (student does not pay the remaining balance) after 20 years +of monthly payments if they were direct federal student loans. +• +Revised Pay As You Earn (REPAYE) repayment plans have payment amounts that are based on income and family +size and calculated as 10% of discretionary income. Eligible loans include direct subsidized and unsubsidized loans, +direct PLUS loans, and some consolidated loans. These loans are forgiven, that is, the student does not pay more, +after 20 or 25 years, provided they were direct federal student loans. +• +Income-Based Repayment (IBR) plans sound like a few of the others and there are similarities in all of them. The +payments are either 10% or 15% of discretionary income, but this plan is meant for those with a relatively high debt. +Every year, income and family size must be updated, and payments are calculated based on those figures. Eligible +loans include direct subsidized and unsubsidized loans, Stafford loans, and PLUS loans made to students, but not +PLUS loans made to parents. +• +Income-Contingent Repayment (ICR) plans have payments that are either 20% of discretionary income or +whatever would be paid if a student were on a fixed payment plan for more than 12 years, whichever is less. Eligible +loans include direct subsidized and unsubsidized, PLUS loans to students, and consolidated loans. +There are many similarities among these repayment plans, and it is easy to misunderstand the nuances of each. +Therefore, be careful entering into any type of repayment contract until you fully understand all the details and +repercussions of the plan you choose. For more detail, see this nerdwallet article "Income-Driven Repayment: Is It Right +for You?" to learn more (https://openstax.org/r/driven_repayment_right)! +EXAMPLE 6.90 +Payments for a REPAYE Program +Warren qualifies for a REPAYE payment plan. His gross income is $32,700. He is single and live in Montana, so the federal +poverty guideline for Warren is $12,000. +1. +What is Warren’s discretionary income? +2. +Under the REPAYE plan, he pays 10% of his discretionary income, but monthly. How much are Warren’s REPAYE +payments? +Solution +1. +The poverty guideline for Warren is $12,000. 150% of that guideline value is +. The gross +income that Warren makes over that $18,000 is his discretionary income. That gross income is $32,700, so his +discretionary income is +. +2. +10% of Warren’s discretionary income is +. He pays monthly, so his monthly payments are +$1,470 divided by 12, or $122.50 per month. +6.9 • Understanding Student Loans +653 + +YOUR TURN 6.90 +Lauren, a Missouri resident, qualifies for a REPAYE payment plan. Her gross income is $39,200. She has one child, so +there are two people in her household. The federal poverty guideline for Lauren is $18,310. +1. What is Lauren’s discretionary income? +2. Under the REPAYE plan, she pays 10% of her discretionary income, but monthly. How much are Lauren’s +REPAYE payments? +Using income to determine payments initially seems excellent. However, if there is no forgiveness at the end of the loan, +then the income driven payment plans can cause problems. For one, your payment may not be sufficient to cover the +interest rate of your loans. In that case, your loan balance actually increases as you make your payments. Eventually, you +are paying for not only your original loan balance, but interest that has been growing and compounding over time. Also, +if the loan term is extended, you may pay more, perhaps a lot more, money over time. You may find yourself in the +position of paying these loans for decades. +With those possible drawbacks, great care must be taken to avoid large problems down the line. +VIDEO +Repayment Plans (https://openstax.org/r/student_loan_repayment_plans) +Student Loan Default and Consequences +The first day a payment is late, the account becomes delinquent. After 90 days, this delinquency is reported to the credit +bureaus, and goes into default. This is serious, as now a credit score is affected, meaning that it will be harder to buy a +car, a home, get a credit card, or a cell phone. Even renting an apartment may be a task not easily overcome. The default +rate for students who do not complete their degree is three times higher than for students who do. +Further, defaulting on a student loan may mean that the borrower loses eligibility for repayment plans, as the balance +and any unpaid interest may become due immediately, and any tax refunds may be withheld and applied to the loan, +and wages may be garnished. One should immediately contact your loan servicer and try to make other arrangements +for repayment if this situation becomes apparent, as different repayment plans are available, if actions are taken quickly. +There are several options that may be open to avoid defaulting. One is called rehabilitation, or is the process in which a +borrower may bring a student loan out of default by adhering to specified repayment requirements, and the other is +consolidation. Certain criteria must be met to enter these programs. +Both of these options are detailed, including the criteria required for eligibility, on the studentaid.gov loan management +page (https://openstax.org/r/default_get_out). +Professionals advise hiring an attorney if one of these paths is chosen. +Check Your Understanding +55. When is the FAFSA typically due? +56. If the college has a program that it advertises will take 6 years, how long is the eligibility period? +57. What is a student’s funding gap? +58. What is a main difference between subsidized federal loans and unsubsidized loans? +59. What is the limit on the total in federal subsidized and unsubsidized loans that can be borrowed for an +undergraduate degree? +60. If a student takes out a private student loan for $9,800 at 5.99% interest but does not begin paying it back for 66 +months, how much is then owed on the loan? +61. How long is the standard repayment plan for student loans? +62. How much of a student’s discretionary income is paid for student loans under the REPAYE payment plan? +SECTION 6.9 EXERCISES +1. What form must be filled out before applying for a student loan? +654 +6 • Money Management +Access for free at openstax.org + +2. What two issues limit the amount of a federal subsidized or unsubsidized student loan a student can take out? +3. What is the limit to the number of years a student can take out a student loan? +4. What types of student loans have deferred payments and do not accrue interest? +5. Which type of student loan is made directly to parents? +6. Which types of student loans require a credit check? +7. What is the limit on the total amount borrowed in federal subsidized and unsubsidized student loans for an +independent undergraduate student? +8. What is the limit on subsidized student loans in the third year of an undergraduate program if the student is a +dependent student? +9. What is the maximum allowed to be borrowed on a PLUS loan? +10. What is the maximum allowed to be borrowed on a private student loan? +11. What is the current interest rate on federal subsidized loans? +12. What feature does a graduated payment plan have? +13. If a student is struggling with paying back their student loans, what can the student do? +14. When a student consolidates federal subsidized and unsubsidized loans, does the interest rate change? Does the +term of the loan change? +15. If a person with federal subsidized and unsubsidized loans enters a pay as you earn program (PAYE), how long will +the person pay on their student loans? +16. How long is it between the time a person is delinquent on a student loan and being in default on the student loan? +17. What is discretionary income? +18. What are the two remedies if a borrower default on their student loans? +19. Toni’s total cost to attend college including all fees, books, and room and board is $18,450. If Toni receives $5,500 +in an instant scholarship from the college and has $2,175 in scholarships, what is Toni’s funding gap? +20. Arthur’s total cost to attend college including all fees, books, and room and board is $31,500. If Arthur receives +$13,500 in an instant scholarship from the college and has $3,475 in private scholarships, what is Arthur’s funding +gap? +21. Amelia is an independent student in her fourth year of an urban studies undergraduate degree. What is the +maximum she can borrow in federal subsidized and unsubsidized student loans that year? +22. Yaroslav is a dependent student in his second year of an undergraduate psychology program. What is the +maximum that he can borrow in federal subsidized and unsubsidized student loans that year? +23. Toni’s total cost to attend her third year of undergraduate school, including all fees, books, and room and board, is +$18,450. Toni receives $5,500 in an instant scholarship from the college and has $2,175 in scholarships. She is a +dependent student. After receiving the maximum federal student loans (subsidized and unsubsidized), and +without her parents taking out a PLUS loan, how much will she need to borrow in private student loans to cover +the funding gap? +24. Arthur’s total cost to attend his second year of his undergraduate program in data science, including all fees, +books, and room and board, is $31,500. Arthur receives $13,500 in an instant scholarship from the college and has +$3,475 in private scholarships. He is a dependent student. After receiving the maximum federal student loans +(subsidized and unsubsidized), and without his parents taking out a PLUS loan, how much will he need to borrow +in private student loans to cover the funding gap? +25. Nantumbwe takes out a private student loan for $7,950 in her second year of her undergraduate health +administration program. She begins paying her loan after 39 months. If her interest rate was 8.75%, how much is +her balance when she begins to pay off her loan? +26. Cameron takes out a private student loan for $11,385 in their third year of their undergraduate communications +program. They begin paying their loan after 27 months. If Cameron’s interest rate was 9.125%, how much is their +balance when they begin to pay off the loan? +6.9 • Understanding Student Loans +655 + +27. Farzana refinances her loans for a reduced interest rate before she begins to pay them off. The total of her +refinanced loan is $39,447.50. The new interest rate is 6.99%. How much are Farzana’s payments if she has a +10-year term on this loan? +28. Carly refinances her loans for a reduced interest rate before she begins to pay them off. The total of her refinanced +loan is $61,332.00. The new interest rate is 7.45%. How much are Carly’s payments if she has a 10-year term on this +loan? +29. Andrew consolidates his federal student loans to make paying them easier. They total $27,582.00. The consolidated +loan is at 4.99% interest. How much are Andrew’s payments if his new term is 15 years? +30. Manuel consolidates his federal student loans to make paying them easier. They total $19,470.00. The consolidated +loan is at 4.99% interest. How much are Manuel’s payments if his term is 10 years? +31. Steven has four members in his Georgia household and makes $53,400 annually. The federal poverty guideline for +him is $27,750. What is Steven’s discretionary income? +32. Akash has three members in his New Jersey household and makes $38,120 annually. The federal poverty guideline +for him is $23,030. What is Akash’s discretionary income? +33. Kateryna has two members in her Iowa household and earns $42,500 annually. The federal poverty guideline for +her is $18,310. She qualifies for a REPAYE repayment plan. What is the maximum monthly payment Kateryna would +pay? +34. Keirstin lives by herself in her New Mexico home and earns $28,530 annually. The federal poverty guideline for her +is $13,590. She qualifies for a REPAYE repayment plan. What is the maximum monthly payment Keirstin would pay? +6.10 Credit Cards +Figure 6.22 Credit cards are both a convenience and a danger. (credit: "Credit Cards" by Sean MacEntee/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Apply for a credit card armed with basic knowledge. +2. +Distinguish between three basic types of credit cards. +3. +Compare and contrast the benefits and drawbacks of credit cards. +4. +Read and understand the basic parts of a credit card statement. +5. +Compute interest, balance due, and minimum payment due for a credit card. +It can be difficult to get along these days without at least one credit card. Most hotels and rental car agencies require +that a credit card is used. There are even a number of retailers and restaurants that no longer accept cash. They make +online purchasing easier. And nothing contributes more to a good credit rating than a solid history of making credit card +payments on time. +Being granted a credits card is a privilege. Used unwisely that privilege can become a curse and the privilege may be +withdrawn. In this section, we will talk about the different types of credit cards and their advantages and disadvantages. +656 +6 • Money Management +Access for free at openstax.org + +The more knowledge a cardholder has about the credit card industry, the better able credit accounts can be managed, +and that knowledge may cause major adjustments to a cardholder’s lifestyle. +All credit cards are not equal, but they all represent consumers borrowing money, usually from a bank, to pay for needs +and “wants.” As such, they are a type of loan, and your repayment may include interest. (You might want to review +Section 6.8, which discusses loans and repayment plans.) +There are many institutions and credit cards to choose from. Use caution as you shop around for a credit card that suits +you. Your top concern is likely the interest rates on purchases and cash advances. But be careful to also read the small +print regarding charges for late payments, and other fees such as an annual fee, where the credit card charges you (the +cardholder) a fee each year for the privilege of using the cards. Many cards charge no such fee, but there are many that +charge modest to heavy fees. Make sure to understand rules for reward programs, where the credit card issuer grants +benefits based on one’s spending. Finally, once one applies for and is granted a credit card, pay attention to the credit +limit the bank offers. Once a company is owed that much money, use of the card for purchases should be curtailed until +some of the debt is paid off. +The interest rate will not matter if the balance is paid every month. When the balance is paid every month, there is +NO INTEREST charged. +Types of Credit Cards +There are basically three types of credit cards: bank-issued credit cards, store-issued credit cards, and travel/ +entertainment credit cards. We will look at all three and explain the good and the bad qualities of each. +Bank-Issued Credit Cards +Perhaps the most widely used credit card type is the bank-issued credit card, like Visa or MasterCard (and even +American Express and Discover cards). These types of cards are an example of revolving credit, meaning that additional +credit is extended before the previous balance is paid—but only up to the assigned credit limit. Bank-issued cards are +considered the most convenient, as they can be used to purchase anything, including apparel, furniture, groceries, fuel +for automobiles, meals, hotel bills, and so on, just as if paying with cash. The interest rates on bank-issued credit cards +are usually lower than those for other credit cards we’ll discuss, and the credit limits are generally higher. Currently, +bank-issued cards have an average 20.09% APR. +Store-Issued Credit Cards +Store-issued credit cards are issued by retailers. One can hardly walk into a store these days without being offered a +discount on purchases if one applies for the store credit card. These cards can only be used in that store or family of +stores that issues the card. However, if a store credit card is associated with Visa, MasterCard, or American Express, then +the card might be used the same way that the bank-issued cards are used. This is called cobranding. The logo of the +bank-issued card will be present on the store card. Many stores offer both types. Like other credit cards, they may come +with an annual fee. +Store credit cards usually charge higher interest rates than bank-issued cards. Currently, store credit cards have an APR +(annual percentage rate) of 24.15%. Any rewards offered by store credit cards are usually limited to purchases made in +their own store, and it typically takes longer to accumulate enough rewards or points to redeem them, whereas +cobranded credit cards offer opportunities to earn rewards on all purchases, regardless of whether purchases are made +in the issuing store or not. +Store credit cards usually offer lower credit limits, at least in the beginning. After being proved to be a responsible credit +card owner, credit limits can be raised. Nevertheless, store credit cards are a good choice for those new to the credit card +industry. If on-time payments are consistently made, it is an excellent way to get started building a credit history. +Travel/Entertainment Cards, or Charge Cards +This is the third type of credit card. The travel and entertainment cards, also known as charge cards, first and +foremost offer very high limits or unlimited credit, but they must be paid in full every month. They generally charge high +annual fees and impose expensive penalties should a payment be late. On the other hand, they typically have longer +grace periods and offer many and various kinds of rewards. +Check out this nerdwallet article about the differences between a charge card and a credit card (https://openstax.org/r/ +cards_charge_cards). +An interest rate will greatly depend on credit score. Responsible use of credit cards will increase a credit score. See +6.10 • Credit Cards +657 + +the WHO KNEW? from The Basics of Loans. +VIDEO +Choosing a Credit Card (https://openstax.org/r/Credit_Card) +WHO KNEW? +Top Travel and Entertainment Cards +At one time, Diner’s Club was the premier entertainment card. To be accepted into the Diner’s Club and be rewarded +with a charge card meant one was special. Today, an American Express card is held with the same reverence as +Diner’s Club was in the past. It was a privilege to own one of these cards. There is a lot of competition going on to +supplant Diner’s Club, as shown by the Chase Sapphire Preferred Card’s (https://openstax.org/r/Chase_Sapphire) +travel rewards and benefits. +Another thing worth mentioning here and something that appears to be an unusual offering is part of the American +Express Gold and higher-level cards. Cardholders can actually open a savings account, buy a CD, or apply for a +personal or business loan from American Express. The company boasts a higher interest rate than what is paid on +traditional savings accounts and CDs. +EXAMPLE 6.91 +Comparing Credit Cards +1. +Which type of credit card is paid off every month so has no interest to be paid, but comes with high fees? +2. +Which type of credit cards are the most widely accepted? +3. +Which type of credit cards are the most limited? +Solution +1. +Charge cards are to be paid off completely each month. +2. +Bank-issued credit cards are the most flexible to use, because they are not limited to which retailers or service +providers accept them. +3. +Store issues credit cards are the most limited, since they only work in that family of stores. +YOUR TURN 6.91 +1. Which type of credit card typically have the highest interest rates? +2. What should be considered other than interest rate when selecting a card? +3. Which type of cards typically have high annual fees? +Credit Card Statements +Cardholders usually receive monthly statements and have 21 days to pay the minimum amount due. The statements +itemize and summarize activity on the credit card for that statement’s billing period. The billing period for a credit card +is generally a month long, but typically does not start and end on the first and last days of the month. The statement will +include the current balance, interest rate, the minimum payment due, and the due date. Be aware, different companies +produce statements that are laid out differently. The information will be clearly labeled though. +The due date is a top concern. Missing a due date is one of the worst things a cardholder can do financially, and this is by +far the biggest downfall of owning a credit card. Not only is the cardholder subject to late fees, but when a payment is +late more than once there is a high probability that the cardholder will be negatively reported to the credit bureaus, +which can quickly erode a credit score. Figure 6.23 shows an excerpt from an actual statement from a Chase Bank Visa +card, based on the current $668.25 balance. +658 +6 • Money Management +Access for free at openstax.org + +Figure 6.23 Credit card statement +Specifically pay attention to the late payment penalty and minimum payment warning statements. stating that if no +other purchases are made and you continue making only the minimum payment, it will take 19 months to pay off the +balance and you will pay $754.00. You can’t say you were not warned. +It is critical that you examine your statement every month because it is always a possibility that your account may have +been compromised. If you should notice fraudulent charges on your statement, notifying the credit card company is +often enough to have those charges researched by the company and removed. The card with the fraudulent charges will +be canceled and a new card with a new account number will be sent to you. +EXAMPLE 6.92 +Reading a Credit Card Statement +On the credit card statement Figure 6.24, identify +1. +The balance due +2. +The minimum required payment +3. +The length of time it takes to pay off the balance by paying the minimum payments and without charging more to +the card +4. +The interest rate for purchases +6.10 • Credit Cards +659 + +Figure 6.24 Credit card account statement +Solution +1. +The balance due is under the payment information heading and is $3,663.23. +2. +The minimum payment due is also under the payment information heading, and is $36.63. +3. +The time to pay off the balance using only minimum payments is below the payment information, and says it takes +2 years and 4 months to pay off the balance. +4. +The interest rate for purchases is toward the bottom of the statement. It is 19.99%. +660 +6 • Money Management +Access for free at openstax.org + +YOUR TURN 6.92 +Referring to the statement above, answer the following: +1. What is the statement period? +2. What is the credit limit? +3. How much in fees were charged? +VIDEO +Reading Credit Card Statements (https://openstax.org/r/Reading_Credit) +Compute Interest, Balance Due, and Minimum Payment Due for a Credit Card +Computing all of these values depends on understanding and computing the average daily balance on a credit card. +Once that is known, the interest, balance due, and minimum payment can be found. +Above all else, if you pay off the entire balance each month, interest is not charged. +Average Daily Balance +Most credit card companies compute interest using the average daily balance method. +To find the average daily balance on your credit card, determine the balance on the card each day of the billing period +(often that month), and take the average. One process to find that average daily balance follows these steps: +1. +Start with a list of transactions with their dates and amounts. +2. +For each day that had transactions, find the total of the transactions for the day. Expenditures are treated as positive +values, payments are treated as negative values. +3. +Create a table containing each day with a different balance. The balance is the previous balance plus the day’s total +transactions. +4. +Add a column for the number of days those balances until the balance changed. +5. +Add a column that contains the balances multiplied by the number of days until the balance changed. +6. +Find the sum of that last column. +7. +Divide the sum by the number of days in the billing period (often the number of days in the month). This is the +average daily balance. +EXAMPLE 6.93 +Computing Average Daily Balance +The billing cycle goes from May 1 to May 31. The balance at the start of the billing cycle is $450.21. The list of +transactions on the card is below. +Date +Activity +Amount +1-May +Billing Date Balance +$450.21 +10-May +Payment +$120.00 +15-May +Groceries +$83.43 +26-May +Auto Parts +$45.12 +26-May +Restaurant +$85.34 +30-May +Shoes +$98.23 +Find the average daily balance for the credit card during the month of May. +6.10 • Credit Cards +661 + +Solution +To find the average daily balance, we use the following steps. +1. +Start with a list of transactions with their dates and amounts. +This list is provided. +2. +For each day that had transactions, find the total of the transactions for the day. +The only day with more than one transaction was May 26. The sum of those transactions is $130.46. Treating the +payment as a negative value, the daily transaction amounts are +Date +Amount +1-May +$450.21 +10-May +-$120.00 +15-May +$83.43 +26-May +$130.46 +30-May +$98.23 +3. +Create a table containing each day with a different balance. +The new table with dates that had different balances is below. +Date +Balance +1-May +$450.21 +10-May +$330.21 +15-May +$413.64 +26-May +$544.10 +30-May +$642.33 +4. +Now, add a column for the number of days those balances until the balance changed. The days until the balance +changes is found by finding the difference in the dates. For instance, from May 15 to May 26 was 11. Adding that +column to the table we have +Date +Balance +Days Until Balance Changes +1-May +$450.21 +9 +10-May +$330.21 +5 +15-May +$413.64 +11 +26-May +$544.10 +4 +30-May +$642.33 +2 +The last entry was 2 since there are 31 days in May. +5. +Add a column that contains the balances multiplied by the number of days until the balance changed. We create the +662 +6 • Money Management +Access for free at openstax.org + +column and multiply the values. +Date +Balance +Days Until Balance Changes +Balance Times Days +1-May +$450.21 +9 +$4,051.89 +10-May +$330.21 +5 +$1,651.05 +15-May +$413.64 +11 +$4,550.04 +26-May +$544.10 +4 +$2,176.40 +30-May +$642.33 +2 +$1,284.66 +6. +Find the sum of that last column. Adding that last column we have a sum of $13,714.04. +7. +There are 31 days in May, so divide the sum by 31, which gives an average of $442.39, which is the average daily +balance. +YOUR TURN 6.93 +1. The billing cycle goes from June 1 to June 30. The previous month’s balance is $563.80. The transactions are in +the table below. +Date +Activity +Amount +1-Jun +Balance +$563.80 +2-Jun +Gasoline +$47.50 +2-Jun +Groceries +$63.42 +15-Jun +Movie +$38.75 +15-Jun +Payment +$250.00 +27-Jun +Pharmacy +$31.21 +28-Jun +Auto fuel +$48.00 +Find the average daily balance for this credit card. +Calculating the Interest for a Credit Card +The interest charged for a credit card is based on the daily interest rate of the card, the number of days in the billing +cycle, and the average daily balance on the card. +FORMULA +The interest charge, +, for a credit card during a billing cycle is +, where ADB is the average daily +balance, +is the annual percentage rate, and +is the number of days in the billing cycle. As before, interest is +6.10 • Credit Cards +663 + +rounded up to the next penny. +EXAMPLE 6.94 +Calculating Interest for a Credit Card Billing Cycle +Compute the interest charged for the credit card based on the given average daily balance (ABD), annual interest rate, +and number of days in the billing cycle. +1. +ADB = $2,765.00, annual interest rate 13.99%, billing cycle of 30 days +2. +ADB = $789.30, annual interest rate 17.99%, billing cycle of 31 days +3. +ADB = $1,037.85, annual interest rate 11.99%, billing cycle of 28 days +Solution +1. +Substituting $2,765.00 for ADB, 0.1399 for +and 30 for +and calculating, we find the interest charge to be +. +2. +Substituting $789.30 for ADB, 0.1799 for +and 31 for +and calculating, we find the interest charge to be +. +3. +Substituting $1,037.85 for ADB, 0.1199 for +and 28 for +and calculating, we find the interest charge to be +. +YOUR TURN 6.94 +Compute the interest charged for the credit card based on the given average daily balance (ABD), annual interest +rate, and number of days in the billing cycle. +1. ADB = $2,135.00, annual interest rate 12.9%, billing cycle of 30 days +2. ADB = $1,589.63, annual interest rate 9.99%, billing cycle of 31 days +3. ADB = $6,803.41, annual interest rate 14.9%, billing cycle of 28 days +WHO KNEW? +Credit Cards Charge Stores Fees +The interest you pay is not the only way a credit card company generates revenue. It also charges fees to the retailers, +online stores, and service providers that allow the consumer, you, to use your credit card to pay them. These are +called processing fees. Currently they typically range from 2.87% to 4.35% of each transaction. That means if you use +your credit card at a store and spend $100.00, the store will have to pay the credit card company somewhere between +$2.87 and $4.35. +One type of processing fee is the interchange fee. Mastercard charges the vendor 1.35% of the sale, plus an +additional percentage up to 3.25%, and a fixed $001 fee for each transaction. +Added to that is an assessment fee. This fee is 0.14% for Visa cards. +Calculating the Balance of a Credit Card +The balance, or sometimes balance due, on a credit card is the previous balance, plus all expenses, minus all payments +and credits, plus the interest on the card. As stated before, if the card was paid off, there is no interest to be paid. +EXAMPLE 6.95 +Calculating the Balance of a Credit Card +Find the balance on the credit card with the given interest charge and balance before interest was charged. The cards +664 +6 • Money Management +Access for free at openstax.org + +were not paid off previously. +1. +Balance before interest is $708.50, interest charge is $8.15 +2. +Balance before interest is $1,395.10, interest charge is $21.32 +Solution +1. +Adding the balance before interest to the interest charge, we find the balance to be $716.65. +2. +Adding the balance before interest to the interest charge, we find the balance to be $1,416.42. +YOUR TURN 6.95 +Find the balance on the credit card with the given interest charge and balance before interest was charged. The +cards were not paid off previously. +1. Balance before interest is $560.00, interest charge is $6.44 +2. Balance before interest is $3,218.00, interest charge is $49.17 +The next example puts all those steps together. +EXAMPLE 6.96 +Find Balance Due from Transactions and Interest Rate +Kaylen’s credit card charges 16.9% annual interest. His current billing period is from November 1 to November 30. The +balance on November 1 was $1,845.23. Use Kaylen’s following transactions to determine his balance due at the end of +the billing cycle. +Date +Activity +Amount +1-Nov +Billing Date Balance +$1,845.23 +3-Nov +Groceries +$78.50 +4-Nov +Tablet +$159.00 +4-Nov +Online Game Purchase +$39.99 +4-Nov +Restaurant +$47.10 +10-Nov +Payment +$300.00 +13-Nov +Gasoline +$58.75 +13-Nov +Clothing +$135.00 +18-Nov +Gift +$30.00 +18-Nov +Restaurant +$21.75 +28-Nov +Gasoline +$43.79 +Solution +The first step is to find Kaylen’s average daily balance. To find the average daily balance, we use the following steps. +1. +Start with a list of transactions with their dates and amounts. +6.10 • Credit Cards +665 + +This list is provided. +2. +For each day that had transactions, find the total of the transactions for the day. The days with more than one +transaction were Nov. 4, Nov. 13, and Nov. 18. Treating the payment on November 10 as a negative value, the daily +transaction amounts are +Date +Amount +1-Nov +$1,845.23 +3-Nov +$78.50 +4-Nov +$246.09 +10-Nov +-$300.00 +13-Nov +$193.75 +18-Nov +$51.75 +28-Nov +$43.79 +3. +Create a table containing each day with a different balance. The new table with dates that had different balances is +below. +Date +Balance +1-Nov +$1,845.23 +3-Nov +$1,923.73 +4-Nov +$2,169.82 +10-Nov +$1,869.82 +13-Nov +$2,063.57 +18-Nov +$2,115.32 +28-Nov +$2,159.11 +4. +Now, add a column for the number of days those balances until the balance changed. The days until the balance +changes is found by finding the difference in the dates. For instance, from May 15 to May 26 was 11. Adding that +column to the table we have +The last entry was 3 since there are 30 days in November (the 28th, 29th, and 30th). +Date +Balance +Days Until Balance Changes +1-Nov +$1,845.23 +2 +3-Nov +$1,923.73 +1 +666 +6 • Money Management +Access for free at openstax.org + +Date +Balance +Days Until Balance Changes +4-Nov +$2,169.82 +6 +10-Nov +$1,869.82 +3 +13-Nov +$2,063.57 +5 +18-Nov +$2,115.32 +10 +28-Nov +$2,159.11 +3 +5. +Add a column that contains the balances multiplied by the number of days until the balance changed. We create the +column and multiply the values. +Date +Balance +Days Until Balance Changes +Balance Times Days +1-Nov +$1,845.23 +2 +$3,690.46 +3-Nov +$1,923.73 +1 +$1,923.73 +4-Nov +$2,169.82 +6 +$13,018.92 +10-Nov +$1,869.82 +3 +$5,609.46 +13-Nov +$2,063.57 +5 +$10,317.85 +18-Nov +$2,115.32 +10 +$21,153.20 +28-Nov +$2,159.11 +3 +$6,477.33 +6. +Find the sum of that last column. Adding that last column we have a sum of $62,190.95. +7. +There are 30 days in November, so divide the sum by 30, which gives an average of $2,073.03, which is the average +daily balance. +With the average daily balance, we can determine the interest that is charged for November. Substituting ADB = +$2,073.03, += 0.169, and += 30 into the formula +and calculating, we find the interest to be +. +This interest is added to the final balance from the table in step 3, $2,159.11, which yields a balance due of $2,101.83. +YOUR TURN 6.96 +1. Angel’s credit card charges 16.9% annual interest. His current billing period is from August 1 to August 31. The +balance on August 1 was $982.45. Use Angel’s following transactions to determine his balance due at the end of +the billing cycle. +6.10 • Credit Cards +667 + +Date +Activity +Amount +1-Aug +Billing Date Balance +$982.45 +5-Aug +Food +$125.31 +13-Aug +Payment +$500.00 +14-Aug +Gasoline +$51.65 +14-Aug +Pizza +$36.99 +14-Aug +Shoes +$89.45 +19-Aug +Electric bill +$178.34 +21-Aug +Internet +$36.99 +21-Aug +Food +$93.45 +30-Aug +Gasoline +$43.18 +Minimum Payment Due +The minimum payment due is the smallest required amount to be paid on a credit card to avoid late fees and penalties, +such as an increased interest rate. The calculations for this may differ from card to card. They also depend in the balance +of the credit card. General guidelines for minimum payment due are: +• +For larger balances (usually over $1,000), the minimum payment will be some percentage of the balance due. +• +For moderate balances (between $25 and $1,000), the minimum would be a specified dollar amount. $25 seems to +be a common value. +• +If the balance is small (under $25 for instance), then the minimum payment is the balance. +Those are just guidelines. Individual cards may vary in these values. +Minimum payments should only be paid if money is short in a given month. The length of time to pay off a credit card +using minimum payments is quite long, and results in paying a lot of interest. It is strongly discouraged. +EXAMPLE 6.97 +Calculate the Minimum Payment Due +The FYA credit card company has the following minimum payment policy. For balances over $1,000, the minimum +payment is 2.5% of the balance due plus fees, but not interest. For balances between $500.00 and $999.99, the minimum +payment is $50.00. For balances $499.99 and under, the minimum payment is $25.00 or the balance due, whichever is +smaller. +In the following, calculate the minimum payment due given the credit card minimum payment policy, the balance due +and fees charged. +1. +Balance due is $1,309.00, no fees +2. +Balance due is $265.50, $35 in fees +3. +Balance due is $784.90, no fees +Solution +1. +The balance is over $1,000, so the minimum payment is 2.5% of the balance due plus fees. 2.5% of the balance due +668 +6 • Money Management +Access for free at openstax.org + +is +. Since there are no fees, the minimum payment due is $32.73. +2. +The balance is under $499.99, so the minimum payment due is $25.00. +3. +The balance is between $500.00 and $999.00, so the minimum payment due is $50.00. +YOUR TURN 6.97 +The CLH credit card company has the following minimum payment policy. For balances over $1,000, the minimum +payment is 1% of the balance due plus interest and fees. For balances between $25.00 and $999.99, the minimum +payment is $25.00 plus any fees. For balances under $25.00, the minimum payment is the balance due plus any fees. +In the following, calculate the minimum payment due given the credit card minimum payment policy, the balance +due and fees charged. +1. Balance due is $2,308.00, billing cycle interest is $24.39, no fees +2. Balance due is $265.50, $59.00 in fees +3. Balance due is $19.90, no fees +Check out this nerdwallet article about minimum payments (https://openstax.org/r/minimum_payment) for more! +Check Your Understanding +63. What are the three main types of credit cards? +64. Which type of credit card typically has the highest interest rate? +65. Name three important pieces of information that are included in a credit card statement. +66. A credit card has an average daily balance of $1,428.50 during a 31-day billing cycle. Find the interest if the interest +rate is 15.9%. +67. The billing cycle goes from June 15 to July 14, or 30 days. The balance at the start of the billing cycle is $3,825.50. +The list of transactions on the card is below. Find the average daily balance. +Date +Activity +Amount +15-Jun +Billing Date Balance +$3,825.50 +20-Jun +Food +$125.31 +20-Jun +Clothing +$345.00 +29-Jun +Payment +$750.00 +1-Jul +Restaurant +$94.80 +6-Jul +Gasoline +$49.75 +6-Jul +Home Repair Supplies +$683.94 +6-Jul +Internet +$49.99 +10-Jul +Cell Phone +$85.00 +10-Jul +Food +$175.24 +68. Find the balance due on a credit card statement if ADB = $487.65, the annual interest rate is 16.8%, the billing cycle +was 30 days, and the balance at the end of the billing cycle, before interest is added, was $689.47. +6.10 • Credit Cards +669 + +SECTION 6.10 EXERCISES +1. Name three main criteria to choose a credit card by. +2. Which type of credit card is most convenient to use? +3. Which type of credit card comes with no preset spending limit? +4. How much interest does a credit card holder pay if they pay off the balance every month? +5. Which type of credit card comes with the highest annual fees? +Use the following credit card statement for the following exercises. +6. What is the minimum payment due? +7. How long will it take to pay the balance if the minimum only is paid and no new purchases are made? +8. How much interest was charged in this billing cycle? +9. What is the balance on the credit card? +10. What is the credit limit on this card? +11. What is the late payment fee for this card? +12. When is the payment due? +13. The billing cycle for a credit card goes from April 1 to April 30. The balance at the start of the billing cycle is +$1,598.00. The list of transactions on the card is below. Find the average daily balance for the billing cycle. +670 +6 • Money Management +Access for free at openstax.org + +Date +Activity +Amount +1-Apr +Billing Date Balance +$1,598.00 +9-Apr +Gasoline +$51.24 +9-Apr +Food +$105.56 +9-Apr +Payment +$675.00 +13-Apr +Camping Trip +$229.75 +21-Apr +Gasoline +$38.45 +22-Apr +Gifts +$148.88 +22-Apr +Food +$49.75 +30-Apr +Gym Payment +$74.99 +14. The billing cycle for a credit card goes from September 1 to September 30. The balance at the start of the billing +cycle is $384.25. The list of transactions on the card is below. Find the average daily balance for the billing cycle. +Date +Activity +Amount +1-Sep +Billing Date Balance +$384.25 +2-Sep +Food +$94.54 +5-Sep +Gasoline +$25.65 +5-Sep +Internet +$39.99 +6-Sep +Payment +$380.00 +9-Sep +Insurance +$174.52 +16-Sep +Food +$83.54 +16-Sep +Day Care +$350.00 +20-Sep +Tires +$2,337.56 +21-Sep +Child Clothing +$27.65 +21-Sep +Gasoline +$31.00 +28-Sep +Television +$299.95 +15. The billing cycle for a credit card goes from October 10 to November 9. The balance at the start of the billing cycle +is $930.50. The list of transactions on the card is below. Find the average daily balance for the billing cycle. +6.10 • Credit Cards +671 + +Date +Activity +Amount +10-Oct +Billing Date Balance +$930.50 +11-Oct +Clothing +$350.00 +14-Oct +Computer +$865.84 +17-Oct +Food +$106.51 +21-Oct +Payment +$700.00 +21-Oct +Restaurant +$134.52 +21-Oct +Hotel +$387.56 +30-Oct +Hockey Game +$76.47 +5-Nov +Memorabilia +$150.00 +5-Nov +Restaurant +$94.45 +6-Nov +Gasoline +$49.19 +16. The billing cycle for a credit card goes from February 15 to March 16 during a non-leap year. The balance at the +start of the billing cycle is $292.82. The list of transactions on the card is below. Find the average daily balance for +the billing cycle. +Date +Activity +Amount +15-Feb +Billing Date Balance +$292.82 +21-Feb +Food +$64.57 +22-Feb +Gasoline +$31.50 +22-Feb +Food +$71.94 +28-Feb +Insurance +$133.25 +28-Feb +Payment +$100.00 +3-Mar +Gasoline +$26.61 +12-Mar +School Trip +$300.00 +In the following exercises, compute the interest charged for the credit card based on the given average daily balance +(ABD), annual interest rate, and number of days in the billing cycle. +17. ADB = $350.00, annual interest rate 14.9%, billing cycle of 30 days. +18. ADB = $4,312.00, annual interest rate 9.99%, billing cycle of 31 days. +19. ADB = $563.38, annual interest rate 17.9%, billing cycle of 30 days. +20. ADB = $1,043.53, annual interest rate 11.9%, billing cycle of 31 days. +672 +6 • Money Management +Access for free at openstax.org + +In the following exercises, find the balance on the credit card with the given interest charge and balance before interest +was charged. The cards were not paid off previously. +21. Balance before interest is $1630.00, interest charge is $16.48. +22. Balance before interest is $621.00, interest charge is $7.81. +23. Balance before interest is $1,380.00, interest charge is $15.35 +24. Balance before interest is $2,774.00, interest charge is $44.05. +25. Alsaggr’s credit card charges 11.9% annual interest. His current billing period is from April 1 to April 30. The +balance on April 1 was $1,598.00. Use Alsaggr’s following transactions to determine his balance due at the end of +the billing cycle. +Date +Activity +Amount +1-Apr +Billing Date Balance +$1,598.00 +9-Apr +Gasoline +$51.24 +9-Apr +Food +$105.56 +9-Apr +Payment +$675.00 +13-Apr +Camping Trip +$229.75 +21-Apr +Gasoline +$38.45 +22-Apr +Gifts +$148.88 +22-Apr +Food +$49.75 +30-Apr +Gym Payment +$74.99 +26. Marisa’s credit card charges 8.9% annual interest. Her current billing period is from September 1 to September 30. +The balance on September 1 was $384.25. Use Marisa’s following transactions to determine her balance due at the +end of the billing cycle. +Date +Activity +Amount +1-Sep +Billing Date Balance +$384.25 +2-Sep +Food +$94.54 +5-Sep +Gasoline +$25.65 +5-Sep +Internet +$39.99 +9-Sep +Insurance +$174.52 +16-Sep +Food +$83.54 +16-Sep +Day Care +$350.00 +20-Sep +Tires +$2,337.56 +21-Sep +Child Clothing +$27.65 +6.10 • Credit Cards +673 + +Date +Activity +Amount +21-Sep +Gasoline +$31.00 +28-Sep +Television +$299.95 +27. Haley’s credit card charges 18.9% annual interest. Her current billing period is from October 10 to November 9. +The balance on October 10 was $930.50. Use Haley’s following transactions to determine her balance due at the +end of the billing cycle. +Date +Activity +Amount +10-Oct +Billing Date Balance +$930.50 +11-Oct +Clothing +$350.00 +14-Oct +Computer +$865.84 +17-Oct +Food +$106.51 +21-Oct +Payment +$700.00 +21-Oct +Restaurant +$134.52 +21-Oct +Hotel +$387.56 +30-Oct +Hockey Game +$76.47 +5-Nov +Memorabilia +$150.00 +5-Nov +Restaurant +$94.45 +6-Nov +Gasoline +$49.19 +28. Pavly’s credit card charges 10.9% annual interest. His current billing period is from February 15 to March 16 in a +non-leap year. The balance on February 15 was $292.82. Use Pavly’s following transactions to determine his +balance due at the end of the billing cycle. +Date +Activity +Amount +15-Feb +Billing Date Balance +$292.82 +21-Feb +Food +$64.57 +22-Feb +Gasoline +$31.50 +22-Feb +Food +$71.94 +28-Feb +Insurance +$133.25 +28-Feb +Payment +$100.00 +674 +6 • Money Management +Access for free at openstax.org + +Date +Activity +Amount +3-Mar +Gasoline +$26.61 +12-Mar +School Trip +$300.00 +6.11 Buying or Leasing a Car +Figure 6.25 The choice to lease or buy is based on cost and other concerns. (credit: “Car Insurance” by Pictures of Money/ +Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Evaluate basics of car purchasing. +2. +Compute purchase payments and identify the related interest cost. +3. +Evaluate the basics of leasing a car. +4. +Identify and contrast the pros and cons of purchasing versus leasing a car. +5. +Investigate the types of car insurance. +6. +Solve application problems involving owning and maintaining a car. +There are people who don’t need a car and won’t purchase one. But for many people, whether or not to have a car is not +a question. Having a car is a basic necessity for these people. +Obtaining a car can be daunting. The models, the features, the additional costs, and finding funding are all steps that +need to be taken. One of the big decisions is whether to buy the car or to lease the car. This section will address some of +the issues associated with each option. +The Basics of Car Purchasing +The biggest questions you will answer before purchasing a car are, what do you want and what do you need? +Does it have to be new? Does it have to be a make and model you are familiar with? Does it have to have assisted +driving? What other details are important to you? For a new vehicle, every feature beyond standard features comes with +additional cost, which leads to the question that constrains all of your decisions about a car. How much can you afford to +spend on a car? +What you can afford must include insurance costs (discussed later in this section) and maintenance and upkeep. Once +you have this in mind, you can search for a car that matches, as closely as possible, what you want and can afford. Most, +if not all, dealers have websites that you can search through to identify the car you want. If new cars are not affordable, +used cars cost less but come with the wear and tear of use. +6.11 • Buying or Leasing a Car +675 + +The sticker price of the car, called the manufacturer’s suggested retail price (MSRP), or the negotiated price you arrive at, +isn’t the end of the cost to buying a car. There are many fees that accompany the purchase of the car, and perhaps even +sales tax. These include but aren’t necessarily limited to the following: +• +the title and registration fee, which includes registering your car with the state, getting the license plate, and +assigning the title of the car to the lender. This cannot be avoided. +• +a destination fee, which covers the cost of delivering the vehicle to the dealer +• +a documentation fee, sometimes referred to a processing fee of handling fee, is the cost of all the paperwork the +dealer did to get you the car +• +a dealer preparation fee, which is for washing the car and other preparation of that sort. You should try to +negotiate that out of the cost of the dealer tries to charge for that +• +extended warranties and maintenance plans, which help cover some of the costs of caring for the car. +• +Sales tax. +You could pay for these immediately, but they are often added to the financing of the car, meaning they become part of +the principal of your loan. +EXAMPLE 6.98 +Total Cost to Purchase a Car +Nichole negotiates with her car dealership so that the price is $21,800. She needs to pay the 6.75% sales tax on the car. +Other fees are $31.00 for title and registration, $1,000 in destination fees, and a documentation fee of $175. What is the +total cost of Nichole’s car? +Solution +We add the car’s sales cost, sales tax and all other fees to arrive at this value. The sales tax is 6.75% on the price she +negotiated, so is +. Adding these up, we have a total cost of +. +YOUR TURN 6.98 +1. Luther negotiates for the price of his car, reaching agreement at $28,975. He needs to pay 8% in sales tax, 2.1% +in ownership tax, a $950 destination fee, processing fees totaling $370, and registration fee of $617. What is the +total cost for Luther’s purchase? +One way to bring down payments on a car is to provide a down payment or a trade in. This is money applied to the +purchase price before financing happens. Be warned, the sales tax applies to the full purchase price! If you reduce the +amount financed, the payments respond by going down. This often becomes part of the negotiating process. +EXAMPLE 6.99 +Total Cost to Purchase a Car with Down Payment +Sophia negotiates a $19,800 price for her new car. The sales tax is 9.5% in her area, and the dealership charges her $300 +in documentation fees. Her title, plates, and registration come to $321.50. The dealership adds to this a destination fee +of $1,100. If she places a down payment of $5,000 on the car, what is the total she will finance for the car? +Solution +The price was $19,800. The sales tax of 9.5% is based on this number. The sales tax comes to +. +Adding all the fees to the price and the sales tax brings the total cost of the car to +. Her down payment is applied to this number, so the $5,000 +is subtracted from $23,402.50. The subtraction yields the amount to be financed, which is $18,402.50. +YOUR TURN 6.99 +1. Carlos buys a car with a negotiated price of $36,250. The sales tax in his region is 6.5%. The dealer charges a +676 +6 • Money Management +Access for free at openstax.org + +$1,200 destination fee and a $450 documentation fee. He must pay for title, registration, and license plates, +which come to $21.50. If he has a $7,500 down payment, how much will he have to finance? +When purchasing a car, the total cost to obtain the car is not the only factor in your monthly price. You will also pay an +interest rate for the loan you obtain. The interest rate you will get is dependent on your credit score (see The Basics of +Loans). But you can choose from different lenders. The dealership will likely offer to finance your car loan. Frequently, +dealerships offer special financing with very low rates. This is to help move inventory, and may indicate their desire to +make sales. This might make negotiating easier. Even if the dealership offers financing, check with your bank or credit +union to determine the interest rates they are offering. To reduce your payments, choose the lowest rate you can find. +Purchase Payments and Interest +Whether or not you buy a new car or a used car, if you finance the purchase, you are taking out a loan. The interest rates +available for used cards are frequently higher than those for new cars. These loan payments work exactly the same way +as other loans do as far as payments are concerned. The payment function comes from The Basics of Loans. The +difference between financing a new car or a used car is that financing a new car typically comes with a lower interest rate +and a longer term that financing a used car. +FORMULA +The payment, +, per period to pay off a loan with beginning principal +is +, where +is +the annual interest rate in decimal form, +is the term in years, and +is the number of payments per year (typically, +loans are paid monthly making += 12). +Note, payment to lenders is always rounded up to the next penny. +Often, the formula takes the form +, where +is the interest rate per period (annual rate +divided by the number of periods per year), and +is the total number of payments to be made. +EXAMPLE 6.100 +New Car Payments +In the following, calculate the monthly payment using the given total to be financed, the interest rate, and the term of +the car loan. +1. +Total to be financed is $31,885, interest rate is 2.9%, for 5 years. +2. +Total to be financed is $22,778, interest rate is 4.5%, for 6 years. +Solution +1. +The amount to be financed is the principal, +, which is $31,885. The rate +is 0.029, and the term is += 5 years. These +are monthly payments, so += 12. Substituting and calculating, we find the monthly payment to be +2. +The amount to be financed is the principal, +, which is $22,778. The rate +is 0.045, and the term is += 6 years. These +are monthly payments, so += 12. Substituting and calculating, we find the monthly payment to be +6.11 • Buying or Leasing a Car +677 + +YOUR TURN 6.100 +In the following, calculate the monthly payment using the given total to be financed, the interest rate, and the term +of the car loan. +1. Total to be financed is $18,325, interest rate is 6.75%, for 4 years. +2. Total to be financed is $41,633, interest rate is 3.9%, for 6 years. +EXAMPLE 6.101 +Used Car Payments +Calculate the monthly payment for the used car if the total to be financed is $16,990, the interest rate is 7.5%, and the +loan term is 3 years. +Solution +The amount to be financed is the principal, +, which is $16,990. The rate +is 0.075, and the term is += 3 years. These are +monthly payments, so += 12. Substituting and calculating, we find the monthly payment to be +YOUR TURN 6.101 +1. Calculate the monthly payment for a car loan that has $21,845 being financed at an interest rate of 6.3% for 4 +years. +The Basics of Leasing a Car +Leasing a car is an alternative to purchasing a car. It is still a loan, and acts like one in many respects. They typically last +either 24 months or 36 months, though other terms are available. Leases also come with mileage limits, frequently +10,000, 12,000, or 15,000 miles per year. When the lease is over, the car is returned to the dealer. At that time there may +be fees that have to be paid, such as for damage to the car or for extra miles driven over the limit. +There are two components to lease costs. One is the monthly payment for the lease. The other is the fees for leasing, +These often are paid before the lease is complete. These include: +678 +6 • Money Management +Access for free at openstax.org + +• +a down payment, which is your initial payment that is applied to the price of the car. It reduces the amount you +finance, much the same as when you purchase a car. It is recommended that this be negotiated away. +• +the acquisition fee, sometimes called the bank fee. This is the money charge for the company to set up the lease. It +is essentially a paperwork fee. It is not likely that this can be negotiated. +• +a security deposit, which might be required. It is about the same as 1 month’s payment for the lease. The deposit is +returned to you if the car is in good shape at the end. This can be negotiated away. +• +disposition fees, which cover the cost the company will incur when they take your car back and are typically +between $200 and $450. +• +the title, registration, and license fees, just as with the purchase of a car. +• +sales tax, which will likely be applied. The sales tax only covers the depreciated portion of the car (more on +depreciation later) in many states. Since this depends on the state in which the car is leased, you should determine +the sales tax rules for where you lease the car. +As you can imagine, this can come to a fairly high dollar amount. +EXAMPLE 6.102 +Cost to Obtain a Lease +Donna wants to lease a Subaru Outback in Eden, New York. Find the total cost of obtaining her lease if there is no down +payment, $175.00 in acquisition fees, a security deposit of $300.00, $350.00 in disposition fees, $102.50 in title and +registration fees, and sales tax of $3,536.05. +Solution +Adding these values together we find the total cost is $4,463.55. +YOUR TURN 6.102 +1. RJ wants to lease a Tahoe in Salt Lake City, Utah. Find the total cost of obtaining his lease if he has a $5,000.00 +down payment, $225 in acquisition fees, a security deposit of $750.00, $500.00 in disposition fees, and sales tax +of $4,092. +PEOPLE IN MATHEMATICS +Zollie Frank +Zollie Frank and Armund Shoen founded one of the original leasing companies, Four Wheels, in 1939. Their company +leased automobiles to corporations. They began by leasing 5 cars to the Petrolager pharmaceutical company in year +one. This saved Petrolager money and provided a steady cash flow to the Four Wheels business. In year two, the +number of cars leased to Petrolager was 75. Their new idea was to lease cars directly to companies for one year. +Previously, such companies might pay for mileage, gas, and a partial down payment. Sadly, the salesmen who were +being so helped often left the company before the car was paid for, and so the company lost the down payment +money. +The lease was for $45 per month per car for one year. +You have some obligations when you lease a car. You must keep the car in good condition, cleaned, maintained, and free +of anything more than minor damage. If the car is in poor condition when the car is returned, you will be responsible for +the cost to bring the car to an acceptable condition. You are also expected to keep the mileage under its limit. If you go +over, you will pay 10 to 25 cents per mile over. +Lease Payments +Lease payments are similar to regular loan payments, but have some other details. Calculating a lease payment involves +knowing the following values: +• +The price of the car. This is the cost you would pay for the car after applying all discounts, incentives, and +6.11 • Buying or Leasing a Car +679 + +negotiations. +• +Residual Value. This is the manufacturer's estimate of the car's value after a set period of time. The residual value is +expressed as a percentage of the manufacturer’s suggested retail price (MSRP). +• +Months. This is the length of the lease. Most leases are either 24- or 36-month leases, but other terms are available. +• +Monthly Depreciation. The monthly depreciation is the difference between the price of the car and the residual +value, divided by the number of months of the lease, and represents the monthly loss of value of the car while it’s +being used. +• +Money Factor (MF). This is the interest rate, but expressed in a different way for a lease. Converting from the +money factor to the annual percentage rate (APR) is done by multiplying the MF by 2400. Naturally, converting an +APR to a MF is done by dividing the APR by 2400. +FORMULA +The monthly depreciation for a car, MD, is +, +is the price paid for the car, +is the residual value of the car, +and +is the number of months of the lease. +The annual percentage rate for a lease is +, where MF is the money factor of the lease. The MF for a +lease is +. +EXAMPLE 6.103 +Monthly Depreciation of a Car +1. +The purchase price of a car is $25,000. Its residual price is $14,500. What is its monthly depreciation for a 36-month +lease? +2. +The purchase price of a car is $30,000. Its residual price is $18,600. What is its monthly depreciation for a 24-month +lease? +Solution +1. +The monthly depreciation formula is +, Substituting $25,000 for +, $14,500 for +, and 36 for +, we find +MD to be +. +2. +The monthly depreciation formula is +, Substituting $30,000 for +, $18,600 for +, and 24 for +, we find +MD to be +. +YOUR TURN 6.103 +1. The purchase price of a car is $27,500. Its residual price is $17,875.00. What is its monthly depreciation for a +24-month lease? +EXAMPLE 6.104 +Converting Between APR and MF +1. +Find the annual percentage rate if the money factor is 0.00001875. +2. +Find the money factor if the APR is 6.25%. +Solution +1. +The APR is the money factor times 2400, so +. Expressed as a +percentage, the APR is 4.5%. +2. +The MF is the APR divided by 2400, so +. +YOUR TURN 6.104 +1. Find the annual percentage rate if the money factor is +. +680 +6 • Money Management +Access for free at openstax.org + +2. Find the money factor if the APR is 7.9%. +Once the values above are found, the payment for the lease can be calculated. +FORMULA +The payment, +, for a lease is +, where +is the price paid for the car, +is the +residual value of the car, +is the number of months of the lease, and MF is the money factor for the lease. +EXAMPLE 6.105 +Calculating Car Lease Payments +Calculate the lease payments for car with the following price, residual price, length of lease, and money factor or APR. +1. +Price is $28,344, residual price is $18,140.16, 24-month lease, money factor is 0.000025. +2. +Price is $22,500, residual price is $13,050, 36-month lease, APR is 7.5%. +Solution +1. +Substituting the values += $28,344, += $18,140.16, += 24 and MF = 0.000025 into the formula and calculating, the +monthly lease payment is +2. +Given the APR, we find the MF which is +. Substituting the values += +$28,344, += $18,140.16, += 24 and the MF into the formula and calculating, the monthly lease payment is +YOUR TURN 6.105 +Calculate the lease payments for car with the following price, residual price, length of lease, and money factor or +APR. +1. Price is $38,750, residual price is $18,140.16, 36-month lease, money factor is 0.000035. +2. Price is $45,600, residual price is $21,312.50, 24-month lease, APR is 11.7%. +Comparing Purchasing and Leasing +When deciding to buy or lease a car, the differences between the two options should be carefully evaluated. The +following is a list of points of comparison between the two. +• +The payments for a lease are likely less than the payment for purchasing. +• +When leasing, you get a new car after the lease term is over, typically 24 or 36 months. Buying the car means the +same car is driven until it is re-sold and a new one bought. Essentially, leasing a car is equivalent to renting a car. +• +The leased car is new, so all warrantees are in force and you drive the car during its best years. When the car is +purchased, it may be kept past its warrantees and may be driven until it is quite old. +• +Each time you lease a new car, all the fees and taxes must be paid again. When buying a car, these fees are only paid +once. +• +Leasing contracts carry restrictions on the mileage you can drive per year, and going over incurs more cost at the +6.11 • Buying or Leasing a Car +681 + +end of the lease. Buying the car means no such mileage limits. +• +When leasing, you are obligated to keep the vehicle in good condition and maintained according to the dealer’s +schedule. Some dealerships will even pay for oil changes over the life of the lease. When the car is purchased, the +upkeep schedule is the choice of the owner. +• +When a car is purchased and kept for long enough, the warranty expires and the owner is responsible for all +maintenance items and repairs. The warrantee for a car won’t expire during the lease term. +• +When a new car is purchased and the loan is paid off the car is still owned by the buyer and may be traded in when +a new car is to be purchased. When leasing, the car is returned to the dealer when the lease term is over. +When deciding between the two, you are choosing between these features. If you aren’t willing to drive an older car or +deal with the upkeep that accompanies an older car, you may want to lease. This means you will need to pay those +beginning costs each time the lease is up. If you want to own the car after the payments are over, then you may want to +buy a car. This means you are paying for all the upkeep after the warrantees expire, but you have no limits on mileage +and own the car at the end. It really depends on your preferences. +EXAMPLE 6.106 +Lease or Buy +In the following, determine if a lease or purchase of a car is better. +1. +Joyce is concerned with large repairs and does not want to deal with them. +2. +Maurice prefers to drive newer cars. +Solution +1. +Since Joyce does not want to deal with repairs, so leasing would be a better choice. This way, the warranty covers +most of the big repairs that could need to be done. +2. +Since Maurice likes to drive new cars, leasing is a better option, since he will lease a new car every 2 to 3 years. +YOUR TURN 6.106 +1. Christopher does a lot of driving, averaging over 20,000 miles per year. +2. John eventually wants to not make car payments. +Car Insurance +Car insurance is meant to cover costs associated with accidents involving cars. Most states (all except New Hampshire +and Virginia) require some insurance. Without insurance, the state may not let you get a license for your car or register +your car. Your state’s requirements can be hard to follow. Fortunately, insurance companies and brokers will make sure +your insurance is sufficient for your state and will warn you if you try to not meet the requirements. Of course, they may +offer more than what is sufficient, so it is your responsibility to determine how much coverage you want, as long as the +minimum insurance requirements are met. The cost of insurance should be accounted for when evaluating the +affordability of buying or leasing a car. +Whether your car is leased or owned, you do need insurance. This contributes to the cost of having the vehicle. Leasing +or owning makes no difference to the insurance company you choose, because they are insuring you based on what you +are driving, your driving record, and other information about you including where you live and your age. These +insurance policies have many components that address different costs that can come from auto accidents. This may +make details confusing, and you may not realize what you are paying for until you must use it. Here is a brief outline of +the different components of auto insurance, many of which are required by the state that issues your driver’s license. +• +Liability insurance is mandatory coverage in most states. Liability insurance covers property damage and injuries +to others should you be found legally responsible for an accident. You are required to have the minimum amount of +coverage, as determined by your state, in both areas. +• +Collision insurance is insurance covering the damage caused to your car if involved in an accident with another +vehicle. +• +Comprehensive insurance is an extra level of coverage if involved in an accident with another vehicle and covers +other things like theft, vandalism, fire, or weather events as outlined in your policy. There is a deductible assigned to +each type of insurance, an amount that you pay out of pocket before your comprehensive coverage takes effect. +Comprehensive insurance is often required if you lease or finance the purchase of a vehicle. +682 +6 • Money Management +Access for free at openstax.org + +• +Uninsured or underinsured motorist insurance: If you are hit by an uninsured or underinsured motorist, this +insurance will help pay medical bills and damage to your car. +• +Medical payments insurance is mandatory in some states and helps pay for medical costs associated with an +accident, regardless of who is at fault. +• +Personal injury protection insurance is coverage for certain medical bills and other expenses due to a car +accident. Other covered expenses may include loss of income or childcare, depending on your policy. +• +Gap insurance is designed to cover the gap between what is owed on the car and what the car is worth in the event +your car is a total loss. +• +Rental reimbursement insurance is coverage for a rental car while your car is under repair resulting from an +accident. +You can also purchase other special insurance policies, such as classic car insurance, new car replacement insurance, and +sound system replacement insurance, to name a few. It is important that you determine exactly what you need, as +insurance policies can be expensive and vary according to your age, driving history, and where you live. +EXAMPLE 6.107 +Types of Insurance +1. +Which component of insurance pays if you are in an accident with a motorist without insurance? +2. +Which component of insurance pays for the remaining principal owed on your car in the case of a total loss? +Solution +1. +Uninsured motorist insurance covers accidents with those who have no insurance. +2. +Gap insurance will cover the gap between what is owed on the car and what it is worth if an accident results in a +total loss. +YOUR TURN 6.107 +1. Which component of insurance covers medical bills in the event of an accident? +2. Which component of insurance pays for costs to others if you have an accident and are found legally +responsible for those damages? +EXAMPLE 6.108 +Monthly Cost of Owning a Car +If your car payment is $287.50 per month and your car insurance is $930 every 6 months, what is the cost of the car per +month when accounting for the insurance? +Solution +The cost of the car including insurance is the monthly payment, $287.50, plus the monthly cost of the insurance. The +insurance cost per month is +since the insurance cost is for every 6 months. Adding those the cost with +insurance is $442.50. +YOUR TURN 6.108 +1. If your car payment is $410.86 per month and your car insurance is $2,190 per year, what is the cost of the car +per month when accounting for the insurance? +Maintaining a Car +Cars are not a buy it and forget it item. They require upkeep, which adds to the cost of owning the car. Tires, brakes, and +wipers need replacing. Oli changes, inspections, so many things other than gasoline. Below is a list of some maintenance +requirements for cars, along with cost and roughly how often they should happen. +6.11 • Buying or Leasing a Car +683 + +Maintenance +Frequency +Cost Range +New Tires +Every 5 years +$25–$300 per tire +Oil Change +Every 3,000–6,000 miles +$35–$75 +Wipers +Every 6–12 months +$20–$40 +Inspection +Annual +$10–$50 +Brake pads +10,000–20,000 miles +$200–$300 +Air Filter +15,000–30,000 miles +$35–$80 +When designing a budget, these expected costs should be accounted for. Extra money per month should be saved in +addition to this budget category, to handle unanticipated, and perhaps very costly, repairs. +EXAMPLE 6.109 +Estella needs to budget for her car maintenance. She expects to buy new tires each 4 years, which will cost her $480 to +replace them all. Oil changes near her cost $49.99, and she believes she will get one every 4 months. Her inspection +costs $15 per year. Wipers for her car cost $95 for all three and she anticipates changing them every year. She drives less +than 30,000 miles per year, so she plans to replace the air filter once per year. The air filter for her car costs $57.50. How +much should she budget per month to cover these costs? +Solution +Her yearly costs are the wipers, inspection, and tires, which total $167.50. Tires will be bought every 5 years, so per year +she should budget $96. Her oil changes, which will happen three times per year, cost $49.99 each, so she’ll spend +$149.97 for the year on oil changes. Adding these up, her yearly budget should include $413.47 for maintenance. +Dividing by 12 gives the monthly budget for maintenance, which is $34.46 (rounded up to the next penny). +YOUR TURN 6.109 +1. Natalie needs to budget for her car maintenance. She expects to buy new tires each 5 years, which will cost her +$390 to replace them all. Oil changes near her cost $59.99, and she believes she will get one every 3 months. Her +inspection costs $25 per year. Wipers for her car cost $115 for all three and she anticipates changing them every +year. She drives less than 30,000 miles per year, so she plans to replace the air filter once per year. The air filter +for her car costs $46.25. How much should she budget per month to cover these costs? +Check Your Understanding +69. What is destination fee? +70. What is a title and registration fee? +71. Calculate the monthly payment if the total to be financed is $34,570, the interest rate is 3.5%, for 5 years. +72. What might be different for a used car loan and a new car loan? +73. What fees are similar between leasing a car and buying a car? +74. If the cost of a car is $30,000, and the residual value of the car after 3 years is $18,000, what is the monthly +depreciation for the car? +75. If the money factor for a lease is 0.000015625, what is the annual interest rate? +76. For people who do not mind driving an older car or maintaining a car, which is preferable, a lease or purchasing? +684 +6 • Money Management +Access for free at openstax.org + +77. What does collision insurance cover? +SECTION 6.11 EXERCISES +1. What are the two main questions to answer before buying a car? +2. What is a documentation fee? +3. What is a dealer preparation fee? +4. What is a down payment? +5. How long do leases typically last? +6. Can you drive a leased car any mileage? +7. What are three fees associated with leasing a car? +8. What is the monthly depreciation for a leased car? +9. How are the money factor and the annual percentage rate related? +10. Name two advantages to leasing over buying a car. +11. Name two advantages to buying a car over leasing a car. +12. What does liability insurance cover? +13. What does collision insurance cover? +14. Many expenses associated with a car can be anticipated. Name three maintenance expenses that can be +anticipated. +In the following exercises, find the total cost to purchase the car. +15. Alexia negotiates a purchase price of $17,850 for her new car. The sales tax in her area is 6.5%. Her license, +plates, and registration come to $285.00. The dealership charges her a $600 destination fee and a $150 +processing fee. How much will she finance in total for the car? +16. Stephanie negotiates a purchase price of $25,670 for her new car. The sales tax in her area is 8.0%. Her license, +plates, and registration come to $389.00. The dealership charges her a $700 destination fee and a $345 +processing fee. How much will she finance in total for the car? +17. Matthew negotiates a purchase price of $35,100 for her new car. The sales tax in his area is 7.25%. His license, +plates, and registration come to $325.00. The dealership charges him a $900 destination fee and a $125 +processing fee. How much will he finance in total for the car? +18. Madisyn negotiates a purchase price of $45,800 for her new car. The sales tax in her area is 7.25%. Her license, +plates, and registration come to $199.00. The dealership charges her a $1,000 destination fee and a $275 +processing fee. How much will she finance in total for the car? +In the following exercises, calculate the car payment based on the total financed and the interest rate. +19. Total to be financed is $36,775, interest rate is 2.75%, for 6 years. +20. Total to be financed is $29,350, interest rate is 3.9%, for 5 years. +21. Total to be financed is $27,180, interest rate is 1.99%, for 7 years. +22. Total to be financed is $15,489, interest rate is 6.75%, for 4 years. +In the following exercises, find the total cost to obtain the lease. +23. A $2,000.00 down payment, $120 in acquisition fees, a security deposit of $350.00, $200.00 in disposition fees, +and sales tax of $2,860. +24. No down payment, $260 in acquisition fees, a security deposit of $450.00, $400.00 in disposition fees, and sales +tax of $3,155.00. +25. A $4,000 down payment, $360 in acquisition fees, a security deposit of $900.00, $1,000.00 in disposition fees, +and sales tax of $4,275. +26. A $7,000.00 down payment, $225 in acquisition fees, a security deposit of $800.00, $675.00 in disposition fees, +and sales tax of $3,673. +In the following exercises, find the monthly depreciation of the car. +27. The purchase price of a car is $34,000. Its residual price is $23,500. What is its monthly depreciation for a +24-month lease? +6.11 • Buying or Leasing a Car +685 + +28. The purchase price of a car is $23,500. Its residual price is $11,750. What is its monthly depreciation for a +36-month lease? +In the following exercises find the APR based on the MF. +29. MF = 0.00004125 +30. MF = +In the following exercises find the MF based on the APR. +31. 8.75% +32. 6.25% +In the following exercises, find the lease payment based on the given information. +33. Price is $41,700, residual price is $27,105, 24-month lease, money factor is 0.000025. +34. Price is $22,165, residual price is $12,855.70, 24-month lease, money factor is 0.0000275. +35. Price is $30,650, residual price is $16,857.50, 36-month lease, APR is 8.25%. +36. Price is $24,800, residual price is $14,384, 36-month lease, APR is 5.85%. +37. Sara needs to budget for her car maintenance. She expects to buy new tires each 3 years, which will cost her $540 +to replace them all. Oil changes near her cost $39.99, and she believes she will get one every 4 months. Her +inspection costs $20 per year. Wipers for her car cost $115 for all three and she anticipates changing them every +year. She drives less than 30,000 miles per year, so she plans to replace the air filter once per year. The air filter for +her car costs $36.25. She plans to get new brake pads every year, which cost her $267. How much should she +budget per month to cover these costs? +38. Mwibeleca needs to budget for his car maintenance. He expects to buy new tires each 3 years, which will cost her +$560 to replace them all. Oil changes near his cost $69.99, and he believes he will get one every 4 months. His +inspection costs $25 per year. Brake pads will cost $215 each year. Wipers for his car cost $143.50 for all three and +he anticipates changing them every year. He drives less than 30,000 miles per year, so he plans to replace the air +filter once per year. The air filter for his car costs $62.88. How much should Mwibeleca budget per month to cover +these costs? +39. John will either lease or buy a car. The total cost to purchase the car is $35,830, and he would finance the car for 5 +years at 2.99%. If he leases, he would pay $3,287.50 for the lease, and then his payments would be based on a +price of $32,750, a residual price of $20,305, 36 months, with a money factor of 0.00001725. Compare the +payments to purchase the car to the payments of the lease plus the lease cost divided by 36. +40. Zachary will either lease or buy a car. The total cost to purchase the car is $22,945, and he would finance the car for +6 years at 1.99%. If he leases, he would pay $2,387.75 for the lease, and then his payments would be based on a +price of $21,350, a residual price of $12,390.30, 36 months, with a money factor of 0.000018375. Compare the +payments to purchase the car to the payments of the lease plus the lease cost divided by 36. +686 +6 • Money Management +Access for free at openstax.org + +6.12 Renting and Homeownership +Figure 6.26 Purchasing a home is a big investment, while renting is a lower cost alternative. (credit: "New construction, +new development house for sale" by jongorey/houseandhammer.com, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Evaluate advantages and disadvantages of renting. +2. +Evaluate advantages and disadvantages of homeownership. +3. +Calculate the monthly payment for a mortgage and related interest cost. +4. +Read and interpret an amortization schedule. +5. +Solve application problems involving affordability of a mortgage. +After renting an apartment for 10 years, you realize that it may be time to purchase a home. Your job is stable, and you +could use more space. It is time to investigate becoming a homeowner. What are the things that you must consider, and +what is the financial benefit of owning as opposed to renting? This section is about the advantages, disadvantages, and +costs of homeownership as opposed to renting. +Advantages and Disadvantages of Renting +When renting, you will likely sign a lease, which is a contract between a renter and a landlord. A landlord is the person +or company that owns property that is rented. The lease will detail your responsibilities, restrictions on activities, +deposits, fees, maintenance, repairs, and rent during the term of the lease. It also defines what your landlord can, and +cannot, do with the property while you occupy the property. +Like leasing a car, there are advantages to renting but also some disadvantages. Some advantages are: +• +Lower cost. +• +Short-term commitment. +• +Little to no maintenance cost. The landlord pays for or performs most maintenance. +• +You need not stay at end of lease. Once the lease term is over (the lease is up), you are not obligated to stay. +• +If renting in an apartment complex, there may be a pool, gym, or community room for renters to use. +Of course, there are disadvantage too: +• +No tax incentives. +• +Housing cost is not fixed. When the lease is up, the rent can change. +• +No equity. When you are done living in a rental, you have built no value. +• +Restrictions on occupants. There may be a limit on how many can live in the apartment. +• +Restrictions on decorating. The property is not yours, so any decorating or improvements need landlord permission. +• +Limits on pets. Permission for pets, and their number and type, will be set forth in the lease. +• +May not be able to remain when lease term is over. The landlord can, at the end of your lease, invite you to leave. +• +The building may be sold, and the new landlord may institute changes to the lease when the previous lease expires. +Renting has fees to be paid at the start of the lease. Typically, when you rent, you will pay first and last months’ rent and +a security deposit. A security deposit is a sum of money that the landlord holds until the renter leaves the rental +property. The deposit will cover repairs for damage to the apartment during the renter’s stay but may be returned if the +apartment is in good condition. If your landlord runs a credit check on you, the landlord may charge you for that. +6.12 • Renting and Homeownership +687 + +Advantages of Buying a Home +The advantages to buying a home mirror the disadvantages of renting, and the disadvantages of home ownership +mirror the advantages of renting. +Some advantages to buying a home are: +• +There are tax incentives. The interest you pay for your mortgage (more on that later) is deductible on your federal +income tax. +• +There are no restrictions on pets or occupants, unless laws in your area specify limits for homes. +• +You can redecorate any way you wish, limited only by the laws in your area. +• +Once your mortgage is set with a fixed-interest rate, your housing cost is fixed. +• +Your home grows equity, that is, the difference between what you owe and what the house is worth grows. You can +use the equity to secure loans, and you recover the equity (and more if you’re fortunate) when you sell the house. +• +As long as you pay your mortgage and maintain the home to the standards of your community, you can stay as long +as you wish. +Some disadvantages to home ownership are: +• +The cost is higher than renting. Mortgages and associated costs are typically higher than rent for a similar living +space. +• +The owner is responsible for upkeep, maintenance, and repairs. These can be extremely costly. +• +The owner cannot walk away from the property. It can be sold, but simply leaving the property, especially if not paid +off yet, has serious consequences. +The big question of affordability looms large over the decision to rent or buy. Renting, strictly from an affordability +viewpoint, comes with much less initial outlay and smaller commitment. If you do not have sufficient income to regularly +save for possibly expensive repairs, or your credit isn’t quite as good as it needs to be, then renting may be the best +choice. Of course, even if you can afford to buy a home, you may choose to rent based on the comparative advantages. +VIDEO +Rent or Buy (https://openstax.org/r/Rent_or_Buy) +Buying a home really involves two buyers. You and the mortgage company. The mortgage company has interest in the +home, as they are providing the funds for the home. They want to protect their investment, and many fees are about the +bank as much as the buyer. They fund a mortgage based on the value they assign the property. Not you. This means they +will want some certainty that the home is sound, and you are a good investment. +WHO KNEW? +Closing Costs +When a home is bought, there are many costs that need to be paid at the time of purchase, which are lumped under +the term closing costs. At the start of 2022, the average closing costs for a single-family home exceeded $6,800. These +costs include: +• +The appraisal fee, which is what is paid to someone to establish the home’s worth. The value of the home to the +bank may differ from what the home is listed for, or what an app tells you the home is worth. It may run +approximately $350. +• +The home inspection fee. The inspection should reveal any problems with the house that will need to be fixed +either before or after you obtain the home. +• +The title search. The is a records search to insure there are no issues with who actually owns the property. It can +cost about 0.5% to 1% of the amount you are financing. +• +Prepaid taxes. You will need to pay about 6 months of taxes at the time of purchase. +• +The credit report fee. This is a fee for checking your credit. You might pay $25 or more for this. +• +The origination fee. This is the price the mortgage company charges you to cover the costs of creating the +mortgage. This could be 0.5% to 1% (or more) of the amount you are borrowing. +• +The application fee. This is just a processing fee and could come to several hundred dollars. +• +The underwriting fee. This covers the cost of verifying your financial qualifications. It could be a flat fee, or some +688 +6 • Money Management +Access for free at openstax.org + +small percentage of the amount financed. Such as 0.5% or 1%. +• +Attorney fees. If you use an attorney, you will have to pay the attorney. +• +State of local fees. This may include a filing fee charged by the county or municipality in which you reside. +That’s a long list, and it is not even complete. When buying, be prepared to see these costs. It can be surprising. But in +the end, you will have equity in the home, which means when you sell your home, you will get some of your money +back. +VIDEO +Closing Costs (https://openstax.org/r/closing_costs) +In the end, you must weigh your options and carefully consider your priorities in choosing to rent or buy a home. +Mortgages +Some people will purchase a home or condo with cash, but the majority of people will apply for a mortgage. A mortgage +is a long-term loan and the property itself is the security. The bank decides the minimum down payment (with your +input), the payment schedule, the duration of the loan, whether the loan can be assumed by another party, and the +penalty for late payments. The title of the home belongs to the bank. +Since a mortgage is a loan, everything about loans from The Basics of Loans holds true, including the formula for the +payments. +Monthly Mortgage Payments +The formula to calculate your monthly payments of principal and interest uses APR as the annual interest rate. +FORMULA +The payment, +, per month to pay down a mortgage with beginning principal +is +, where +is the annual interest rate in decimal form and +is the number of years of +the payment. +Note, payment to lenders is always rounded up to the next penny. +To find the total amount of your payments over the life of the loan, multiply your monthly payments by the number of +payments. +EXAMPLE 6.110 +30-Year Mortgage at 4.8% Interest +Evan buys a house. His 30-year mortgage comes to $132,650 with 4.8% interest. Find Evan’s monthly payments. +Solution +Using the information above, += $132,650, += 0.048 and += 30. Substituting those values into the formula +and calculating, we find the payment is +6.12 • Renting and Homeownership +689 + +His mortgage payment is $695.97. +YOUR TURN 6.110 +1. Paulo buys a house. His 20-year mortgage comes to $153,899 with 4.21% interest. Find Evan’s monthly +payments. +To find the total amount of your payments over the life of the loan, multiply your monthly payments by the number of +payments. This can be useful information, but not too many people reach the end of their mortgage. They tend to move +before the mortgage is paid off. +FORMULA +The total paid, +, on an +year mortgage with monthly payments +is +. +EXAMPLE 6.111 +30-Year Mortgage at 5.35% Interest +Cassandra buys a house. Her 30-year mortgage comes to $99,596 with 5.35% interest. If Cassandra pays off the +mortgage over those 30 years, how much will she have paid in total? +Solution +To find the total paid over the life of the mortgage, use the formula +. To calculate this, the payment +must be found. Using the information above, += $99,596, += 0.0535 and += 30. Substituting those values into the +formula +and calculating, we find the payment is +Using the mortgage payment of $556.16 and += 30 years in the formula +, the total that Cassandra will +pay for the mortgage is $200,217.60. +YOUR TURN 6.111 +1. Arthur buys a house. Their 15-year mortgage comes to $225,879 with 4.91% interest. If Arthur pays off the +mortgage over those 15 years, how much will they have paid in total? +With the principal of the mortgage and how much total is paid over the life of the mortgage, the cost of financing can be +found by subtracting the principal of the mortgage from the total paid over the life of the mortgage. +FORMULA +The cost of financing a mortgage, CoF, is +where +is the mortgage’s starting principal and +is the total +690 +6 • Money Management +Access for free at openstax.org + +paid over the life of the mortgage. +EXAMPLE 6.112 +30-Year Mortgage at 5.35% Interest +Cassandra buys a house. Her 30-year mortgage comes to $99,596 with 5.35% interest. What was Cassandra’s cost of +financing? +Solution +In Example 6.111, we found that the total Cassandra will pay for the $99,569 mortgage is $200,217.60. Subtracting those +we find the cost of financing +. +YOUR TURN 6.112 +1. Arthur buys a house. Their 15-year mortgage comes to $225,879 with 4.91% interest. What was Arthur’s cost of +financing? +WHO KNEW? +Private Mortgage Insurance (PMI) +When you purchase a home, you will have to pay a down payment. This means you have money tied to the property, +which lenders believe makes you less likely to walk away from a property. The amount of the down payment will be +decided between you and the mortgage company. However, if your down payment is less than 20% of the property +value, you will be required to pay private mortgage insurance (PMI). This is insurance you pay for so that the +mortgage company is protected if you default on the loan. It often comes to between 0.5% and 2.25% of the original +loan amount. It increases your monthly payment. Once you reach 20% of the loan value, you can request that the PMI +be dropped. Even if you do not request cancelling the PMI, it will eventually and automatically be dropped. +For more, see this article on ways to get rid of PMI (https://openstax.org/r/mortgage_insurance). +Reading and Interpreting Amortization Tables +Amortization tables were addressed in The Basics of Loans. They are most frequently encountered when analyzing +mortgages. +The amortization table for a 30-year mortgage is quite long, containing 360 rows. A full table will not be reproduced +here. We can, though, read information from a portion of an amortization table. +EXAMPLE 6.113 +Amortization Table for a 30-Year, $165,900 Mortgage +Figure 6.27 shows a portion of an amortization table for a 30-year, $165,900 mortgage. Use that table to answer the +following questions. +1. +What is the interest rate? +2. +How much are the payments? +3. +How much of payment 175 goes to principal? +4. +How much of payment 180 goes to interest? +5. +What’s the remaining balance on the mortgage after payment 170? +6.12 • Renting and Homeownership +691 + +Figure 6.27 Amortization table +Solution +1. +Reading at the top of the table, we see the interest rate is 5.61%. +2. +Reading from the top of the table or from the column labeled Payment, we see the payments are $953.44 per +month. +3. +In the row for payment 175, we see that the amount that goes to principal is $400.44. +4. +In the row for payment 180, we see that the amount that goes to interest is $543.56. +5. +In the row for payment 170, we see the remaining balance is $119,873.35. +YOUR TURN 6.113 +Below is a portion of an amortization table for a 30-year, $228,320 mortgage. Use that table to answer the following +questions. +692 +6 • Money Management +Access for free at openstax.org + +Amortization table +1. What is the interest rate? +2. How much are the payments? +3. How much of payment 235 goes to principal? +4. How much of payment 215 goes to interest? +5. What’s the remaining balance on the mortgage after payment 227? +6. At what payment does the amount that is applied to mortgage finally exceed the amount applied to interest? +Escrow Payments +The last few examples have looked at mortgage payments, which cover the principal and interest. However, when you +take out a mortgage, the payment is sometimes much higher than that. This is because your mortgage company also +has you pay into an escrow account, which is a savings account maintained by the mortgage company. +Your insurance payments will be set by your insurer and the mortgage company will pay them on time for you from your +escrow account. Your property taxes are set by where you live and are typically a percentage of your property’s assessed +value. The assessed value is the estimation of the value of your home and does not necessary reflect the purchase or +resale value of the home. Your property taxes will also be paid on time by the mortgage company from your escrow +account. +For example, in Kalamazoo, Michigan, the effective tax rate for property is 1.69% of the assessed value of the home. +These escrow payments, which cover bills for the home, can increase the monthly payments for your home well beyond +the basic principal and interest payment. +EXAMPLE 6.114 +Adding Escrow Payments to Mortgage Payments +Jenna decides to purchase a home, with mortgage of $108,450 at 6% interest for 30 years. The assessed value of her +home is $75,600. Her property taxes come to 5.7% of her assessed value. Jenna also has to pay her home insurance +every 6 months, which is $744 per six months. How much, including escrow, will Jenna pay per month? +Solution +Using the payment function to find her mortgage payments, +, with += $108,405, += +6.12 • Renting and Homeownership +693 + +0.06, and += 30, her payments are +Jenna also pays into escrow 1/12 of her property taxes per month. Her property taxes are 5.7% of the assessed value of +$75,600, which comes to +. This is an annual tax, so she pays 1/12 of that each month, or +$359.10. Jenna’s home insurance is $744 per 6 months, so each month she pays $124.00 for insurance. Adding these +together, her monthly payment is +. This is quite a bit more than the $649.95 +for the principal and interest. +YOUR TURN 6.114 +1. Destiny decides to purchase a home, with mortgage of $159,195.50 at 5.75% interest for 30 years. The assessed +value of her home is $100,000. Her property taxes come to 5.42% of her assessed value. Destiny also has to pay +her home insurance every 6 months, and that comes to $843 per 6 months. How much, including escrow, will +Destiny pay per month? +Check Your Understanding +78. Does renting or buying have tax advantages? +79. Which has more restrictions, renting or buying? +80. Which has housing cost that does not change? +81. What is the name given to a loan for a home? +82. What are the monthly payments for a 30-year mortgage of $108,993 with 6.14% interest. +83. What is the cost of financing for a 30-year mortgage of $108,993 with 6.14% interest if the mortgage is paid off? +84. Consider the following amortization table. What is the amount of payment 141 that goes to principal? +694 +6 • Money Management +Access for free at openstax.org + +85. What is the name of the account that the mortgage company holds your taxes and insurance in? +SECTION 6.12 EXERCISES +In the following exercises, indicate if the advantage listed is for renting or buying a home. +1. Short-term commitment. +2. Tax advantage. +3. Freedom to remodel. +4. Builds equity. +5. Cost is lower. +6. You do not pay for repairs. +7. No pet restrictions +8. More flexibility to move. +9. Housing cost is fixed. +10. May have other amenities. +In the following exercises, find the mortgage payment for the given loan amount, interest rate, and term. +11. Loan amount is $78,560, interest rate is 5.87%, 30-year mortgage. +12. Loan amount is $125,800, interest rate is 6.5%, 30-year mortgage. +13. Loan amount is $96,400, interest rate is 4.9%, 15-year mortgage. +14. Loan amount is $267,450, interest rate is 5.25%, 20-year mortgage. +In the following exercises, find the total paid on the mortgage if it is fully paid through the term. +15. Loan amount is $78,560, interest rate is 5.87%, 30-year mortgage. +16. Loan amount is $125,800, interest rate is 6.5%, 30-year mortgage. +17. Loan amount is $96,400, interest rate is 4.9%, 15-year mortgage. +18. Loan amount is $267,450, interest rate is 5.25%, 20-year mortgage. +In the following exercises, find the cost of financing for the mortgages if they are fully paid. +19. Loan amount is $78,560, interest rate is 5.87%, 30-year mortgage. +20. Loan amount is $125,800, interest rate is 6.5%, 30-year mortgage. +21. Loan amount is $96,400, interest rate is 4.9%, 15-year mortgage. +22. Loan amount is $267,450, interest rate is 5.25%, 20-year mortgage. +In the following exercises, use the amortization table to answer the question. +23. What is the term of the mortgage? +24. How much of payment 165 applies to interest? +6.12 • Renting and Homeownership +695 + +25. What is the remaining balance after payment 155? +26. How much total interest was paid after payment 149? +In the following exercises, use the amortization schedule to answer the question. +27. What is the interest rate for the mortgage? +28. How much of payment 110 applies to principal? +29. What is the remaining balance after payment 94? +30. How much total interest was paid after payment 111? +In the following exercises, find the total monthly payment including both the mortgage payment and the escrow +payment. +31. Mortgage of $87,690 at 6.2% interest for 30 years. Assessed value of the home is $75,600. Property taxes come +to 5.65% of assessed value. Home insurance of $815 paid every 6 months. +32. Mortgage of $143,900 at 5.05% interest for 30 years. Assessed value of the home is $90,150. Property taxes +come to 5.88% of assessed value. Home insurance of $924 paid every 6 months. +33. Mortgage of $65,175 at 6.48% interest for 30 years. Assessed value of the home is $62,800. Property taxes +come to 6.75% of assessed value. Home insurance of $558 paid every 6 months. +34. Mortgage of $245,950 at 5.35% interest for 30 years. Assessed value of the home is $156,500. Property taxes +come to 6.41% of assessed value. Home insurance of $972 paid every 6 months. +For the following exercises, read the following: Fifteen-year mortgage compared to 30-year mortgage. Mortgage +interest rates are often higher for 30-year mortgages than 15-year mortgages. However, the payments for 15-year +mortgages are considerably higher. The following exercises explore the difference between a 15- and 30-year mortgage +for a mortgage of $100,000. +35. The 15-year mortgage interest rate is 5.65%. +a. +Find the payment. +b. +Determine the total that would be paid if the mortgage was completed. +c. +Find the cost of financing for this mortgage. +36. The 30-year mortgage rate is 6.4%. +a. +Find the payment. +b. +Determine the total that would be paid if the mortgage was completed. +c. +Find the cost of financing for this mortgage. +37. How different are the payments, the total paid, and the cost to finance? +38. Summarize the answer in the previous question. +For the following exercises, read the following: Fifteen-year mortgage compared to 30-year mortgage. A 15-year +696 +6 • Money Management +Access for free at openstax.org + +mortgage comes with advantages, the biggest being the home is paid off much sooner, and equity is built much more +quickly. Mortgage interest rates are often higher for 30-year mortgages than 15-year mortgages. However, the +payments for 15-year mortgages are considerably higher. The following exercises explore the difference between a 15- +and 30-year mortgage for a mortgage of $200,000. +39. The 15-year mortgage interest rate is 5.6%. +a. +Find the payment. +b. +Determine the total that would be paid if the mortgage was completed. +c. +Find the cost of financing for this mortgage. +40. The 30-year mortgage rate is 6.25%. +a. +Find the payment. +b. +Determine the total that would be paid if the mortgage was completed. +c. +Find the cost of financing for this mortgage. +41. How different are the payments, the total paid, and the cost to finance? +42. Summarize the answer in the previous question. +6.13 Income Tax +Figure 6.28 Federal income tax is a concern for most US citizens. (credit: "1040 US tax form" by Marco Verch Professional +Photographer/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Determine gross, adjusted gross, and taxable income. +2. +Apply exemptions, deductions, and credits to basic income tax calculations. +3. +Compute FICA tax. +4. +Solve tax application problems for working students. +Before the start of the American Civil War in 1861, most of the country’s revenue came from tariffs on trade and excise +taxes. However, this fell far short of the high cost of the war. Because of this, the federal government enacted the nation’s +first income tax with the Revenue Act of 1861, which created the Internal Revenue Service as we know it today. +No one likes paying income tax, but it is a reality of life. In this section, we will learn about Form 1040, the U.S. Individual +Income Tax Return, and ways to prepare for tax time. +The U.S. tax code may change from year to year. Because of this, this section includes examples of how taxes, +deductions, and exemptions might be computed. The types of income, deductions, and exemptions that are used in the +examples are used in the current tax code. +6.13 • Income Tax +697 + +Gross, Adjusted Gross, and Taxable Income +Your income drives how much you pay in taxes. The more you earn, the more you are likely to pay. But your income +alone is not the full story. When you add all the money you earned from your job, freelance work, interest from savings, +and other sources, you have your gross income. If you are an employee, your income from your job will be reported on +a W-2, which is sent to you by your employer. Income from freelance work will be reported on a 1099-MISC form, and is +sent by the company that paid you. Income from interest is reported on a 1099-INT form and comes from the entity that +paid the interest. +Before you determine how much you owe in taxes, you will make certain adjustments to that gross income. You will +deduct, or subtract, some of income from the gross income. That’s your adjusted gross income, or AGI. That is still not +what you are taxed on. Next, you need to apply exemptions to your income. These are pieces of income that the +government does not tax. After that is done, you reach your taxable income. We will look at each of these parts of the +taxable income. +WHO KNEW? +Gifts and Winning +Money given as a gift my be taxed if the gift amount is high enough. If you win $50,000 in the lottery, that money is +taxed as income. If you give a family member a large cash gift, that gift will be subject to a tax provided that the gift +exceeds the federally set limits. +You will notice that your paycheck already has taxes taken out of it. Your employer will withhold some of your income, +sending it directly to the federal, state, and local governments. It is an estimate of how much you will owe in income tax. +In the end, it reduces how much you will pay when your taxes are due. If they withhold too much income, you will receive +the extra they withheld in the form of a refund. +EXAMPLE 6.115 +Computing Gross Income +Roger is preparing to do his taxes. He worked two jobs, with reported income of $27,500 and $13,200. His two CDs +yielded $327 together. He also won a split club raffle for $2,000. What was Roger’s gross earnings? +Solution +This is the sum of his wages, winnings and interest earned. Add these Roger’s gross income was $43,027. +YOUR TURN 6.115 +1. Chloe prepares for her taxes by collecting her W-2 and her 1099-MISC documents. The wages from her regular +job were $41,780. She did some freelance work and earned $5,500 from that job. What is Chloe’s gross income? +Your adjusted gross income (AGI) is computed before your taxes are determined. It begins with the gross income, and +then subtracts from that income any deduction. Deductions are expenditures on your part that the government won’t +tax. These deductions include money deposited into tax-deferred investments, and mortgage interest that you paid, +charitable contributions if you made any, medical bills over a threshold, medical insurance under certain circumstances, +and property taxes. If you add all these up, and they are all legal deductions, the sum is subtracted from your gross +income, leaving the AGI. +EXAMPLE 6.116 +Compute Adjusted Gross Income +Chandra’s gross income is $58,400. She deposited $3,000 into her tax-deferred retirement account for the year. Her +mortgage interest paid was $1,250 for the year, and her property taxes came to $4,200. What is her AGI? +698 +6 • Money Management +Access for free at openstax.org + +Solution +To get Chandra ’s AGI, total her deductions and subtract that total from her gross income. The total of her deductions is +. She subtracts that total from her gross income, making her AGI +. +YOUR TURN 6.116 +1. Isabelle’s gross income is $93,450. She paid $7,840 in mortgage interest and $3,810 in property taxes. She also +donated $1,500 to her favorite charity. What is Isabelle’s AGI? +Remember that your AGI is not your taxable income. Exemptions need to be subtracted from the AGI to reach your +taxable income. Exemptions are income that the government does not tax. Some examples of exempt income are +disbursements from health savings accounts for qualified medical expenses, bond interest, some IRA distributions, and +gifts given that are under $16,000. Note that exemptions are different from deductions: exemptions are excused +incomes, whereas deductions are excused expenditures. +EXAMPLE 6.117 +Taxable Income +Yelizaveta’s AGI is $75,490. However, she gave a $2,000 gift to her mother. What is her taxable income? +Solution +Taxable income is AGI minus any exemptions. Gifts below $16,000 are exempted, so her taxable income is +. +YOUR TURN 6.117 +1. Ammie has an AGI of $43,100. However, $3,400 was a disbursement from her health savings account. What is her +taxable income? +Tax Credits +Another piece of the tax puzzle is tax credits. This is money subtracted from the tax you owe. +Tax credits are very different from deductions or exemptions. Deductions and exemptions are taken away from your +gross income before the tax you owe is calculated. A tax credit, is subtracted, dollar for dollar, from your tax bill. Once +the tax you owe is calculated, subtract the any tax credits from that calculated tax. +Some of the tax credits are refundable. This means that if subtracting them from your tax results in a negative number, +you receive a tax refund. For more details, see this article about tax credits (https://openstax.org/r/taxcredit.asp). +The federal government has placed income limits and restrictions and on those eligible to receive tax credits because +their value is so high. Here is a partial list of tax credits that you might qualify for: +• +Earned income credit is a refundable tax credit for low- to moderate-income workers and ranges from $560 to +$6,935 depending on dependents and income. This is refundable +• +American opportunity credit is a credit taken by parents who have children enrolled in college at least half time +and pursuing a degree. This credit is worth $2,500 per student for the first 4 years of undergraduate school, subject +to income limits. This is a refundable tax credit. +• +Lifetime learning credit is a credit is equivalent to 20% of educational expenses, up to $2,000 per year, subject to +income limits. There is no cap to how many years you can apply for this credit. +• +Child tax credit is worth $2,000 per child under the age of 17 if that child lives at home at least half the year, subject +to income limits. This is a refundable tax credit. +• +Child and dependent care tax credit was designed to help pay for child care while the parent works. The amount +of the credit is dependent on your income. However, the maximum amount that can be received is, in 2022, $4,000 +for one eligible person, or $8,000 for two or more qualifying people. A dependent qualifies if they are a child under +6.13 • Income Tax +699 + +13 years old, a spouse who is unable to care for themselves, or some other qualifying person. This is a refundable +tax credit. +• +Premium tax credit was created by the Affordable Care Act, and it is one that is received by many people +throughout the year. In essence it is a health insurance premium subsidy. The amount of the credit is based on your +income and the price of health insurance in your area. This is a refundable tax credit. +EXAMPLE 6.118 +Apply a Tax Credit +Kaitlyn has calculated the tax she owes to be $5,200. However, she receives an earned income tax credit of $1,715. How +much does Kaitlyn owe after applying the earned income tax credit? +Solution +The amount of taxes Katilyn owes is her calculated tax of $5,200 minus the credit she receives. The amount she owes in +taxes is +. +YOUR TURN 6.118 +1. Antonio’s taxes owed based on his income are $3,950. However, he qualifies for the earned income tax credit of +$1,925 and a child tax credit of $2,000. How much does Antonio owe after applying the tax credits? +EXAMPLE 6.119 +Tax Credits and a Refund +Chanajah calculated their tax owed, which came to $4,300. They have an earned income tax credit of $2,190, a child tax +credit of $2,000, and a child and dependent care tax credit of $4,000. How much tax does Chanajah owe, or how much +will they get in a refund? +Solution +Adding Chanajah’s tax credits together, we find their total to be $8,190. That is more than the tax they owe, which was +$4,300. Each of those credits is refundable, which means they will receive a refund. Subtracting the credits from the tax +owed yields +. This is negative, so represents a refund of $3,890. +YOUR TURN 6.119 +1. Ismail owed $2,350 in taxes. He has a child tax credit of $2,000 and an earned income tax credit of $1,640. After +applying the tax credit, how much does Ismail owe or how much does he get in a refund? +VIDEO +Deductions Versus Credits (https://openstax.org/r/Versus_Credits) +Computing FICA Taxes +FICA stands for the Federal Insurance Contributions Act of 1935. FICA taxes are used solely to fund Social Security and +Medicare and are separate from federal income tax. It amounts to 7.65% of your gross pay, which is withheld from your +paycheck automatically. Your employer is required to match the 7.65% amount. Of the 7.65%, 6.2% goes to Social +Security (SSI), and 1.45% goes to Medicare. +As of 2022, SSI tax only applies to the first $147,000 of earnings. Any gross income above that is not taxed for social +security. This limit changes every year. +Medicare tax, on the other hand, applies to the entirety of your gross income. +700 +6 • Money Management +Access for free at openstax.org + +EXAMPLE 6.120 +Computed FICA Taxes +McKenzi earned $2,700 in gross income, before taxes, in a given 2-week period. How much does she owe in FICA taxes, +and how much of that is for SSI? +Solution +The FICA tax is 7.65% of her gross earnings. 7.65% of her $2,700 is $206.55. Also, the SSI is 6.2% of her income, or +$167.40 +YOUR TURN 6.120 +1. Arianne earned $3,200 over a 2-week pay period. How much FICA tax does she pay, and how much of that is for +SSI? +EXAMPLE 6.121 +Social Security Tax with Higher Income +Renard earned $195,000 in wages for the year. How much in SSI taxes does Renard owe for the year? +Solution +Since Renard earned more than the taxable limit of $147,000 dollars, he only pays the 6.2% SSI tax on $147,000. This +comes to $9,114. +YOUR TURN 6.121 +1. Andy earned $169,450 in wages this year. How much in SSI taxes does Andy owe? +Calculating Your Income Tax +Your income tax bill and your income tax rate are based on your taxable income. The tax system in the United States is +progressive, meaning that the tax rates are marginal so the higher your taxable income the higher the tax rate you will +pay. Taxable income is broken into brackets, or ranges of income. Each bracket has a different tax rate. The tax brackets +and rates for single filers as of 2022 are given below: +Bracket +Lower Income Limit +Upper Income Limit +Tax Rate +1 +0 +$10,275 +10% +2 +$10,276 +$41,775 +12% +3 +$41,776 +$89,075 +22% +4 +$89,076 +$170,050 +24% +5 +$170,051 +$215,950 +32% +Table 6.3 Federal Income Tax Brackets for Single Filers, 2021–2022 +(data source: https://www.irs.gov/newsroom/irs-provides-tax-inflation- +adjustments-for-tax-year-2022) +6.13 • Income Tax +701 + +Bracket +Lower Income Limit +Upper Income Limit +Tax Rate +6 +$215,951 +$539,900 +35% +7 +$539,901 +37% +Table 6.3 Federal Income Tax Brackets for Single Filers, 2021–2022 +(data source: https://www.irs.gov/newsroom/irs-provides-tax-inflation- +adjustments-for-tax-year-2022) +So if your taxable income is $76,500 and you are filing as a single filer, your tax bill will be 22% of that $76,500, right? +Wrong. Your income is split among those brackets and the money in each bracket is taxed at that bracket’s tax rate. +Seems confusing. Here is a list of steps to follow to find the tax owed. +Step 1: Find the bracket for the income. +Step 2: For each bracket below the income bracket, the tax from that bracket is: +Step 2a: Find the difference between the upper limit of that bracket and upper limit of the next lower bracket. If this is +bracket 1, use 0 as the upper limit of the previous bracket. +Step 2b: The tax from that bracket is the bracket tax rate applied to the difference from Step 2a. +Step 3: For the bracket that the income belongs to, find the income minus the lower limit for the bracket. +Step 4: The tax for the bracket of the income is tax rate for that bracket applied to the difference found in Step 3. +Step 5: Add these various tax values to get the total income tax. +There are various tax brackets, and the rates may change in any given year. The income limits may also change. For all +examples going forward, we will use the single filer tax brackets, even if those brackets are not appropriate (e.g., married +or head of household filers). +EXAMPLE 6.122 +Income Tax on Taxable Income +Faith has a taxable income of $103,650. How much income tax does Faith owe? +Solution +Step 1: Faith’s income belongs to the 4th tax bracket, $89,076 to $170,050. So the taxes from the first three brackets +follow steps 2a and 2b. +For bracket 1: +Step 2a: The upper limit is $10,275, the upper limit of the previous bracket was 0, so the difference is $10,275. +Step 2b: For the first bracket, she owes 10% of $10,275, or $1,027.50. +For bracket 2: +Step 2a: The upper limit is $41,775, the upper limit from the previous bracket was $10,275, so the difference is $31,500 +Step 2b: For the second bracket, she owes 12% of $31,500, or $3,780. +For bracket 3: +Step 2a: The upper limit is $89,075, the upper limit from the previous bracket is $41,175, so the difference was $47,300 +Step 2b: For the third bracket, she owes 22% of $47,300, or $10,406. +Step 3: For bracket 4, her income is $103,650, the upper limit of the previous bracket is $89,075. The difference of those +is +. +Step 4: The tax he owes for that bracket is 24% of the difference, which is $3,498. +Step 5: The total in taxes that Emmanuel owes is the sum of the taxes found above, or +702 +6 • Money Management +Access for free at openstax.org + +YOUR TURN 6.122 +1. Jenna’s taxable income is $73,500. Find how much income tax she owes. +Remember that your employer will estimate how much tax you will owe and withholds it from paychecks. This +means you may have already paid some, if not all and more, of the income tax you owe. +VIDEO +Tax Brackets (https://openstax.org/r/Brackets) +EXAMPLE 6.123 +Finding Income Tax Owed +Emmanuel is preparing his taxes. His W-2 from work shows gross income for the year of $95,250. He also has a +1099-MISC for some freelance art work he did, amounting to $7,500. Emmanuel also deposited $4,500 into a tax- +deferred retirement plan. He paid $7,920 in mortgage interest for the year and $3,740 in property taxes. He also +qualified for $4,000 in child tax credits. Based on this information, how much tax does Emmanuel owe or how much does +he get in a refund? +Solution +We need to know Emmanuel’s taxable income, based on gross income, deductions, and exemptions. +His gross income is the amount from his W-2 and his 1099-MISC. Adding these gives a gross income of +. +Subtracting his deductions will yield his AGI. His deductions are $7,920 for mortgage interest, $3,740 for property taxes, +and $4,500 deposited into his retirement account. Adding these, his total deductions are $16,160. Subtracting the +deductions from the gross income, we find his AGI to be $86,590. +Emmanuel seems to have no exemptions, so his AGI and his taxable income are the same. +We find the taxes Emmanuel owes using the process outlined above. +Step 1: His income belongs to the third tax bracket, $41,776 to $89,0875. So the taxes from the first two brackets follow +steps 2a and 2b. +For bracket 1: +Step 2a: The upper limit is $10,275, the upper limit of the previous bracket was 0, so the difference is $10,275. +Step 2b: For the first bracket, he owes 10% of $10,275, or $1,027.50. +For bracket 2: +Step 2a: The upper limit is $41,775, the upper limit from the previous bracket was $10,275, so the difference is $31,500 +Step 2b: For the second bracket, he owes 12% of $31,500, or $3,780. +Step 3: For bracket 3, his income is $86,590, the upper limit of the previous bracket is $41,775. The difference of those is +. +Step 4: The tax he owes for that bracket is 22% of the difference, which is $9,859.30. +Step 5: The total in taxes that Emmanuel owes is the sum of the taxes found above, or +. +Once his taxes are computed, he subtracts his tax credits. His only tax credit is $4,000. The total he owes in taxes is +. +6.13 • Income Tax +703 + +YOUR TURN 6.123 +1. Jacob is preparing his taxes. His W-2 from work shows gross income for the year of $39,885. He also has a +1099-INT for a savings account, amounting to $378. Jacob also deposited $2,500 into a tax-deferred retirement +plan. He also qualified for $2,000 in child tax credits and $4,000 in child and dependent care tax credit. Based on +this information, how much tax does Jacob owe or how much does he receive in a refund? +Check Your Understanding +86. Gross income consists of what? +87. Taxable income is found after what is done to gross income? +88. If the gross income is $50,000 and there are three deductions of $2,000, $3,000, and $8,000, what is the adjusted +gross income? +89. When is a tax credit applied? +90. Do you pay 6.2% for SSI tax for all your earnings? +91. For a taxable income of $41,800, how much income tax is owed? +92. If $5,600 is owed in taxes but there are $4,000 in tax credits, how much income tax is owed? +SECTION 6.13 EXERCISES +1. What is gross income? +2. How is adjusted gross income derived from gross income? +3. Is there a difference between AGI and taxable income? If so, what is the difference? +4. How are deductions different from tax credits? +5. What are the two components of the FICA tax? +6. What taxes do employers match? +7. If you are in the 32% tax bracket, is your tax computed on your full income? +8. What does it mean for a tax credit to be refundable? +In the following exercises, compute the gross income. +9. Wages from W-2 are $32,800, freelance income from a 1099-MISC is $1,050, and a gift given to a family member +of $2,760. +10. Wages from W-2 are $59,380, interest income from 1099-INT is $1,500, freelance income from a 1099-MISC is +$3,500, and a split club raffle winning of $6,755. +11. Wages from job 1 on W-2 are $36,200, wages from job 2 on W-2 are $21,400, interest income from a 1099-INT is +$374. +12. Wages from W-2 are $121,450, interest from a 1099-INT is $3,400. +In the following exercises find the adjusted gross income, AGI, based on gross income and deductions. +13. Gross income was $65,700, $2,280 deposited in a tax-deferred IRA, mortgage interest was $4,715, property +taxes were $3,065. +14. Gross income was $183,200, $5,000 deposited in a tax-deferred IRA, $7,300 in charitable contributions, $8,350 +in mortgage interest, and $6,900 in property taxes. +15. Gross income was $31,200, $1,500 deposited in a tax-deferred IRA account. +16. Gross income was $41,500, $1,250 deposited in a tax-deferred IRA account, $4,210 in mortgage interest, $2,980 +in property tax. +In the following exercises, find the taxable income based on AGI and exemptions. +17. AGI of $34,560, $2,500 disbursement from a health savings account for a qualifying medical expense. +18. AGI of $56,750, gift given to a family member of $8,000, $550 in bond interest. +19. AGI of $120,940, gift given to a family member of $15,000, bond interest of $4,500. +20. AGI of $28,450, gift given to a family member of $2,000. +704 +6 • Money Management +Access for free at openstax.org + +In the following exercises, find the taxes owed or refund received based on income tax bill and tax credits. +21. Tax bill of $5,300, child tax credit of $4,000, earned income tax credit of $1,630. +22. Tax bill is $17,300, child tax credit of $4,000, child and dependent care tax credit $8,000, lifetime learning credit +of $1,000. +23. Tax bill of $5,205, child tax credit of $6,000, earned income tax credit of $2,450. +24. Tax bill of $11,300, child tax credit of $2,000, earned income tax credit of $650, child and dependent care credit +of $4,000. +25. Tax bill of $13,750, child tax credit of $2,000, earned income tax credit of $780, child and dependent care credit +of $2,000. +In the following exercises, determine the SSI tax and the total FICA tax for the given incomes. +26. Earning for pay period were $3,500. +27. Earnings for pay period were $1,400. +28. Earnings for pay period were $3,150. +29. Earnings for the year were $135,000. +30. Earnings for the year were $203,400. +In the following exercises, find the income tax owed using the taxable income and the tax table below, for a person +filing single. +Bracket +Lower Income Limit +Upper Income Limit +Tax Rate +1 +0 +$10,275 +10% +2 +$10,276 +$41,775 +12% +3 +$41,776 +$89,075 +22% +4 +$89,076 +$170,050 +24% +5 +$170,051 +$215,950 +32% +6 +$215,951 +$539,900 +35% +7 +$539,901 +37% +31. Taxable income of $36,250. +32. Taxable income of $63,500. +33. Taxable income of $209,450. +34. Taxable income of $92,250. +35. Alexandra is preparing her taxes. Her W-2 from work shows gross income for the year of $51,300. She also has a +1099-INT for interest on a savings account for $910. Alexandra also deposited $2,750 into a tax-deferred +retirement plan. She paid $2,150 in mortgage interest for the year and $2,060 in property taxes. Based on this +information, how much tax does Alexandra owe or how much does she get in a refund? +36. Tymoteusz is preparing his taxes. His W-2 from work shows gross income for the year of $47,680. He also has a +1099-MISC for some freelance consulting he did, amounting to $1,800. Tymoteusz deposited $3,000 into a tax- +deferred retirement plan. He also qualified for $2,000 in child tax credits and $4,000 in child and dependent care +tax credits. Based on this information, how much tax does Tymoteusz owe or how much does he get in a refund? +6.13 • Income Tax +705 + +Chapter Summary +Key Terms +6.1 Understanding Percent +• +Percent +• +Fractional form +• +Decimal form +• +Total +• +Base +• +Percent of the total +• +Part +• +Amount +6.2 Discounts, Markups, and Sales Tax +• +Discount +• +Cost +• +Markup +• +Retail price +6.3 Simple Interest +• +Interest +• +Principal +• +Annual percentage rate +• +Simple interest +• +Term +• +Due +• +Origination date +• +Payoff amount +• +Future value +• +Partial payment +• +Present value +6.4 Compound Interest +• +Compound interest +• +Effective annual yield +6.5 Making a Personal Budget +• +Budget +• +Necessary expenses +• +Fixed expenses +• +Variable expenses +• +50-30-20 budget philosophy +6.6 Methods of Savings +• +Savings account +• +1099 form +• +Certificate of deposit +• +Money market account +• +Return on investment +• +Ordinary annuity +6.7 Investments +• +Bonds +• +Maturity date +• +Stocks +• +Dividend +• +Mutual fund +706 +6 • Chapter Summary +Access for free at openstax.org + +• +Prospectus +• +Issue price +• +Shares +• +Stock table +• +Individual retirement account +• +Roth IRA +• +401(k) +6.8 The Basics of Loans +• +Fixed interest rate +• +Variable interest rate +• +Installment loan +• +Loan amortization +• +Revolving credit +• +Amortization table +• +Cost of finance +6.9 Understanding Student Loans +• +FAFSA +• +College funding gap +• +Subsidized loan +• +Unsubsidized loan +• +Parent loan for undergraduate students +• +Private student loan +• +School-channel loan +• +Direct-to-consumer loan +• +Standard repayment plan +• +Federal consolidation +• +Refinancing +• +Private consolidation +• +Graduated repayment plan +• +Extended repayment plan +• +Discretionary income +• +Pay as you earn (PAYE) repayment plan +• +Revised pay as you earn (REPAYE) repayment plan +• +Income-based (IBR) repayment plan +• +Income-contingent (ICR) repayment plan +• +Delinquent +• +Default +• +Rehabilitation +6.10 Credit Cards +• +Reward program +• +Annual fee +• +Credit limit +• +Bank-issued credit card +• +Store-issued credit card +• +Travel and entertainment cards +• +Charge cards +• +Billing period +• +Balance +• +Minimum payment +• +Average daily balance +6.11 Buying or Leasing a Car +• +Title and registration fees +• +Destination fee +• +Documentation fee +6 • Chapter Summary +707 + +• +Dealer preparation fee +• +Extended warranty +• +Down payment +• +Acquisition fee +• +Security deposit +• +Disposition fees +• +Liability insurance +• +Collision insurance +• +Comprehensive insurance +• +Uninsured or underinsured motorist insurance +• +Medical payment insurance +• +Personal injury insurance +• +Gap insurance +• +Rental reimbursement insurance +6.12 Renting and Homeownership +• +Lease +• +Landlord +• +Mortgage +• +Escrow account +• +Assessed value +6.13 Income Tax +• +Gross income +• +Adjusted gross income +• +Exemption +• +Taxable income +• +Deduction +• +Tax credit +• +Earned income credit +• +American opportunity credit +• +Lifetime learning credit +• +Child tax credit +• +Child and dependent care tax credit +• +Premium tax credit +Key Concepts +6.1 Understanding Percent +• +Know what a percent is as a fraction, a decimal, and as a part of the whole. +• +Use the percent equation to find any of the three values that are related by the equation. +• +Apply the percent equation in applications. +6.2 Discounts, Markups, and Sales Tax +• +Discounts are markdowns from an original price. +• +Mark-ups are increases to the price paid by a retailer to cover their costs. +• +be able to calculate the markup based on a percentage of the cost +• +Sales taxes vary from state to state and often county to county. +• +Retail prices, sales prices and percent discounts can be calculated if the other two values are known. +• +Original costs, retail prices, and percent markup can be calculated if the other two values are known. +• +In calculations, sales tax acts like a markup. +6.3 Simple Interest +• +Interest is money that is paid by a borrower for the privilege of borrowing the money. +• +Simple interest is computed by substituting the principal, interest rate, and number of years into the formula +• +The payoff for a loan is the amount of principal remaining on a loan plus the interest that accumulated on the loan +since the last payment. +708 +6 • Chapter Summary +Access for free at openstax.org + +• +The future value of an investment yielding simple interest is the original principal plus the interest earned on the +investment. +• +When making a partial payment, some of the payment pays off all the accumulated interest, while the remainder of +the payment is applied to the principal of the loan. +• +Finding the present value of an investment is used to determine how much should be invested now in order to +achieve a specific goal. +6.4 Compound Interest +• +Compound interest means that the interest earned during one period will earn interest in later periods. Essentially, +the amount of the principal grows from period to period. +• +The important values in computing compound interest are the interest rate, the principal, the length of time the +investment, and the number of times the investment is compounded. +• +Compound interest has minimal impact early, but later has a very large impact. +• +You can determine how much to invest today in order to reach a goal for some time later. +• +Compound interest can be translated into an effective annual yield, which allows for comparison between +investment options. +6.5 Making a Personal Budget +• +A budget is a set of guidelines for how to allocate your income. +• +Budgeting helps to plan for many of life’s expenses +• +Budgets are used to compare income to expenses. When expenses exceed income, changes have to be made. +• +Budgets can help evaluate the affordability of life changes. +• +One guideline for setting a budget is the 50-30-20 budget philosophy. The guidelines suggest that 50% of income is +allocated to necessary expenses, 30% to expenses that wants, and 20% to savings and other debt reduction. +6.6 Methods of Savings +• +There are three main types of savings accounts, saving accounts, certificates of deposit (CD), and money market +accounts. +• +Savings account are very risk free, and so yield low interest rates. +• +The differences in the three types of savings accounts relate to their convenience. +• +Savings account typically have a lower interest rate that money market accounts, which typically have lower interest +rates than CDs. +• +Ordinary annuities more accurately reflect how we save, in that money is deposited repeatedly over time. +• +Spreadsheet software, such as Google Sheets, have built in functions that can be used to quickly calculate both the +future value of an ordinary annuity account, but also the payment necessary to reach a goal using an ordinary +annuity. +6.7 Investments +• +There are many different investments with different returns and risks. +• +Bonds are loans form the purchaser to the entity selling the bond. +• +Bonds have some tax benefits, low to no risk, and a low return. +• +Stocks represent part ownership in a company. As such, stock holders share in the profits, and losses, of the +company. +• +Information, including price, P/E, yearly highs and lows, and dividend amount can be found in online stock tables +available on many websites. +• +Mutual funds represent collections of professionally administered investment vehicles. Have shares in a mutual +fund has lower risk than ownership of stocks. +• +Retirement accounts employ some of the same strategies as mutual funds, in that they spread the risk and are +professionally managed. +• +IRAs and Roth IRAs differ on when taxes are paid on the money, and who can use them. Roth IRAs have income +limits while traditional IRAs do not. +6.8 The Basics of Loans +• +There are many reasons for a loan, but primarily it is taken out for a large expense when cash is not available. +• +Each payment for an installment loan consists of an interest portion and a principal portion. +• +There is a formula to calculate the payment necessary to pay off a loan in installments. +• +Amortization schedules, or tables, show how each payment is applied to principal and interest. It also includes other +details such as remaining balance and total interest paid. +6 • Chapter Summary +709 + +• +Loans often have other fees associated with them such as origination fees or application fees. The total of the +interest paid and the fees is the cost of finance. +6.9 Understanding Student Loans +• +The FAFSA must be filled out each year that a student wishes to borrow for. +• +A student’s funding gap determines how much they need in loans to pay for college. +• +Federal subsidized student loans defer payments until after graduation and interest does not accrue on these loans. +• +Unsubsidized student loans defer payment until after graduation but interest begins accruing as soon as the loan +finds are disbursed. +• +There are both yearly and aggregate limits for student loans to prevent over-borrowing, among other reasons. +• +Federal direct loans have a low interest rate set by the government, but other student loans have varying rates of +interest set by the banks. +• +The standard repayment plan lasts 10 years and is made up of monthly payments. +• +Consolidating or refinancing student loans merges many student loans into one loan. +• +If only federal loans are consolidated, the interest rate is the same as the individual loans, currently set at 4.99%. +• +If other loans are refinanced together, the interest rate may be lower with the new loan. +• +Other repayment plans are available. Such a plan may have payment that start small and grow as the loan is paid +off, or it may have a longer term, or may be based on the discretionary income of the student. +• +Being delinquent on a student loan is a precursor to being in default. Making payments in a timely fashion allows +the student to avoid this situation. +6.10 Credit Cards +• +Credit cards can be a flexible way to pay for almost anything, but can become a financial hazard if used unwisely. +• +When deciding which credit card to apply for, evaluate the interest rate, fees (annual and late), reward programs and +credit limit. Be sure they meet your criteria. +• +Paying off the balance of your credit card every month will control your spending and will never result in paying +interest. +• +Credit card statements hold all important information about your credit card, including payment, balances, charges +and billing cycle dates. +• +Although the minimum payment is attractive precisely because it is so small, paying only the minimum results is a +long payoff term and higher interest costs. +6.11 Buying or Leasing a Car +• +There are many factors to consider when choosing to buy or lease a car. +• +The cost of the car is increased by a number of fees and sales tax. +• +There are advantages to buying a car and advantages to leasing a car. The decision between the two depends on +the preference of the buyer. +• +Insurance covers costs associate with accidents. It is made up of various components. +• +The costs of owning a car, including insurance and maintenance, should be a part of the budgeting process. +• +Budgeting for unexpected repairs can ease the stress of encountering large repair bill. +6.12 Renting and Homeownership +• +There are many points of comparison between renting and buying a house. +• +Before deciding to buy a house, you should carefully consider all the responsibilities that come with home +ownership. +• +Renting comes with more restrictions on the renter, but with fewer costs and is easier to move from. +• +Owning a house has more costs but has more freedom, plus the owner creates equity. +• +Mortgages are loans, and payments are calculated in the same way as any other loan. +• +Amortization tables help a homeowner understand the mortgage and how the payments are applied to the +principal and interest. +• +In addition to paying the amount financed for a mortgage, the monthly payment will include an escrow payment, +which covers insurance and taxes. +6.13 Income Tax +• +Federal income tax is based on income after certain adjustments. +• +Gross income is income from all sources, including gifts and winnings. +• +Before taxes are calculated, the taxable income is found by subtracting deductions and exemptions from gross +income. +710 +6 • Chapter Summary +Access for free at openstax.org + +• +Income tax is progressive, increasing in rate as income increases. +• +Being in the 32% tax bracket means some of your income is taxed at 10%, some at 12%, some at 22%, some at 24%, +and the rest at 32%. +• +Income in each tax bracket is taxed at that bracket’s rate, which means in 2022 the first $10,275 earned is taxed at +10% only. +• +Tax credits are subtracted from the taxes that are owed. +• +Some tax credits are refundable, which means they can make the amount you owe negative, which results in a +refund. +Videos +6.1 Understanding Percent +• +Finding Percent of a Total (https://openstax.org/r/solve_percent_problem1) +• +Finding the Total from the Percent and the Part (https://openstax.org/r/solve_percent_problem2) +• +Finding the Percent When the Total and the Part Are Known (https://openstax.org/r/solve_percent_problem3) +6.2 Discounts, Markups, and Sales Tax +• +Computing Price Based on a Percent Off Coupon (https://openstax.org/r/Computing_Price_Based) +• +Finding Sales Tax Percentage (https://openstax.org/r/Finding_Sales_Tax) +6.4 Compound Interest +• +Compound Interest (https://openstax.org/r/compound_interest_beginners) +• +Compare Simple Interest to Interest Compounded Annually (https://openstax.org/r/ +compare_simple_compound_interest1) +• +Compare Simple Interest and Compound Interest for Different Number of Periods Per Year (https://openstax.org/r/ +compare_simple_compound_interest2) +6.5 Making a Personal Budget +• +Creating a Budget (https://openstax.org/r/creating_budget) +• +50-30–20 Budget Philosophy (https://openstax.org/r/50-30-20_budgeting_rule) +6.6 Methods of Savings +• +Return on Investment, ROI (https://openstax.org/r/This_video) +• +Future Value Using Google Sheets (https://openstax.org/r/cell_references) +6.7 Investments +• +Bonds (https://openstax.org/r/investing_basics_bonds) +• +Reading Stock Summary Online (https://openstax.org/r/Reading_Stock) +• +Mutual Funds (https://openstax.org/r/investing_basics_mutual_funds) +• +401(k) Accounts (https://openstax.org/r/401(k)s) +6.8 The Basics of Loans +• +Credit Scores Explained (https://openstax.org/r/credit_scores_explained) +• +Reading an Amortization Table (https://openstax.org/r/Amortization_Table) +6.9 Understanding Student Loans +• +Types of Student Loans (https://openstax.org/r/Student_Loans) +• +Repayment Plans (https://openstax.org/r/student_loan_repayment_plans) +6.10 Credit Cards +• +Choosing a Credit Card (https://openstax.org/r/Credit_Card) +• +Reading Credit Card Statements (https://openstax.org/r/Reading_Credit) +6.12 Renting and Homeownership +• +Rent or Buy (https://openstax.org/r/Rent_or_Buy) +• +Closing Costs (https://openstax.org/r/closing_costs) +6 • Chapter Summary +711 + +6.13 Income Tax +• +Deductions Versus Credits (https://openstax.org/r/Versus_Credits) +• +Tax Brackets (https://openstax.org/r/Brackets) +Formula Review +6.1 Understanding Percent +part = percent x total +6.2 Discounts, Markups, and Sales Tax +6.3 Simple Interest +6.4 Compound Interest +6.6 Methods of Savings +6.7 Investments +712 +6 • Chapter Summary +Access for free at openstax.org + +6.8 The Basics of Loans +6.9 Understanding Student Loans +6.10 Credit Cards +6.11 Buying or Leasing a Car +6.12 Renting and Homeownership +Projects +Creating Your Future Budget +In this project, you will create a budget based on a job you are likely to have after you graduate. +1. +Go online and research the average starting salary for the profession you are studying for. Use at least two sources. +Be sure to record the web address from your search. +2. +Approximate your monthly take-home pay. You may use the SmartAsset (https://openstax.org/r/ +paycheck_calculator) website to estimate this. +3. +Use the 50-30-20 budget philosophy to determine how much you should budget for needs, wants and savings, or +extra debt reduction. +4. +Create a list of likely expenses. This list must include rent/mortgage, utilities, food, and school loan repayment. You +may also want to include car payments, gasoline, and other items. +5. +Categorize each expense as need, want, or savings. +6. +Using the amounts found in step 3, decide how much to allocate to each of your expenses. It may help to quickly +research how much rent is where you want to live. +7. +Discuss the choices you had to make, and why you prioritized some expenses over others. +Interest Rate and Time: What Is the Relationship? +The interplay between interest rate and time for an annuity is not easily seen. How the amount that must be deposited +per compounding period, +, changes based on the time and interest rate would be useful to understand. In this +project, you will explore this relationship. We will use a fixed future value of += $1,000,000 and a fixed number of +periods per year, 12 (monthly compounding). With those, we’ll find various annuity payments that must be made to +6 • Chapter Summary +713 + +reach the goal. +The annual interest rate for the investment is in the top row. The number of years for the investment is in the left +column. In each cell (or box), find the monthly payment necessary to reach the goal of $1,000,000. +Annual Interest Rate +1.5% +2.0% +3.0% +5.0% +7.5% +10.0% +Number of Years +10 +15 +20 +30 +40 +45 +Describe how the interest rates and number of years impact the payment necessary to reach the goal of $1,000,000. +Finding a Home +In this project you will identify a home you like, and then estimate the costs associated with that home. +1. +Find a home in your region that you would like to buy using an online search of listings in your area. Zillow is a good +place to begin. +2. +Find the asking price for this home. Assume you would pay that price. +3. +Find an estimation for closing costs in your area. Assume you finance those costs also. +4. +Estimate the taxes to be paid on the home per year. It is likely that the online listing of the home has an estimate for +the taxes for the house. +5. +Use Google to determine the average homeowner’s insurance cost in your region. +6. +Use the internet to determine the average interest rate for a 30-year mortgage. +7. +Find how much you would pay per month, based on the answers to the previous questions, including the escrow +payments for taxes and insurance. +8. +Assume you will pay $50 per $100,000 borrowed in PMI. Add this to the monthly payment. +714 +6 • Chapter Summary +Access for free at openstax.org + +Chapter Review +Understanding Percent +1. Convert 0.45 to percent form. +2. What is 70% of 200? +3. Rewrite the fraction +as a percentage. +4. 20 is what percent of 500? +Discounts, Markups, and Sales Tax +5. An item with a retail price of $250.00 is on sale for 30% off. What is the sale price of the item? Round to the nearest +penny. +6. The sales price of an item is $38.50 after a 40% discount. What was the retail price of the item? Round to the +nearest penny. +7. A retailer buys an item for $1,000.00 and has a 65% markup. What is the retail price of the item? Round to the +nearest penny. +8. The sale price of an item is $46.00 and the retail price is $65.00. What is the percent discount? Round to two +decimal places. +9. The retail price of an item is $345.38 and sales tax in the region is 8.25%. How much is the sales tax? +Simple Interest +10. $3,000 is invested in a 5-year CD earning 2.25% interest. How much is the CD worth in 5 years? +11. A $4,000.00 simple interest loan is taken out for 3 years at 12.5%. How much is owed when the loan comes due? +12. A $10,000 loan with annual simple interest of 14.9% is taken out for 90 days. How much is due in 90 days? +13. A simple interest loan for $20,000 is taken out at 13.9% annual interest rate. A partial payment of $13,500.00 is +made 30 days into the loan period. After this payment, what will the remaining balance be? +14. Find the present value of an investment with future value of $15,000 if the investment earns 3.55% simple interest +for 10 years? +Compound Interest +15. Find the future value after 20 years of $15,000 deposited in an account bearing 4.26% interest compounded +quarterly. +16. What is the effective annual yield for an account bearing 3.21% interest compounded quarterly? Round to two +decimal places if necessary. +17. Find the present value of $100,000 in an account bearing 5.25% interest compounded daily after 30 years. +Making a Personal Budget +18. In budgeting, what is the difference between an expense that is a need and an expense that is a want? +19. Apply the 50-30-20 budget philosophy to a monthly income of $6,000.00. +20. Create the budget for a person with the following income and bills, and determine how much the income exceeds +or falls short of the expenses: +Job = $5,250, Side job = $550, rent = $1,150.00, utilities = $150.00, internet = $39.99, student loans = $375.00, food += $550.00, bus and subway = $112.00, credit cards = $200.00, entertainment = $400.00, clothing = $150.00. +21. In question 20, the budget for a person was created. Can the person afford to buy a car if the monthly payments +will be $365.50, monthly car insurance will be $114.75, and gasoline will be $200.00? +Methods of Savings +22. Why is a CD less flexible than a money market account? +23. Which is likely to have the highest interest rate, a savings account, CD, or money market account? +24. $3,000 is deposited in a money market account bearing 3.88% interest compounded monthly. How much is the +account worth in 15 years? +25. Refer to question 24. What is the return on investment for the money market account? Round to two decimal +6 • Chapter Summary +715 + +places, if necessary. +26. $300 is deposited monthly in an ordinary annuity that bears 6.5% interest compounded monthly. How much is in +the annuity after 30 years? +27. How much must be deposited quarterly in an account bearing 4.89% interest compounded quarterly for 30 years +so the account has a future value of $500,000? +Investments +28. Why is investing in a stock riskier than in a mutual fund? +29. Which investment offers high rates of return with reduced risk? +30. Which investment offer a fixed rate of return over time? +31. Which is the riskiest investment? +32. How much is earned on a 10-year bond with issue price of $1,000 that pays 3.9% annually? +33. Henri’s employer matches up to 6% of salary for IRA contributions. Henri’s annual income is $52,500. If Henri +wants to deposit $3,000 annually in his IRA, how much in matching funds does the employer deposit in the IRA? +34. How much must be deposited quarterly in a mutual fund that is expected to earn 11.2% interest compounded +quarterly if the account is to be worth $1,000,000 after 38 years? +35. Lorraine purchases stocks for $5,000. The stocks paid dividends by reinvesting the dividends in stock. She sold the +stock for $13,750 after 6 years. What was Lorraine’s annual return on that investment? Round to two decimal +places if necessary. +36. What are two ways stocks earn money? +The Basics of Loans +37. What is an installment loan? +38. An installment loan with monthly payments has an outstanding balance of $3,560. If the annual interest rate is 9%, +how much interest will be paid that month? +39. A loan for $42,000 is taken out for 6 years at 7.5% interest. What are the payments for that loan? +40. The total interest paid on a loan was $4,500. The loan had an origination fee of $400.00, a $125.00 processing fee, +and a filing fee of $250.00. How much was the cost of financing for that loan? +41. Using the amortization table below, what is the remaining balance after payment 25? +716 +6 • Chapter Summary +Access for free at openstax.org + +Understanding Student Loans +42. If a college program takes 5 years, how long is the student’s eligibility period for student loans? +43. What is the funding gap for a student with cost of college equal to $35,750 and non-loan financial aid of $24,150? +44. What is the maximum amount of federal subsidized and unsubsidized loans for a dependent student in their third +year of an undergraduate data science program? +45. A single person has gross income of $28,500. Their poverty guideline is $12,000. What is their discretionary +income? +Credit Cards +46. The billing cycle for a credit card goes from March 15 to April 14. The balance at the start of the billing cycle is +$450.00. The list of transactions on the card is below. Find the average daily balance for the billing cycle. +6 • Chapter Summary +717 + +Date +Activity +Amount +15-Mar +Billing Date Balance +$450.00 +21-Mar +Movie +$50.00 +28-Mar +Gasoline +$65.00 +31-Mar +Snacks +$15.50 +1-Apr +Dinner +$63.60 +1-Apr +Payment +$300.00 +1-Apr +Gasoline +$48.90 +9-Apr +Plane Tickets +$288.50 +47. The average daily balance for Greg’s last credit card statement was $1,403.50. The card charges 15.9% interest. If +the billing cycle for that statement was 30 days, how much interest is Greg charged? +48. The billing cycle for a credit card goes from March 15 to April 14. The balance at the start of the billing cycle is +$450.00. The list of transactions on the card is below. The interest rate is 15.9%. What is the balance due at the end +of the billing cycle? +Date +Activity +Amount +15-Mar +Billing Date Balance +$450.00 +21-Mar +Movie +$50.00 +28-Mar +Gasoline +$65.00 +31-Mar +Snacks +$15.50 +1-Apr +Dinner +$63.60 +1-Apr +Payment +$300.00 +1-Apr +Gasoline +$48.90 +9-Apr +Plane Tickets +$288.50 +Buying or Leasing a Car +49. Name three common fees when leasing a car. +50. Name three common fees when buying a car. +51. If the cost of a car is $35,500, and the residual value of the car after 3 years is $27,800, what is the monthly +depreciation for the car? +52. What is the annual interest rate if the money factor is 0.000045? +53. What is the payment for a lease on a car if the price of the car is $29,900, the residual price of the car is $16,500, +the lease is for 24 months, and the APR is 8.9%? +718 +6 • Chapter Summary +Access for free at openstax.org + +Renting and Homeownership +54. Name two advantages of renting a residence. +55. Name two advantages of buying a home. +56. What is PMI? +57. What are the monthly payments for a 25-year mortgage for $165,000 at 4.8% interest, not including escrow? +58. What are the monthly payments, including escrow, for a 25-year mortgage for $165,000 at 4.8% interest, if the +property taxes are $5,200 annually, the homeowner’s insurance is $825 every 6 months, and the PMI is $45 per +month? +Income Tax +59. What is the maximum gift that a person can receive that is an exemption on federal income taxes? +60. Find the gross income if a person earns $51,500 at their full-time job, they earn $5,300 for contracted services, and +$5,000 in gambling winnings. +61. Find adjusted gross income for a person with a gross income of $67,850, if they had a $2,000 gift, they paid $1,000 +in mortgage interest, $4,250 in property taxes, and they contributed $750 to a qualifying charity. +62. How much SSI is paid by a person with an annual income of $183,500? +63. How much in FICA taxes are paid from a person’s paycheck with gross income of $3,450? +64. How much in federal income tax does a person filing single pay if their taxable income is $205,240? Use the tax +table below. +Bracket +Lower Income Limit +Upper Income Limit +Tax Rate +1 +0 +$10,275 +10% +2 +$10,276 +$41,775 +12% +3 +$41,776 +$89,075 +22% +4 +$89,076 +$170,050 +24% +5 +$170,051 +$215,950 +32% +6 +$215,951 +$539,900 +35% +7 +$539,901 +37% +Chapter Test +1. 45 is what percent of 180? +2. Find 47% of 31. Round to two decimal places. +3. What is the sale price of a $90 shirt if there is a 25% discount for the shirt? +4. A small boutique buys purses for $27.50 per purse. What is the boutique’s retail price for the purses if they add a +90% markup? +5. What was the discount on a computer is the retail price was $1,200 and the sale price is $950? Round to two +decimal places if necessary. +6. A 15% simple interest loan of $2,000 is taken out for 75 days. How much is owed when the loan is due? +7. What is the future value of a $15,000 investment made at 4.65% interest compounded yearly for 10 years? +8. Apply the 50-30-20 budget philosophy to a monthly income of $3,920. +6 • Chapter Summary +719 + +9. $2,500 is deposited in an account that bears 5.25% interest compounded quarterly for 15 years. What is the return +on investment for that investment? Round to two decimal places if necessary. +10. How much per month must be paid into an ordinary annuity so its future value is $150,000 after 10 years if the +annuity earns 3.85% interest compounded monthly? +11. Bethanie’s employer matches up to 7% of annual salary for deposits into the company sponsored IRA. Bethanie +earns $62,400 annually. She deposits $8,000 annually inter her IRA. How much does the company deposit in +Bethanie’s IRA? +12. If $360 is deposited monthly in a mutual fund that is expected to earn 9.35% compounded monthly, how much will +the mutual fund be worth after 23 years? +13. A $14,250, 4-year loan is taken out at 9.15% interest. What are the monthly payments for the loan? +14. Using the amortization table below, how much total interest has been paid after payment 20? +15. What is the maximum amount of federal subsidized and unsubsidized loans per year for an independent student +in a graduate program? +720 +6 • Chapter Summary +Access for free at openstax.org + +16. Braden takes out a private student loan for $6,250 in his second year of his undergraduate IT program. He begins +paying the loan after 39 months. If Braden’s interest rate was 8%, how much is his balance when he begins to pay +off the loan? +17. The billing cycle for Therese’s credit card is July 7 to August 6 (31 days). The interest rate on the card is 17.9%. The +balance on her card at the start of the billing cycle is $925.00. The table below has her transactions for the billing +cycle. What is her balance due at the end of the cycle? +Date +Activity +Amount +7-Jul +Billing Date Balance +$925.00 +10-Jul +Child Care +$850.00 +10-Jul +Gasoline +$51.00 +12-Jul +Food +$135.50 +18-Jul +Payment +$750.00 +18-Jul +Pizza +$35.00 +18-Jul +Gift +$25.00 +21-Jul +Food +$121.75 +30-Jul +Gasoline +$47.50 +1-Aug +Department store +$173.00 +4-Aug +Food +$98.00 +18. The cost of a car is $45,750. The residual price after 2 years is $28,822.50. What is the monthly depreciation for the +car? +19. Find the payment for a 3-year lease on a car if the price of the car is $30,000, the residual price of the car is +$20,000, and the APR is 8.5%. +20. What are the monthly payments, excluding escrow, for a 20-year mortgage for $131,500 at 5.15% interest? +21. What are the monthly payments, including escrow, for a 20-year mortgage for $131,500 at 5.15% interest, with +property taxes of $3,170 per year and homeowner’s insurance of $715 every 6 months? +22. If a person earns $2700 gross on a paycheck, how much is taken out in FICA taxes? +23. Find the taxable income for a person who has $81,200 in earnings, $1,250 reported on a 1099-INT for a savings +account, $5,400 reported on a 1099-MISC for a contract job they did, paid $5,250 in mortgage interest, paid $4,700 +in property taxes, and has $4,200 in bond interest. +24. Find the federal income tax owed by a person filing single who has a taxable income of $114,750. Use the table +below to determine the taxes. +Bracket +Lower Income Limit +Upper Income Limit +Tax Rate +1 +0 +$10,275 +10% +2 +$10,276 +$41,775 +12% +6 • Chapter Summary +721 + +Bracket +Lower Income Limit +Upper Income Limit +Tax Rate +3 +$41,776 +$89,075 +22% +4 +$89,076 +$170,050 +24% +5 +$170,051 +$215,950 +32% +6 +$215,951 +$539,900 +35% +7 +$539,901 +37% +25. If a person owes $20,000 in federal income tax, but qualifies for $4,000 in child tax credit and $8,000 for child and +dependent care credit, how much federal tax does the person actually pay? +722 +6 • Chapter Summary +Access for free at openstax.org + +Figure 7.1 Roulette is a game whose outcomes are based entirely on the concept of probability. (credit: modification of +work "Roulette wheel" by Håkan Dahlström/Flickr, CC BY 2.0) +Chapter Outline +7.1 The Multiplication Rule for Counting +7.2 Permutations +7.3 Combinations +7.4 Tree Diagrams, Tables, and Outcomes +7.5 Basic Concepts of Probability +7.6 Probability with Permutations and Combinations +7.7 What Are the Odds? +7.8 The Addition Rule for Probability +7.9 Conditional Probability and the Multiplication Rule +7.10 The Binomial Distribution +7.11 Expected Value +Introduction +Casinos are big business; according to the American Gaming Association, commercial casinos in the United States +brought in over $43 billion in revenue in 2019. Casinos must walk a fine line in order to be profitable. Their customers +must lose more money than they win, on average, in order to stay in business. But if the chances of a single customer +winning more money than they lose is too small, people will stop coming in the door to play the games. +In this chapter, we'll study the techniques a casino must use to determine how likely it is that a customer will win a +particular game, and then how the casino decides how much money a winner will rake in so that the customers are +happy, but the casino also turns a profit in the long run. In order to figure out those likelihoods, we have to be able to +somehow consider every possible outcome of these games. For example, in a game that involves players receiving 5 +cards from a deck of 52, there are 2,598,960 possibilities for each player. We'll start off this chapter by learning how to +count those possible outcomes. +7 +PROBABILITY +7 • Introduction +723 + +7.1 The Multiplication Rule for Counting +Figure 7.2 The Multiplication Rule for Counting allows us to compute more complicated probabilities, like drawing two +aces from a deck. (credit: “Pair of Aces – Poker” by Poker Photos/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Apply the Multiplication Rule for Counting to solve problems. +One of the first bits of mathematical knowledge children learn is how to count objects by pointing to them in turn and +saying: “one, two, three, …” That’s a useful skill, but when the number of things that we need to count grows large, that +method becomes onerous (or, for very large numbers, impossible for humans to accomplish in a typical human lifespan). +So, mathematicians have developed short cuts to counting big numbers. These techniques fall under the mathematical +discipline of combinatorics, which is devoted to counting. +Multiplication as a Combinatorial Short Cut +One of the first combinatorial short cuts to counting students learn in school has to do with areas of rectangles. If we +have a set of objects to be counted that can be physically arranged into a rectangular shape, then we can use +multiplication to do the counting for us. Consider this set of objects (Figure 7.3): +Figure 7.3 +Certainly we can count them by pointing and running through the numbers, but it’s more efficient to group them (Figure +7.4). +Figure 7.4 +If we group the balls by 4s, we see that we have 6 groups (or, we can see this arrangement as 4 groups of 6 balls). Since +multiplication is repeated addition (i.e., +), we can use this grouping to quickly see that there +are 24 balls. +Let’s generalize this idea a little bit. Let’s say that we’re visiting a bakery that offers customized cupcakes. For the cake, +we have three choices: vanilla, chocolate, and strawberry. Each cupcake can be topped with one of four types of frosting: +vanilla, chocolate, lemon, and strawberry. How many different cupcake combinations are possible? We can think of +laying out all the possibilities in a grid, with cake choices defining the rows and frosting choices defining the columns +724 +7 • Probability +Access for free at openstax.org + +(Figure 7.5). +Figure 7.5 +Since there are 3 rows (cakes) and 4 columns (frostings), we have +possible combinations. This is the reasoning +behind the Multiplication Rule for Counting, which is also known as the Fundamental Counting Principle. This rule says +that if there are +ways to accomplish one task and +ways to accomplish a second task, then there are +ways to +accomplish both tasks. We can tack on additional tasks by multiplying the number of ways to accomplish those tasks to +our previous product. +EXAMPLE 7.1 +Using the Multiplication Rule for Counting +Every card in a standard deck of cards has two identifying characteristics: a suit (clubs, diamonds, hearts, or spades; +these are indicated by these symbols, respectively: +, +, +, +) and a rank (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and +king; the letters A, J, Q, and K are used to represent the words). Each possible pair of suit and rank appears exactly once +in the deck. How many cards are in the standard deck? +Solution +Since there are 4 suits and 13 ranks, the number of cards must be +(Figure 7.6). +7.1 • The Multiplication Rule for Counting +725 + +Figure 7.6 Standard Deck of Cards, Sorted by Rank and Suit (credit: "Playing Cards, USS Arkansas" by Naval History & +Heritage Command/Flickr, CC BY 2.0) +YOUR TURN 7.1 +1. Joe’s Pizza Shack offers pizzas with 4 different types of crust and a choice of 15 toppings. How many different +one-topping pizzas can be made at Joe’s? +EXAMPLE 7.2 +Using the Multiplication Rule for Counting for 4 Groups +The University Combinatorics Club has 31 members: 8 seniors, 7 juniors, 5 sophomores, and 11 first-years. How many +possible 4-person committees can be formed by selecting 1 member from each class? +Solution +Since we have 8 choices for the senior, 7 choices for the junior, 5 for the sophomore, and 11 for the first-year, there are +different ways to fill out the committee. +YOUR TURN 7.2 +1. The menu for Joe’s Pizza Shack offers pizzas with 4 different types of crust and a choice of 15 toppings. Suppose +that Joe’s also offers a choice of 3 sauces and 2 cheese blends. How many different one-topping pizzas can be +made at Joe’s now? +EXAMPLE 7.3 +Using the Multiplication Rule for Counting for More Groups +The standard license plates for vehicles in a certain state consist of 6 characters: 3 letters followed by 3 digits. There are +26 letters in the alphabet and 10 digits (0 through 9) to choose from. How many license plates can be made using this +format? +Solution +Since there are 26 different letters and 10 different digits, the total number of possible license plates is +726 +7 • Probability +Access for free at openstax.org + +. +YOUR TURN 7.3 +1. At a certain college, ID cards are issued to all students, faculty, and staff. These cards have unique ID codes for +each person: a letter to indicate the person’s status (S for students, F for faculty, and E for staff), followed by 5 +digits and finally 3 letters (these letters can be anything). How many different ID codes can be created using this +scheme? +Check Your Understanding +1. A website that lets you build custom belts has 18 different buckles and 30 different straps. How many different +belts can be made using those materials? +2. A chain of chicken restaurants offers a combo that includes your choice of 3 or 5 chicken strips, along with your +choice of side dish. If there are 7 side dishes, how many different ways are there to build this combo meal? +3. When you flip a coin, there are 2 possible outcomes: heads and tails. Let’s say you flip a coin 10 times, and after +each you write down the result of the flip (H for heads, T for tails). How many different results (strings of 10 +characters, where each is either an H or a T) are possible? +4. A T-shirt company allows shoppers to customize their shirts in several ways. There are 5 sizes, 8 shirt colors, 4 +designs, and 5 design colors. How many different shirts can be made? +5. Josephine is trying to build her class schedule for next semester. Because of her work schedule, she has only 4 +class periods that can work for her, and she must take 4 classes. If there are 15 classes that she could take during +the first period, 18 during the second, 12 during the third, and 8 during the fourth, how many different schedules +could Josephine build? +SECTION 7.1 EXERCISES +An ice-cream parlor sells 26 different flavors of ice cream. A basic sundae has one scoop of any flavor of ice cream, your +choice of one of 3 sauces, and any one of 8 different toppings. +1. How many different basic sundaes are possible? +2. The ice-cream parlor also sells a medium sundae. The options are the same except it starts with 2 scoops of ice +cream, which can be the same flavor or different flavors. How many different medium sundaes are there? +3. The ice-cream parlor also sells a large sundae. The choice of a large sundae allows you to choose any 3 scoops +of ice cream, any 2 sauces (they can be the same, or you can choose 2 different ones), and any 3 toppings (that +might be 3 servings of the same topping, or 2 servings of one topping and a single serving of another, or 3 +different toppings). How many different large sundaes are possible? +4. A company that builds custom computers offers 4 hard drive sizes, 4 memory sizes, 3 graphics cards, and 3 display +options. How many computer configurations do they offer, if customers choose one of each customization? +5. A video game allows users to customize their avatars. There are 12 hair styles that users may choose from, as well +as 5 hair colors, 8 skin tones, 24 shirts, 12 pants, and 8 shoes. How many different avatars are possible? +6. A small company has 3 divisions: Sales, Research and Development, and Manufacturing. One person from each +division will be chosen to create an advisory board for the management group. If there are 8 people in Sales, 15 in +Research and Development, and 48 in Manufacturing, how many different compositions of the advisory board are +possible? +7. A multiple-choice quiz has 5 questions, each of which has 4 possible answers. How many different ways are there +to respond to this quiz? +8. The teacher decides to make the quiz from above a little harder by offering 5 responses on each of the 5 questions. +How many ways are there to respond to this quiz? +9. In the United States, radio and television broadcast stations are assigned unique identifiers known as call signs. +Call signs consist of 4 letters. The first is either K or W (generally speaking, stations with a K call sign are west of +7.1 • The Multiplication Rule for Counting +727 + +the Mississippi River and stations with a W call sign are east of the river, though there are several exceptions to this +rule); the remaining 3 letters can be anything. How many possible call signs are there under this system? +10. Little sister has asked big brother to play a new game she’s invented. It uses a modified deck of cards with 3 suits +and only the numbered cards (those with rank 2 through 10). How many cards are in her deck? +11. The board game Mastermind has 2 players. One of them is designated the codemaker who creates a code that +consists of a series of 4 colors (indicated in the game with 4 colored pegs), which may contain repeats. The other +player, who is the codebreaker, tries to guess the code. If there are 6 colors that the codemaker can use to make +the code, how many possible codes can be made? +7.2 Permutations +Figure 7.7 We can use permutations to calculate the number of different orders of finish in an Olympic swimming heat. +(credit: “London 2012 Olympics Park Stratford London” by Gary Bembridge/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Use the Multiplication Rule for Counting to determine the number of permutations. +2. +Compute expressions containing factorials. +3. +Compute permutations. +4. +Apply permutations to solve problems. +Swimming events are some of the most popular events at the summer Olympic Games. In the finals of each event, 8 +swimmers compete at the same time, making for some exciting finishes. How many different orders of finish are +possible in these events? In this section, we’ll extend the Multiplication Rule for Counting to help answer questions like +this one, which relate to permutations. A permutation is an ordered list of objects taken from a given population. The +length of the list is given, and the list cannot contain any repeated items. +Applying the Multiplication Rule for Counting to Permutations +In the case of the swimming finals, one possible permutation of length 3 would be the list of medal winners (first, +second, and third place finishers). A permutation of length 8 would be the full order of finish (first place through eighth +place). Let’s use the Multiplication Rule for Counting to figure out how many of each of these permutations there are. +EXAMPLE 7.4 +Using the Multiplication Rule for Counting to Find the Number of Permutations +The final heat of Olympic swimming events features 8 swimmers (or teams of swimmers). +1. +How many different podium placements (first place, second place, and third place) are possible? +2. +How many different complete orders of finish (first place through eighth place) are possible? +728 +7 • Probability +Access for free at openstax.org + +Solution +1. +Let’s start with the first place finisher. How many options are there? Since 8 swimmers are competing, there are 8 +possibilities. Once that first swimmer completes the race, there are 7 swimmers left competing for second place. +After the second finisher is decided, there are 6 swimmers remaining who could possibly finish in third place. Thus, +there are 8 possibilities for first place, 7 for second place, and 6 for third place. The Multiplication Rule for Counting +then tells us there are +different ways the winners’ podium can be filled out. +2. +To look at the complete order of finish, we can continue the pattern we can see in part 1 of this example: There are 5 +possibilities for fourth place, 4 for fifth place, 3 for sixth place, 2 for seventh place, and then just 1 swimmer is left to +finish in eighth place. Using the Multiplication Rule for Counting, we see that there are +possible orders of finish. +YOUR TURN 7.4 +1. You have a hand of 5 cards (that happen to create what’s called a royal flush in the game of poker): 10 , J , Q , +K , and A . Into how many different orders can you put those cards? +Factorials +The pattern we see in Example 7.4 occurs commonly enough that we have a name for it: factorial. +For any positive whole number +, we define the factorial of +(denoted +and read " +factorial") to be the product of +every whole number less than or equal to +. We also define 0! to be equal to one. We will use factorials in a couple of +different contexts, so let's get some practice doing computations with them. +EXAMPLE 7.5 +Computing Factorials +Compute the following: +1. +2. +3. +Solution +1. +2. +There are two ways to approach this calculation. The first way is to compute the factorials first, then divide: +However, there is an easier way! You may notice in the second step that there are several terms that can be +canceled; that’s always the case whenever we divide factorials. In this case, notice that we can rewrite the numerator +like this: +With that in mind, we can proceed this way by canceling out the 6!: +That’s much easier! +3. +Let’s approach this one using our canceling technique. When we see two factorials in either the numerator or +denominator, we should focus on the larger one first. So: +7.2 • Permutations +729 + +YOUR TURN 7.5 +Compute the following: +1. +2. +3. +Permutations +As we’ve seen, factorials can pop up when we’re computing permutations. In fact, there is a formula that we can use to +make that connection explicit. Let’s define some notation first. If we have a collection of +objects and we wish to create +an ordered list of +of the objects (where +), we’ll call the number of those permutations +(read “the number +of permutations of +objects taken +at a time”). We formalize the formula we'll use to compute permutations below. +FORMULA +If you wondered why we defined +earlier, it was to make formulas like this one work; if we have +objects and want +to order all of them (so, we want the number of permutations of +objects taken +at a time), we get +. Next, we’ll get some practice computing these permutations. +EXAMPLE 7.6 +Computing Permutations +Find the following numbers: +1. +The number of permutations of 12 objects taken 3 at a time +2. +The number of permutations of 8 objects taken 5 at a time +3. +The number of permutations of 32 objects taken 2 at a time +Solution +1. +2. +3. +YOUR TURN 7.6 +Find the following numbers: +1. The number of permutations of 6 objects taken 2 at a time +2. The number of permutations of 14 objects taken 4 at a time +3. The number of permutations of 19 objects taken 3 at a time +EXAMPLE 7.7 +Applying Permutations +1. +A high school graduating class has 312 students. The top student is declared valedictorian, and the second-best is +named salutatorian. How many possible outcomes are there for the valedictorian and salutatorian? +730 +7 • Probability +Access for free at openstax.org + +2. +In the card game blackjack, the dealer’s hand of 2 cards is dealt with 1 card faceup and 1 card facedown. If the game +is being played with a single deck of (52) cards, how many possible hands could the dealer get? +3. +The University Combinatorics Club has 3 officers: president, vice president, and treasurer. If there are 18 members +of the club, how many ways are there to fill the officer positions? +Solution +1. +This is the number of permutations of 312 students taken 2 at a time, and +. +2. +We want the number of permutations of 52 cards taken 2 at a time, and +. +3. +Here we’re looking for the number of permutations of 18 members taken 3 at a time, and +. +YOUR TURN 7.7 +1. One of the big draws at this year’s state fair is the pig race. There are 15 entrants, and prizes are given to the top +three finishers. How many different combinations of top-three finishes could there be? +WHO KNEW? +Very Big Permutations +Permutations involving relatively small sets of objects can get very big, very quickly. A standard deck contains 52 +cards. So, the number of different ways to shuffle the cards—in other words, the number of permutations of 52 +objects taken 52 at a time—is +(written out, that’s an 8 followed by 67 zeroes). The estimated age of the +universe is only about +seconds. So, if a very bored all-powerful being started shuffling cards at the instant +the universe began, it would have to have averaged at least +shuffles per second since the +beginning of time to have covered every possible arrangement of a deck of cards. That means the next time you pick +up a deck of cards and give it a good shuffle, it’s almost certain that the particular arrangement you created has never +been created before and likely never will be created again. +Check Your Understanding +6. Compute 5!. +7. Compute +. +8. Compute +. +9. Compute +. +10. The standard American edition of the board game Monopoly has a deck of 15 orange Chance cards. In how many +different ways could the first 4 Chance cards drawn in a game appear? +SECTION 7.2 EXERCISES +For the following exercises, give a whole number that’s equal to the given expression. +1. 3! +2. 9! +3. +4. +5. +6. +7. +8. +9. +10. +7.2 • Permutations +731 + +11. +12. +13. +14. +The following exercises are about the card game euchre, which uses a partial standard deck of cards: It only has the +cards with ranks 9, 10, J, Q, K, and A for a total of 24 cards. Some variations of the game use the 8s or the 7s and 8s, but +we’ll stick with the 24-card version. +15. A euchre hand contains 5 cards. How many ways are there to receive a 5-card hand (where the order in which +the cards are received matters, i.e., 9 , J , +, +, +is different from +J , 9 , +, +? +16. After all 4 players get their hands, the remaining 4 cards are placed facedown in the center of the table. How +many arrangements of 4 cards are there from this deck? +17. Euchre is played with partners. How many ways are there for 2 partners to receive 5-card hands (where the +order in which the cards are received matters)? +18. How many different arrangements of the full euchre deck are possible (i.e., how many different shuffles are +there)? +The following exercises involve a horse race with 13 entrants. +19. How many possible complete orders of finish are there? +20. An exacta bet is one where the player tries to predict the top two finishers in order. How many possible exacta +bets are there for this race? +21. A trifecta bet is one where the player tries to predict the top three finishers in order. How many possible trifecta +bets are there for this race? +22. A superfecta bet is one where the player tries to predict the top four finishers in order. How many possible +superfecta bets are there for this race? +7.3 Combinations +Figure 7.8 Combinations help us count things like the number of possible card hands, when the order in which the cards +were drawn doesn’t matter. (credit: “IMG_3177” by Zanaca/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Distinguish between permutation and combination uses. +2. +Compute combinations. +3. +Apply combinations to solve applications. +In Permutations, we studied permutations, which we use to count the number of ways to generate an ordered list of a +given length from a group of objects. An important property of permutations is that the order of the list matters: The +results of a race and the selection of club officers are examples of lists where the order is important. In other situations, +the order is not important. For example, in most card games where a player receives a hand of cards, the order in which +the cards are received is irrelevant; in fact, players often rearrange the cards in a way that helps them keep the cards +organized. +Combinations: When Order Doesn’t Matter +In situations in which the order of a list of objects doesn’t matter, the lists are no longer permutations. Instead, we call +732 +7 • Probability +Access for free at openstax.org + +them combinations. +EXAMPLE 7.8 +Distinguishing Between Permutations and Combinations +For each of the following situations, decide whether the chosen subset is a permutation or a combination. +1. +A social club selects 3 members to form a committee. Each of the members has an equal share of responsibility. +2. +You are prompted to reset your email password; you select a password consisting of 10 characters without repeats. +3. +At a dog show, the judge must choose first-, second-, and third-place finishers from a group of 16 dogs. +4. +At a restaurant, the special of the day comes with the customer’s choice of 3 sides taken from a list of 6 possibilities. +Solution +1. +Since there is no distinction among the responsibilities of the 3 committee members, the order isn’t important. So, +this is a combination. +2. +The order of the characters in a password matter, so this is a permutation. +3. +The order of finish matters in a dog show, so this is a permutation. +4. +A plate with mashed potatoes, peas, and broccoli is functionally the same as a plate with peas, broccoli, and mashed +potatoes, so this is a combination. +YOUR TURN 7.8 +Decide whether the following represent permutations or combinations: +1. On Halloween, you give each kid who comes to your door 3 pieces of candy, taken randomly from a candy +dish. +2. Your class is going on a field trip, but there are too many people for one vehicle. Your instructor chooses half +the class to take the first vehicle. +Counting Combinations +Permutations and combinations are certainly related, because they both involve choosing a subset of a large group. Let’s +explore that connection, so that we can figure out how to use what we know about permutations to help us count +combinations. We’ll take a basic example. How many ways can we select 3 letters from the group A, B, C, D, and E? If +order matters, that number is +. That’s small enough that we can list them all out in the table below. +ABC +ABD +ABE +ACB +ACD +ACE +ADB +ADC +ADE +AEB +AEC +AED +BAC +BAD +BAE +BCA +BCD +BCE +BDA +BDC +BDE +BEA +BEC +BED +CAB +CAD +CAE +CBA +CBD +CBE +CDA +CDB +CDE +CEA +CEB +CED +DCA +DAC +DAE +DBA +DBC +DBE +DCA +DCB +DCE +DEA +DEB +DEC +EAB +EAC +EAD +EBA +EBC +EBD +ECA +ECB +ECD +EDA +EDB +EDC +7.3 • Combinations +733 + +Now, let’s look back at that list and color-code it so that groupings of the same 3 letters get the same color, as shown in +Figure 7.9: +Figure 7.9 +After color-coding, we see that the 60 cells can be seen as 10 groups (colors) of 6. That’s no coincidence! We’ve already +seen how to compute the number of permutations using the formula To compute the number of combinations, let’s +count them another way using the Multiplication Rule for Counting. We’ll do this in two steps: +Step 1: Choose 3 letters (paying no attention to order). +Step 2: Put those letters in order. +The number of ways to choose 3 letters from this group of 5 (A, B, C, D, E) is the number of combinations we’re looking +for; let’s call that number +(read “the number of combinations of 5 objects taken 3 at a time”). We can see from our +chart that this is ten (the number of colors used). We can generalize our findings this way: remember that the number of +permutations of +things taken +at a time is +. That number is also equal to +, and so it must be the +case that +. Dividing both sides of that equation by +gives us the formula below. +FORMULA +EXAMPLE 7.9 +Using the Combination Formula +Compute the following: +1. +2. +3. +Solution +1. +2. +3. +734 +7 • Probability +Access for free at openstax.org + +YOUR TURN 7.9 +Compute the following: +1. +2. +3. +EXAMPLE 7.10 +Applying the Combination Formula +1. +In the card game Texas Hold’em (a variation of poker), players are dealt 2 cards from a standard deck to form their +hands. How many different hands are possible? +2. +The board game Clue uses a deck of 21 cards. If 3 people are playing, each person gets 6 cards for their hand. How +many different 6-card Clue hands are possible? +3. +Palmetto Cash 5 is a game offered by the South Carolina Education Lottery. Players choose 5 numbers from the +whole numbers between 1 and 38 (inclusive); the player wins the jackpot of $100,000 if the randomizer selects those +numbers in any order. How many different sets of winning numbers are possible? +Solution +1. +A standard deck has 52 cards, and a hand has 2 cards. Since the order doesn’t matter, we use the formula for +counting combinations: +2. +Again, the order doesn’t matter, so the number of combinations is: +3. +There are 38 numbers to choose from, and we must pick 5. Since order doesn’t matter, the number of combinations +is: +YOUR TURN 7.10 +1. At a charity event with 58 people in attendance, 3 raffle winners are chosen. All receive the same prize, so +order doesn’t matter. How many different groups of 3 winners can be chosen? +2. A sorority with 42 members needs to choose a committee with 4 members, each with equal responsibility. +How many committees are possible? +The notation and nomenclature used for the number of combinations is not standard across all sources. You’ll +sometimes see +instead of +. Sometimes you’ll hear that expression read as “ +choose ” as shorthand for +“the number of combinations of +objects taken +at a time.” +PEOPLE IN MATHEMATICS +Early Eastern Mathematicians +Although combinations weren’t really studied in Europe until around the 13th century, mathematicians of the Middle +and Far East had already been working on them for hundreds of years. The Indian mathematician known as Pingala +had described them by the second century BCE; Varāhamihira (fl. sixth century) and Halayudha (fl. 10th century) +extended Pingala’s work. In the ninth century, a Jain mathematician named Mahāvīra gave the formula for +7.3 • Combinations +735 + +combinations that we use today. +In 10th-century Baghdad, a mathematician named Al-Karaji also knew formulas for combinations; though his work is +now lost, it was known to (and repeated by) Persian mathematician Omar Khayyam, whose work survives. Khayyam is +probably best remembered as a poet, with his Rubaiyat being his most famous work. +Meanwhile, in 11th-century China, Jia Xian also was working with combinations, as was his 13th-century successor +Yang Hui. +It is not known whether the discoveries of any of these men were known in the other regions, or if the Indians, +Persians, and Chinese all came to their discoveries independently. We do know that mathematical knowledge and +sometimes texts did get passed along trade routes, so it can’t be ruled out. +EXAMPLE 7.11 +Combining Combinations with the Multiplication Rule for Counting +The student government at a university consists of 10 seniors, 8 juniors, 6 sophomores, and 4 first-years. +1. +How many ways are there to choose a committee of 8 people from this group? +2. +How many ways are to choose a committee of 8 people if the committee must consist of 2 people from each class? +Solution +1. +There are 28 people to choose from, and we need 8. So, the number of possible committees is +. +2. +Break the selection of the committee members down into a 4-step process: Choose the seniors, then choose the +juniors, then the sophomores, and then the first-years, as shown in the table below: +Class +Number of Ways to Choose Committee Representatives +senior +junior +sophomore +first-year +The Multiplication Rule for Counting tells us that we can get the total number of ways to complete this task by +multiplying together the number of ways to do each of the four subtasks. So, there are +possible committees with these restrictions. +YOUR TURN 7.11 +1. How many ways are there to choose a hand of 6 cards from a standard deck with the constraint that 3 are +, 2 +are +, and 1 is +? +Check Your Understanding +11. Suppose you want to count the number of ways that you can arrange the apps on the home screen on your phone. +Should you use permutations or combinations? +12. Your little brother is packing up for a family vacation, but there’s only room for 3 of his toys. If you want to know +how many possible groups of toys he can bring, should you use permutations or combinations? +13. Compute +. +14. Compute +. +736 +7 • Probability +Access for free at openstax.org + +15. You’re planning a road trip with some friends. Though you have 6 friends you’d consider bringing along, you only +have room for 3 other people in the car. How many different possibilities are there for your road trip squad? +16. You’re packing for a trip, for which you need 3 shirts and 3 skirts. If you have 8 shirts and 5 skirts that would work +for the trip, how many different ways are there to pack for the trip? +SECTION 7.3 EXERCISES +For the following exercises, decide whether the situation describes a permutation or a combination. +1. You’re packing for vacation, and you need to pick 5 shirts. +2. You and your friends are about to play a game, and you need to decide who will have the first turn, second turn, +and so on. +3. You are watching your favorite reality show, and you want to know how many possibilities there are for the +order of finish for the top three. +4. You are going to be working in groups of 4 with your classmates, and you want to know how many possibilities +there are for the composition of your group. +For the following exercises, express your answers as whole numbers. +5. +6. +7. +8. +9. +10. +11. +12. +13. +14. +15. +16. +17. In most variations of the card game poker, a hand consists of 5 cards, where the order doesn’t matter. How many +different poker hands are there? +18. A professor starts each class by choosing 3 students to present solutions to homework problems to the class. If +there are 41 students in the class, in how many different ways can the professor make those selections? +19. An election for at-large members of a school board has 7 candidates; 3 will be elected. How many different ways +can those 3 seats be filled? +20. There are 20 contestants on a reality TV show; at the end of the first episode, 10 are eliminated. How many +different groups of eliminated contestants are possible? +21. At a horse race, bettors can place a bet called an exacta box. For this bet, the player chooses 2 horses; if those +horses finish first and second (in either order), the player wins. In a race with 12 horses in the field, how many +possible exacta box bets are there? +The following exercises are about the card game euchre, which uses a partial standard deck of cards: it only has the +cards with ranks 9, 10, J, Q, K, and A (for a total of 24 cards). Some variations of the game use the 8s or the 7s and 8s, +but we’ll stick with the 24-card version. +22. A euchre hand contains 5 cards. How many ways are there to receive a 5-card hand (where the order in which +the cards are received doesn’t matter, i.e., 9 , J , +, +, +is the same as +J , 9 , +, +)? +23. After all 4 players get their hands, the remaining 4 cards are placed face down in the center of the table. How +many different groups of 4 cards are there from this deck? +24. Euchre is played with partners. How many ways are there for 2 partners to receive 5-card hands (where, as +above, the order doesn’t matter)? Hint: After the first person gets their cards, there are +cards left +for the second person. +You and 5 of your friends are at an amusement park, and are about to ride a roller coaster. The cars have room for 6 +people arranged in 3 rows of 2, so you and your friends will perfectly fill one car. +25. How many ways are there to choose the 2 people in the front row? +26. Assuming the front row has been selected, how many ways are there to choose the 2 people in the middle row? +7.3 • Combinations +737 + +27. Assuming the first 2 rows have been selected, how many ways are there to choose the 2 people in the back +row? +28. Using the Multiplication Rule for Counting and your answers to the earlier parts of this exercise, how many +ways are there for your friends to sort yourselves into rows to board the roller coaster? +The University Combinatorics Club has 18 members. Four of them will be selected to form a committee. +29. How many different committees of 4 are possible, assuming all of the duties are shared equally? +30. Instead of sharing responsibility equally, one person will be chosen to be the committee chair. How many +different committees are possible? Count these by selecting a chair first, then selecting the remaining 3 +members of the committee from the remaining club members and use the Multiplication Rule for Counting. +Show your work. +31. Let’s count the number of committees with chairs a different way: First, choose 4 people for the committee (as +in the first question), then choose 1 of the 4 to be chair. Show your work. Do you get the same number? +Powerball® is a multistate lottery game, which costs $2 to play. Players fill out a ticket by choosing 5 numbers between +1 and 69 (these are the white balls) and then a single number between 1 and 26 (this is the Powerball). +32. How many different ways are there to choose the white balls? Players who match these 5 numbers exactly (but +not the Powerball) win $1 million. +33. How many ways are there to choose the Powerball? Players who correctly pick the Powerball win $4. +34. How many ways are there to play the game altogether? Players who match all 5 white balls and the Powerball +win (or share) the grand prize. (The grand prize starts at $40 million; if no players win the grand prize, the value +goes up for the next drawing. The highest value it has ever reached is $1.586 billion!) How many ways are there +to fill out a single Powerball ticket? +35. You are in charge of programming for a music festival. The festival has a main stage, a secondary stage, and +several smaller stages. There are 40 bands confirmed for the festival. Five of those will play the main stage, and 8 +will play the secondary stage. How many ways are there for you to allocate bands to these 2 stages? +7.4 Tree Diagrams, Tables, and Outcomes +Figure 7.10 In genetics, the characteristics of an offspring organism depends on the characteristics of its parents. (credit: +“Pea Plant” by Maria Keays/Flickr, CC BY 2.0)) +Learning Objectives +After completing this section, you should be able to: +1. +Determine the sample space of single stage experiment. +2. +Use tables to list possible outcomes of a multistage experiment. +3. +Use tree diagrams to list possible outcomes of a multistage experiment. +In the 19th century, an Augustinian friar and scientist named Gregor Mendel used his observations of pea plants to set +out his theory of genetic propagation. In his work, he looked at the offspring that resulted from breeding plants with +different characteristics together. For applications like this, it is often insufficient to only know in how many ways a +738 +7 • Probability +Access for free at openstax.org + +process might end; we need to be able to list all of the possibilities. As we’ve seen, the number of possible outcomes can +be very large! Thus, it’s important to have a strategy that allows us to systematically list these possibilities to make sure +we don’t leave any out. In this section, we’ll look at two of these strategies. +Single Stage Experiments +When we are talking about combinatorics or probability, the word “experiment” has a slightly different meaning than it +does in the sciences. Experiments can range from very simple (“flip a coin”) to very complex (“count the number of +uranium atoms that undergo nuclear fission in a sample of a given size over the course of an hour”). Experiments have +unknown outcomes that generally rely on something random, so that if the experiment is repeated (or replicated) the +outcome might be different. No matter what the experiment, though, analysis of the experiment typically begins with +identifying its sample space. +The sample space of an experiment is the set of all of the possible outcomes of the experiment, so it’s often expressed +as a set (i.e., as a list bound by braces; if the experiment is “randomly select a number between 1 and 4,” the sample +space would be written +). +EXAMPLE 7.12 +Finding the Sample Space of an Experiment +For each of the following experiments, identify the sample space. +1. +Flip a coin (which has 2 faces, typically called “heads” and “tails”) and note which face is up. +2. +Flip a coin 10 times and count the number of heads. +3. +Roll a 6-sided die and note the number that is on top. +4. +Roll two 6-sided dice and note the sum of the numbers on top. +Solution +1. +If we use “H” to denote “heads is facing up” and “T” to denote “tails is facing up”, then the sample space is {H, T}. +2. +It’s possible (though unlikely) that there will be no heads flipped; the outcome in that case would be “0.” It’s also +possible (more likely, but still quite unlikely) that only one flip will result in heads. Any other whole number is +possible, up to the maximum: We’re flipping the coin 10 times, so we can’t get any more than 10 heads. So, the +sample space is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. +3. +There are 6 numbers on the die: 1, 2, 3, 4, 5, and 6. So, the sample space for a single roll of the die is {1, 2, 3, 4, 5, 6}. +4. +If we roll 2 dice, the smallest possible sum we could get is +and the biggest is +. Every other +whole number between those two is possible. So, the sample space is {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. +YOUR TURN 7.12 +Identify the sample space of each experiment. +1. You draw a card from a standard deck and note its suit. +2. You draw a card from a standard deck and note its rank. +3. You roll a 4-sided die and note the number on the bottom. (A 4-sided die is shaped like a pyramid, so when it +comes to rest, there’s no single side facing up.) +4. You roll three 4-sided dice and note the sum of the numbers on the bottom. +Multistage Experiments +Some experiments have more complicated sample spaces because they occur in stages. These stages can occur in +succession (like drawing cards one at a time) or simultaneously (rolling 2 dice). Sample spaces get more complicated as +the complexity of the experiment increases, so it’s important to choose a systematic method for identifying all of the +possible outcomes. The first method we’ll discuss is the table. +Using Tables to Find Sample Spaces +Tables are useful for finding the sample space for experiments that meet two criteria: (1) The experiment must have only +two stages, and (2) the outcomes of each stage must have no effect on the outcomes of the other. When the stages do +not affect each other, we say the stages are independent. Otherwise, the stages are dependent and so we can’t use +tables; we’ll look at a method for analyzing dependent stages soon. +7.4 • Tree Diagrams, Tables, and Outcomes +739 + +EXAMPLE 7.13 +Determining Independence +Decide whether the two stages in these experiments are independent or dependent. +1. +You flip a coin and note the result, and then flip the coin again and note the result. +2. +You draw 2 cards from a standard deck (52 cards), one at a time. +Solution +1. +No matter what happens on the first flip, the second flip has the same sample space: {H, T} (You’ll sometimes hear +the phrase “The coin has no memory”). So, these stages are independent. +2. +Let’s say that the first card you draw is A . The sample space for the second draw consists of all the cards except A +(since that card is no longer in the deck, you can’t draw it again). If instead that first card was 2 , the sample space +for the second draw is different: it’s every card except 2 . Since the sample space for the second card changes +based on the result of the first draw, these stages are dependent. +YOUR TURN 7.13 +Decide whether the two stages in these experiments are dependent or independent. +1. You’re getting dressed to go to a party, and you plan to wear a blouse and a skirt. You choose the blouse first, +then the skirt (assume that you’d be comfortable wearing any of your skirts with any of your blouses). +2. On further reflection, you realize that some of your skirts clash with some of your blouses. So, you choose the +blouse first, and then choose a skirt that goes with your chosen blouse. +If you have a two-stage experiment with independent stages, a table is the most straightforward way to identify the +sample space. To build a table, you list the outcomes of one stage of the experiment along the top of the table and the +outcomes of the other stage down the side. The cells in the interior of the table are then filled using the outcomes +associated with each cell’s row and column. Let’s look at an example. +EXAMPLE 7.14 +Using Tables to Identify Sample Spaces +Identify the sample spaces of these experiments using tables. +1. +You roll two dice: one 4-sided and one 6-sided. +2. +You’re in an ice-cream shop, and you’re going to get a single scoop of ice cream with a topping. The flavors of ice +cream you’re considering are vanilla, chocolate, and rocky road; the toppings are fudge, whipped cream, and +sprinkles. +3. +The pea plants you’re breeding have two possible pod colors: green and yellow. These colors are decided by a +particular gene, which comes in two types: “G” for green, and “g” for yellow (In genetics, capital letters usually +denote dominant genes, while lower-case letters denote recessive genes). Each plant has two genes. If you breed a +Gg pea plant with a gg plant, the offspring plant will get one gene from each parent. What are the possible +outcomes? +Solution +1. +Step 1: Make the outline of a table, with the results of the 4-sided roll on one side and the results of the 6-sided roll +on the other. In practice, it doesn’t matter which you choose; for this example, we’ll put the 4-sided results on top +(labeling the columns of the chart) and the 6-sided results on the side (labeling the rows of the chart) as shown in +the following table: +740 +7 • Probability +Access for free at openstax.org + +4-Sided Roll +1 +2 +3 +4 +6-Sided Roll +1 +2 +3 +4 +5 +6 +Step 2: Fill in the results in each cell of the table below. Use the notation of an ordered pair, with the row label first. +4-Sided Roll +1 +2 +3 +4 +6-Sided Roll +1 +(1,1) +(1,2) +(1,3) +(1,4) +2 +(2,1) +(2,2) +(2,3) +(2,4) +3 +(3,1) +(3,2) +(3,3) +(3,4) +4 +(4,1) +(4,2) +(4,3) +(4,4) +5 +(5,1) +(5,2) +(5,3) +(5,4) +6 +(6,1) +(6,2) +(6,3) +(6,4) +So, for example, (3,2) represents the outcome where the 6-sided roll results in a 3, and the 4-sided roll gives us a 2. +Thus, the sample space of the experiment is {(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), +(4,2), (4,3), (4,4), (5,1), (5,2), (5,3), (5,4), (6,1), (6,2), (6,3), (6,4)}. +2. +Step 1: Let’s put the flavors on the rows and the toppings on the columns of the following table: +Toppings +fudge +whipped cream +sprinkles +Flavors +vanilla +chocolate +rocky road +Step 2: We can fill in the cells of the table below with the resulting combinations. +7.4 • Tree Diagrams, Tables, and Outcomes +741 + +Toppings +fudge +whipped cream +sprinkles +Flavors +vanilla +vanilla with fudge +vanilla with whipped cream +vanilla with sprinkles +chocolate +chocolate with fudge +chocolate with whipped cream +chocolate with sprinkles +rocky road +rocky road with fudge +rocky road with whipped cream +rocky road with sprinkles +So, the sample space is {vanilla with fudge, vanilla with whipped cream, vanilla with sprinkles, chocolate with fudge, +chocolate with whipped cream, chocolate with sprinkles, rocky road with fudge, rocky road with whipped cream, +rocky road with sprinkles}. +3. +Step 1: We’ll put the parents’ genes (P1 and P2) as labels on the rows and columns of the following table: +P2 +g +g +P1 +G +g +Step 2: We’ll fill in the offspring’s gene composition, listing parent 1’s gene first in the table below. +P2 +g +g +P1 +G +Gg +Gg +g +gg +gg +Thus, the sample space is {Gg, Gg, gg, gg}. (Diagrams like this, which allow us to identify the genotypes of offspring, +are called Punnett squares in honor of Reginald Punnett (1875–1967), who first used them in the context of +genetics.) +YOUR TURN 7.14 +1. Use a table to identify the sample space of an experiment in which you flip a coin and roll a 6-sided die. +Using Tree Diagrams to Identify Sample Spaces +In experiments where there are more than two stages, or where the stages are dependent, a tree diagram is a helpful +tool for systematically identifying the sample space. Tree diagrams are built by first drawing a single point (or node), +then from that node we draw one branch (a short line segment) for each outcome of the first stage. Each branch gets its +own node at the other end (which we typically label with the corresponding outcome for that branch); from each of +these, we draw another branch for each outcome of the second stage, assuming that the outcome of the first stage +matches the branch we were on. If there are other stages, we can continue from there by continuing to add branches +and nodes. This sounds really complicated, but it’s easier to understand through an example. +742 +7 • Probability +Access for free at openstax.org + +EXAMPLE 7.15 +Using a Tree Diagram to Identify a Sample Space +Use a tree diagram to find the sample spaces of each of the following experiments: +1. +You flip a coin 3 times, noting the outcome of each flip in order. +2. +You flip a coin. If the result is heads, you roll a 4-sided die. If it’s tails, you roll a 6-sided die. +3. +You are planning to go on a hike with a group of friends. There are 3 trails to consider: Abel Trail, Borel Trail, and +Condorcet Trail. One of your friends, Jess, requires a wheelchair; if she joins you, the group couldn’t handle the rocky +Condorcet Trail. +Solution +1. +Step 1: Let’s start by placing our first node (Figure 7.11). +Figure 7.11 +Step 2: We’ll add two branches, one for each outcome of the first coin flip, and label them (Figure 7.12). +Figure 7.12 +Step 3: We’re ready for stage two of the experiment: another coin flip. At each node, we add in branches that +represent those outcomes (Figure 7.13). +Figure 7.13 +Finally, we can add another set of branches for the outcomes of the third stage (Figure 7.14). +7.4 • Tree Diagrams, Tables, and Outcomes +743 + +Figure 7.14 +(These final nodes are called leaves.) +Step 4: We can write down the outcomes in the sample space by tracing the path out to each leaf, writing down the +outcome at each node we pass through. For example, this leaf (Figure 7.15): +Figure 7.15 +is reached via this path (Figure 7.16): +744 +7 • Probability +Access for free at openstax.org + +Figure 7.16 +Step 5: We’ll label that leaf as "HTH" (Figure 7.17), since the path passes through nodes labeled H, T, and H on its +way out to our leaf. +Figure 7.17 +Step 6: We can label the remaining leaves using the same method (Figure 7.18). +7.4 • Tree Diagrams, Tables, and Outcomes +745 + +Figure 7.18 +The sample space is the labels on the leaves: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. +2. +Step 1: We’ll start with our initial node (Figure 7.19). +Figure 7.19 +Step 2: We’ll add in branches for the outcomes of the first stage (Figure 7.20), which is the coin flip. +Figure 7.20 +Step 3: The second stage of the experiment depends on the outcome of the first stage. +a. +If the outcome of the first stage was H, then we roll a 4-sided die. So, only on the node for H, we’ll add in the +outcomes of a 4-sided die roll (Figure 7.21). +746 +7 • Probability +Access for free at openstax.org + +Figure 7.21 +b. +If the outcome of the first stage was T, then we roll a 6-sided die. So, we’ll add those branches to the node for T +(Figure 7.22). +Figure 7.22 +Step 4: We can label the leaves to get the sample space (Figure 7.23). +7.4 • Tree Diagrams, Tables, and Outcomes +747 + +Figure 7.23 +The sample space is: {H1, H2, H3, H4, T1, T2, T3, T4, T5, T6}. +c. +Step 1: Let’s label the trails A, B, and C for ease of labeling. Even though the trails are listed first in the exercise, we +can’t use the trail choice as our first stage: the trails available to us depend on whether Jess is able to join the trip. +So, the first stage is whether Jess joins us (J) or not (N) (Figure 7.24). +Figure 7.24 +Step 2: We list the appropriate trails on each branch (Figure 7.25). +748 +7 • Probability +Access for free at openstax.org + +Figure 7.25 +So, the sample space is {JA, JB, NA, NB, NC}. +YOUR TURN 7.15 +1. You have a modified deck of cards containing only J , Q , and K . You draw 2 cards without replacing them +(where order matters). Use a tree diagram to identify the sample space. +Check Your Understanding +17. You flip a coin 6 times and note the number of heads. What is the sample space of this experiment? +18. You are ordering a combo meal at a restaurant. The meal comes with either 8 or 12 chicken nuggets, and your +choice of crinkle fries, curly fries, or onion rings. Create a table to help you identify the sample space containing +your combo meal possibilities. +19. You need one more class to fill out your schedule for next semester. You want to take either History 101 (H), +English 220 (E), or Sociology 112 (S). There are two professors teaching the history class: Anderson (A) and Burr (B); +one professor teaching the English class: Carter (C); and three people teaching sociology: Johnson (J), Kirk (K), and +Lambert (L). Create a tree diagram that helps you identify all your options. +20. Identify the sample space from Exercise 18. +21. Identify the sample space from Exercise 19. +SECTION 7.4 EXERCISES +In the following exercises, you are rolling a special 6-sided die that has both a colored letter and a colored number on +each face. The faces are labeled with: a red 1 and a blue A, a red 1 and a green A, an orange 1 and a green B, an orange +2 and a red C, a purple 3 and a brown D, an orange 4 and a blue E. You will roll the die once and take note of something +about the face that is showing. Identify the sample space: +1. You note the number. +2. You note the letter. +3. You note the color of the number. +4. You note the color of the letter. +5. You note the number and letter. +6. You note the number and its color. +7. You note the letter and its color. +In Example 7.13, we used a table to identify the possible genetic result when two pea plants are bred; offspring plants +get one of the two genes from each parent. Use tables to identify the sample space of offspring for the parents listed in +7.4 • Tree Diagrams, Tables, and Outcomes +749 + +the following exercises: +8. GG and gg +9. GG and Gg +10. Gg and Gg +You’re visiting a pasta bar, where you have your choice of pasta types (cavatappi, ziti, or penne) and sauce (marinara, +alfredo, or pesto). +11. Make a table that shows the sample space for your choices. +12. Write out the sample space for the pasta bar. +You have a new art print that you’d like to get framed. You have 3 good choices for the material of the frame: oak, +maple, and cherry. You’ll also choose a framing mat, for which there are 4 possible colors that will work with your print: +plum, lilac, periwinkle, and violet. +13. Make a table that shows the sample space for your choices. +14. Write out the sample space for the framing choices. +You are shopping for a new laptop. The brand you’re considering offers laptops with screen sizes 11”, 13”, 14”, 15”, and +17”. It also offers five memory choices: 4GB, 8GB, 12GB, 16GB, and 32GB. +15. Make a table that shows the sample space for your choices. +16. Write out the sample space for the laptop choices. +For the following exercises, decide whether the described two-stage experiments have independent or dependent +stages. +17. Siobhan and Tristan are trying to decide where they will have dinner. Siobhan wants to go to Antoine’s, Burger +Hut, or the Chowder Palace. Tristan prefers Burger Hut, Chowder Palace, or Duck Duck Taco. They will flip a coin +to decide who gets to choose, then that person selects a restaurant from their list. +18. On the TV game show The Price Is Right, contestants play games to try to win prizes. One of these games is +called “Let ‘Em Roll.” In this game, players roll five 6-sided dice. These dice each have 3 faces labeled with a car; +the other 3 faces are labeled with prize money amounts (one each of $500, $1,000, and $1,500). Players get to +roll all 5 dice, and then have the opportunity to win additional rolls. If all 5 dice show a car, the contestant wins a +new car. If not, the contestant can use any additional rolls to reroll the dice with prize money showing to try to +win the car (or, they can take the total amount of money showing on the dice and end the game). Josh is playing +this game, and he has 2 rolls total (these are the two stages of the experiment; the number of cars showing on +the dice will be the reported outcome of each stage). +19. In the game Rock, Paper, Scissors, each of 2 players secretly chooses 1 of the 3 title objects. The players reveal +their choices simultaneously using hand signals. If the players choose the same object, the game is a tie; if they +choose different objects, the winner is determined by the following rules: rock beats scissors, scissors beats +paper, paper beats rock. Jim and Eva are gearing up to play the game. The two stages of the experiment are +Jim’s and Eva’s choices. +20. Paul is at the racetrack and is about to place a daily double bet, where he will try to predict the winner of 2 +consecutive races. The stages of the experiment are the two races. +21. Mishka is looking for a song to play on a jukebox. The machine requires the user to choose an artist first, then +choose a song from among that artist’s songs. The stages of the experiment are choosing the artist and +choosing the song. +John is playing a game that involves flipping a coin and rolling dice. The coin flip happens first. If the outcome is heads, +the player rolls a 4-sided die. If the outcome is tails, the player rolls a 6-sided die. +22. Create a tree diagram to display the possible outcomes of this game. +23. Give the sample space for the game. +In the casino game roulette, a wheel with colored and numbered pockets is spun. At the same time, a marble is spun in +the opposite direction in such a way that after a minute or so the marble drops into one of the pockets. Players try to +guess the color (red, black, or green) or the number that will appear. +24. Draw a tree diagram that shows the possible color outcomes for 3 spins of the wheel. +25. Give the sample space for the three-stage experiment. +26. Siobhan and Tristan are trying to decide where they will have dinner. Siobhan wants to go to Antoine’s, Burger Hut, +or the Chowder Palace. Tristan prefers Burger Hut, Chowder Palace, or Duck Duck Taco. They will flip a coin to +decide who gets to choose, then that person selects a restaurant from their list. Use an appropriate method to +identify the sample space of possible outcomes, which include the person who chooses and the restaurant. +27. On the TV game show The Price Is Right, contestants play games to try to win prizes. One of these games is called +750 +7 • Probability +Access for free at openstax.org + +“Let ‘Em Roll.” In this game, players roll five 6-sided dice. These dice each have 3 faces labeled with a car; the other +3 faces are labeled with prize money amounts (one each of $500, $1000, and $1500). Players get to roll all 5 dice, +then have the opportunity to win additional rolls. If all 5 dice show a car, the contestant wins a new car. If not, the +contestant can use any additional rolls to reroll the dice with prize money showing to try to win the car (or, they can +take the total amount of money showing on the dice and end the game). Kathleen is playing this game, and she +has 3 rolls to try to win the car. On her first roll, 3 of the dice showed cars. Use an appropriate method to find the +sample space of the number of cars showing after each of the following rolls. Give those outcomes as ordered +pairs: (number of cars after the second roll, number of cars after the third roll). +28. In the game Rock, Paper, Scissors, each of 2 players chooses 1 of the 3 title objects. The players reveal their choices +simultaneously using hand signals. Jim and Eva are gearing up to play the game. List the sample space of all the +possible outcomes of the first round of their game as ordered pairs of the form (Jim’s choice, Eva’s choice). +7.5 Basic Concepts of Probability +Figure 7.26 When you roll two dice, some outcomes (like rolling a sum of seven) are more likely than others (rolling a +sum of twelve). (credit: “Dice Isn’t Just A Game; It's A Way of Life” by Leah Love/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Define probability including impossible and certain events. +2. +Calculate basic theoretical probabilities. +3. +Calculate basic empirical probabilities. +4. +Distinguish among theoretical, empirical, and subjective probability. +5. +Calculate the probability of the complement of an event. +It all comes down to this. The game of Monopoly that started hours ago is in the home stretch. Your sister has the dice, +and if she rolls a 4, 5, or 7 she’ll land on one of your best spaces and the game will be over. How likely is it that the game +will end on the next turn? Is it more likely than not? How can we measure that likelihood? This section addresses this +question by introducing a way to measure uncertainty. +Introducing Probability +Uncertainty is, almost by definition, a nebulous concept. In order to put enough constraints on it that we can +mathematically study it, we will focus on uncertainty strictly in the context of experiments. Recall that experiments are +processes whose outcomes are unknown; the sample space for the experiment is the collection of all those possible +outcomes. When we want to talk about the likelihood of particular outcomes, we sometimes group outcomes together; +for example, in the Monopoly example at the beginning of this section, we were interested in the roll of 2 dice that might +fall as a 4, 5, or 7. A grouping of outcomes that we’re interested in is called an event. In other words, an event is a subset +of the sample space of an experiment; it often consists of the outcomes of interest to the experimenter. +Once we have defined the event that interests us, we can try to assess the likelihood of that event. We do that by +7.5 • Basic Concepts of Probability +751 + +assigning a number to each event ( +) called the probability of that event ( +). The probability of an event is a number +between 0 and 1 (inclusive). If the probability of an event is 0, then the event is impossible. On the other hand, an event +with probability 1 is certain to occur. In general, the higher the probability of an event, the more likely it is that the event +will occur. +EXAMPLE 7.16 +Determining Certain and Impossible Events +Consider an experiment that consists of rolling a single standard 6-sided die (with faces numbered 1-6). Decide if these +probabilities are equal to zero, equal to one, or somewhere in between. +1. +2. +3. +4. +5. +6. +Solution +Let's start by identifying the sample space. For one roll of this die, the possible outcomes are {1, 2, 3, 4, 5,6}. We can use +that to assess these probabilities: +1. +We see that 4 is in the sample space, so it’s possible that it will be the outcome. It’s not certain to be the outcome, +though. So, +. +2. +Notice that 7 is not in the sample space. So, +. +3. +Every outcome in the sample space is a positive number, so this event is certain. Thus, +. +4. +Since +is not in the sample space, +. +5. +Some outcomes in the sample space are even numbers (2, 4, and 6), but the others aren’t. So, +. +6. +Every outcome in the sample space is a single-digit number, so +. +YOUR TURN 7.16 +Jorge is about to conduct an experiment that consists of flipping a coin 4 times and recording the number of heads +. Decide if these probabilities are equal to zero, equal to one, or somewhere in between. +1. +2. +3. +Three Ways to Assign Probabilities +The probabilities of events that are certain or impossible are easy to assign; they’re just 1 or 0, respectively. What do we +do about those in-between cases, for events that might or might not occur? There are three methods to assign +probabilities that we can choose from. We’ll discuss them here, in order of reliability. +Method 1: Theoretical Probability +The theoretical method gives the most reliable results, but it cannot always be used. If the sample space of an +experiment consists of equally likely outcomes, then the theoretical probability of an event is defined to be the ratio of +the number of outcomes in the event to the number of outcomes in the sample space. +FORMULA +For an experiment whose sample space +consists of equally likely outcomes, the theoretical probability of the +event +is the ratio +752 +7 • Probability +Access for free at openstax.org + +where +and +denote the number of outcomes in the event and in the sample space, respectively. +EXAMPLE 7.17 +Computing Theoretical Probabilities +Recall that a standard deck of cards consists of 52 unique cards which are labeled with a rank (the whole numbers from 2 +to 10, plus J, Q, K, and A) and a suit ( , +, +, or +). A standard deck is thoroughly shuffled, and you draw one card at +random (so every card has an equal chance of being drawn). Find the theoretical probability of each of these events: +1. +The card is +. +2. +The card is a +. +3. +The card is a king (K). +Solution +There are 52 cards in the deck, so the sample space for each of these experiments has 52 elements. That will be the +denominator for each of our probabilities. +1. +There is only one +in the deck, so this event only has one outcome in it. Thus, +. +2. +There are 13 +in the deck, so +. +3. +There are 4 cards of each rank in the deck, so +. +YOUR TURN 7.17 +You are about to roll a fair (meaning that each face has an equal chance of landing up) 6-sided die, whose faces are +labeled with the numbers 1 through 6. Find the theoretical probabilities of each outcome. +1. You roll a 4. +2. You roll a number greater than 2. +3. You roll an odd number. +It is critical that you make sure that every outcome in a sample space is equally likely before you compute +theoretical probabilities! +EXAMPLE 7.18 +Using Tables to Find Theoretical Probabilities +In the Basic Concepts of Probability, we were considering a Monopoly game where, if your sister rolled a sum of 4, 5, or 7 +with 2 standard dice, you would win the game. What is the probability of this event? Use tables to determine your +answer. +Solution +We should think of this experiment as occurring in two stages: (1) one die roll, then (2) another die roll. Even though +these two stages will usually occur simultaneously in practice, since they’re independent, it’s okay to treat them +separately. +Step 1: Since we have two independent stages, let’s create a table (Figure 7.27), which is probably the most efficient +method for determining the sample space. +7.5 • Basic Concepts of Probability +753 + +Figure 7.27 +Now, each of the 36 ordered pairs in the table represent an equally likely outcome. +Step 2: To make our analysis easier, let’s replace each ordered pair with the sum (Figure 7.28). +Figure 7.28 +Step 3: Since the event we’re interested in is the one consisting of rolls of 4, 5, or 7. Let’s shade those in (Figure 7.29). +754 +7 • Probability +Access for free at openstax.org + +Figure 7.29 +Our event contains 13 outcomes, so the probability that your sister rolls a losing number is +. +YOUR TURN 7.18 +1. If you roll a pair of 4-sided dice with faces labeled with the numbers 1 through 4, what is the probability of rolling +a sum of 6 or 7? +EXAMPLE 7.19 +Using Tree Diagrams to Compute Theoretical Probability +If you flip a fair coin 3 times, what is the probability of each event? Use a tree diagram to determine your answer +1. +You flip exactly 2 heads. +2. +You flip 2 consecutive heads at some point in the 3 flips. +3. +All 3 flips show the same result. +Solution +Let’s build a tree to identify the sample space (Figure 7.30). +7.5 • Basic Concepts of Probability +755 + +Figure 7.30 +The sample space is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}, which has 8 elements. +1. +Flipping exactly 2 heads occurs three times (HHT, HTH, THH), so the probability is +. +2. +Flipping 2 consecutive heads at some point in the experiment happens 3 times: HHH, HHT, THH. So, the probability +is +. +3. +There are 2 outcomes that all show the same result: HHH and TTT. So, the probability is +. +YOUR TURN 7.19 +You have a modified deck of cards containing only 3 , 4 , and 5 . You draw 2 two cards without replacing them +(where order matters). What is the probability of each event? +1. The first card drawn is 3 . +2. The first card drawn has a lower number than the second card. +3. One of the cards drawn is 4 . +PEOPLE IN MATHEMATICS +Gerolamo Cardano +The first known text that provided a systematic approach to probabilities was written in 1564 by Gerolamo Cardano +(1501–1576). Cardano was a physician whose illegitimate birth closed many doors that would have otherwise been +open to someone with a medical degree in 16th-century Italy. As a result, Cardano often turned to gambling to help +ends meet. He was a remarkable mathematician, and he used his knowledge to gain an edge when playing at cards +or dice. His 1564 work, titled Liber de ludo aleae (which translates as Book on Games of Chance), summarized +everything he knew about probability. Of course, if that book fell into the hands of those he played against, his +advantage would disappear. That’s why he never allowed it to be published in his lifetime (it was eventually published +in 1663). Cardano made other contributions to mathematics; he was the first person to publish the third degree +analogue of the Quadratic Formula (though he didn’t discover it himself), and he popularized the use of negative +numbers. +756 +7 • Probability +Access for free at openstax.org + +Method 2: Empirical Probability +Theoretical probabilities are precise, but they can’t be found in every situation. If the outcomes in the sample space are +not equally likely, then we’re out of luck. Suppose you’re watching a baseball game, and your favorite player is about to +step up to the plate. What is the probability that he will get a hit? +In this case, the sample space is {hit, not a hit}. That doesn’t mean that the probability of a hit is +, since those outcomes +aren’t equally likely. The theoretical method simply can’t be used in this situation. Instead, we might look at the player’s +statistics up to this point in the season, and see that he has 122 hits in 531 opportunities. So, we might think that the +probability of a hit in the next plate appearance would be about +. When we use the outcomes of previous +replications of an experiment to assign a probability to the next replication, we’re defining an empirical probability. +Empirical probability is assigned using the outcomes of previous replications of an experiment by finding the ratio of the +number of times in the previous replications the event occurred to the total number of previous replications. +Empirical probabilities aren’t exact, but when the number of previous replications is large, we expect them to be close. +Also, if the previous runs of the experiment are not conducted under the exact set of circumstances as the one we’re +interested in, the empirical probability is less reliable. For instance, in the case of our favorite baseball player, we might +try to get a better estimate of the probability of a hit by looking only at his history against left- or right-handed pitchers +(depending on the handedness of the pitcher he’s about to face). +WHO KNEW? +Probability and Statistics +One of the broad uses of statistics is called statistical inference, where statisticians use collected data to make a guess +(or inference) about the population the data were collected from. Nearly every tool that statisticians use for inference +is based on probability. Not only is the method we just described for finding empirical probabilities one type of +statistical inference, but some more advanced techniques in the field will give us an idea of how close that empirical +probability might be to the actual probability! +EXAMPLE 7.20 +Finding Empirical Probabilities +Assign an empirical probability to the following events: +1. +Jose is on the basketball court practicing his shots from the free throw line. He made 47 out of his last 80 attempts. +What is the probability he makes his next shot? +2. +Amy is about to begin her morning commute. Over her last 60 commutes, she arrived at work 12 times in under half +an hour. What is the probability that she arrives at work in 30 minutes or less? +3. +Felix is playing Yahtzee with his sister. Felix won 14 of the last 20 games he played against her. How likely is he to win +this game? +Solution +1. +Since Jose made 47 out of his last 80 attempts, assign this event an empirical probability of +. +2. +Amy completed the commute in under 30 minutes in 12 of the last 60 commutes, so we can estimate her probability +of making it in under 30 minutes this time at +. +3. +Since Felix has won 14 of the last 20 games, assign a probability for a win this time of +. +YOUR TURN 7.20 +1. Jessie is in charge of quality control at a factory manufacturing SUVs. Today, she’s checking the placement of the +taillight housing. Of the last thousand units off the line, 13 had faulty placement. What empirical probability +might Jesse assign to the next vehicle coming off the line having bad placement? +7.5 • Basic Concepts of Probability +757 + +WORK IT OUT +Buffon’s Needle +A famous early question about probability (posed by Georges-Louis Leclerc, Comte de Buffon in the 18th century) had +to do with the probability that a needle dropped on a floor finished with wooden slats would lay across one of the +seams. If the distance between the slats is exactly the same length as the needle, then it can be shown using calculus +that the probability that the needle crosses a seam is +. Using toothpicks or matchsticks (or other uniformly long and +narrow objects), assign an empirical probability to this experiment by drawing parallel lines on a large sheet of paper +where the distance between the lines is equal to the length of your dropping object, then repeatedly dropping the +objects and noting whether the object touches one of the lines. Once you have your empirical probability, take its +reciprocal and multiply by 2. Is the result close to +? +Method 3: Subjective Probability +In cases where theoretical probability can’t be used and we don’t have prior experience to inform an empirical +probability, we’re left with one option: using our instincts to guess at a subjective probability. A subjective probability is +an assignment of a probability to an event using only one’s instincts. +Subjective probabilities are used in cases where an experiment can only be run once, or it hasn’t been run before. +Because subjective probabilities may vary widely from person to person and they’re not based on any mathematical +theory, we won’t give any examples. However, it’s important that we be able to identify a subjective probability when we +see it; they will in general be far less accurate than empirical or theoretical probabilities. +EXAMPLE 7.21 +Distinguishing among Theoretical, Empirical, and Subjective Probabilities +Classify each of the following probabilities as theoretical, empirical, or subjective. +1. +An eccentric billionaire is testing a brand new rocket system. He says there is a 15% chance of failure. +2. +With 4 seconds to go in a close basketball playoff game, the home team need 3 points to tie up the game and send +it to overtime. A TV commentator says that team captain should take the final 3-point shot, because he has a 38% +chance of making it (greater than every other player on the team). +3. +Felix is losing his Yahtzee game against his sister. He has one more chance to roll 2 dice; he’ll win the game if they +both come up 4. The probability of this is about 2.8%. +Solution +1. +This experiment has never been run before, so the given probability is subjective. +2. +Presumably, the commentator has access to each player’s performance statistics over the entire season. So, the +given probability is likely empirical. +3. +Rolling 2 dice results in a sample space with equally likely outcomes. This probability is theoretical. (We’ll learn how +to calculate that probability later in this chapter.) +YOUR TURN 7.21 +Classify each of the following probabilities as theoretical, empirical, or subjective. +1. You have entered a raffle with 500 entrants. Your probability of winning is 0.2%. +2. Your little brother takes the bus to school each morning. On the first day of school, you believe that the +probability that the bus arrives between 7:15 AM and 7:30 AM is about 80%. +3. Your little brother takes the bus to school each morning. On the last day of school, you believe that the +probability that the bus arrives between 7:15 AM and 7:30 AM is about 73%. +758 +7 • Probability +Access for free at openstax.org + +WHO KNEW? +Benford’s Law +In 1938, Frank Benford published a paper (“The law of anomalous numbers,” in Proceedings of the American +Philosophical Society) with a surprising result about probabilities. If you have a list of numbers that spans at least a +couple of orders of magnitude (meaning that if you divide the largest by the smallest, the result is at least 100), then +the digits 1–9 are not equally likely to appear as the first digit of those numbers, as you might expect. Benford arrived +at this conclusion using empirical probabilities; he found that 1 was about 6 times as likely to be the initial digit as 9 +was! +New Probabilities from Old: Complements +One of the goals of the rest of this chapter is learning how to break down complicated probability calculations into easier +probability calculations. We’ll look at the first of the tools we can use to accomplish this goal in this section; the rest will +come later. +Given an event +, the complement of +(denoted +) is the collection of all of the outcomes that are not in +. (This is +language that is taken from set theory, which you can learn more about elsewhere in this text.) Since every outcome in +the sample space either is or is not in +, it follows that +. So, if the outcomes in +are equally likely, +we can compute theoretical probabilities +and +. Then, adding these last two equations, we get +Thus, if we subtract +from both sides, we can conclude that +. Though we performed this +calculation under the assumption that the outcomes in +are all equally likely, the last equation is true in every situation. +FORMULA +How is this helpful? Sometimes it is easier to compute the probability that an event won’t happen than it is to compute +the probability that it will. To apply this principle, it’s helpful to review some tricks for dealing with inequalities. If an +event is defined in terms of an inequality, the complement will be defined in terms of the opposite inequality: Both the +direction and the inclusivity will be reversed, as shown in the table below. +If +is defined with: +then +is defined with: +7.5 • Basic Concepts of Probability +759 + +EXAMPLE 7.22 +Using the Formula for Complements to Compute Probabilities +1. +If you roll a standard 6-sided die, what is the probability that the result will be a number greater than one? +2. +If you roll two standard 6-sided dice, what is the probability that the sum will be 10 or less? +3. +If you flip a fair coin 3 times, what is the probability that at least one flip will come up tails? +Solution +1. +Here, the sample space is {1, 2, 3, 4, 5, 6}. It’s easy enough to see that the probability in question is +, because there +are 5 outcomes that fall into the event “roll a number greater than 1.” Let’s also apply our new formula to find that +probability. Since +is defined using the inequality +, then +is defined using +. Since there’s only one +outcome (1) in +, we have +. Thus, +. +2. +In Example 7.18, we found the following table of equally likely outcomes for rolling 2 dice (Figure 7.31): +Figure 7.31 +Here, the event +is defined by the inequality +. Thus, +is defined by +. There are three outcomes +in +: two 11s and one 12. Thus, +. +3. +In Example 7.15, we found the sample space for this experiment consisted of these equally likely outcomes: {HHH, +HHT, HTH, HTT, THH, THT, TTH, TTT}. Our event +is defined by +, so +is defined by +. The only outcome +in +is the first one on the list, where zero tails are flipped. So, +. +YOUR TURN 7.22 +1. If you roll a pair of 4-sided dice with faces labeled 1 through 4, what is the probability that the sum of the +resulting numbers will be greater than 3? Hint: You found this sample space in an earlier Your Turn. +Check Your Understanding +You have two coins: a nickel and a quarter. You flip them both. Find the probabilities of these events: +22. Both come up heads. +23. The quarter comes up heads. +24. You get one heads and one tails. +25. You get three tails. +Decide whether the given probabilities were most likely derived theoretically, empirically, or subjectively. +26. A poker player has a 16% chance of making a hand called a flush on the next card. +27. Your friend Jacob tells you that there’s a 20% chance he’ll get married in the next 5 years. +28. Ashley has a coin that they think might not be fair, so they flip it 100 times and note that the result was heads 58 +760 +7 • Probability +Access for free at openstax.org + +times. So, Ashley says the probability of flipping heads on that coin is about 58%. +If you flip a fair coin 50 times, the probability of getting 20 or fewer heads is about 10.1% (a fact we’ll learn how to verify +later). +29. If +is the event “number of heads is 20 or fewer”, describe the event +using an inequality. +30. Find +. +SECTION 7.5 EXERCISES +For the following exercises, we are considering two special 6-sided dice. Each face is labeled with a number and a letter: +the first die has faces 1A, 1B, 2A, 2C, 4A, 4E; the second has faces 1A, 1A, 2A, 2B, 3A, 3C. Assume that each face has an +equal probability of landing face up. +1. Use a table to identify the sample space of the experiment in which we roll both dice and note the sum of the +two numbers that are showing. +2. What is the probability that we roll a sum less than 8? +3. What is the probability that we roll a sum larger than 8? +4. What is the probability that we roll a sum less than or equal to 2? +5. What is the probability that we roll a sum greater than 2? +6. What is the probability that we roll an even sum? +7. What is the probability that we roll an odd sum? +8. Use a table to identify the sample space of the experiment in which we roll both dice and note the two letters +that are showing. +9. What is the probability that no As are showing? +10. What is the probability that at least one A is showing? +11. What is the probability that two As are showing? +12. What is the probability that two of the same letter are showing? +13. What is the probability that the letter on the first die comes alphabetically before the letter on the second die? +14. What is the probability that two vowels are showing? +15. What is the probability that two consonants are showing? +16. What is the probability that one consonant and one vowel are showing? +In the following Exercises, decide whether the given probability was likely determined theoretically, empirically, or +subjectively. +17. Carolyn is breeding two pea plants, and notes that 8 of the 25 offspring plants have yellow peas. So, she +concludes that there’s a 32% chance that an offspring of these two plants will have yellow peas. +18. At the beginning of the semester, Malik estimates there’s a 30% chance that he’ll earn As in all his classes. +19. Abbie is deciding where she will attend college in the fall. Right now, she thinks there’s an 80% chance she’ll +attend an in-state school. +20. If you roll three standard 6-sided dice, the probability that the sum will be 8 is +. +21. According to the app he uses to play the game, Jason has won 18 of the last 100 games of solitaire he’s played. +So, the probability that he wins the next one is about 18%. +22. Jim and Anne are both in a club with 10 members. If 3 people are chosen at random to form a committee, then +the probability that both Jim and Anne are chosen is +. +For the following exercises, use the following table of the top 15 players by number of plate appearances (PA) in the +2019 Major League Baseball season to assign empirical probabilities to the given events. A plate appearance is a +batter’s opportunity to try to get a hit. The other columns are runs scored (R), hits (H), doubles (2B), triples (3B), home +runs (HR), walks (BB), and strike outs (SO). +Name +Team +PA +R +H +2B +3B +HR +BB +SO +Marcus Semien +OAK +747 +123 +187 +43 +7 +33 +87 +102 +Whit Merrifield +KCR +735 +105 +206 +41 +10 +16 +45 +126 +Ronald Acuna Jr. +ATL +715 +127 +175 +22 +2 +41 +76 +188 +7.5 • Basic Concepts of Probability +761 + +Name +Team +PA +R +H +2B +3B +HR +BB +SO +Jonathan Villar +BAL +714 +111 +176 +33 +5 +24 +61 +176 +Mookie Betts +BOS +706 +135 +176 +40 +5 +29 +97 +101 +Rhys Hoskins +PHI +705 +86 +129 +33 +5 +29 +116 +173 +Jorge Polanco +MIN +704 +107 +186 +40 +7 +22 +60 +116 +Rafael Devers +BOS +702 +129 +201 +54 +4 +32 +48 +119 +Ozzie Albies +ATL +702 +102 +189 +43 +8 +24 +54 +112 +Eduardo Escobar +ARI +699 +94 +171 +29 +10 +35 +50 +130 +Xander Bogaerts +BOS +698 +110 +190 +52 +0 +33 +76 +122 +José Abreu +CHW +693 +85 +180 +38 +1 +33 +36 +152 +Pete Alonso +NYM +693 +103 +155 +30 +2 +53 +72 +183 +Freddie Freeman +ATL +692 +113 +176 +34 +2 +38 +87 +127 +Alex Bregman +HOU +690 +122 +164 +37 +2 +41 +119 +83 +23. Mookie Betts gets a home run in his next plate appearance. +24. Xander Bogaerts strikes out in his next plate appearance. +25. Jonathan Villar gets a hit in his next plate appearance. +26. Rhys Hoskins gets a walk in his next plate appearance. +27. José Abreu scores a run in his next plate appearance. +28. Eduardo Escobar hits a triple in his next plate appearance. +29. Whit Merrifield hits a double in his next plate appearance. +30. Ronald Acuna Jr. gets an extra-base hit (double, triple, or home run) in his next plate appearance. +762 +7 • Probability +Access for free at openstax.org + +7.6 Probability with Permutations and Combinations +Figure 7.32 Bingo and many lottery games depend on selecting one or more numbers at random from a list; often this is +done by drawing numbered balls from a bin. (credit: “Redundant Bingo Balls” by Greg Clarke/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Calculate probabilities with permutations. +2. +Calculate probabilities with combinations. +In our earlier discussion of theoretical probabilities, the first step we took was to write out the sample space for the +experiment in question. For many experiments, that method just isn’t practical. For example, we might want to find the +probability of drawing a particular 5-card poker hand. Since there are 52 cards in a deck and the order of cards doesn’t +matter, the sample space for this experiment has +possible 5-card hands. Even if we had the patience +and space to write them all out, sorting through the results to find the outcomes that fall in our event would be just as +tedious. +Luckily, the formula for theoretical probabilities doesn’t require us to know every outcome in the sample space; we just +need to know how many outcomes there are. In this section, we’ll apply the techniques we learned earlier in the chapter +(The Multiplication Rule for Counting, permutations, and combinations) to compute probabilities. +Using Permutations to Compute Probabilities +Recall that we can use permutations to count how many ways there are to put a number of items from a list in order. If +we’re looking at an experiment whose sample space looks like an ordered list, then permutations can help us to find the +right probabilities. +EXAMPLE 7.23 +Using Permutations to Compute Probabilities +1. +In horse racing, an exacta bet is one where the player tries to predict the top two finishers in particular race in order. +If there are 9 horses in a race, and a player decided to make an exacta bet at random, what is the probability that +they win? +2. +You are in a club with 10 people, 3 of whom are close friends of yours. If the officers of this club are chosen at +random, what is the probability that you are named president and one of your friends is named vice president? +3. +A bag contains slips of paper with letters written on them as follows: A, A, B, B, B, C, C, D, D, D, D, E. If you draw 3 +slips, what is the probability that the letters will spell out (in order) the word BAD? +7.6 • Probability with Permutations and Combinations +763 + +Solution +1. +Since order matters for this situation, we’ll use permutations. How many different exacta bets can be made? Since +there are 9 horses and we must select 2 in order, we know there are +possible outcomes. That’s the size of +our sample space, so it will go in the denominator of the probability. Since only one of those outcomes is a winner, +the numerator of the probability is 1. So, the probability of randomly selecting the winning exacta bet is +. +2. +There are 10 people in the club, and 2 will be chosen to be officers. Since the order matters, there are +different ways to select officers. Next, we must figure out how many outcomes are in our event. We’ll use the +Multiplication Rule for Counting to find that number. There is only 1 choice for president in our event, and there are +3 choices for vice president. So, there are +outcomes in the event. Thus, the probability that you will serve +as president with one of your friends as vice president is +. +3. +There are 12 slips of paper in the bag, and 3 will be drawn. So, there are +possible outcomes. Now, we’ll +compute the number of outcomes in our event. The first letter drawn must be a B, and there are 3 of those. Next +must come an A (2 of those) and then a D (4 of those). Thus, there are +outcomes in our event. So, the +probability that the letters drawn spell out the word BAD is +. +YOUR TURN 7.23 +1. Another bag of letters contains C, C, C, C, D, D, I, I, I, T, T, T, Y, Y, Y, Y. What is the probability that 4 letters chosen +at random will spell, in order, CITY? +Combinations to Computer Probabilities +If the sample space of our experiment is one in which order doesn’t matter, then we can use combinations to find the +number of outcomes in that sample space. +EXAMPLE 7.24 +Using Combinations to Compute Probabilities +1. +Palmetto Cash 5 is a game offered by the South Carolina Education Lottery. Players choose 5 numbers from the +whole numbers between 1 and 38 (inclusive); the player wins the jackpot of $100,000 if the randomizer selects those +numbers in any order. If you buy one ticket for this game, what is the probability that you win the top prize by +choosing all 5 winning numbers? +2. +There’s a second prize in the Palmetto Cash 5 game that a player wins if 4 of the player's 5 numbers are among the +5 winning numbers. What’s the probability of winning the second prize? +3. +Scrabble is a word-building board game. Players make hands of 7 letters by selecting tiles with single letters printed +on them blindly from a bag (2 tiles have nothing printed on them; these blanks can stand for any letter). Players use +the letters in their hands to spell out words on the board. Initially, there are 100 tiles in the bag. Of those, 44 are (or +could be) vowels (9 As, 12 Es, 9 Is, 8 Os, 4 Us, and 2 blanks; we’ll treat Y as a consonant). What is the probability that +your initial hand has no vowels? +Solution +1. +There are 38 numbers to choose from, and the order of the 5 we pick doesn’t matter. So, there are +outcomes in the sample space. Only one outcome is in our winning event, so the probability of winning is +. +2. +As in part 1 of this example,, there are 501,492 outcomes in the sample space. The tricky part here is figuring out +how many outcomes are in our event. To qualify, the outcome must contain 4 of the 5 winning numbers, plus one +losing number. There are +ways to choose the 4 winning numbers, and there are +losing +numbers. So, using the Multiplication Rule for Counting, there are +outcomes in our event. Thus, the +probability of winning the second prize is +, which is about 0.00033. +3. +The number of possible starting hands is +. There are +consonants in the +bag, so the number of all-consonant hands is +. Thus, the probability of drawing all consonants +is +. +764 +7 • Probability +Access for free at openstax.org + +YOUR TURN 7.24 +1. At a charity event with 58 people in attendance, 3 raffle winners are chosen. All receive the same prize, so +order doesn’t matter. You are attending with 4 of your friends. What is the probability that at least one of you +or your friends wins a raffle prize? Hint: Find the probability that none of you wins, and use the formula for +complements. +2. If you draw a hand of 5 cards from a standard deck, what is the probability that 2 cards are +and 3 cards are +? +Check Your Understanding +For the following exercises, you are drawing Scrabble tiles without replacement from a bag containing the letters A, C, +E, E, I, N, N, S, S, W. +31. What is the probability that you draw (in order) the letters W-I-N? +32. What is the probability that you draw (in order) the letters W-I-S-E? +33. What is the probability that you draw (in order) the letters S-E-E-N? +34. What is the probability that you draw (in any order) the letters W-I-N? +35. What is the probability that you draw (in any order) the letters W-I-S-E? +36. What is the probability that you draw (in any order) the letters S-E-E-N? +SECTION 7.6 EXERCISES +The following exercises deal with our version of the game blackjack. In this card game, players are dealt a hand of two +cards from a standard deck. The dealer’s cards are dealt with the second card face up, so the order matters; the other +players’ hands are dealt entirely face down, so order doesn’t matter. The goal of the game is to build a hand whose +point value is as close as possible to 21 without going over. The point values of each card are as follows: numbered +cards are worth the number on the face (for example, +is worth 8 points); jacks, queens, and kings are each worth 10 +points, and aces are worth either 1 or 11 points (the player can choose). Players whose hands are worth less than 21 +points may ask to be dealt additional cards one at a time until they either go over 21 points or they choose to stop. +1. What is the probability that a player (not the dealer) is dealt an initial hand worth 21 points? This can only +happen with an ace and a card worth 10 points (10, J, Q, or K). +2. What is the probability that the dealer is dealt an initial hand worth 21 points, with an ace showing? +3. What is the probability that a player is dealt 2 cards worth 10 points each? +4. What is the probability that a player is dealt an initial hand with an 8 and a 3? +5. What is the probability that a player is dealt an initial hand with two 8s? +6. What is the probability that a player is dealt 2 +? +7. In some versions of the game, a player wins automatically if they draw a hand of 5 cards that doesn’t go over 21 +points. One way this can happen is if they draw 5 cards, all of which are A, 2, 3, or 4. What is the probability of +drawing 5 cards from that collection? +In horse racing, a trifecta bet is one where the player tries to predict the top three finishers in order. In the following +exercises, find the probability of choosing a winning trifecta bet at random when the field contains the given number of +horses. +8. 6 horses +9. 8 horses +10. 10 horses +In the following exercises, you are about to draw Scrabble tiles from a bag without replacement; the bag contains the +letters A, A, C, E, E, E, L, L, N, O, R, S, S, S, T, X. +11. What is the probability of drawing the letters E-A-R, in order? +12. What is the probability of drawing the letters E-A-R, in any order? +13. What is the probability of drawing the letters S-E-A-L, in order? +14. What is the probability of drawing the letters S-E-A-L, in any order? +15. What is the probability of drawing the letters L-A-S-S, in order? +16. What is the probability of drawing the letters L-A-S-S, in any order? +17. What is the probability of drawing 3 tiles that are all vowels? +18. What is the probability of drawing 3 tiles that are all consonants? +19. What is the probability of drawing 4 tiles in the pattern vowel-consonant-vowel-consonant, in order? +7.6 • Probability with Permutations and Combinations +765 + +20. What is the probability of drawing 2 vowels and 2 consonants, in any order? +21. What is the probability of drawing at least 1 vowel when drawing four tiles? (Hint: use the Formula for +Complements.) +22. What is the probability of drawing at least 1 consonant when drawing four tiles? +The following exercises involve the board game Clue, which involves a deck of 21 cards: 6 suspects, 6 weapons, and 9 +rooms. At the beginning of the game, 1 card of each of the 3 types is secretly removed from the deck (the object of the +game is to identify those 3 cards). The remaining 18 cards are dealt out to the players. Assuming there are 3 players, +each player gets 6 cards. Find the probabilities of a player being dealt hands with the given characteristics. +23. All 6 cards are rooms. +24. 5 cards are suspects (the sixth can be anything). +25. None of the cards are rooms. +26. None of the cards are suspects. +27. 3 cards are suspects and 3 are weapons. +28. There are 2 cards of each type. +29. There are 3 rooms, 2 suspects, and 1 weapon. +30. There are 4 rooms and 5 suspects. +7.7 What Are the Odds? +Figure 7.33 Scratch-off lottery tickets, as well as many other games, represent the likelihood of winning using odds. +(credit: “My Scratch-off Winnings” by Shoshanah/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Compute odds. +2. +Determine odds from probabilities. +3. +Determine probabilities from odds. +A particular lottery instant-win game has 2 million tickets available. Of those, 500,000 win a prize. If there are 500,000 +winners, then it follows that there are 1,500,000 losing tickets. When we evaluate the risk associated with a game like +this, it can be useful to compare the number of ways to win the game to the number of ways to lose. In the case of this +game, we would compare the 500,000 wins to the 1,500,000 losses. In other words, there are 3 losing tickets for every +winning ticket. Comparisons of this type are the focus of this section. +Computing Odds +The ratio of the number of equally likely outcomes in an event +to the number of equally likely outcomes not in the +event +is called the odds for (or odds in favor of) the event. The opposite ratio (the number of outcomes not in the +event to the number in the event +to the number in the event +is called the odds against the event. +766 +7 • Probability +Access for free at openstax.org + +Both odds and probabilities are calculated as ratios. To avoid confusion, we will always use fractions, decimals, or +percents for probabilities, and we’ll use colons to indicate odds. The rules for simplifying fractions apply to odds, +too. Thus, the odds for winning a prize in the game described in the section opener are +and the odds against winning a prize are +. These would often be described in words as “the odds of winning are +one to three in favor” or “the odds of winning are three to one against.”. +Notice that, while probabilities must always be between zero and one inclusive, odds can be any (non-negative) +number, as we’ll see in the next example. +EXAMPLE 7.25 +Computing Odds +1. +If you roll a fair 6-sided die, what are the odds for rolling a 5 or higher? +2. +If you roll two fair 6-sided dice, what are the odds against rolling a sum of 7? +3. +If you draw a card at random from a standard deck, what are the odds for drawing a +? +4. +If you draw 2 cards at random from a standard deck, what are the odds against them both being +? +Solution +1. +The sample space for this experiment is {1, 2, 3, 4, 5, 6}. Two of those outcomes are in the event “roll a five or +higher,” while four are not. So, the odds for rolling a five or higher are +. +2. +In Example 7.18, we found the sample space for this experiment using the following table (Figure 7.34): +Figure 7.34 +There are 6 outcomes in the event “roll a sum of 7,” and there are 30 outcomes not in the event. So, the odds against +rolling a 7 are +. +3. +There are 13 +in a standard deck, and +others. So, the odds in favor of drawing a +are +. +4. +There are +ways to draw 2 +, and +ways to draw 2 cards that are not both +. So, the +odds against drawing 2 +are +. +YOUR TURN 7.25 +You roll a pair of 4-sided dice with faces labeled 1 through 4. +1. What are the odds for rolling a sum greater than 3? +2. What are the odds against both dice giving the same number? +7.7 • What Are the Odds? +767 + +Odds as a Ratio of Probabilities +We can also think of odds as a ratio of probabilities. Consider again the instant-win game from the section opener, with +500,000 winning tickets out of 2,000,000 total tickets. If a player buys one ticket, the probability of winning is +, and the probability of losing is +. Notice that the ratio of the probability of winning to the +probability of losing is +, which matches the odds in favor of winning. +FORMULA +For an event +, +We can use these formulas to convert probabilities to odds, and vice versa. +EXAMPLE 7.26 +Converting Probabilities to Odds +Given the following probabilities of an event, find the corresponding odds for and odds against that event. +1. +2. +Solution +1. +Using the formula, we have: +(Note that in the last step, we simplified by multiplying both terms in the ratio by 5.) +Since the odds for +are +, the odds against +must be +. +2. +Again, we’ll use the formula: +(In the last step, we simplified by dividing both terms in the ratio by 0.17.) +It follows that the odds against +are approximately +. +YOUR TURN 7.26 +1. If the probability of an event +is 80%, find the odds for and the odds against +. +Now, let’s convert odds to probabilities. Let’s say the odds for an event are +. Then, using the formula above, we +have +. Converting to fractions and solving for +, we get: +768 +7 • Probability +Access for free at openstax.org + +Let’s put this result in a formula we can use. +FORMULA +If the odds in favor of +are +, then +. +EXAMPLE 7.27 +Converting Odds to Probabilities +Find +if +: +1. +The odds of +are +in favor +2. +The odds of +are +against +Solution +1. +Using the formula we just found, we have +. +2. +If the odds against are +, then the odds for are +. Thus, using the formula, +. +YOUR TURN 7.27 +Find +if +: +1. The odds of +are +against +2. The odds of +are +in favor +Some places, particularly state lottery websites, will use the words “odds” and “probability” interchangeably. Never +assume that the word “odds” is being used correctly! Compute one of the odds/probabilities yourself to make sure +you know how the word is being used! +Check Your Understanding +For the following exercises, you are rolling a 6-sided die with 3 orange faces, 2 green faces, and 1 blue face. +37. What are the odds in favor of rolling a green face? +38. What are the odds against rolling a blue face? +39. What are the odds in favor of rolling an orange face? +40. What are the odds in favor of an event with probability +? +41. What are the odds against an event with probability +? +42. What is the probability of an event with odds +against? +43. What is the probability of an event with odds +in favor? +SECTION 7.7 EXERCISES +For the following exercises, find the probabilities of events with the given odds in favor. +7.7 • What Are the Odds? +769 + +1. +2. +3. +4. +5. +6. +7. +8. +For the following exercises, find the probabilities of events with the given odds against. +9. +10. +11. +12. +13. +14. +15. +16. +In the following exercises, find the odds in favor of events with the given probabilities. Give your answer as a ratio of +whole numbers. If neither of those two numbers is 1, also give an answer as a ratio involving both 1 and a number +greater than or equal to 1 (for example, the odds +and +can be reduced to +and +). +17. +18. +19. +20. +21. +22. +23. +24. +In the following exercises, find the odds against events with the given probabilities. Give your answer as a ratio of +whole numbers. If neither of those two numbers is 1, also give an answer as a ratio involving both 1 and a number +greater than or equal to 1 (for example, the odds +and +can be reduced to +and +). +25. +26. +27. +28. +29. +30. +31. +32. +In the following exercises, you are drawing from a deck containing only these 10 cards: +, +, +, +, +, +, +, +, +, +. +33. Let +be the event “draw an ace.” +a. +What is the probability of +? +b. +What are the odds in favor of +? +c. +What are the odds against +? +34. Let +be the event “draw a +”. +a. +What is the probability of +? +b. +What are the odds in favor of +? +c. +What are the odds against +? +35. Let +be the event “draw two +(without replacement).” +a. +What is the probability of +? +770 +7 • Probability +Access for free at openstax.org + +b. +What are the odds in favor of +? +c. +What are the odds against +? +7.8 The Addition Rule for Probability +Figure 7.35 Students can be sorted using a variety of possible categories like class year, major, whether they are a varsity +athlete, and so forth. (credit: “Multicultural Mashup Melds Languages, Cultures at COD 36” by COD Newsroom/Flickr, CC +BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify mutually exclusive events. +2. +Apply the Addition Rule to compute probability. +3. +Use the Inclusion/Exclusion Principle to compute probability. +Up to this point, we have looked at the probabilities of simple events. Simple events are those with a single, simple +characterization. Sometimes, though, we want to investigate more complicated situations. For example, if we are +choosing a college student at random, we might want to find the probability that the chosen student is a varsity athlete +or in a Greek organization. This is a compound event: there are two possible criteria that might be met. We might +instead try to identify the probability that the chosen student is both a varsity athlete and in a Greek organization. In this +section and the next, we’ll cover probabilities of two types of compound events: those build using “or” and those built +using “and.” We’ll deal with the former first. +Mutual Exclusivity +Before we get to the key techniques of this section, we must first introduce some new terminology. Let’s say you’re +drawing a card from a standard deck. We’ll consider 3 events: +is the event “the card is a +,” +is the event “the card is a +10,” and +is the event “the card is a +.” If the card drawn is +, then +and +didn’t occur, but +did. If the card drawn is +instead +, then +didn’t occur, but both +and +did. +We can see from these examples that, if we are interested in several possible events, more than one of them can occur +simultaneously (both +and +, for example). But, if you think about all the possible outcomes, you can see that +and +can never occur simultaneously; there are no cards in the deck that are both +and +. Pairs of events that cannot both +occur simultaneously are called mutually exclusive. Let’s go through an example to help us better understand this +concept. +EXAMPLE 7.28 +Identifying Mutually Exclusive Events +Decide whether the following events are mutually exclusive. If they are not mutually exclusive, identify an outcome that +would result in both events occurring. +1. +You are about to roll a standard 6-sided die. +is the event “the die shows an even number” and +is the event “the +die shows an odd number.” +2. +You are about to roll a standard 6-sided die. +is the event “the die shows an even number” and +is the event “the +die shows a number less than 4.” +3. +You are about to flip a coin 4 times. +is the event “at least 2 heads are flipped” and +is the event “fewer than 3 tails +are flipped.” +7.8 • The Addition Rule for Probability +771 + +Solution +1. +Let’s look at the outcomes for each event: +and +. There are no outcomes in common, so +and +are mutually exclusive. +2. +Again, consider the outcomes in each event: +and +. Since the outcome 2 belongs to both +events, these are not mutually exclusive. +3. +Suppose the results of the 4 flips are HTTH. Then at least 2 heads are flipped, and fewer than 3 tails are flipped. That +means that both +and +occurred, and so these events are not mutually exclusive. +YOUR TURN 7.28 +Suppose you’re about to draw one card from a deck containing only these 10 cards: +, +, +, +, +, +, +, +, +, +. Decide whether these events are mutually exclusive: +1. +is the event “the card is an ace” and +is the event “the card is a king.” +2. +is the event “the card is a +” and +is the event “the card is an ace.” +3. +is the event “the card is a +” and +is the event “the card is a king.” +The Addition Rule for Mutually Exclusive Events +If two events are mutually exclusive, then we can use addition to find the probability that one or the other event occurs. +FORMULA +If +and +are mutually exclusive events, then +. +Why does this formula work? Let’s consider a basic example. Suppose we’re about to draw a Scrabble tile from a bag +containing A, A, B, E, E, E, R, S, S, U. What is the probability of drawing an E or an S? Since 3 of the tiles are marked with E +and 2 are marked with S, there are 5 tiles that satisfy the criteria. There are ten tiles in the bag, so the probability is +. Notice that the probability of drawing an E is +and the probability of drawing an S is +; adding those +together, we get +. Look at the numerators in the fractions involved in the sum: the 3 represents the +number of E tiles and the 2 is the number of S tiles. This is why the Addition Rule works: The total number of outcomes in +one event or the other is the sum of the numbers of outcomes in each of the individual events. +EXAMPLE 7.29 +Using the Addition Rule +For each of the given pairs of events, decide if the Addition Rule applies. If it does, use the Addition Rule to find the +probability that one or the other occurs. +1. +You are rolling a standard 6-sided die. Event +is “roll an even number” and event +is “roll a 3.” +2. +You are drawing a card at random from a standard 52-card deck. Event +is “draw a +” and event +is “draw a king.” +3. +You are rolling a pair of standard 6-sided dice. Event +is “roll an odd sum” and event +is “roll a sum of 10.” The +table we constructed in Example 7.18 might help. +772 +7 • Probability +Access for free at openstax.org + +Figure 7.36 +Solution +1. +Since 3 is not an even number, these events are mutually exclusive. So, we can use the Addition Rule: since +and +, we get +. +2. +If the card drawn is +, then both +and +occur. So, they aren’t mutually exclusive, and the Addition Rule doesn’t +apply. +3. +Since 10 is not odd, these events are mutually exclusive. Since +and +, the Addition Rule gives +us +. +YOUR TURN 7.29 +Suppose you’re about to draw one card from a deck containing only these 10 cards: +, +, +, +, +, +, +, +, +, +. If appropriate, use the Addition Rule to find the probability that one or +the other of these events occurs: +1. +is the event “the card is an ace” and +is the event “the card is a king.” +2. +is the event “the card is a +” and +is the event “the card is an ace.” +3. +is the event “the card is a +” and +is the event “the card is a king.” +Finding Probabilities When Events Aren’t Mutually Exclusive +Let’s return to the example we used to explore the Addition Rule: We’re about to draw a Scrabble tile from a bag +containing A, A, B, E, E, E, R, S, S, U. Consider these events: +is “draw a vowel” and +is “draw a letter that comes after L +in the alphabet.” Since there are 6 vowels, +. There are 4 tiles with letters that come after L alphabetically, so +. What is +? If we blindly apply the Addition Rule, we get +, which would mean that the +compound event +or +is certain. However, it’s possible to draw a B, in which case neither +nor +happens. Where’s +the error? +The events are not mutually exclusive: the outcome U belongs to both events, and so the Addition Rule doesn’t apply. +However, there’s a way to extend the Addition Rule to allow us to find this probability anyway; it’s called the Inclusion/ +Exclusion Principle. In this example, if we just add the two probabilities together, the outcome U is included in the sum +twice: It’s one of the 6 outcomes represented in the numerator of +, and it’s one of the 4 outcomes represented in the +numerator of +. So, that particular outcome has been “double counted.” Since it has been included twice, we can get a +true accounting by excluding it once: +. We can generalize this idea to a formula that we can apply to +find the probability of any compound event built using “or.” +7.8 • The Addition Rule for Probability +773 + +FORMULA +Inclusion/Exclusion Principle: If +and +are events that contain outcomes of a single experiment, then +. +It’s worth noting that this formula is truly an extension of the Addition Rule. Remember that the Addition Rule requires +that the events +and +are mutually exclusive. In that case, the compound event +is impossible, and so +. So, in cases where the events in question are mutually exclusive, the Inclusion/Exclusion Principle +reduces to the Addition Rule. +EXAMPLE 7.30 +Using the Inclusion/Exclusion Principle +Suppose we have events +, +, and +, associated with these probabilities: +Compute the following: +1. +2. +3. +Solution +1. +Using the Inclusion/Exclusion Principle, we get: +2. +Again, we’ll apply the Inclusion/Exclusion Principle: +3. +Applying the Inclusion/Exclusion Principle one more time: +YOUR TURN 7.30 +You are about to roll a special 6-sided die that has both a colored letter and a colored number on each face. The +faces are labeled with: a red 1 and a blue A, a red 1 and a green A, an orange 1 and a green B, an orange 2 and a red +C, a purple 3 and a brown D, an orange 4 and a blue E. Find the probabilities of these events: +1. The number is orange or even. +2. The letter is green or an A. +3. The number is even or the letter is green. +774 +7 • Probability +Access for free at openstax.org + +Check Your Understanding +You are about to draw a card at random from a deck containing only these 10 cards: +, +, +, +, +, +, +, +, +, +. Compute the following probabilities: +44. You draw an ace or a king. +45. You draw a +or a +. +46. You draw an ace or a +. +47. You draw a jack or a +. +48. You draw a jack or a +. +49. You draw a king or a +. +SECTION 7.8 EXERCISES +For the following exercises, we are considering a special 6-sided die, with faces that are labeled with a number and a +letter: 1A, 1B, 2A, 2C, 4A, and 4E. You are about to roll this die once. +1. What is the probability of rolling a 1 or a 2? +2. What is the probability of rolling a 4 or a B? +3. What is the probability of rolling an even number or a consonant? +4. What is the probability of rolling a 2 or an E? +5. What is the probability of rolling an odd number or a vowel? +6. What is the probability of rolling an odd number or a consonant? +In the following exercises, you are drawing a single card from a standard 52-card deck. +7. What is the probability that you draw a +or a +? +8. What is the probability that you draw a +or a 5? +9. What is the probability that you draw a 2 or a 3? +10. What is the probability that you draw a card with an even number on it? +11. What is the probability that you draw a card with an even number on it or a +? +12. What is the probability that you draw an ace or a king? +13. What is the probability that you draw a face card (king, queen, or jack)? +14. What is the probability that you draw a face card or a +? +For the following exercises, use the table provided here, which breaks down the enrollment at a certain liberal arts +college by class year and area of study: +Class Year +First-Year +Sophomore +Junior +Senior +Totals +Area Of Study +Arts +138 +121 +148 +132 +539 +Humanities +258 +301 +275 +283 +1117 +Social Science +142 +151 +130 +132 +555 +Natural Science/Mathematics +175 +197 +203 +188 +763 +Totals +713 +770 +756 +735 +2974 +15. What is the probability that a randomly selected student is a first-year or sophomore? +16. What is the probability that a randomly selected student is a junior or an arts major? +17. What is the probability that a randomly selected student is majoring in the social sciences or the natural +sciences/mathematics? +18. What is the probability that a randomly selected student is a social science major or a sophomore? +19. What is the probability that a randomly selected student is a senior or is a humanities major? +20. What is the probability that a randomly selected student is majoring in the arts or humanities? +The following exercises are about the casino game roulette. In this game, the dealer spins a marble around a wheel +that contains 38 pockets that the marble can fall into. Each pocket has a number (each whole number from 0 to 36, +7.8 • The Addition Rule for Probability +775 + +along with a double zero) and a color (0 and 00 are both green; the other 36 numbers are evenly divided between black +and red). Players make bets on which number (or groups of numbers) they think the marble will land on. The figure +shows the layout of the numbers and colors, as well as some of the bets that can be made. +Roulette Table (credit: "American Roulette Table Layout" by Film8ker/Wikimedia Commons, Public Domain) +What is the probability of winning at least one of the following pairs of bets on a single spin of the wheel? +21. First dozen (wins if any of the numbers 1–12 come up) or second dozen (wins on 13–24) +22. Red (wins on any of the 18 red numbers) or black (wins on any of the 18 black numbers) +23. Even (wins on any even number 2–36; 0 and 00 both lose this bet) or red +24. Middle column (the numbers 2, 5, 8, 11, …, 35) or black +25. Middle column or red +26. Right column (the numbers 3, 6, 9, …, 36) or black +27. Right column or red +28. Odd or black +29. Even or black +30. The street bet (a bet on 3 numbers that make up a row on the table) on 1, 2, 3 or odd +31. The street bet on 1, 2, 3 or even +32. The corner bet (a bet on 4 numbers that form a square on the table) on 1, 2, 4, 5 or first dozen +33. The corner bet on 1, 2, 4, 5 or second dozen +34. The basket bet (which wins on 0, 00, 1, 2, 3) or red +35. The basket bet or black +776 +7 • Probability +Access for free at openstax.org + +7.9 Conditional Probability and the Multiplication Rule +Figure 7.37 If you roll two dice by throwing them one at a time, the face showing on the first die will affect the possible +outcomes for the sum of the two dice. (credit: “dice” by Ciarán Archer/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Calculate conditional probabilities. +2. +Apply the Multiplication Rule for Probability to compute probabilities. +Back in Example 7.18, we constructed the following table (Figure 7.38) to help us find the probabilities associated with +rolling two standard 6-sided dice: +Figure 7.38 +For example, 3 of these 36 equally likely outcomes correspond to rolling a sum of 10, so the probability of rolling a 10 is +. However, if you choose to roll the dice one at a time, the probability of rolling a 10 will change after the first die +comes to rest. For example, if the first die shows a 5, then the probability of rolling a sum of 10 has jumped to +—the +event will occur if the second die also shows a 5, which is 1 of 6 equally likely outcomes for the second die. If instead the +first die shows a 3, then the probability of rolling a sum of 10 drops to 0—there are no outcomes for the second die that +will give us a sum of 10. +Understanding how probabilities can shift as we learn new information is critical in the analysis of our second type of +compound events: those built with “and.” This section will explain how to compute probabilities of those compound +events. +Conditional Probabilities +When we analyze experiments with multiple stages, we often update the probabilities of the possible final outcomes or +the later stages of the experiment based on the results of one or more of the initial stages. These updated probabilities +are called conditional probabilities. +7.9 • Conditional Probability and the Multiplication Rule +777 + +In other words, if +is a possible outcome of the first stage in a multistage experiment, then the probability of an event +conditional on +(denoted +, read “the probability of +given +”) is the updated probability of +under the +assumption that +occurred. +In the example that opened this section, we might consider rolling two dice as a multistage experiment: rolling one, then +the other. If we define +to be the event “roll a sum of 10,” +to be the event “first die shows 5,” and +to be the event +“first die shows 3,” then we computed +, +, and +. +EXAMPLE 7.31 +Computing Conditional Probabilities +1. +April is playing a coin-flipping game with Ben. She will flip a coin 3 times. If the coin lands on heads more than tails, +April wins; if it lands on tails more than heads, Ben wins. Let +be the event “April wins,” +be “first flip is heads,” +and +be “first flip is tails.” Compute +, +, and +. +2. +You are about to draw 2 cards without replacement from a deck containing only these 10 cards: +, +, +, +, +, +, +, +, +, +. We’ll define the following events: +is “both cards are the same rank,” +is “first card is an +ace,” and +is “first card is a king.” Compute +and +. +3. +Jim’s sock drawer contains 5 black socks and 3 blue socks. To avoid waking his partner, Jim doesn’t want to turn the +lights on, so he puts on 2 socks at random. Let +be the event “Jim’s 2 socks match,” let +be the event “the sock on +Jim’s left foot is black,” and let +be the event “the sock on Jim’s left foot is blue.” Compute +, +, and +. +Solution +1. +Step 1. The sample space is {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. The event +consists of the first 4 of those +outcomes: HHH, HHT, HTH, and THH. Thus, +. +Step 2. Now, let’s compute +. We are assuming the result of the first flip is heads. That leaves us with 4 +possible outcomes: HHH, HHT, HTH, and HTT. Of those, April wins 3 (HHH, HHT, HTH) and loses one (HTT). So, +. +Step 3. If the result of the first flip is instead tails, the 4 possible outcomes are THH, THT, TTH, and TTT. Of those, +April wins 1 (THH) and loses 3 (THT, TTH, TTT). So, +. +2. +Step 1. If the event +happens, then 1 of the 4 aces is drawn first; the remaining cards in the deck are 3 aces, 2 +kings, 2 queens, and 2 jacks. In order for the event +to occur, the second card drawn has to be an ace. Since there +are 3 aces among the remaining 9 cards, +. +Step 2. If the event +happens instead, then the first card drawn is a king. That leaves 4 aces, 1 king, 2 queens, and +2 jacks in the deck. Under the assumption that the first card is a king, the event +will occur only if the second card +is also a king. Since only one of the remaining 9 cards is a king, we have +. +3. +Step 1. We can view the event +as a compound event using “or”: both socks are blue or both socks are black. Let’s +compute the probability that both socks are blue using combinations. We’re choosing 2 socks from a group of 8; 3 of +the 8 are blue. So, +. Similarly, +. Therefore, since +these events are mutually exclusive, we can use the Addition Rule: +. +Step 2. If the sock on Jim’s left foot is black (i.e., +occurred), then there are 4 remaining black socks of the 7 in the +drawer. So, +. +Step 3. If the sock on Jim’s left foot is blue ( +occurred), then there are 2 blue socks among the 7 remaining in the +drawer. So, +. +YOUR TURN 7.31 +You are about to roll a special 6-sided die that has both a colored letter and a colored number on each face. The +faces are labeled with: a red 1 and a blue A, a red 1 and a green A, an orange 1 and a green B, an orange 2 and a red +C, a purple 3 and a brown D, an orange 4 and a blue E. Find the given conditional probabilities: +778 +7 • Probability +Access for free at openstax.org + +1. +2. +3. +In Tree Diagrams, Tables, and Outcomes, we introduced the concept of dependence between stages of a multistage +experiment. We stated at the time that two stages were dependent if the result of one stage affects the other stage. +We explained that dependence in terms of the sample space, but sometimes that dependence can be a little more +subtle; it’s more properly understood in terms of conditional probabilities. Two stages of an experiment are +dependent if +for some outcome of the second stage +and outcome of the first stage +. +WHO KNEW? +Protecting Bombers in World War II +In his book How Not to Be Wrong, Jordan Ellenberg recounts this anecdote: During World War II, the American +military wanted to add additional armor plating to bomber aircraft, in order to reduce the chances that they get shot +down. So, they collected data on planes after returning from missions. The data showed that the fuselage, wings, and +fuel system had many more bullet holes (per unit area) than the engine compartments, so the military brass wanted +to add additional armor to the parts of the plane that were hit most often. Luckily, before they added the armor to the +planes, they asked for a second opinion. Abraham Wald, a Jewish mathematician who had fled the rising Nazi regime, +pointed out that it was far more important that the armor plating be added to areas where there were fewer bullet +holes. Why? The planes they were studying had already completed their missions, so the military was essentially +looking at conditional probabilities: the probability of suffering a bullet strike, given that the plane made it back +safely. More bullet holes in an area on the plane indicated that was a region that wasn’t as important for the plane’s +survival! +Compound Events Using “And” and the Multiplication Rule +For multistage experiments, the outcomes of the experiment as a whole are often stated in terms of the outcomes of the +individual stages. Commonly, those statements are joined with “and.” For example, in the sock drawer example just +above, one outcome might be “the left sock is black and the right sock is blue.” As with “or” compound events, these +probabilities can be computed with basic arithmetic. +FORMULA +Multiplication Rule for Probability: If +and +are events associated with the first and second stages of an experiment, +then +. +In The Addition Rule for Probability, we considered probabilities of events connected with “and” in the statement of +the Inclusion/Exclusion Principle. These two scenarios are different; in the statement of the Inclusion/Exclusion +Principle, the events connected with “and” are both events associated with the same single-stage experiment (or the +same stage of a multistage experiment). In the Multiplication Rule, we’re looking at events associated with different +stages of a multistage experiment. +EXAMPLE 7.32 +Using the Multiplication Rule for Probability +You are president of a club with 10 members: 4 seniors, 3 juniors, 2 sophomores, and 1 first-year. You need to choose 2 +members to represent the club on 2 college committees. The first person selected will be on the Club Awards Committee +and the second will be on the New Club Orientation Committee. The same person cannot be selected for both. You +decide to select these representatives at random. +7.9 • Conditional Probability and the Multiplication Rule +779 + +1. +What is the probability that a senior is chosen for both positions? +2. +What is the probability that a junior is chosen first and a sophomore is chosen second? +3. +What is the probability that a sophomore is chosen first and a senior is chosen second? +Solution +1. +We need the probability that a senior is chosen first and a senior is chosen second. These are two stages of a +multistage experiment, so we’ll apply the Multiplication Rule for Probability: +Since there are 4 seniors among the 10 members, +. Next, assuming a senior is +chosen first, there are 3 seniors among the 9 remaining members. So, +. Putting this all together, we get +. +2. +There are 3 juniors among the 10 members, so +. Assuming a junior is chosen first, there +are 2 sophomores among the remaining 9 members, so +. Thus, +using the Multiplication Rule for Probability, we have +. +3. +The probability that a sophomore is chosen first is +, and the probability that a senior is chosen second given +that a sophomore was chosen first is +. Thus, using the Multiplication Rule for Probability, we have: +. +YOUR TURN 7.32 +You’re drawing 2 cards in order from a deck containing only the cards +, +, +, +, +, +, +, and +. +Compute the following: +1. +2. +3. +WORK IT OUT +The Birthday Problem +One of the most famous problems in probability theory is the Birthday Problem, which has to do with shared +birthdays in a large group. To make the analysis easier, we’ll ignore leap days, and assume that the probability of +being born on any given date is +. Now, if you have 366 people in a room, we’re guaranteed to have at least one +pair of people who share a single birthday. Imagine filling the room by first admitting someone born on January 1, +then someone born on January 2, and so on… The 365th person admitted would be born on December 31. If you add +one more person to the room, that person’s birthday would have to match someone else’s. +Let’s look at the other end of the spectrum. If you choose two people at random, what is the probability that they +share a birthday? As with many probability questions, this is best addressed by find out the probability that they do +not share a birthday. The first person’s birthday can be anything (probability 1), and the second person’s birthday can +be anything other than the first person’s birthday (probability +). The probability that they have different birthdays +is +. So, the probability that they share a birthday is +. +What if we have three people? The probability that they all have different birthdays can be obtained by extending our +previous calculation: The probability that two people have different birthdays is +, so if we add a third to the mix, +the probability that they have a different birthday from the other two is +. So, the probability that all three have +different birthdays is +, and thus the probability that there’s a shared birthday in the group is +. +The big question is this: How many people do we need in the room to have the probability of a shared birthday +780 +7 • Probability +Access for free at openstax.org + +greater than +? Make a guess, then with a partner keep adding hypothetical people to the group and computing +probabilities until you get there! +It is often useful to combine the rules we’ve seen so far with the techniques we used for finding sample spaces. In +particular, trees can be helpful when we want to identify the probabilities of every possible outcome in a multistage +experiment. The next example will illustrate this. +EXAMPLE 7.33 +Using Tree Diagrams to Help Find Probabilities +The board game Clue uses a deck of 21 cards: 6 suspects, 6 weapons, and 9 rooms. Suppose you are about to draw 2 +cards from this deck. There are 6 possible outcomes for the draw: 2 suspects, 2 weapons, 2 rooms, 1 suspect and 1 +weapon, 1 suspect and 1 room, or 1 weapon and 1 room. What are the probabilities for each of these outcomes? +Solution +Step 1: Let’s start by building a tree diagram that illustrates both stages of this experiment. Let’s use S, W, and R to +indicate drawing a suspect, weapon, and room, respectively (Figure 7.39). +Figure 7.39 +Step 2: We want to start computing probabilities, starting with the first stage. The probability that the first card is a +suspect is +. The probability that the first card is a weapon is the same: +. Finally, the probability that the first card +is a room is +. +7.9 • Conditional Probability and the Multiplication Rule +781 + +Step 3: Let’s incorporate those probabilities into our tree: label the edges going into each of the nodes representing the +first-stage outcomes with the corresponding probabilities (Figure 7.40). +Figure 7.40 +Note that the sum of the probabilities coming out of the initial node is 1; this should always be the case for the +probabilities coming out of any node! +Step 4: Let’s look at the case where the first card is a suspect. There are 3 edges emanating from that node (leading to +the outcomes SS, SW, and SR). We’ll label those edges with the appropriate conditional probabilities, under the +assumption that the first card is a suspect. First, there are 5 remaining suspect cards among the 20 left in the deck, so +. Using similar reasoning, we can compute +and +. +Step 5: Checking our work, we see that the sum of these 3 probabilities is again equal to 1. Let’s add those to our tree +(Figure 7.41). +782 +7 • Probability +Access for free at openstax.org + +Figure 7.41 +Step 6: Let’s continue filling in the conditional probabilities at the other nodes, always checking to make sure the sum of +the probabilities coming out of any node is equal to 1 (Figure 7.42). +7.9 • Conditional Probability and the Multiplication Rule +783 + +Figure 7.42 +Step 7: We can compute the probability of landing on any final node by multiplying the probabilities along the path we +would take to get there. For example, the probability of drawing a suspect first and a weapon second (i.e., ending up on +the node labeled “SW”) is +, as illustrated in Figure 7.43. +784 +7 • Probability +Access for free at openstax.org + +Figure 7.43 +Step 8: Let’s fill in the rest of the probabilities (Figure 7.44). +7.9 • Conditional Probability and the Multiplication Rule +785 + +Figure 7.44 +Step 9: A helpful feature of tree diagrams is that the final outcomes are always mutually exclusive, so the Addition Rule +can be directly applied. For example, the probability of drawing one suspect and one room (in any order) would be +. We can find the probabilities of the other outcomes in a similar fashion, as shown in +the following table: +Outcome +Probability +2 suspects +2 weapons +2 rooms +1 suspect and 1 weapon +1 suspect and 1 room +1 weapon and 1 room +786 +7 • Probability +Access for free at openstax.org + +Checking once again, the sum of these 6 probabilities is 1, as expected. +YOUR TURN 7.33 +1. You are about to perform the following two-stage experiment. First, you will flip a coin. If the result is heads, roll +a standard 6-sided die. If the result of the coin flip is tails, roll a modified 6-sided die with faces labeled 1, 1, 1, 2, +2, 3. Use a tree diagram to find the probability of rolling each of the numbers from 1 to 6. +WORK IT OUT +The Monty Hall Problem +On the original version of the game show Let’s Make a Deal, originally hosted by Monty Hall and now hosted by +Wayne Brady, one contestant was chosen to play a game for the grand prize of the day (often a car). Here’s how it +worked: On the stage were three areas concealed by numbered curtains. The car was hidden behind one of the +curtains; the other two curtains hid worthless prizes (called “Zonks” on the show). The contestant would guess which +curtain concealed the car. To build tension, Monty would then reveal what was behind one of the other curtains, which +was always one of the Zonks (Since Monty knew where the car was hidden, he always had at least one Zonk curtain +that hadn’t been chosen that he could reveal). Monty then turned to the contestant and asked: “Do you want to stick +with your original choice, or do you want to switch your choice to the other curtain?” What should the contestant do? +Does it matter? +With a partner or in a small group, simulate this game. You can do that with a small candy (the prize) hidden under +one of three cups, or with three playing cards (just decide ahead of time which card represents the “Grand Prize”). +One person plays the host, who knows where the prize is hidden. Another person plays the contestant and tries to +guess where the prize is hidden. After the guess is made, the host should reveal a losing option that wasn’t chosen by +the contestant. The contestant then has the option to stick with the original choice or switch to the other, unrevealed +option. Play about 20 rounds, taking turns in each role and making sure that both contestant strategies (stick or +switch) are used equally often. After each round, make a note of whether the contestant chose “stick” or “switch” and +whether the contestant won or lost. Find the empirical probability of winning under each strategy. Then, see if you +can use tree diagrams to verify your findings. +Check Your Understanding +For the following exercises, you are rolling two 6-sided dice, each of which has 3 orange faces, 2 green faces, and 1 blue +face. +50. What is the probability of rolling 2 orange faces? +51. What is the probability of rolling 2 green faces? +52. What is the probability of rolling 1 orange and 1 green face (in any order)? +For the following exercises, you are about to draw 2 cards at random (without replacement) from a deck containing +only these 10 cards: +, +, +, +, +, +, +, +, +, +. +53. What is the probability of drawing 2 aces? +54. What is the probability of drawing an ace first and a king second? +55. What is the probability of drawing a +and a +(in any order)? +SECTION 7.9 EXERCISES +For the following exercises, we are considering a special 6-sided die, with faces that are labeled with a number and a +letter: 1A, 1B, 2A, 2C, 4A, and 4E. You are about to roll this die twice. +1. What is the probability of rolling two 1s? +2. What is the probability of rolling two vowels? +3. What is the probability of rolling an even number first and an odd number second? +4. What is the probability of rolling an even number and an odd number in any order? +7.9 • Conditional Probability and the Multiplication Rule +787 + +5. What is the probability of rolling a consonant first and a 1 second? +6. What is the probability of rolling one number less than 3 and one number greater than 3, in any order? +In the following exercises, you are about to draw Scrabble tiles from a bag; the bag contains the letters A, A, C, E, E, E, L, +L, N, O, R, S, S, S, T, X. +7. If you draw 1 tile at random, compute +a. +b. +8. If you draw 1 tile at random, compute: +a. +b. +9. If you draw 2 tiles with replacement, compute +. +10. If you draw 2 tiles without replacement, compute +. +11. If you draw 2 tiles with replacement, compute +. +12. If you draw 2 tiles without replacement, compute +. +13. If you draw 2 tiles with replacement, compute +. +14. If you draw 2 tiles without replacement, compute +. +15. If you draw 2 tiles with replacement, compute +. +16. If you draw 2 tiles without replacement, compute +. +For the following exercises, use the table provided, which breaks down the enrollment at a certain liberal arts college by +class year and area of study. +Class Year +First-Year +Sophomore +Junior +Senior +Totals +Area Of Study +Arts +138 +121 +148 +132 +539 +Humanities +258 +301 +275 +283 +1117 +Social Science +142 +151 +130 +132 +555 +Natural Science/Mathematics +175 +197 +203 +188 +763 +Totals +713 +770 +756 +735 +2974 +17. Compute the probability that a randomly selected student is a sophomore, given that they are majoring in the +arts. +18. Compute the probability that a randomly selected student is majoring in the arts, given that they are a +sophomore. +19. If two seniors are chosen at random, compute the probability that both are social science majors. Give your +answer as a decimal, rounded to 5 decimal places. +20. If two humanities majors are chosen at random, compute the probability that the first is a senior and the +second is a junior. Give your answer as a decimal, rounded to 5 decimal places. +21. If two natural science/mathematics majors are chosen at random, compute the probability that one is a +sophomore and one is a senior (in any order). Give your answer as a decimal, rounded to 5 decimal places. +22. If two students are chosen at random, compute the probability that one is an arts major and one is a social +science major, in any order. Give your answer as a decimal, rounded to 5 decimal places. +In the following exercises deal with the game “Punch a Bunch,” which appears on the TV game show The Price Is Right. +In this game, contestants have a chance to punch through up to 4 paper circles on a board; behind each circle is a card +with a dollar amount printed on it. There are 50 of these circles; the dollar amounts are given in this table: +788 +7 • Probability +Access for free at openstax.org + +Dollar Amount +Frequency +$25,000 +1 +$10,000 +2 +$5,000 +4 +$2,500 +8 +$1,000 +10 +$500 +10 +$250 +10 +$100 +5 +Contestants are shown their selected dollar amounts one at a time, in the order selected. After each is revealed, the +contestant is given the option of taking that amount of money or throwing it away in favor of the next amount. (You +can watch the game being played in the video Playing “Punch a Bunch.” (https://openstax.org/r/ +Playing_Punch_a_Bunch)) Jeremy is playing “Punch a Bunch” and gets 2 punches. +23. What is the probability that both punches are worth less than $1,000? +24. What is the probability that both punches are worth more than $2,500? +25. What is the probability that the second punch is worth more than the first punch, given that the first punch was +worth $250? +26. What is the probability that the second punch is worth more than the first punch, given that the first punch was +worth $1,000? +27. What is the probability that the second punch is worth less than the first punch, given that the first punch was +worth $250? +28. What is the probability that the second punch is worth less than the first punch, given that the first punch was +worth $1,000? +29. What is the probability that both punches are worth $100? +30. What is the probability that both punches are worth the same amount? +In the following exercises, we consider two baseball teams playing a best-of-three series (meaning the first team to win +two games wins the series). Team A is a little bit better than Team B, so we expect Team A will win 55% of the time. +31. What is the probability that Team A wins the series given that Team B wins the first game? +32. What is the probability that Team B wins the series given that Team B wins the first game? +33. What is the probability that Team B wins the series given that Team A wins the first game? +34. What is the probability that Team A wins the series given that Team A wins the first game? +35. Build a tree diagram that shows all possible outcomes of the series. Label the edges with appropriate +probabilities. +36. What is the probability that Team A wins the series? +37. If instead Team A has a 75% chance of winning each game, what is the probability that Team A wins the series? +38. If instead Team A has a 90% chance of winning each game, what is the probability that Team A wins the series? +7.9 • Conditional Probability and the Multiplication Rule +789 + +7.10 The Binomial Distribution +Figure 7.45 If one baseball team has a 65% chance of beating another in any single game, what’s the likelihood that they +win a best-of-seven series? (credit: “baseball game” by Britt Reints/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify binomial experiments. +2. +Use the binomial distribution to analyze binomial experiments. +It’s time for the World Series, which determines the champion for this season in Major League Baseball. The scrappy Los +Angeles Angels are facing the powerhouse Cincinnati Reds. Computer models put the chances of the Reds winning any +single game against the Angels at about 65%. The World Series, as its name implies, isn’t just one game, though: it’s +what’s known as a “best-of-seven” contest: the teams play each other repeatedly until one team wins 4 games (which +could take up to 7 games total, if each team wins three of the first 6 matchups). If the Reds truly have a 65% chance of +winning a single game, then the probability that they win the series should be greater than 65%. Exactly how much +bigger? +If you have the patience for it, you could use a tree diagram like we used in Example 7.33 to trace out all of the possible +outcomes, find all the related probabilities, and add up the ones that result in the Reds winning the series. Such a tree +diagram would have +final nodes, though, so the calculations would be very tedious. Fortunately, we have tools +at our disposal that allow us to find these probabilities very quickly. This section will introduce those tools and explain +their use. +Binomial Experiments +The tools of this section apply to multistage experiments that satisfy some pretty specific criteria. Before we move on to +the analysis, we need to introduce and explain those criteria so that we can recognize experiments that fall into this +category. Experiments that satisfy each of these criteria are called binomial experiments. A binomial experiment is an +experiment with a fixed number of repeated independent binomial trials, where each trial has the same probability of +success. +Repeated Binomial Trials +The first criterion involves the structure of the stages. Each stage of the experiment should be a replication of every +other stage; we call these replications trials. An example of this is flipping a coin 10 times; each of the ten flips is a trial, +and they all occur under the same conditions as every other. Further, each trial must have only two possible outcomes. +These two outcomes are typically labeled “success” and “failure,” even if there is not a positive or negative connotation +associated with those outcomes. Experiments with more than two outcomes in their sample spaces are sometimes +reconsidered in a way that forces just two outcomes; all we need to do is completely divide the sample space into two +parts that we can label “success” and “failure.” For example, your grade on an exam might be recorded as A, B, C, D, or F, +but we could instead think of the grades A, B, C, and D as “success” and a grade of F as “failure.” Trials with only two +outcomes are called binomial trials (the word binomial derives from Latin and Greek roots that mean “two parts”). +Independent Trials +The next criterion that we’ll be looking for is independence of trials. Back in Tree Diagrams, Tables, and Outcomes, we +said that two stages of an experiment are independent if the outcome of one stage doesn’t affect the other stage. +Independence is necessary for the experiments we want to analyze in this section. +Fixed Number of Trials +Next, we require that the number of trials in the experiment be decided before the experiment begins. For example, we +790 +7 • Probability +Access for free at openstax.org + +might say “flip a coin 10 times.” The number of trials there is fixed at 10. However, if we say “flip a coin until you get 5 +heads,” then the number of trials could be as low as 5, but theoretically it could be 50 or a 100 (or more)! We can’t apply +the tools from this section in cases where the number of trials is indeterminate. +Constant Probability +The next criterion needed for binomial experiments is related to the independence of the trials. We must make sure that +the probability of success in each trial is the same as the probability of success in every other trial. +EXAMPLE 7.34 +Identifying Binomial Experiments +Decide whether each of the following is a binomial experiment. For those that aren’t, identify which criterion or criteria +are violated. +1. +You roll a standard 6-sided die 10 times and write down the number that appears each time. +2. +You roll a standard 6-sided die 10 times and write down whether the die shows a 6 or not. +3. +You roll a standard 6-sided die until you get a 6. +4. +You roll a standard 6-sided die 10 times. On the first roll, we define “success” as rolling a 4 or greater. After the first +roll, we define “success” as rolling a number greater than the result of the previous roll. +Solution +1. +Since we’re noting 1 of 6 possible outcomes, the trials are not binomial. So, this isn’t a binomial experiment. +2. +We have 2 possible outcomes (“6” and “not 6”), the trials are independent, the probability of success is the same +every time, and the number of trials is fixed. This is a binomial experiment. +3. +Since the number of trials isn’t fixed (we don’t know if we’ll get our first 6 after 1 roll or 20 rolls or somewhere in +between), this isn’t a binomial experiment. +4. +Here, the probability of success might change with every roll (on the first roll, that probability is +; if the first roll is a +6, the probability of success on the next roll is zero). So, this is not a binomial experiment. +YOUR TURN 7.34 +Decide whether the following experiments are binomial experiments: +1. Draw a card from a well-shuffled deck, note its suit, and replace it. Repeat this process 5 times. +2. Draw 5 cards from a well-shuffled deck and count the number of +. +3. Draw a card from a well-shuffled deck, note whether it is a +or not, and replace it. Repeat this process 5 +times. +4. Draw cards from a well-shuffled deck until you get a +. +The Binomial Formula +If we flip a coin 100 times, you might expect the number of heads to be around 50, but you probably wouldn’t be +surprised if the actual number of heads was 47 or 52. What is the probability that the number of heads is exactly 50? Or +falls between 45 and 55? It seems unlikely that we would get more than 70 heads. Exactly how unlikely is that? +Each of these questions is a question about the number of successes in a binomial experiment (flip a coin 100 times, +“success” is flipping heads). We could theoretically use the techniques we’ve seen in earlier sections to answer each of +these, but the number of calculations we’d have to do is astronomical; just building the tree diagram that represents this +situation is more than we could complete in a lifetime; it would have +final nodes! To put that number +in perspective, if we could draw 1,000 dots every second, and we started at the moment of the Big Bang, we’d currently +be about 0.00000003% of the way to drawing out those final nodes. Luckily, there’s a shortcut called the Binomial +Formula that allows us to get around doing all those calculations! +FORMULA +Binomial Formula: Suppose we have a binomial experiment with +trials and the probability of success in each trial is +. Then: +7.10 • The Binomial Distribution +791 + +. +We can use this formula to answer one of our questions about 100 coin flips. What is the probability of flipping exactly 50 +heads? In this case, +, +, and +, so +. Unfortunately, +many calculators will balk at this calculation; that first factor ( +) is an enormous number, and the other two factors +are very close to zero. Even if your calculator can handle numbers that large or small, the arithmetic can create serious +errors in rounding off. +TECH CHECK +Luckily, spreadsheet programs have alternate methods for doing this calculation. In Google Sheets, we can use the +BINOMDIST function to do this calculation for us. Open up a new sheet, click in any empty cell, and type +“=BINOMDIST(50,100,0.5,FALSE)” followed by the Enter key. The cell will display the probability we seek; it’s about 8%. +Let’s break down the syntax of that function in Google Sheets: enter “=BINOMDIST( , +, +, FALSE)” to find the +probability of +successes in +trials with probability of success +. +EXAMPLE 7.35 +Using the Binomial Formula +1. +Find the probability of rolling a standard 6-sided die 4 times and getting exactly one 6 without using technology. +2. +Find the probability of rolling a standard 6-sided die 60 times and getting exactly ten 6s using technology. +3. +Find the probability of rolling a standard 6-sided die 60 times and getting exactly eight 6s using technology. +Solution +1. +We’ll apply the Binomial Formula, where +, +, and +: +2. +Here, +, +, and +. In Google Sheets, we’ll enter “=BINOMDIST(10, 60, 1/6, FALSE)” to get our result: +0.137. +3. +This experiment is the same as in Exercise 2 of this example; we’re simply changing the number of successes from +10 to 8. Making that change in the formula in Google Sheets, we get the probability 0.116. +YOUR TURN 7.35 +Compute the probabilities (rounded to 3 decimal places) of the following events related to rolling a standard 4-sided +die (with faces labeled 1, 2, 3, and 4): +1. You roll the die 10 times and get exactly four 2s. +2. You roll the die 20 times and get exactly four 2s. +3. You roll the die 30 times and get exactly four 2s. +The Binomial Distribution +If we are interested in the probability of more than just a single outcome in a binomial experiment, it’s helpful to think of +the Binomial Formula as a function, whose input is the number of successes and whose output is the probability of +observing that many successes. Generally, for a small number of trials, we’ll give that function in table form, with a +792 +7 • Probability +Access for free at openstax.org + +complete list of the possible outcomes in one column and the probability in the other. +For example, suppose Kristen is practicing her basketball free throws. Assume Kristen always makes 82% of those shots. +If she attempts 5 free throws, then the Binomial Formula gives us these probabilities: +Shots Made +Probability +0 +0.000189 +1 +0.004304 +2 +0.0392144 +3 +0.1786432 +4 +0.4069096 +5 +0.3707398 +A table that lists all possible outcomes of an experiment along with the probabilities of those outcomes is an example of +a probability density function (PDF). A PDF may also be a formula that you can use to find the probability of any +outcome of an experiment. +Because they refer to the same thing, some sources will refer to the Binomial Formula as the Binomial PDF. +If we want to know the probability of a range of outcomes, we could add up the corresponding probabilities. Going back +to Kristen’s free throws, we can find the probability that she makes 3 or fewer of her 5 attempts by adding up the +probabilities associated with the corresponding outcomes (in this case: 0, 1, 2, or 3): +The probability that the outcome of an experiment is less than or equal to a given number is called a cumulative +probability. A table of the cumulative probabilities of all possible outcomes of an experiment is an example of a +cumulative distribution function (CDF). A CDF may also be a formula that you can use to find those cumulative +probabilities. +Cumulative probabilities are always associated with events that are defined using +. If other inequalities are used to +define the event, we must restate the definition so that it uses the correct inequality. +Here are the PDF and CDF for Kristen’s free throws: +Shots Made +Probability +Cumulative +0 +0.000189 +0.000189 +1 +0.004304 +0.004493 +2 +0.0392144 +0.0437073 +3 +0.1786432 +0.2223506 +4 +0.4069096 +0.6292602 +5 +0.3707398 +1 +7.10 • The Binomial Distribution +793 + +TECH CHECK +Google Sheets can also compute cumulative probabilities for us; all we need to do is change the “FALSE” in the +formulas we used before to "TRUE." +EXAMPLE 7.36 +Using the Binomial CDF +Suppose we are about to flip a fair coin 50 times. Let +represent the number of heads that result from those flips. Use +technology to find the following: +1. +2. +3. +4. +5. +Solution +1. +The event here is defined by +, which is the inequality we need to have if we want to use the Binomial CDF. In +Google Sheets, we’ll enter “=BINOMDIST(22, 50, 0.5, TRUE)” to get our answer: 0.2399. +2. +This event uses the wrong inequality, so we need to do some preliminary work. If +, that means +(because +has to be a whole number). So, we’ll enter “=BINOMDIST(25, 50, 0.5, TRUE)” to find +. +3. +The inequality associated with this event is pointing in the wrong direction. If +is the event +, that means +that +contains the outcomes {29, 30, 31, 32, 33, …}. Thus, +must contain the outcomes {…, 25, 26, 27, 28}. In +other words, +is defined by +. Since it uses +, we can find +using “=BINOMDIST(28, 50, 0.5, TRUE)”: +0.8389 So, using the formula for probabilities of complements, we have +4. +As in part 3, this inequality is pointing in the wrong direction. If +is the event +, then +contains the +outcomes {20, 21, 22, 23, …}. That means +contains the outcomes {…, 16, 17, 18, 19}, and so +is defined by +. So, we can find +using “=BINOMDIST(19, 50, 0.5, TRUE)”: 0.0595. Finally, using the formula for +probabilities of complements, we get: +5. +If +, that means we are interested in the outcomes {21, 22, 23, 24}. This doesn’t look like any of the +previous situations, but there is a way to find this probability using the Binomial CDF. We need to put everything in +terms of “less than or equal to,” so we’ll first note that all of our outcomes are less than or equal to 24. But we don’t +want to include values that are less than or equal to 20. So, we have three events: let +be the event defined by +(note that we’re trying to find +). Let +be defined by +, and let +be defined by +. Of +these three events, +contains the most outcomes. If +occurs, then either +or +must have occurred. Moreover, +and +are mutually exclusive. Thus, +, by the Addition Rule. Solving for the probability that we +want, we get +YOUR TURN 7.36 +You are about to roll a standard 6-sided die 20 times. Let +denote a success, which will be rolling a number greater +794 +7 • Probability +Access for free at openstax.org + +than 4. Find the probabilities of the following events, rounded to 4 decimal places: +1. +2. +3. +4. +5. +Finally, we can answer the question posed at the beginning of this section. Remember that the Reds are facing the +Angels in the World Series, which is won by the team who is first to win 4 games. The Reds have a 65% chance to win any +game against the Angels. So, what is the probability that the Reds win the World Series? At first glance, this is not a +binomial experiment: The number of games played is not fixed, since the series ends as soon as one team wins 4 games. +However, we can extend this situation to a binomial experiment: Let’s assume that 7 games are always played in the +World Series, and the winner is the team who wins more games. In a way, this is what happens in reality; it’s as though +the first team to lose 4 games (and thus cannot win more than the other team) forfeits the rest of their games. So, we +can treat the actual World Series as a binomial experiment with seven trials. If +is the number of games won by the +Reds, the probability that the Reds win the World Series is +. Using the techniques from the last example, we +get +. +PEOPLE IN MATHEMATICS +Abraham de Moivre +Abraham de Moivre was born in 1667 in France to a Protestant family. Though he was educated in Catholic schools, +he remained true to his faith; in 1687, he fled with his brother to London to escape persecution under the reign of +King Louis XIV. Once he arrived in England, he supported himself as a freelance math tutor while he conducted his +own research. Among his interests was probability; in 1711, he published the first edition of The Doctrine of Chances: +A Method of Calculating the Probabilities of Events in Play. This book was the second textbook on probability (after +Cardano’s Liber de ludo aleae). De Moivre discovered an important connection between the binomial distribution and +thenormal distribution (an important concept in statistics; we’ll explore that distribution and its connection to the +binomial distribution in Chapter 8). De Moivre also discovered some properties of a new probability distribution that +later became known as the Poisson distribution. +Check Your Understanding +You are rolling a 6-sided die with 3 orange faces, 2 green faces, and 1 blue face. +56. If you roll the die 5 times and note the color showing on each roll, is this a binomial experiment? +57. If you roll the die 5 times and count the number times you roll a green face, is this a binomial experiment? +58. If you count how many times you roll the die until you get a blue face, is this a binomial experiment? +Suppose you’re rolling the same colored 6-sided die 10 times. Let +, +, and +represent the number of times the die +lands with an orange, green, and blue side up, respectively. Find these probabilities (round to 4 decimal places): +59. +60. +61. +SECTION 7.10 EXERCISES +In the following exercises, decide whether the described experiments are binomial experiments. For those that are not, +explain why they aren’t. +1. A golfer practices putts from 1 foot, 2 feet, 3 feet, 4 feet, and 5 feet; “success” is defined as making the putt. +2. A game designer rolls a pair of dice 100 times and counts the number of times the sum is at least 10. +3. A student who is completely unprepared for a multiple-choice pop quiz guesses on all 10 questions. There are 4 +choices for each of the first 5 questions and 5 choices for each of the last 5 questions. “Success” is defined as +answering the question correctly. +4. A baseball player is practicing pitching; he throws pitches until he gets 50 strikes. +5. A statistician stops 20 college students at random outside a dining hall and notes their class year. +7.10 • The Binomial Distribution +795 + +6. An employee at a bowling alley watches each patron’s first ball and counts how many are strikes over the course +of his shift. +In the following exercises, you have an 8-sided die with 4 faces colored orange, 3 colored red, and 1 colored yellow. +You’re going to roll the die 100 times: Let +be the number of times a yellow face is showing, +be the number of times +a red face is showing, and +be the number of times an orange face is showing. Find the given probabilities, rounded +to 4 decimal places: +7. +8. +9. +10. +11. +12. +13. +14. +15. +16. +The following exercises are about series of games, where Team A faces Team B in a best-of series (like the World +Series). Find the probability that Team A wins the series in each of the following scenarios: +17. +best-of-5 series +18. +best-of-7 series +19. +best-of-15 series +20. +, best-of-31 series +21. +, best-of-101 series +22. +, best-of-5 series +23. +, best-of-7 series +24. +, best-of-15 series +25. +, best-of-31 series +26. +, best-of-101 series +27. Give the table of the PDF for flipping a fair coin 5 times and counting the heads. Do the calculations without using +technology. +28. Give the table of the CDF for flipping a fair coin 5 times and counting the heads. Do the calculations without using +technology. +The following exercises are about the casino game roulette. In this game, the dealer spins a marble around a wheel +that contains 38 pockets that the marble can fall into. Each pocket has a number (each whole number from 0 to 36, +along with a “double zero”) and a color (0 and 00 are both green; the other 36 numbers are evenly divided between +black and red). Players make bets on which number (or groups of numbers) they think the marble will land on. The +figure shows the layout of the numbers and colors, as well as some of the bets that can be made. +796 +7 • Probability +Access for free at openstax.org + +Roulette Table (credit: "American Roulette Table Layout" by Film8ker/Wikimedia Commons, Public Domain) +29. If a player bets $1 on red and wins, the player gets $2 back (the original $1 bet plus $1 winnings). What is the +probability that the player wins more than they lose if the player bets on red on 5 consecutive spins? +30. What is the probability that the player wins more than they lose if the player bets on red on 15 consecutive +spins? +31. What is the probability that the player wins more than they lose if the player bets on red on 30 consecutive +spins? +32. What is the probability that the player wins more than they lose if the player bets on red on 100 consecutive +spins? +33. What is the probability that the player wins more than they lose if the player bets on red on 200 consecutive +spins? +34. What is the probability that the player wins more than they lose if the player bets on red on 1,000 consecutive +spins? +35. What is the probability that the player wins more than they lose if the player bets on red on 5,000 consecutive +spins? +7.11 Expected Value +Figure 7.46 The concept of expected value allows us to analyze games that involve randomness, like Roulette. (credit: +“Roulette Table and Roulette Wheel in a Casino with People betting on numbers” by Marco Verch/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Calculate the expected value of an experiment. +2. +Interpret the expected value of an experiment. +3. +Use expected value to analyze applications. +The casino game roulette has dozens of different bets that can be made. These bets have different probabilities of +7.11 • Expected Value +797 + +winning but also have different payouts. In general, the lower the probability of winning a bet is, the more money a +player wins for that bet. With so many options, is there one bet that’s “smarter” than the rest? What’s the best play to +make at a roulette table? In this section, we’ll develop the tools we need to answer these questions. +Expected Value +Many experiments have numbers associated with their outcomes. Some are easy to define; if you roll 2 dice, the sum of +the numbers showing is a good example. In some card games, cards have different point values associated with them; +for example, in some forms of the game rummy, aces are worth 15 points; 10s, jacks, queens, and kings are worth 10; +and all other cards are worth 5. The outcomes of casino and lottery games are all associated with an amount of money +won or lost. These outcome values are used to find the expected value of an experiment: the mean of the values +associated with the outcomes that we would observe over a large number of repetitions of the experiment. (See +Conditional Probability and the Multiplication Rule for more on means.) +That definition is a little vague; How many is “a large number?” In practice, it depends on the experiment; the number +has to be large enough that every outcome would be expected to appear at least a few times. For example, if we’re +talking about rolling a standard 6-sided die and we note the number showing, a few dozen replications should be +enough that the mean would be representative. Since the probability of each outcome is +, we would expect to see each +outcome about 8 times over the course of 48 replications. However, if we’re talking about the Powerball lottery, where +the probability of winning the jackpot is about +, we would need several billion replications to ensure that every +outcome appears a few times. Luckily, we can find the theoretical expected value before we even run the experiment the +first time. +FORMULA +Expected Value: If +represents an outcome of an experiment and +represents the value of that outcome, then +the expected value of the experiment is: +, +where +is the “sum,” meaning we add up the results of the formula that follows over all possible outcomes. +EXAMPLE 7.37 +Finding Expected Values +Find the expected values of the following experiments. +1. +Roll a standard 6-sided die and note the number showing. +2. +Roll two standard 6-sided dice and note the sum of the numbers showing. +3. +Draw a card from a well-shuffled standard deck of cards and note its rummy value (15 for aces; 10 for tens, jacks, +queens, and kings; 5 for everything else). +Solution +1. +Step 1: Let’s start by writing out the PDF table for this experiment. +Value +Probability +1 +2 +3 +4 +798 +7 • Probability +Access for free at openstax.org + +Value +Probability +5 +6 +Step 2: To find the expected value, we need to find +for each possible outcome in the table below. +Value +Probability +1 +2 +3 +4 +5 +6 +Step 3: We add all of the values in that last column: +. So, the expected value of a +single roll of a die is 3.5. +2. +Back in Example 7.18, we made this table of all of the equally likely outcomes (Figure 7.47): +Figure 7.47 +Step 1: Let’s use Figure 7.47 to create the PDF for this experiment, as shown in the following table: +7.11 • Expected Value +799 + +Value +Probability +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Step 2: We can multiply each row to find +as shown in the following table: +Value +Probability +2 +3 +4 +5 +6 +7 +8 +9 +10 +800 +7 • Probability +Access for free at openstax.org + +Value +Probability +11 +12 +Step 3: We can add the last column to get the expected value: +. +So, the expected value is 7. +3. +Step 1: Let’s make a PDF table for this experiment. There are 3 events that we care about, so let’s use those events in +the table below: +Event +Probability +{A} +{10, J, Q, K} +{2, 3, 4, 5, 6, 7, 8, 9} +Step 2: Let’s add a column to the following table for the values of each event: +Event +Probability +Value +{A} +15 +{10, J, Q, K} +10 +{2, 3, 4, 5, 6, 7, 8, 9} +5 +Step 3: We’ll add the column for the product of the values and probabilities to the table below: +Event +Probability +Value +{A} +15 +{10, J, Q, K} +10 +{2, 3, 4, 5, 6, 7, 8, 9} +5 +Step 4: We’ll find the sum of the last column: +. Thus, the expected Rummy value of a +randomly selected card is about 7.3. +7.11 • Expected Value +801 + +YOUR TURN 7.37 +1. Find the expected value of the number showing when you roll a special 6-sided die with faces {1, 1, 2, 3, 5, 8}. +2. Find the expected value of the number of heads showing if you flip a coin 3 times. +3. You are about to play a game, where you flip a coin 3 times. If all 3 flips result in heads, you win $20. If you get +2 heads, you win $10. If you flip 1 or 0 heads, you win nothing. What is the expected value of your winnings? +Let’s make note of some things we can learn from Example 7.37. First, as Exercises 1 and 3 demonstrate, the expected +value of an experiment might not be a value that could come up in the experiment. Remember that the expected value is +interpreted as a mean, and the mean of a collection of numbers doesn’t have to actually be one of those numbers. +Second, looking at Exercise 1, the expected value (3.5) was just the mean of the numbers on the faces of the die: +. This is no accident! If we break that fraction up using the addition in the numerator, we get +, which we can rewrite as +. That’s exactly the +computation we did to find the expected value! In fact, expected values can always be treated as a special kind of mean +called a weighted mean, where the weights are the probabilities associated with each value. When the probabilities are +all equal, the weighted mean is just the regular mean. +Interpreting Expected Values +As we noted, the expected value of an experiment is the mean of the values we would observe if we repeated the +experiment a large number of times. (This interpretation is due to an important theorem in the theory of probability +called the Law of Large Numbers.) Let’s use that to interpret the results of the previous example. +EXAMPLE 7.38 +Interpreting Expected Values +Interpret the expected values of the following experiments. +1. +Roll a standard 6-sided die and note the number showing. +2. +Roll 2 standard 6-sided dice and note the sum of the numbers showing. +3. +Draw a card from a well-shuffled standard deck of cards and note its Rummy value (15 for aces; 10 for tens, jacks, +queens, and kings; 5 for everything else). +Solution +1. +If you roll a standard 6-sided die many times, the mean of the numbers you roll will be around 3.5. +2. +If you roll a pair of standard 6-sided dice many times, the mean of the sums of the numbers you roll will be about 7. +3. +If you draw a card from a well-shuffled deck many times, the mean of the Rummy values of the cards would be +around 7.3. +YOUR TURN 7.38 +1. Interpret the expected value of the number showing when you roll a special 6-sided die with faces {1, 1, 2, 3, +5, 8}. +2. Interpret the expected value of the number of heads showing if you flip a coin 3 times. +3. You are about to play a game where you flip a coin 3 times. If all 3 flips result in heads, you win $20. If you get +2 heads, you win $10. If you flip 1 or 0 heads, you win nothing. Interpret the expected value of your winnings. +WHO KNEW? +Pascal’s Wager +The French scholar Blaise Pascal (1623–1662) was among the earliest mathematicians to study probabilities, and was +the first to accurately describe and compute expected values. In his book Pensées (Thoughts), he turned the analysis +of expected values to his belief in the Christian God. He said that there is no way for people to establish the +802 +7 • Probability +Access for free at openstax.org + +probability that God exists, but since the “winnings” on a bet that God exists (and that you then lead your life +accordingly) are essentially infinite, the expected value of taking that bet is always positive, no matter how unlikely it +is that God exists. +Using Expected Value +Now that we know how to find and interpret expected values, we can turn our attention to using them. Suppose +someone offers to play a game with you. If you roll a die and get a 6, you get $10. However, if you get a 5 or below, you +lose $1. Is this a game you’d want to play? Let’s look at the expected value: The probability of winning is +and the +probability of losing is +, so the expected value is +. That means, on average, you’ll +come out ahead by about 83 cents every time you play this game. It’s a great deal! On the other hand, if the winnings for +rolling a 6 drop to $3, the expected value becomes +, meaning you should expect to +lose about 33 cents on average for every time you play. Playing that game is not a good idea! In general, this is how +casinos and lottery corporations make money: Every game has a negative expected value for the player. +WHO KNEW? +Expected Values in Football +In the 21st century, data analytics tools have revolutionized the way sports are coached and played. One tool in +particular is used in football at crucial moments in the game. When a team faces a fourth down (the last possession in +a series of four possessions, a fairly common occurrence), the coach faces a decision: Run one play to try to gain a +certain number of yards, or kick the ball away to the other team. Here’s the interesting part of the decision: If the +team “goes for it” and runs the play and they are successful, then they keep possession of the ball and can continue in +their quest to score more points. If they are unsuccessful, then they lose possession of the ball, giving the other team +an opportunity to score points. If, instead, the team punts, or kicks the ball away, then the other team gets possession +of the ball, but in a worse position for them than if the original team goes for it and fails. To analyze this situation, +data analysts have generated empirical probabilities for every fourth down situation, and computed the expected +value (in terms of points) for each decision. Coaches frequently use those calculations when they decide which option +to take! +PEOPLE IN MATHEMATICS +Pierre de Fermat and Blaise Pascal +In 1654, a French writer and amateur mathematician named Antoine Gombaud (who called himself the Chevalier du +Mére) reached out to his gambling buddy Blaise Pascal to answer a question that he’d read about called the “problem +of points.” The question goes like this: Suppose you’re playing a game that is scored using points, and the first person +to earn 5 points is the winner. The game is interrupted with the score 4 points to 2. If the winner stood to win $100, +how should the prize money be divided between the players? Certainly the person who is 1 point away from victory +should get more, but how much more? +We have developed tools in this section to answer this question. At its heart, it’s a question about conditional +probabilities and expected value. At the time that Pascal first started thinking about it, though, those ideas hadn’t yet +been invented. Pascal reached out to a colleague named Pierre de Fermat, and over the course of a couple of months, +their correspondence with each other would eventually solve the problem. In the process, they first described +conditional probabilities and expected values! +Apart from their work in probability, these men are famous for other work in mathematics (and, in Pascal’s case, +philosophy and physics). Fermat is remembered for his work in geometry and in number theory. After his death, the +statement of what came to be called “Fermat’s Last Theorem” was discovered scribbled in the margin of a book, with +the note that Fermat had discovered a “marvelous proof that this margin is too small to contain.” The theorem says +that any equation of the form +has no positive integer solutions if +. No proof of that theorem was +7.11 • Expected Value +803 + +discovered until 1994, when Andrew Wiles used computers and new branches of geometry to finally prove the +theorem! +Pascal is remembered for the “arithmetical triangle” that is named for him (though he wasn’t the first person to +discover it; see the section on the binomial distribution for more), as well as work in geometry. In physics, Pascal +worked on hydrodynamics and air pressure (the SI unit for pressure is named for him), and in philosophy, Pascal +advocated for a mathematical approach to philosophical problems. +EXAMPLE 7.39 +Using Expected Values +In the casino game keno, a machine chooses at random 20 numbers between 1 and 80 (inclusive) without replacement. +Players try to predict which numbers will be chosen. Players don’t try to guess all 20, though; generally, they’ll try to +predict between 1 and 10 of the chosen numbers. The amount won depends on the number of guesses they made and +the number of guesses that were correct. +1. +At one casino, a player can try to guess just 1 number. If that number is among the 20 selected, the player wins $2; +otherwise, the player loses $1. What is the expected value? +2. +At the same casino, if a player makes 2 guesses and they’re both correct, the player wins $14; otherwise, the player +loses $1. What is the expected value? +3. +Players can also make 3 guesses. If 2 of the 3 guesses are correct, the player wins $1. If all 3 guesses are correct, the +player wins $42. Otherwise, the player loses $1. What is the expected value? +4. +Which of these games is the best for the player? Which is the best for the casino? +Solution +1. +There are 20 winning numbers out of 80, so if we try to guess one of them, the probability of guessing correctly is +. The probability of losing is then +, and so the expected value is +. +2. +There are +winning choices out of +total ways to choose 2 numbers. So, the probability of +winning is +and the probability of losing is +. So, the expected value of the game is +. +3. +Step 1: Let’s start with the big prize. There are +ways to correctly guess 3 winning numbers out of +ways to guess three numbers total. That means the probability of winning the big prize is +. +Step 2: Let’s find the probability of the second prize. The denominator is the same: 82,160. Let’s figure out the +numerator. To win the second prize, the player must pick 2 of the 20 winning numbers and one of the 60 losing +numbers. The number of ways to do that can be found using the Multiplication Rule for Counting: there are +ways to pick 2 winning numbers and 60 ways to pick 1 losing number, so there are +ways to win the second prize. So, the probability of winning that second prize is +. +Step 3: Since the overall probability of winning is +, the probability of losing must +be +. So, the expected value is +. +4. +The bet that’s the best for the player is the one with the highest expected value for the player, which is guessing two +numbers. The best one for the casino is the one with the lowest expected value for the player, which is guessing one +number. +YOUR TURN 7.39 +The casino game craps involves rolling 2 standard 6-sided dice. While the main game involves repeated rolls of the +dice, players can also bet on the outcomes of single rolls. Find the expected values of the following three $1 bets. +Then decide which bet is the best for the player and which bet is the best for the casino. +1. Players can bet on the next roll of the dice being a sum of 7. Winners get $4 and losers lose $1. +2. Players can bet on the next roll of the dice being a sum of 12. Winners get $30 and losers lose $1 +804 +7 • Probability +Access for free at openstax.org + +3. Players can bet that the next roll of the dice will be “any craps” (a sum of 2, 3, or 12). Winners get $7 and losers +lose $1. +WORK IT OUT +Make Your Own Lottery +By yourself or with a partner, devise your own lottery scheme. Assume you would have access to one or more +machines that choose numbers randomly. What will a lottery draw look like? How many numbers are players +choosing from? How many will be drawn? Will they be drawn with replacement or without replacement? What +conditions must be met for a player to win first or second (or more!) prize? Once you’ve decided that, decide the +payoff structure for winners, and how much the game will cost to play. Try to make the game enticing enough that +people will want to play it, but with enough negative expected value that the lottery will make money. Aim for the +expected value to be about −0.25 times the cost of playing the game. +Check Your Understanding +You are about to roll a 20-sided die with faces labeled as follows: 5 faces have a 1, 6 faces have a 3, 4 faces have a 5, 3 +faces have a 7, and 2 faces have a 9. +62. What is the expected value of the number showing on the die after it’s rolled? +63. Interpret your answer. +You are about to play a game in which you draw 3 ping-pong balls without replacement from a barrel. The barrel +contains 6 green balls and 4 red balls. If all 3 of your selections are green, you win $5. If 2 of the 3 are green, you win +$1. If 2 or more of your selections are red, you lose $5. +64. What is the expected value of this game? +65. Interpret your answer. +66. Is it advantageous to you to play the game? How do you know? +SECTION 7.11 EXERCISES +You roll a standard 6-sided die and win points equal to the square of the number shown. +1. What’s the expected value of the number of points you win? +2. Interpret your answer. +In the classic board game The Game of Life, players have the chance to play the market. A spinner with 10 equally likely +spaces is spun to choose a random number. If the result is 3 or less, the player loses $25,000. If the result is 7 or more, +the player wins $50,000. If the result is a 4, 5, or 6, the player doesn’t win or lose anything. +3. What is the expected value of playing the market? +4. Interpret the answer. +The Game of Life players also occasionally have the opportunity to speculate. Players choose any 2 of the 10 numbers +on the spinner and then give it a spin. If one of their numbers is chosen, they win $140,000; if not, they lose $10,000. +5. What is the expected value of this speculation? +6. Interpret your answer. +7. Which is better for The Game of Life players: playing the market or speculating? How do you know? +A charitable organization is selling raffle tickets as a fundraiser. They intend to sell 5,000 tickets at $10 each. One ticket +will be randomly selected to win the grand prize of a new car worth $35,000. +8. What is the expected value of a single ticket? +9. Interpret your answer. +10. The organization is worried they won’t be able to sell all the tickets, so they announce that, in addition to the +grand prize, they will offer 10 second prizes of $500 in cash. What is the new expected value of a single ticket? +11. Interpret your answer. +In the following exercises involve randomly selecting golf balls from a bucket. The bucket contains 4 yellow balls +(numbered 1-4) and 6 white balls (numbered 1-6). +12. If you draw a single ball, what is the expected number of yellow balls selected? +7.11 • Expected Value +805 + +13. Suppose you draw 2 balls with replacement. +a. +Give a PDF table for the possible outcomes for the number of yellow balls selected. +b. +What is the expected number of yellow balls selected? +14. Suppose you draw 2 balls without replacement. +a. +Give a PDF table for the possible outcomes for the number of yellow balls selected. +b. +What is the expected number of yellow balls selected? +15. Suppose you draw 3 balls with replacement. +a. +Give a PDF table for the possible outcomes for the number of yellow balls selected. +b. +What is the expected number of yellow balls selected? +16. Suppose you draw 3 balls without replacement. +a. +Give a PDF table for the possible outcomes for the number of yellow balls selected. +b. +What is the expected number of yellow balls selected? +17. If you draw a single ball, what is the expected value of the number on the ball? +18. Suppose you draw 2 balls with replacement. +a. +Give a PDF table for the possible outcomes for the sum of the numbers on the selected balls. +b. +What is the expected sum of the numbers on the balls? +19. Suppose you draw 2 balls without replacement. +a. +Give a PDF table for the possible outcomes for the sum of the numbers on the selected balls. +b. +What is the expected sum of the numbers on the balls? +The following exercises deal with the game “Punch a Bunch,” which appears on the TV game show The Price Is Right. In +this game, contestants have a chance to punch through up to 4 paper circles on a board; behind each circle is a card +with a dollar amount printed on it. There are 50 of these circles; the dollar amounts are given in this table: +Dollar Amount +Frequency +$25,000 +1 +$10,000 +2 +$5,000 +4 +$2,500 +8 +$1,000 +10 +$500 +10 +$250 +10 +$100 +5 +Contestants are shown their selected dollar amounts one at a time, in the order selected. After each is revealed, the +contestant is given the option of taking that amount of money or throwing it away in favor of the next amount. (You +can watch the game being played in the video Playing "Punch a Bunch." (https://openstax.org/r/ +Playing_Punch_a_Bunch)) Anita is playing “Punch a Bunch” and gets 2 punches. +20. If Anita got $500 on her first punch, what’s the expected value of her second punch? +21. If Anita got $500 on her first punch, should she throw out her $500 and take the results of her second punch? +How do you know? +22. If Anita got $1,000 on her first punch, what’s the expected value of her second punch? +23. If Anita got $1,000 on her first punch, should she throw out her $1,000 and take the results of her second +punch? How do you know? +24. If Anita got $2,500 on her first punch, what’s the expected value of her second punch? +806 +7 • Probability +Access for free at openstax.org + +25. If Anita got $2,500 on her first punch, should she throw out her $2,500 and take the results of her second +punch? How do you know? +The following exercises are about the casino game roulette. In this game, the dealer spins a marble around a wheel +that contains 38 pockets that the marble can fall into. Each pocket has a number (each whole number from 0 to 36, +along with a “double zero”) and a color (0 and 00 are both green; the other 36 numbers are evenly divided between +black and red). Players make bets on which number (or groups of numbers) they think the marble will land on. The +figure shows the layout of the numbers and colors, as well as some of the bets that can be made. +Roulette Table (credit: "American Roulette Table Layout" by Film8ker/Wikimedia Commons, Public Domain) +26. If a player makes a $1 bet on a single number, they win $35 if that number comes up, but lose $1 if it doesn’t. +What is the expected value of this bet? +27. Interpret your answer to the previous question. +28. Suppose a player makes the $1 bet on a single number in 5 consecutive spins. What is the expected value of +this series of bets? (Hint: use the Binomial Distribution.) +29. Interpret your answer to the previous question. +30. If a player makes a $10 bet on first dozen, they win $20 if one of the numbers 1–12 comes up but lose $10 +otherwise. What is the expected value of this bet? +31. Interpret your answer to the previous question. +32. Suppose a player makes the $10 bet on first dozen in 4 consecutive spins. What is the expected value of that +series of bets? +33. Interpret your answer to the previous question. +34. If a player makes a $10 basket bet, they win $60 if 0, 00, 1, 2, or 3 come up but lose $10 otherwise. What is the +expected value of this bet? +35. Interpret your answer to previous question. +36. Which is better for the player: a $10 first dozen bet or a $10 basket bet? How do you know? +7.11 • Expected Value +807 + +Chapter Summary +Key Terms +7.1 The Multiplication Rule for Counting +• +combinatorics +• +Multiplication Rule for Counting (Fundamental Counting Principle) +7.2 Permutations +• +permutation +• +factorial +7.3 Combinations +• +combination +7.4 Tree Diagrams, Tables, and Outcomes +• +experiment +• +replication +• +sample space +• +independent/dependent +7.5 Basic Concepts of Probability +• +event +• +probability +• +theoretical probability +• +empirical probability +• +subjective probability +7.7 What Are the Odds? +• +odds (for/against) +7.8 The Addition Rule for Probability +• +mutually exclusive +7.9 Conditional Probability and the Multiplication Rule +• +conditional probability +7.10 The Binomial Distribution +• +binomial experiment +• +probability density function (PDF) +• +cumulative distribution function (CDF) +7.11 Expected Value +• +expected value +Key Concepts +7.1 The Multiplication Rule for Counting +• +The Multiplication Rule for Counting is used to count large sets. +7.2 Permutations +• +Using the Multiplication Rule for Counting to enumerate permutations. +• +Simplifying and computing expressions involving factorials. +• +Using factorials to count permutations. +7.3 Combinations +• +Permutations are used to count subsets when order matters; combinations work when order doesn't matter. +• +Combinations can also be computed using factorials. +808 +7 • Chapter Summary +Access for free at openstax.org + +7.4 Tree Diagrams, Tables, and Outcomes +• +We identify the sample space of an experiment by identifying all of its possible outcomes. +• +Tables can help us find a sample space by keeping the possible outcomes organized. +• +Tree diagrams provide a visualization of the sample space of an experiment that involves multiple stages. +7.5 Basic Concepts of Probability +• +The theoretical probability of an event is the ratio of the number of equally likely outcomes in the event to the +number of equally likely outcomes in the sample space. +• +Empirical probabilities are computed by repeating the experiment many times, and then dividing the number of +replications that result in the event of interest by the total number of replications. +• +Subjective probabilities are assigned based on subjective criteria, usually because the experiment can’t be repeated +and the outcomes in the sample space are not equally likely. +• +The probability of the complement of an event is found by subtracting the probability of the event from one. +7.6 Probability with Permutations and Combinations +• +We use permutations and combinations to count the number of equally likely outcomes in an event and in a sample +space, which allows us to compute theoretical probabilities. +7.7 What Are the Odds? +• +Odds are computed as the ratio of the probability of an event to the probability of its compliment. +7.8 The Addition Rule for Probability +• +The Addition Rule is used to find the probability that one event or another will occur when those events are mutually +exclusive. +• +The Inclusion/Exclusion Principle is used to find probabilities when events are not mutually exclusive. +7.9 Conditional Probability and the Multiplication Rule +• +Conditional probabilities are computed under the assumption that the condition has already occurred. +• +The Multiplication Rule for Probability is used to find the probability that two events occur in sequence. +7.10 The Binomial Distribution +• +Binomial experiments result when we count the number of successful outcomes in a fixed number of repeated, +independent trials with a constant probability of success. +• +The binomial distribution is used to find probabilities associated with binomial experiments. +• +Probability density functions (PDFs) describe the probabilities of individual outcomes in an experiment; cumulative +distribution functions (CDFs) give the probabilities of ranges of outcomes. +7.11 Expected Value +• +The expected value of an experiment is the sum of the products of the numerical outcomes of an experiment with +their corresponding probabilities. +• +The expected value of an experiment is the most likely value of the average of a large number of replications of the +experiment. +Formula Review +7.2 Permutations +• +7.3 Combinations +• +The formula for counting combinations is: +7.5 Basic Concepts of Probability +• +For an experiment whose sample space +consists of equally likely outcomes, the theoretical probability of the +event +is the ratio +where +and +denote the number of outcomes in the event and in the +7 • Chapter Summary +809 + +sample space, respectively. +• +7.7 What Are the Odds? +• +For an event +, +• +If the odds in favor of +are +, then +. +7.8 The Addition Rule for Probability +• +If +and +are mutually exclusive events, then +. +• +If +and +are events that contain outcomes of a single experiment, then +. +7.9 Conditional Probability and the Multiplication Rule +• +If +and +are events associated with the first and second stages of an experiment, then +. +7.10 The Binomial Distribution +• +Suppose we have a binomial experiment with +trials and the probability of success in each trial is +. Then: +7.11 Expected Value +• +If +represents an outcome of an experiment and +represents the value of that outcome, then the expected +value of the experiment is: +where +stands for the sum, meaning we add up the results of the formula that follows over all possible outcomes. +Projects +1. +The Binomial Distribution is one of many examples of a discrete probability distribution. Other examples include the +Geometric, Hypergeometric, Multinomial, Poisson, and Negative Binomial Distributions. Choose one of these +distributions, and find out what makes it different from the Binomial Distribution. In what situations can it be +applied? How is it used? Once you have an idea of how it’s used, write a series of five questions like the ones in this +chapter that can be answered with that distribution, and find the answers. +2. +Binomial is a word that also comes up in algebra; the word describes polynomials with two terms. At first glance, +there isn’t much to indicate that these two uses of the word are related, but it turns out there is a connection. +Explore the connection between the Binomial Distribution and the algebraic concept of binomial expansion, (the +process of multiplying out expressions like +for a positive whole number +). Search for a connection with the +mathematical object known as Pascal’s Triangle. +3. +Hazard is a dice game that was mentioned in Chaucer’s Canterbury Tales. It was a popular game of chance played in +taverns and coffee houses well into the 18th century; its popularity at the time of the foundation of probability +theory means that it was a common example in early texts on finding expected values and probabilities. Find the +rules of the game, and get some practice playing it. Then, analyze the choices that the caster gets to make, and +decide which is most advantageous, using the language of expected values. +810 +7 • Chapter Summary +Access for free at openstax.org + +Chapter Review +The Multiplication Rule for Counting +1. You are booking a round trip flight for vacation. If there are 4 outbound flight options and 7 return flight options, +how many different options do you have? +2. You are putting together a social committee for your club. You’d like broad representation, so you will choose one +person from each class. If there are 8 seniors, 12 juniors, 10 sophomores, and 6 first-years, how many committees +are possible? +3. The Big Breakfast Platter at Jimbo’s Sausage Haus gives you your choice of 4 flavors of sausage, 5 preparations for +eggs, 3 different potato options, and 4 different breads. If you choose one of each, how many different Big +Breakfast Platters can be selected? +4. The multiple-choice quiz you’re about to take has 10 questions with 4 choices for each. How many ways are there +to fill out the quiz? +Permutations +5. Compute +. +6. Compute +. +7. Compute +8. Compute +. +9. Compute +. +10. Compute +. +11. As you plan your day, you see that you have 6 tasks on your to-do list. You’ll only have time for 5 of those. How +many schedules are possible for you today? +12. As captain of your intramural softball team, you are responsible for setting the 10-person batting order for the +team. If there are 12 people on the team, how many batting orders are possible? +Combinations +13. If you’re trying to decide which 4 of your 12 friends to invite to your apartment for a dinner party, are you using +permutations or combinations? +14. If you’re trying to decide which of your guests sits where at your table, are you using permutations or +combinations? +15. Compute +. +16. Compute +. +17. How many ways are there to draw a hand of 8 cards from a deck of 16 cards? +18. In a card game with 4 players and a deck of 12 cards, how many ways are there to deal out the four 3-card hands? +Tree Diagrams, Tables, and Outcomes +19. If you draw a card at random from a standard 52-card deck and note its suit, what is the sample space? +20. If you draw 2 cards at random from a standard 52-card deck and note the 2 suits (without paying attention to the +order), what is the sample space? +21. If you draw 2 Scrabble tiles in order without replacement from a bag containing E, E, L, S, what is the sample +space? +22. If you draw 2 Scrabble tiles without replacement and ignoring order from a bag containing E, E, L, S, what is the +sample space? +23. If you draw 2 Scrabble tiles without replacement and ignoring order from a bag containing E, E, L, S, what is the +sample space? +24. If you draw 2 Scrabble tiles with replacement and ignoring order from a bag containing E, E, L, S, what is the +sample space? +7 • Chapter Summary +811 + +Basic Concepts of Probability +25. If you read that the probability of flipping 10 heads in a row is +, is that probability most likely theoretical, +empirical, or subjective? +26. If someone tells you that there is a 40% chance that a Democrat wins the U.S. Presidential election in 2132, is that +probability most likely theoretical, empirical, or subjective? +27. If your professor says that you have a 20% chance of getting an A in her class because 20% of her students +historically have earned As, is that probability most likely theoretical, empirical, or subjective? +In the following exercises, you are about to roll a standard 12-sided die (with faces labeled 1–12). +28. What is the probability of rolling a negative number? +29. What is the probability of rolling a number less than 20? +30. What is the probability of rolling an 11? +31. What is the probability of rolling a number less than 7? +32. What is the probability of not rolling an 11? +33. What is the probability of rolling a multiple of 4? +34. Over the last 30 years, it has rained 12 times on May 1. What empirical probability would you assign to the +event "it rains next May 1"? +Probability with Permutations and Combinations +In the following exercises, you’re drawing cards from a special deck of cards containing +, +, +, +, +, +, +, +, +, +. +35. If you draw 4 cards without replacement, what is the probability of drawing a 2, 3, 4, and 5 in order? +36. If you draw 4 cards without replacement, what is the probability of drawing a 2, 3, 4, and 5 in any order? +37. If you draw 3 cards without replacement, what is the probability that you draw a +, a +, and a +, in order? +38. If you draw 3 cards without replacement, what is the probability that you draw 2 +and 1 +, in any order? +What Are the Odds? +39. If you roll a standard 20-sided die (with faces numbered 1–20), what are the odds against rolling a number less +than 5? +40. If you roll a standard 20-sided die (with faces numbered 1–20), what are the odds in favor of rolling greater than a +5? +41. If +, what are the odds in favor of +? +42. If +, what are the odds against +? +The Addition Rule for Probability +In the following exercises, you’re drawing a single card from a special deck of cards containing +, +, +, +, +, +, +, +, +, +. +43. What is the probability of drawing a 2 or a 3? +44. What is the probability of drawing a +or a +? +45. What is the probability of drawing a 2 or a +? +46. What is the probability of drawing an even number or a +? +Conditional Probability and the Multiplication Rule +In the following exercises, you’re drawing from a special deck of cards containing +, +, +, +, +, +, +, +, +, +. +47. If you draw a single card, what is: +a. +b. +c. +48. If you draw a single card, what is: +a. +b. +c. +In the following exercises, you are playing the following game that involves rolling 2 dice, one at a time. First, you roll a +812 +7 • Chapter Summary +Access for free at openstax.org + +standard 6-sided die. If the result is a 4 or less, your second roll uses a standard 4-sided die. If the result of the first roll +is a 5 or 6, your second roll uses a standard 6-sided die. Find these probabilities: +49. +50. +51. +52. +The Binomial Distribution +In the following exercises, decide whether the described experiment is a binomial experiment. If it is, identify the +number of trials and the probability of success in each trial. If it isn’t, explain why it isn’t. +53. Draw 5 cards with replacement from a standard deck and count the number of +. +54. Draw 5 cards without replacement from a standard deck and count the number of +. +55. Draw cards from a standard deck and count how many cards are chosen before the first +appears. +In the following exercises, you are about to roll a standard 20-sided die. Round answers to 4 decimal places. +56. Suppose you are going to roll the die 4 times. Give a full PDF table for the number of times a number greater +than 16 appears. +57. If you roll the die 10 times, what is the probability that a number between 1 and 5 (inclusive) comes up exactly +once? +58. If you roll the die 40 times, what is the probability that 20 comes up fewer than 2 times? +59. If you roll the die 40 times, what is the probability that 20 comes up 4 or more times? +60. If you roll the die 100 times, what is the probability that the number of times the die lands on something less +than or equal to 7 is between 30 and 35 (inclusive)? +61. If you roll the die 100 times, what is the probability that the number of times the die lands on something less +than or equal to 7 is exactly 36? +62. If you roll the die 100 times, what is the probability that the die lands on 20 between 5 and 8 times, inclusive? +Expected Value +63. You are playing a game where you roll a pair of standard 6-sided dice. You win $32 if you get a sum of 12, and +lose $1 otherwise. What is the expected value of this game? +64. Interpret your answer. +You are playing a game where you roll a standard 12-sided die 4 times. If you roll 12 four times, you win $1,000. If you +roll 12 three times, you win $100. If you roll 12 twice, you win $10. If you roll 12 one time, you don’t win or lose +anything. If you roll don’t roll a single 12, you lose $1. +65. What is the expected value of this game? +66. Interpret your answer. +67. Which game would be better to play? Why? +Chapter Test +Each of the following exercises involve drawing a Scrabble tile from a bag. These tiles are labeled with a letter and a +point value, as follows: A(1), C(3), D(2), E(1), E(1), J(8), K(5), O(1), R(1), R(1). +1. How many ways are there to draw a vowel and then a consonant from the bag? +2. How many ways are there to draw a tile worth an even number of points and then a tile worth an odd number +of points from the bag? +3. How many ways are there to draw 4 tiles from the bag without replacement, if order matters? +4. How many ways are there to draw 4 consonants from the bag without replacement, if order matter? +5. How many ways are there to draw 4 tiles from the bag with replacement, if order does not matter? +6. How many ways are there to draw 4 consonants from the bag with replacement, if order does not matter? +7. Give the sample space of the experiment that asks you to draw 2 tiles from the bag with replacement and note +their point values, where order doesn’t matter. Give the outcomes as ordered pairs. +8. Give the sample space of the experiment that asks you to draw 2 tiles from the bag with replacement and note +their point values, where order doesn’t matter. Give the outcomes as ordered pairs. +9. If you draw a single tile from the bag, what is the probability that it’s an E? +10. If you draw a single tile from the bag, what is the probability that it’s not an A? +11. If you draw 3 tiles from the bag without replacement, what is the probability that they spell RED, in order? +12. If you draw 3 tiles from the bag without replacement, what is the probability that they spell RED, in any order? +13. What are the odds against drawing a vowel? +14. Use your answer to question 12 to find the odds against drawing three tiles without replacement and being +7 • Chapter Summary +813 + +able to spell RED. +15. If you draw one tile, what is the probability of drawing a J or a K? +16. If you draw one tile, what is the probability that it’s a vowel or that it’s worth more than 4 points? +17. Suppose you’re about to draw one tile from the bag. Find +and +. +18. If you draw 2 tiles with replacement, what is the probability of drawing a consonant first and then a vowel? +19. If you draw 2 tiles without replacement, what is the probability of drawing a consonant first and then a vowel? +20. If you draw 10 tiles with replacement, what is the probability that you draw exactly 3 vowels? Round to 3 +decimal places. +21. If you draw 100 tiles with replacement, what is the probability that you draw fewer than 35 vowels? Round to 4 +decimal places. +22. Find and interpret the expected number of points on the tile, assuming you draw 1 tile from the bag. +23. Find the expected sum of points on 2 tiles, selected without replacement. +24. If your friend offers you a bet where they pay you $10 if you draw a vowel from the bag, but you owe them $5 if +you draw a consonant, should you take it? How do you know? +814 +7 • Chapter Summary +Access for free at openstax.org + +Figure 8.1 Statistics can be used to decide on a fair salary for sports stars. (credit: “Jasmine Powell goes up for a shot in a +game against Jacksonville” by Lorie Shaull/Flickr, CC BY 2.0) +Chapter Outline +8.1 Gathering and Organizing Data +8.2 Visualizing Data +8.3 Mean, Median and Mode +8.4 Range and Standard Deviation +8.5 Percentiles +8.6 The Normal Distribution +8.7 Applications of the Normal Distribution +8.8 Scatter Plots, Correlation, and Regression Lines +Introduction +Before the 2021 WNBA season, professional basketball player Candace Parker signed a contract with the Chicago Sky, +which entitled her to a salary of $190,000. This amount was the 23rd highest in the league at the time. How did the +team’s management decide on her salary? They likely considered some intangible qualities, like her leadership skills. +However, much of their deliberations probably took into account her performance on the court. For example, Parker led +the league in rebounds in the 2020 season (214 of them) and scored 14.7 points per game (which ranked her 18th +among all WNBA players). Further, Parker brought 13 seasons of experience to the team. All of these factors played a +role in deciding the terms of her contract. +Estimating the value of one variable (like salary) based on other, measurable variables (points per game, experience, +rebounds, etc.) is among the most important applications of statistics, which is the mathematical field devoted to +gathering, organizing, summarizing, and making decisions based on data. +8 +STATISTICS +8 • Introduction +815 + +8.1 Gathering and Organizing Data +Figure 8.2 Surveys are commonly used to gather data. (credit: “survey” by Donnell King/Flickr, CC0 1.0 Public Domain) +Learning Objectives +After completing this section, you should be able to: +1. +Distinguish among sampling techniques. +2. +Organize data using an appropriate method. +3. +Create frequency distributions. +When a polling organization wants to try to establish which candidate will win an upcoming election, the first steps are +to write questions for the survey and to choose which people will be asked to respond to the survey. These can seem like +simple steps, but they have far-reaching implications in the analysis the pollsters will later carry out. The process by +which samples (or groups of units from which we collect data) are chosen can strongly affect the data that are collected. +Units are anything that can be measured or surveyed (such as people, animals, objectives, or experiments) and data are +observations made on units. +One of the most famous failures of good sampling occurred in the first half of the twentieth century. The Literary Digest +was among the most respected magazines of the early twentieth century. Despite the name, the Digest was a weekly +newsmagazine. Starting in 1916, the Digest conducted a poll to try to predict the winner of each US Presidential election. +For the most part, their results were good; they correctly predicted the outcome of all five elections between 1916 and +1932. In 1936, the incumbent President Franklin Delano Roosevelt faced Kansas governor Alf Landon, and once again the +Digest ran their famous poll, with results published the week before the election. Their conclusion? Landon would win in +a landslide, 57% to 43%. Once the actual votes had been counted, though, Roosevelt ended up with 61% of the popular +vote, 18% more than the poll predicted. What went wrong? +The short answer is that the people who were chosen to receive the survey (over ten million of them!) were not a good +representation of the population of voting adults. The sample was chosen using the Digest's own base of subscribers as +well as publicly available lists of people that were likely adults (and therefore eligible to vote), mostly phone books and +vehicle registration records. The pollsters then mailed every single person on these lists a survey. Around a quarter of +those surveys were returned; this constituted the sample that was used to make the Digest’s disastrously incorrect +prediction. However, the Digest made an error in failing to consider that the election was happening during the Great +Depression, and only the wealthy had disposable income to spend on telephone lines, automobiles, and magazine +subscriptions. Thus, only the wealthy were sent the Digest’s survey. Since Roosevelt was extremely popular among +poorer voters, many of Roosevelt’s supporters were excluded from the Digest’s sample. +Another more complicated factor was the low response rate; only around 25% of the surveys were returned. This created +what’s called a non-response bias. +Sampling and Gathering Data +The Digest's failure highlights the need for what is now considered the most important criterion for sampling: +randomness. This randomness can be achieved in several ways. Here we cover some of the most common. +A simple random sample is chosen in a way that every unit in the population has an equal chance of being selected, +and the chances of a unit being selected do not depend on the units already chosen. An example of this is choosing a +group of people by drawing names out of a hat (assuming the names are well-mixed in the hat). +816 +8 • Statistics +Access for free at openstax.org + +A systematic random sample is selected from an ordered list of the population (for example, names sorted +alphabetically or students listed by student ID). First, we decide what proportion of the population will be in our sample. +We want to express that proportion as a fraction with 1 in the numerator. Let’s call that number D. Next, we’ll choose a +random number between one and D. The unit at that position will go into our sample. We’ll find the rest of our sample by +choosing every Dth unit in the list, starting with our random number. +To walk through an example, let’s say we want to sample 2% of the population: +. (Note: If the number in +the denominator isn’t a whole number, we can just round it off. This part of the process doesn’t have to be precise.) We +can then use a random number generator to find a random number between 1 and 50; let's use 31. In our example, our +sample would then be the units in the list at positions 31, 81 (31 + 50), 131 (81 + 50), and so forth. +A stratified sample is one chosen so that particular groups in the population are certain to be represented. Let’s say you +are studying the population of students in a large high school (where the grades run from 9th to 12th), and you want to +choose a sample of 12 students. If you use a simple or systematic random sample, there’s a pretty good chance that +you’ll miss one grade completely. In a stratified sample, you would first divide the population into groups (the strata), +then take a random sample within each stratum (that’s the singular form of “strata”). In the high school example, we +could divide the population into grades, then take a random sample of three students within each grade. That would get +us to the 12 students we need while ensuring coverage of each grade. +A cluster sample is a sample where clusters of units are chosen at random, instead of choosing individual units. For +example, if we need a sample of college students, we may take a list of all the course sections being offered at the +college, choose three of them at random (the sections are the clusters), and then survey all the students in those +sections. A sample like this one has the advantage of convenience: If the survey needs to be administered in person, +many of your sample units will be located in one place at the same time. +EXAMPLE 8.1 +Random Sampling +For each of the following situations, identify whether the sample is a simple random sample, a systematic random +sample, a stratified random sample, a cluster random sample, or none of these. +1. +A postal inspector wants to check on the performance of a new mail carrier, so she chooses four streets at random +among those that the carrier serves. Each household on the selected streets receives a survey. +2. +A hospital wants to survey past patients to see if they were satisfied with the care they received. The administrator +sorts the patients into groups based on the department of the hospital where they were treated (ICU, pediatrics, or +general), and selects patients at random from each of those groups. +3. +A quality control engineer at a factory that makes smartphones wants to figure out the proportion of devices that +are faulty before they are shipped out. The phones are currently packed in boxes for shipping, each of which holds +20 devices. The engineer wants to sample 100 phones, so he selects five crates at random and tests every phone in +those five crates. +4. +A newspaper reporter wants to write a story on public perceptions on a project that will widen a congested street. +She stands on the side of the street in question and interviews the first five people she sees there. +5. +An executive at a streaming video service wants to know if her subscribers would support a second season of a new +show. She gets a list of all the subscribers who have watched at least one episode of the show, and uses a random +number generator to select a sample of 50 people from the list. +6. +An agent for a state’s Department of Revenue is in charge of selecting 100 tax returns for audit. He has a list of all of +the returns eligible for audit (about 12,000 in all), sorted by the taxpayer’s ID number. He asks a computer to give +him a random number between 1 and 120; it gives him 15. The agent chooses the 15th, 135th, 255th, 375th, and +every 120th return after that to be audited. +Solution +To decide which type of random sample is being used in each of these, we need to focus on how the randomization is +being incorporated. +1. +The surveys are being given to households, so households are the units in this case. But households aren’t being +chosen randomly; instead, streets are being chosen at random. These form clusters of units, so this is a cluster +random sample. +2. +In this case, the administrator isn’t selecting patients at random from the entire list of patients. Instead, she is +choosing at random from the patients who were in each of the departments (ICU, pediatrics, general) separately. +The departments form strata, so this is a stratified random sample. +8.1 • Gathering and Organizing Data +817 + +3. +The engineer is testing whether the phones are faulty, so those are the units. But the random process is being used +to select the crates of phones. Those crates form clusters, so this is a cluster random sample. +4. +The reporter isn’t using a random process at all, so this sample doesn’t belong to any of the types we have been +talking about. A sample like this one is sometimes described as a convenience sample, and shouldn’t be used in a +statistical setting. +5. +The executive is choosing her sample completely at random from the full population, so this is a simple random +sample. +6. +The agent is choosing from the full population, but is only choosing the first unit for the sample at random; the rest +are chosen by skipping down the list systematically. Thus, this is a systematic random sample. +YOUR TURN 8.1 +For each of the following situations, identify whether the sample is a simple random sample, a systematic random +sample, a stratified random sample, a cluster random sample, or none of these. +1. The chairperson of the University Chess Club is trying to decide on a time for the club’s regular meetings, so +she emails all of the members of the club to find their preferences. +2. The registrar at a small college wants to use a survey to determine if their office could do a better job of +serving students. They choose three students at random from each major to take the survey. +3. A civic club is organizing a raffle as a fundraiser. To determine the three winners, each of the tickets is put into +a large drum, then the tickets are thoroughly mixed. A blindfolded club member pulls three tickets out of the +drum. +PEOPLE IN MATHEMATICS +George Gallup +Figure 8.3 George Gallup was a founder of survey sampling techniques, and his legacy lives on to this day. (credit: +"George Gallup at the National Press Club, Washington, D.C., 1969" by Bernard Gotfryd/Library of Congress Prints & +Photographs Division, public domain) +George Gallup (1901–1984) rose to fame in 1936 when his prediction of the percentage of the vote going to each +candidate in that year’s U.S. Presidential election was more accurate than the one published in Literary Digest, and he +did so using a sample that was much smaller than the Digest. He even took it one step farther, predicting with high +accuracy the erroneous results of the poll that the Literary Digest would end up publishing! Gallup’s theories on public +opinion polling essentially created that field. In 1948, Gallup’s reputation took a bit of a hit, when he famously, but +incorrectly, predicted that Thomas Dewey would beat incumbent Harry Truman in that year’s Presidential election. +Over the following decades, however, public trust in Gallup’s polls recovered and even steadily increased. The +company Gallup founded (https://openstax.org/r/Gallup) continues to conduct daily public opinion polls, as well as +provides consulting services for businesses. +818 +8 • Statistics +Access for free at openstax.org + +Organizing Data +Once data have been collected, we turn our attention to analysis. Before we analyze, though, it’s useful to reorganize the +data into a format that makes the analysis easier. For example, if our data were collected using a paper survey, our raw +data are all broken down by respondent (represented by an individual response sheet). To perform an analysis on all the +responses to an individual question, we need to first group all the responses to each question together. The way we +organize the data depends on the type of data we’ve collected. +There are two broad types of data: categorical and quantitative. Categorical data classifies the unit into a group (or +category). Examples of categorical data include a response to a yes-or-no question, or the color of a person’s eyes. +Quantitative data is a numerical measure of a property of a unit. Examples of quantitative data include the time it takes +for a rat to run through a maze or a person’s daily calorie intake. We’ll look at each type of data in turn when considering +how best to organize. +Categorical Data Organization +The best way to organize categorical data is using a categorical frequency distribution. A categorical frequency +distribution is a table with two columns. The first contains all the categories present in the data, each listed once. The +second contains the frequencies of each category, which are just a count of how often each category appears in the data. +EXAMPLE 8.2 +Creating a Categorical Frequency Distribution +A teacher records the responses of the class (28 students) on the first question of a multiple choice quiz, with five +possible responses (A, B, C, D, and E): +A +A +C +A +B +B +A +E +A +C +A +A +A +C +E +A +B +A +A +C +A +B +E +E +A +A +C +C +Create a categorical frequency distribution that organizes the responses. +Solution +Step 1: For each possible response, count the number of times that response appears in the data. In the responses for +this class, “A” appears 14 times, “B” 4 times, “C” 6 times, “D” 0 times, and “E” 4 times. +Step 2: Make a table with two columns. The first column should be labeled so that the reader knows what the responses +mean, and the second should be labeled “Frequency.” +Response to First Question +Frequency +A +14 +B +4 +C +6 +D +0 +E +4 +Step 3: Check your work. If you add up your frequencies, you should get the same number as the total number of +responses. Twenty-eight students answered that first question, and +. +8.1 • Gathering and Organizing Data +819 + +YOUR TURN 8.2 +1. Students in a statistics class who were asked to provide their majors provided the data below: +Undecided +Biology +Biology +Sociology +Political Science +Sociology +Undecided +Undecided +Undecided +Biology +Biology +Education +Biology +Biology +Political Science +Political Science +Create a categorical frequency distribution to organize these responses. +Quantitative Data +We have a couple of options available for organizing quantitative data. If there are just a few possible responses, we can +create a frequency distribution just like the ones we made for categorical data above. For example, if we’re surveying a +group of high school students and we ask for each student’s age, we’ll likely only get whole-number responses between +13 and 19. Since there are only around seven (and likely fewer) possible responses, we can treat the data as if they’re +categorical and create a frequency distribution as before. +EXAMPLE 8.3 +Creating a Quantitative Frequency Distribution +Attendees of a conflict resolution workshop are asked how many siblings they have. The responses are as follows: +1 +0 +1 +1 +2 +0 +3 +1 +1 +4 +1 +2 +0 +1 +3 +1 +2 +1 +2 +4 +1 +0 +1 +3 +0 +1 +2 +2 +1 +5 +Create a frequency distribution to organize the responses. +Solution +Step 1: Count the number of times you see each unique response: “0” appears 5 times, “1” appears 13 times, “2” appears +6 times, “3” appears 3 times, “4” appears twice, and “5” appears once. +Step 2: Make a table with two columns. The first column should be labeled so that the reader knows what the responses +mean, and the second should be labeled “Frequency.” Then fill in the results of our count. +Number of Siblings +Frequency +Number of Siblings +Frequency +0 +5 +3 +3 +1 +13 +4 +2 +2 +6 +5 +1 +Step 3: Check your work. If you add up your counts, you should get the same number as the total number of responses. +Looking back at the raw data, there were 30 responses, and +. +820 +8 • Statistics +Access for free at openstax.org + +YOUR TURN 8.3 +1. A question on a community survey asked each respondent to give the number of people who shared their +residence, and the data from the responses was as follows: +1 +3 +2 +2 +1 +3 +3 +4 +2 +2 +2 +4 +1 +1 +2 +3 +1 +1 +5 +2 +1 +4 +3 +2 +1 +2 +2 +1 +3 +1 +3 +3 +4 +1 +4 +2 +2 +2 +1 +4 +Create a frequency distribution to organize the responses. +If there are many possible responses, a frequency distribution table like the ones we’ve seen so far isn’t really useful; +there will likely be many responses with a frequency of one, which means the table will be no better than looking at the +raw data. In these cases, we can create a binned frequency distribution. A binned frequency distribution groups the +data into ranges of values called bins, then records the number of responses in each bin. +For example, if we have height data for individuals measured in centimeters, we might create bins like 150–155 cm, +155–160 cm, and so forth (making sure that every data value falls into a bin). We must be careful, though; in this +scenario, it’s not clear which bin would contain a response of 155 cm. Usually, responses on the edge of a bin are placed +in the higher bin, but it’s good practice to make that clear. In cases where responses are rounded off, you can avoid this +issue by leaving a gap between the bins that couldn’t contain any responses. In our example, if the measurements were +all rounded off to the nearest centimeter, we could make bins like 150–154 cm, 155–159 cm, etc. (since a response like +154.2 isn’t possible). We’ll use this method going forward. How do we decide what the boundaries of our bins should be? +There’s no one right way to do that, but there are some guidelines that can be helpful. +1. +Every data value should fall into exactly one bin. For example, if the lowest value in our data is 42, the lowest bin +should not be 45–49. +2. +Every bin should have the same width. Note that if we shift the upper limits of our bins down a bit to avoid +ambiguity (like described above), we can’t simply subtract the lower limit from the upper limit to get the bin width; +instead, we subtract the lower limit of the bin from the lower limit of the next bin. For example, if we’re looking at +GPAs rounded to the nearest hundredth, we might choose bins like 2.00–2.24, 2.25–2.49, 2.50–2.74, etc. These bins +all have a width of 0.25. +3. +If the minimum or maximum value of the data falls right on the boundary between two bins, then it’s OK to bend +the rule just a little in order to avoid having an additional bin containing just that one value. We’ll see an example of +this in just a moment. +4. +If we have too many or too few bins, it can be difficult to get a good sense of the distribution. Seven or eight bins is +ideal, but that’s not a firm rule; anything between five and twelve is fine. We often choose the number of bins so +that the widths are round numbers. +EXAMPLE 8.4 +Creating a Binned Frequency Distribution +The GPAs of students enrolled in an advanced sociology class are listed in the following table. At this institution, 4.00 is +the maximum possible GPA. +3.93 +3.43 +2.87 +2.51 +2.70 +1.91 +2.32 +2.85 +3.06 +3.03 +3.49 +1.84 +3.72 +2.56 +1.99 +3.40 +3.74 +3.23 +1.98 +3.05 +1.43 +2.90 +1.20 +3.72 +3.56 +3.07 +2.58 +4.00 +2.79 +3.81 +2.60 +3.69 +2.88 +3.34 +1.51 +3.63 +3.45 +1.89 +2.30 +2.98 +3.04 +2.70 +Create a binned frequency distribution for the data. +Solution +Step 1: Identify the max and min values in your bins. Looking at the dataset, you can see that the lowest value is 1.20, +8.1 • Gathering and Organizing Data +821 + +and the highest is 4.00. +Step 2: Get a rough idea of bin widths. Aim for seven or eight bins, give or take a couple. For eight bins, the minimum +width can be found by taking the difference between the largest and smallest data values and dividing by the number of +bins: +If we use 0.35 for our widths, starting at our minimum value of 1.20, we’ll get bins with these boundaries: 1.20, 1.55, +1.90, 2.25, 2.60, 2.95, 3.30, 3.65, 4.00. +Step 3: Consider the context of the values. Because these are GPAs, there are natural breaks at 2.00 and 3.00 that are +important. (People like whole numbers!) Since 0.35 is very close to +, let’s use that for our bin width instead, and make +sure that whole numbers fall on the boundaries. That means our first bin needs to start at 1.00 and go up to 1.33 to +make sure our minimum value is included. The next bin will run from 1.34 to 1.66, and so forth. +Step 4: Create the distribution table. We start our distribution table by filling in the bins: +GPA Range +Frequency +GPA Range +Frequency +GPA Range +Frequency +1.00–1.33 +2.00–2.33 +3.00–3.33 +1.34–1.66 +2.34–2.66 +3.34–3.66 +1.67–1.99 +2.67–2.99 +3.67–4.00 +Notice that the last bin doesn’t follow the pattern; since our maximum data value is right on the upper boundary of that +last bin, this is a case where we can bend that rule just a little to avoid creating a bin for 4.00–4.33 (which wouldn’t really +make sense in the context of these GPAs anyway, since 4.00 is the maximum possible GPA). +Step 5: Complete the table with the frequencies. Finish the table by counting the number of data values that fall in +each bin, and recording them in the frequency column: +GPA Range +Frequency +GPA Range +Frequency +GPA Range +Frequency +1.00–1.33 +1 +2.00–2.33 +2 +3.00–3.33 +6 +1.34–1.66 +2 +2.34–2.66 +4 +3.34–3.66 +7 +1.67–1.99 +5 +2.67–2.99 +8 +3.67–4.00 +7 +Step 6: Check your work. Add up the frequencies to make sure all the data values are included. We started with forty- +two data values, and +. +YOUR TURN 8.4 +1. The following table displays the ages of a sample of customers who have shopped at a new boutique. +56 +39 +35 +32 +26 +53 +55 +47 +70 +43 +33 +33 +43 +41 +26 +40 +31 +34 +33 +53 +Create a binned frequency distribution to summarize these data. +822 +8 • Statistics +Access for free at openstax.org + +Check Your Understanding +For the following problems, decide whether randomization is being used in the selection of these samples. If it is, +identify the type of random sample (simple, systematic, cluster, or stratified). +1. High school guidance counselors want to know the proportion of the school’s seniors who intend to apply for +college. They choose four senior homerooms at random, then visit each one and ask every student in those +homerooms whether they intend to apply. +2. A quality control technician wants to ensure that the sandals being made in his factory are up to specifications, +so they check the first five pairs they see coming off the line. +3. A college athletic department wants to check up on the mental wellness of its student-athletes. The department +wants to ensure every varsity sport is represented, so they survey three randomly selected members of each +team. +4. The purchasing manager for a chain of bookstores wants to make sure they’re buying the right types of books to +put on the shelves, so they take a sample of 20 books that customers bought in the last five days and record the +genres. Use the raw data below to create a categorical frequency distribution. +Nonfiction +Young Adult +Romance +Cooking +Young Adult +Young Adult +Thriller +Young Adult +Nonfiction +True Crime +Romance +Nonfiction +Thriller +True Crime +Romance +True Crime +Thriller +Romance +Young Adult +Young Adult +5. A survey of college students asked how many courses those students were currently taking. Create a quantitative +frequency distribution to summarize the raw data given below: +3 +4 +4 +3 +5 +4 +4 +3 +2 +3 +5 +5 +3 +3 +4 +3 +2 +4 +3 +3 +4 +3 +5 +3 +3 +3 +2 +3 +1 +3 +4 +3 +6. The World Bank provides data on every country in the world. The following is a sample of twenty-five countries, +along with the number of cell phone subscriptions registered in that country per hundred residents. Create a +binned frequency distribution for the cell phone data. +Country +Cell +Country +Cell +Cameroon +83.7 +Benin +78.5 +Vanuatu +82.5 +Eritrea +13.7 +Georgia +140.7 +Mauritania +92.2 +Kazakhstan +146.6 +Czech Republic +119 +Bermuda +105.9 +Qatar +151.1 +Russia +157.9 +Pakistan +73.4 +Hungary +113.5 +Egypt +105.5 +(source (https://data.worldbank.org/)) +8.1 • Gathering and Organizing Data +823 + +Country +Cell +Country +Cell +Costa Rica +180.2 +Nepal +123.2 +Algeria +111 +Turkey +96.4 +Somalia +48.3 +Congo +43.5 +Fiji +114.2 +Venezuela +78.5 +El Salvador +156.5 +Germany +133.6 +Angola +44.7 +(source (https://data.worldbank.org/)) +SECTION 8.1 EXERCISES +For the following exercises, data are collected on a sample of items found in a grocery store. Classify each of these +datasets obtained from that sample as being categorical or quantitative. +1. Price +2. Calories per serving +3. Whether the product is gluten-free +4. Package weight +5. Country of origin +For the following exercises, decide whether random samples are being selected. If they are, decide whether they are +simple, systematic, cluster, or stratified. +6. A newspaper asks its readers to answer an online poll about proposed zoning changes in their city. +7. An electronics retailer uses a computer to randomly select customers in its rewards club to take a survey about +their interest in a new product. +8. The student affairs office at a university wants to make sure students who live on campus are satisfied with +their access to laundry facilities. They select five students at random from each residence hall to take the survey. +9. A professor wants to gauge how much time her students spend on homework, so she asks that question of +each student who comes to her office hours that day. +10. The management at a restaurant wants feedback about its new menu. They choose ten tables at random, and +survey each person seated at that table. +11. The transit authority in a large city wants to know about usage on a particular train route. They choose a +number between 1 and 5 at random, and get 4. They then count the number of people on the fourth train to +pass through the station, and then count every fifth train after that. +12. A candidate for a seat in the U.S. Congress wants to learn which issues are most important to her potential +constituents. She chooses 50 people at random from each zip code in her district to survey. +For the following exercises, you have been tasked with surveying a sample of 100 registered voters who live in your +town. You have access to a spreadsheet containing the following data on every registered voter: name, address, age, +phone number. The spreadsheet also can generate a unique random number for each person. +13. Describe how you might choose a simple random sample from this population. +14. Describe how you might choose a stratified random sample from this population to ensure that all age groups +are represented. +15. Assume that there are 50,000 registered votes on your list. Describe how you might choose a systematic +random sample from this population. +16. A sample of students was asked, “Which social media platform, if any, do you use most frequently?” The raw +responses are given in this table: +824 +8 • Statistics +Access for free at openstax.org + +None +Twitter +Snapchat +Snapchat +Twitter +Facebook +Instagram +Snapchat +Twitter +None +Snapchat +Instagram +Instagram +Facebook +None +Instagram +Snapchat +Twitter +Snapchat +Instagram +Instagram +Twitter +Snapchat +Twitter +Facebook +None +Instagram +Instagram +Twitter +Instagram +Create a categorical frequency distribution to summarize these data. +17. A sample of students at a large university were asked whether they were full-time students living on campus (Full- +Time Residential, FTR), full-time students who commuted (FTC), or part-time students (PT). The raw data are in the +table below: +FTR +FTR +FTC +PT +FTR +PT +FTR +FTC +FTC +PT +FTC +FTC +PT +FTR +FTC +PT +FTR +FTC +FTC +FTR +FTR +PT +FTC +FTC +FTC +PT +FTR +PT +FTC +FTC +FTR +PT +Give the categorical frequency distribution for these data. +18. A survey of students in a math class asked for the respondents’ birth months. The table below lists the responses: +Dec +Feb +Apr +Sep +Nov +Dec +Aug +Feb +Feb +Sep +Oct +Feb +Jun +Jan +Jul +May +May +Jan +Mar +Feb +Nov +Oct +Apr +Oct +Aug +Jan +May +Jan +Give the categorical frequency distribution of the birth months. +19. Students in a statistics class were asked how many countries (besides their home countries) they had visited. The +table below gives the raw responses: +0 +2 +1 +1 +3 +2 +0 +0 +0 +2 +1 +1 +0 +1 +1 +0 +2 +0 +1 +0 +1 +0 +2 +0 +1 +1 +0 +0 +1 +0 +Create a frequency distribution to summarize the data. +20. The following table contains the top 25 receivers (by number of receptions) in the NFL during the 2020 season, +along with their teams and the number of fumbles each made over the course of the season: +Player +Team +Fumbles +Player +Team +Fumbles +Stefon Diggs +BUF +0 +Calvin Ridley +ATL +1 +Davante Adams +GNB +1 +Robert Woods +LAR +2 +DeAndre Hopkins +ARI +3 +Justin Jefferson +MIN +1 +Darren Waller +LVR +2 +Diontae Johnson +PIT +2 +(source (http://www.pro-football-reference.com)) +8.1 • Gathering and Organizing Data +825 + +Player +Team +Fumbles +Player +Team +Fumbles +Travis Kelce +KAN +1 +Tyreek Hill +KAN +1 +Allen Robinson +CHI +0 +Terry McLaurin +WAS +1 +Keenan Allen +LAC +3 +Alvin Kamara +NOR +1 +Tyler Lockett +SEA +1 +D.K. Metcalf +SEA +1 +JuJu Smith-Schuster +PIT +3 +Cole Beasley +BUF +0 +Robby Anderson +CAR +1 +Brandin Cooks +HOU +0 +Amari Cooper +DAL +0 +J.D. McKissic +WAS +3 +Cooper Kupp +LAR +1 +Tyler Boyd +CIN +1 +Curtis Samuel +CAR +1 +(source (http://www.pro-football-reference.com)) +Create a frequency distribution for the number of fumbles made by these players. +21. A public opinion poll about an upcoming election asked respondents, “How many political advertisements do you +recall seeing on television in the last 24 hours?” The responses were as follows +6 +2 +5 +5 +2 +2 +4 +1 +3 +0 +1 +2 +1 +6 +2 +5 +2 +4 +8 +6 +3 +3 +4 +2 +5 +3 +4 +2 +2 +3 +Create a frequency distribution for these data. +For the following exercises, use the following table of data on the top 15 receivers (by number of receptions) in the NFL +during the 2020 season: +Player +Team +Age +Receptions +Yards +Yds/Rec +TD +Long +Stefon Diggs +BUF +27 +127 +1535 +12.1 +8 +55 +Davante Adams +GNB +28 +115 +1374 +11.9 +18 +56 +DeAndre Hopkins +ARI +28 +115 +1407 +12.2 +6 +60 +Darren Waller +LVR +28 +107 +1196 +11.2 +9 +38 +Travis Kelce +KAN +31 +105 +1416 +13.5 +11 +45 +Allen Robinson +CHI +27 +102 +1250 +12.3 +6 +42 +Keenan Allen +LAC +28 +100 +992 +9.9 +8 +28 +Tyler Lockett +SEA +28 +100 +1054 +10.5 +10 +47 +(source (http://www.pro-football-reference.com)) +826 +8 • Statistics +Access for free at openstax.org + +Player +Team +Age +Receptions +Yards +Yds/Rec +TD +Long +JuJu Smith-Schuster +PIT +24 +97 +831 +8.6 +9 +31 +Robby Anderson +CAR +27 +95 +1096 +11.5 +3 +75 +Amari Cooper +DAL +26 +92 +1114 +12.1 +5 +69 +Cooper Kupp +LAR +27 +92 +974 +10.6 +3 +55 +Calvin Ridley +ATL +26 +90 +1374 +15.3 +9 +63 +Robert Woods +LAR +28 +90 +936 +10.4 +6 +56 +Justin Jefferson +MIN +21 +88 +1400 +15.9 +7 +71 +(source (http://www.pro-football-reference.com)) +22. Make a binned frequency distribution for receiving yards (“Yards”) using bins of width 200. +23. Make another binned frequency distribution for receiving yards (“Yards”), but this time use bins of width 250. +24. Make a binned frequency distribution for number of yards per reception (“Yds/Rec”), using bins of width 1. +25. Make a binned frequency distribution for longest reception (“Long”), using bins of width 10. +8.2 Visualizing Data +Figure 8.4 Data visualizations can help people quickly understand important features of a dataset. (credit: "Group of +diverse people having a business meeting" by Rawpixel Ltd/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Create charts and graphs to appropriately represent data. +2. +Interpret visual representations of data. +3. +Determine misleading components in data displayed visually. +Summarizing raw data is the first step we must take when we want to communicate the results of a study or experiment +to a broad audience. However, even organized data can be difficult to read; for example, if a frequency table is large, it +can be tough to compare the first row to the last row. As the old saying goes: a picture is worth a thousand words (or, in +this case, summary statistics)! Just as our techniques for organizing data depended on the type of data we were looking +8.2 • Visualizing Data +827 + +at, the methods we’ll use for creating visualizations will vary. Let’s start by considering categorical data. +Visualizing Categorical Data +If the data we’re visualizing is categorical, then we want a quick way to represent graphically the relative numbers of +units that fall in each category. When we created the frequency distributions in the last section, all we did was count the +number of units in each category and record that number (this was the frequency of that category). Frequencies are nice +when we’re organizing and summarizing data; they’re easy to compute, and they’re always whole numbers. But they can +be difficult to understand for an outsider who’s being introduced to your data. +Let’s consider a quick example. Suppose you surveyed some people and asked for their favorite color. You communicated +your results using a frequency distribution. Jerry is interested in data on favorite colors, so he reads your frequency +distribution. The first row shows that twelve people indicated green was their favorite color. However, Jerry has no way of +knowing if that’s a lot of people without knowing how many people total took your survey. Twelve is a pretty significant +number if only twenty-five people took the survey, but it’s next to nothing if you recorded a thousand responses. For that +reason, we will often summarize categorical data not with frequencies, but with proportions. The proportion of data +that fall into a particular category is computed by dividing the frequency for that category by the total number of units in +the data. +Proportions can be expressed as fractions, decimals, or percentages. +EXAMPLE 8.5 +Finding Proportions +Recall Example 8.2, in which a teacher recorded the responses on the first question of a multiple choice quiz, with five +possible responses (A, B, C, D, and E). The raw data was as follows: +A +A +C +A +B +B +A +E +A +C +A +A +A +C +E +A +B +A +A +C +A +B +E +E +A +A +C +C +We computed a frequency distribution that looked like this: +Response to First Question +Frequency +A +14 +B +4 +C +6 +D +0 +E +4 +Now, let's compute the proportions for each category. +Solution +Step 1: In order to compute a proportion, we need the frequency (which we have in the table above) and the total +number of units that are represented in our data. We can find that by adding up the frequencies from all the categories: +. +Step 2: To find the proportions, we divide the frequency by the total. For the first category (“A”), the proportion is +828 +8 • Statistics +Access for free at openstax.org + +We can compute the other proportions similarly, filling in the rest of the table: +Response to First Question +Frequency +Proportion +A +14 +B +4 +C +6 +D +0 +E +4 +Step 3: Check your work: If you add up your proportions, you should get 1 (if you’re using fractions or decimals) or 100% +(if you’re using percentages). In this case, +If you need to round off the results of the computations to get your percentages or decimals, then the sum might +not be exactly equal to 1 or 100% in the end due to that rounding error. +YOUR TURN 8.5 +1. In Your Turn 8.2, students in a statistics class were asked to provide their majors. Those results are again listed +below: +Undecided +Biology +Biology +Sociology +Political Science +Sociology +Undecided +Undecided +Undecided +Biology +Biology +Education +Biology +Biology +Political Science +Political Science +You created a frequency distribution: +Major +Frequency +Biology +6 +Education +1 +Political Science +3 +8.2 • Visualizing Data +829 + +Major +Frequency +Sociology +2 +Undecided +4 +Now, find the proportions associated with each category. Express your answers as percentages. +Now that we can compute proportions, let’s turn to visualizations. There are two primary visualizations that we’ll use for +categorical data: bar charts and pie charts. Both of these data representations work on the same principle: If proportions +are represented as areas, then it’s easy to compare two proportions by assessing the corresponding areas. Let’s look at +bar charts first. +Bar Charts +A bar chart is a visualization of categorical data that consists of a series of rectangles arranged side-by-side (but not +touching). Each rectangle corresponds to one of the categories. All of the rectangles have the same width. The height of +each rectangle corresponds to either the number of units in the corresponding category or the proportion of the total +units that fall into the category. +EXAMPLE 8.6 +Building a Bar Chart +In Example 8.5, we computed the following proportions: +Response to First Question +Frequency +Proportion +A +14 +50% +B +4 +14.3% +C +6 +21.4% +D +0 +0% +E +4 +14.3% +Draw a bar chart to visualize this frequency distribution. +Solution +Step 1: To start, we’ll draw axes with the origin (the point where the axes meet) at the bottom left: +Figure 8.5 +Step 2: Next, we’ll place our categories evenly spaced along the bottom of the horizontal axis. The order doesn’t really +matter, but if the categories have some sort of natural order (like in this case, where the responses are labeled A to E), it’s +best to maintain that order. We'll also label the horizontal axis: +830 +8 • Statistics +Access for free at openstax.org + +Figure 8.6 +Step 3: Now, we have a decision to make: Will we use frequencies to define the height of our rectangles, or will we use +proportions? Let’s try it both ways. First, let’s use frequencies. Notice that our frequencies run from zero to 14; this will +correspond to the scale we put on the vertical axis. If we put a tick mark for every whole number between 0 and 14, the +result will be pretty crowded; let’s instead put a mark on the multiples of 3 or 5: +Figure 8.7 +Step 4: Now, let’s draw in the first rectangle. The frequency associated with “A” is 14. So we’ll go to 14 on the vertical axis, +and place a mark at that height above the “A” label: +Figure 8.8 +Step 5: Then, draw vertical lines straight down from the edges of your mark to make a rectangle: +Figure 8.9 +8.2 • Visualizing Data +831 + +Step 6: Finally, we can build the rest of the rectangles, making sure that the bases all have the same length of the base = +width of the rectangle, and the rectangles don’t touch. Notice that, since the frequency for “D” is zero, that category has +no rectangle (but we’ll leave a space there so the reader can see that there is a category with frequency zero). Here’s the +result: +Figure 8.10 +Step 7: That’s it! Now, let’s use proportions instead of frequencies. We'll label the vertical axis with evenly spaced +numbers that run the full range of the percentages in our table: 0% to 50%. We can divide that into five equal parts (so +that each has width 10%), and use that to label our vertical axis: +Figure 8.11 +Step 8: Then, we can fill in the rectangles just as we did before. The height of the “A” rectangle is 50%, the “B” rectangle +goes up to 14.3%, “C” goes to 21.4%, there is no rectangle for “D” (since its proportion is 0%), and the “E” rectangle also +goes up to 14.3%: +Figure 8.12 +Step 9: Notice that the rectangles are basically identical in our two final bar charts. That’s no coincidence! Bar charts that +use proportions and those that use frequencies will always look identical (which is why it doesn’t really matter much +which option you choose). Here’s why: look at the bars for “B” and “C”. The frequencies for these are 4 and 6 respectively. +Notice that 6 is 50% bigger than 4 (since +), which means that the “C” bar will be 50% higher than the “B” bar. +Now look at the same bars using proportions: since +, the bar for “C” will be 50% higher than the bar +for “B.” The same relationships hold for the other bars, too. +832 +8 • Statistics +Access for free at openstax.org + +Figure 8.13 +YOUR TURN 8.6 +1. The students in a statistics class were asked to provide their majors. The computed proportions for each of the +categories are as follows: +Major +Frequency +Proportion +Biology +6 +37.5% +Education +1 +6.3% +Political Science +3 +18.8% +Sociology +2 +12.5% +Undecided +4 +25% +Create a bar graph to visualize these data. Use percentages to label the vertical axis. +In practice, most graphs are now made with computers. You can use Google Sheets (https://openstax.org/r/ +spreadsheet), which is available for free from any web browser. +VIDEO +Make a Simple Bar Graph in Google Sheets (https://openstax.org/r/Google_Sheet) +Now that we’ve explored how bar graphs are made, let’s get some practice reading bar graphs. +EXAMPLE 8.7 +Reading Bar Graphs +The bar graph shown gives data on 2020 model year cars available in the United States. Analyze the graph to answer the +following questions. +8.2 • Visualizing Data +833 + +Figure 8.14 (data source: consumerreports.org/cars) +1. +What proportion of available cars were sports cars? +2. +What proportion of available cars were sedans? +3. +Which categories of cars each made up less than 5% of the models available? +Solution +1. +The bar for sports cars goes up to 10%, so the proportion of models that are considered sports cars is 10%. +2. +The bar corresponding to sedan goes up past 30% but not quite to 35%. It looks like the proportion we want is +between 33% and 34%. +3. +We’re looking for the bars that don’t make it all the way to the 5% line. Those categories are hatchback and wagon. +YOUR TURN 8.7 +The bar graph shows the region of every institution of higher learning in the United States (except for the service +academies, like West Point). +Analyze the bar chart to answer the following questions. +1. Which region contains the largest number of institutions of higher learning? +2. What proportion of all institutions of higher learning can be found in the Southwest? +3. Which regions each have under 5% of the total number of institutions of higher learning? +834 +8 • Statistics +Access for free at openstax.org + +WORK IT OUT +Candy Color: Frequency and Distribution +M&Ms, Skittles, and Reese’s Pieces are all candies that have pieces that are uniformly shaped, but which have +different colors. Do the colors in each bag appear with the same frequency? Get a bag of one of these candies and +make a bar chart to visualize the color distribution. +Pie Charts +A pie chart consists of a circle divided into wedges, with each wedge corresponding to a category. The proportion of the +area of the entire circle that each wedge represents corresponds to the proportion of the data in that category. Pie +charts are difficult to make without technology because they require careful measurements of angles and precise circles, +both of which are tasks better left to computers. +VIDEO +Create Pie Charts Using Google Sheets (https://openstax.org/r/Creating_Pie_Chart) +Pie charts are sometimes embellished with features like labels in the slices (which might be the categories, the +frequencies in each category, or the proportions in each category) or a legend that explains which colors correspond to +which categories. When making your own pie chart, you can decide which of those to include. The only rule is that there +has to be some way to connect the slices to the categories (either through labels or a legend). +EXAMPLE 8.8 +Making Pie Charts +Use the data that follows to generate a pie chart. +Type +Percent +Type +Percent +SUV +43.6% +Minivan +5.5% +Sedan +33.6% +Hatchback +3.6% +Sports +10.0% +Wagon +3.6% +Table 8.1 (data source: +www.consumerreports.org/cars) +Solution +First, enter the chart above into a new sheet in Google Sheets. Next, click and drag to select the full table (including the +header row). Click on the “Insert” menu, then select “Chart.” The result may be a pie chart by default; if it isn’t, you can +change it to a pie chart using the “Chart type” drop-down menu in the Chart Editor. +8.2 • Visualizing Data +835 + +Figure 8.15 (data source: consumerreports.org/cars) +You can choose to use a legend to identify the categories, as well as label the slices with the relevant percentages. +YOUR TURN 8.8 +1. In Your Turn 8.6, you created a bar chart using data on reported majors from students in a class. Here are those +proportions again (sorted from largest to smallest): +Major +Proportion +Biology +37.5% +Undecided +25.0% +Political Science +18.8% +Sociology +12.5% +Education +6.3% +Create a pie graph using those data. +PEOPLE IN MATHEMATICS +Florence Nightingale +Florence Nightingale (1820–1910) is best remembered today for her contributions in the medical field; after +witnessing the horrors of field hospitals that tended to the wounded during the Crimean War, she championed +reforms that encouraged sanitary conditions in hospitals. For those efforts, she is today considered the founder of +modern nursing. +836 +8 • Statistics +Access for free at openstax.org + +Figure 8.16 Florence Nightingale's significant contribution to the field of statistical graphics cannot be understated. +(credit: "Florence Nightingale" by Library of Congress Prints and Photographs Division/http://hdl.loc.gov/loc.pnp/ +pp.print, public domain) +Nightingale is also remembered for her contributions in statistics, especially in the ways we visualize data. She +developed a version of the pie chart that is today known as a polar area diagram, which she used to visualize the +causes of death among the soldiers in the war, highlighting the number of preventable deaths the British Army +suffered in that conflict. +In 1859, the Royal Statistical Society honored her for her contributions to the discipline by electing her to join the +organization. She was the first woman to be so honored. She was later named an honorary member of the American +Statistical Association. Nightingale's status as a revered pioneer in both nursing and statistics is a complex one, +because some of her writings and opinions demonstrate a colonialist mindset and disregard for those who lost their +lives and lands at the hands of the British. Her core statistical writings indicated that she felt superior to the +Indigenous people she was treating. Members of both fields continue to debate her near-iconic role. +Visualizing Quantitative Data +There are several good ways to visualize quantitative data. In this section, we’ll talk about two types: stem-and-leaf plots +and histograms. +Stem-and-Leaf Plots +Stem-and-leaf plots are visualization tools that fall somewhere between a list of all the raw data and a graph. A stem- +and-leaf plot consists of a list of stems on the left and the corresponding leaves on the right, separated by a line. The +stems are the numbers that make up the data only up to the next-to-last digit, and the leaves are the final digits. There is +one leaf for every data value (which means that leaves may be repeated), and the leaves should be evenly spaced across +all stems. These plots are really nothing more than a fancy way of listing out all the raw data; as a result, they shouldn’t +be used to visualize large datasets. +This concept can be difficult to understand without referencing an example, so let’s first look at how to read a stem-and- +leaf plot. +EXAMPLE 8.9 +Reading a Stem-and-Leaf Plot +A collector of trading cards records the sale prices (in dollars) of a particular card on an online auction site, and puts the +results in a stem-and-leaf plot: +0 +5 8 9 +1 +0 0 0 3 4 4 5 5 5 5 6 9 9 +Table 8.2 +8.2 • Visualizing Data +837 + +2 +0 0 0 0 5 5 9 9 +3 +0 0 0 5 5 +4 +0 0 5 +5 +6 +0 +Table 8.2 +Answer the following questions about the data: +1. +How many prices are represented? +2. +What prices represent the five most expensive cards? The five least expensive? +3. +What is the full set of data? +Solution +1. +Each leaf (the numbers on the right side of the bar) represents one data value. So, on the first row (which looks like +0 | 5 8 9), there are three data values (one for each leaf: 5, 8, and 9). The next row has thirteen leaves, then eight, +five, three, zero, and one. Adding those up, we get +data points or prices. +2. +The most expensive card is the last one listed. Its stem is 6 and its leaf is 0, so the price is $60. There are no leaves +associated with the 5 stem, so there were no cards sold for $50 to $59. The next most expensive cards are then on +the 4 stem: $45, $40, and $40 (remember, repeated leaves mean repeated values in the dataset). So, we have our +four most expensive cards. The fifth would be on the next stem up. The biggest leaf on the 3 stem is a 5, so the fifth- +most expensive card sold for $35. +As for the five least-expensive cards, the smallest stem is 0, with leaves 5, 8, and 9. So, the three least expensive +cards sold for $5, $8, and $9 (notice that we don’t write down that leading 0 from the stem in the tens place). The +next two least-expensive cards will be the two smallest leaves on the next stem: $10 and $10. +3. +The full list of data is: 5, 8, 9, 10, 10, 10, 13, 14, 14, 15, 15, 15, 15, 16, 19, 19, 20, 20, 20, 24, 25, 25, 29, 29, 30, 30, 30, +35, 35, 40, 40, 45, 60. +YOUR TURN 8.9 +The stem-and-leaf plot below shows data collected from a sample of employed people who were asked how far (in +miles) they commute each day: +0 +4 6 7 +1 +0 0 0 2 2 2 4 5 8 8 +2 +0 5 5 5 +3 +0 0 5 5 6 +4 +5 +0 +6 +0 +1. How many data points are represented? +838 +8 • Statistics +Access for free at openstax.org + +2. What are the three longest and shortest commutes? +3. What is the full list of data? +Stem-and-leaf plots are useful in that they give us a sense of the shape of the data. Are the data evenly spread out over +the stems, or are some stems “heavier” with leaves? Are the heavy stems on the low side, the high side, or somewhere in +the middle? These are questions about the distribution of the data, or how the data are spread out over the range of +possible values. +Some words we use to describe distributions are uniform (data are equally distributed across the range), symmetric +(data are bunched up in the middle, then taper off in the same way above and below the middle), left-skewed (data are +bunched up at the high end or larger values, and taper off toward the low end or smaller values), and right-skewed (data +are bunched up at the low end, and taper off toward the high end). See below figures. +Looking back at the stem-and-leaf plot in the previous example, we can see that the data are bunched up at the low end +and taper off toward the high end; that set of data is right-skewed. Knowing the distribution of a set of data gives us +useful information about the property that the data are measuring. +Now that we have a better idea of how to read a stem-and-leaf plot, we’re ready to create our own. +EXAMPLE 8.10 +Constructing a Stem-and-Leaf Plot +An entomologist studying crickets recorded the number of times different crickets (of differing species, genders, etc.) +chirped in a one-minute span. The raw data are as follows: +89 +97 +82 +102 +84 +99 +93 +103 +120 +91 +115 +105 +89 +109 +107 +89 +104 +82 +106 +92 +101 +109 +116 +103 +100 +91 +85 +104 +104 +106 +Construct a stem-and-leaf plot to visualize these results. +Solution +Step 1: Before we can create the plot, we need to sort the data in order from smallest to largest: +8.2 • Visualizing Data +839 + +82 +82 +84 +85 +89 +89 +89 +91 +91 +92 +93 +97 +99 +100 +101 +102 +103 +103 +104 +104 +104 +105 +106 +106 +107 +109 +109 +115 +116 +120 +Step 2: Next, we identify the stems. To do that, we cut off the final digit of each number, which leaves us with stems of 8, +9, 10, 11, and 12. Arrange the stems vertically, and add the bar to separate these from the leaves: +8 +9 +10 +11 +12 +Step 3: Write down the leaves on the right side of the bar, giving just the final digit (that we cut off to make the stems) of +each data value. List these in order, and make sure they line up vertically: +8 +2 2 4 5 9 9 9 +9 +1 1 2 3 7 9 +10 +0 1 3 3 4 4 4 5 6 6 7 9 9 +11 +5 6 +12 +0 +Table 8.3 +YOUR TURN 8.10 +1. This table gives the records of the Major League Baseball teams at the end of the 2019 season: +Team +Wins +Losses +Team +Wins +Losses +HOU +107 +55 +PHI +81 +81 +LAD +106 +56 +TEX +78 +84 +NYY +103 +59 +SFG +77 +85 +MIN +101 +61 +CIN +75 +87 +Table 8.4 (source: http://www.mlb.com) +840 +8 • Statistics +Access for free at openstax.org + +Team +Wins +Losses +Team +Wins +Losses +ATL +97 +65 +CHW +72 +89 +OAK +97 +65 +LAA +72 +90 +TBR +96 +66 +COL +71 +91 +CLE +93 +69 +SDP +70 +92 +WSN +93 +69 +PIT +69 +93 +STL +91 +71 +SEA +68 +94 +MIL +89 +73 +TOR +67 +95 +NYM +86 +76 +KCR +59 +103 +ARI +85 +77 +MIA +57 +105 +BOS +84 +78 +BAL +54 +108 +CHC +84 +78 +DET +47 +114 +Table 8.4 (source: http://www.mlb.com) +Create a stem-and-leaf plot for the number of wins. +As we mentioned above, stem-and-leaf plots aren’t always going to be useful. For example, if all the data in your dataset +are between 20 and 29, then you’ll just have one stem, which isn’t terribly useful. (Although there are methods like stem +splitting for addressing that particular problem, we won’t go into those at this time.) On the other end of the spectrum, +the data may be so spread out that every stem has only one leaf. (This problem can sometimes be addressed by +rounding off the data values to the tens, hundreds, or some other place value, then using that place for the leaves.) +Finally, if you have dozens or hundreds (or more) of data values, then a stem-and-leaf plot becomes too unwieldy to be +useful. Fortunately, we have other tools we can use. +Histograms +Histograms are visualizations that can be used for any set of quantitative data, no matter how big or spread out. They +differ from a categorical bar chart in that the horizontal axis is labeled with numbers (not ranges of numbers), and the +bars are drawn so that they touch each other. The heights of the bars reflect the frequencies in each bin. Unlike with +stem-and-leaf plots, we cannot recreate the original dataset from a histogram. However, histograms are easy to make +with technology and are great for identifying the distribution of our data. Let’s first create one histogram without +technology to help us better understand how histograms work. +EXAMPLE 8.11 +Constructing a Histogram +In Example 8.10, we built a stem-and-leaf plot for the number of chirps made by crickets in one minute. Here are the raw +data that we used then: +8.2 • Visualizing Data +841 + +89 +97 +82 +102 +84 +99 +115 +105 +89 +109 +107 +89 +101 +109 +116 +103 +100 +91 +93 +103 +120 +91 +85 +104 +104 +82 +106 +92 +104 +106 +Construct a histogram to visualize these results. +Solution +Step 1: Add data to bins. Histograms are built on binned frequency distributions, so we’ll make that first. Luckily, the +stem-and-leaf plot we made earlier can help us do this much more quickly: +8 +2 2 4 5 9 9 9 +9 +1 1 2 3 7 9 +10 +0 1 2 3 3 4 4 4 5 6 6 7 9 9 +11 +5 6 +12 +0 +If we’re using bins of width 10, we can compute the frequencies by counting the numbers of leaves associated with the +corresponding stem: +Bin +Frequency +80-89 +7 +90-99 +6 +100-109 +14 +110-119 +2 +120-129 +1 +(Note that, when we made binned frequency diagrams in the last module, we noted that if the biggest data value was +right on the border between two bins, it was OK to lump it in with the lower bin. That’s not recommended when building +histograms, so the data value 120 is all alone in the 120-129 bin.) +Step 2: Create the axes. On the horizontal axis, start labeling with the lower end of the first bin (in this case, 80), and go +up to the higher end of the last bin (120). Mark off the other bin boundaries, making sure they’re all evenly spaced. On +the vertical axis, start with zero and go up at least to the greatest frequency you see in your bins (14 in this example), +making sure that the labels you make are evenly spaced and that the difference between those numbers is the same. +Let’s count off our vertical axis by threes: +842 +8 • Statistics +Access for free at openstax.org + +Figure 8.17 +Step 3: Draw in the bars. Remember that the bars of a histogram touch, and that the heights are determined by the +frequency. So, the first bar will cover 80 to 90 on the horizontal axis, and have a height of 7: +Figure 8.18 +Now, we can fill in the others: +Figure 8.19 +Step 4: Let’s compare the histogram we just created to the stem-and-leaf plot we made earlier: +Figure 8.20 +Notice that the leaves on the rotated stem-and-leaf plot match the bars on our histogram! We can view stem-and-leaf +plots as sideways histograms. But, as we’ll see soon, we can do much more with histograms. +8.2 • Visualizing Data +843 + +YOUR TURN 8.11 +1. In Your Turn 10, you made a stem-and-leaf plot of the number of wins for each MLB team in 2019, using this set +of data: +Team +Wins +Losses +HOU +107 +55 +LAD +106 +56 +NYY +103 +59 +MIN +101 +61 +ATL +97 +65 +OAK +97 +65 +TBR +96 +66 +CLE +93 +69 +WSN +93 +69 +STL +91 +71 +MIL +89 +73 +NYM +86 +76 +ARI +85 +77 +BOS +84 +78 +CHC +84 +78 +PHI +81 +81 +TEX +78 +84 +SFG +77 +85 +CIN +75 +87 +CHW +72 +89 +LAA +72 +90 +COL +71 +91 +Table 8.5 +844 +8 • Statistics +Access for free at openstax.org + +Team +Wins +Losses +SDP +70 +92 +PIT +69 +93 +SEA +68 +94 +TOR +67 +95 +KCR +59 +103 +MIA +57 +105 +BAL +54 +108 +DET +47 +114 +Table 8.5 +Create a histogram for the number of wins. Use bins of width 10, starting with a bin for 40-49 (so that your +histogram reflects the stem-and-leaf plot you made earlier). +Now that we’ve seen the connection between stem-and-leaf plots and histograms, we are ready to look at how we can +use Google Sheets to build histograms. +VIDEO +Make a Histogram Using Google Sheets (https://openstax.org/r/CreatingHistograms) +Let’s use Google Sheets to create a histogram for a large dataset. +EXAMPLE 8.12 +Creating a Histogram in Google Sheets +The data in “AvgSAT” (https://openstax.org/r/Chapter8_Data-Sets) contains the average SAT score for students attending +every institution of higher learning in the US for which data is available. Create a histogram in Google Sheets of the +average SAT scores. Use bins of width 50. Are the data uniformly distributed, symmetric, left-skewed, or right-skewed? +Solution +Using the procedure described in the video above, we get this: +8.2 • Visualizing Data +845 + +Figure 8.21 (data source: https://data.ed.gov) +The data are fairly symmetric, but slightly right-skewed. +YOUR TURN 8.12 +1. The file “InState” (https://openstax.org/r/Chapter8_Data-Sets) contains in-state tuition costs (in dollars) for every +institution of higher learning in the United States for which data is available (data from data.ed.gov). Create a +histogram in Google Sheets of in-state tuition costs. Choose a bin size that you think works well. Are the data +uniformly distributed, symmetric, left-skewed, or right-skewed? +Bar Charts for Labeled Data +Sometimes we have quantitative data where each value is labeled according to the source of the data. For example, in +the Your Turn above, you looked at in-state tuition data. Every value you used to create that histogram was associated +with a school; the schools are the labels. In YOUR TURN 8.11, you found a histogram of the wins of every Major League +Baseball team in 2019. Each of those win totals had a label: the team. If we’re interested in visualizing differences among +the different teams, or schools, or whatever the labels are, we create a different version of the bar graph known as a bar +chart for labeled data. +These graphs are made in Google Sheets in exactly the same way as regular bar graphs. The only change is that the +vertical axis will be labeled with the units for your quantitative data instead of just “Frequency.” +EXAMPLE 8.13 +Building a Bar Chart for Labeled Data +The following table shows the gross domestic product (GDP) for the United States for the years 2010 to 2019: +Year +GDP (in $ trillions) +Year +GDP (in $ trillions) +2010 +14.992 +2015 +18.225 +2011 +15.543 +2016 +18.715 +2012 +16.197 +2017 +19.519 +Table 8.6 (source: https://data.worldbank.org) +846 +8 • Statistics +Access for free at openstax.org + +Year +GDP (in $ trillions) +Year +GDP (in $ trillions) +2013 +16.785 +2018 +20.580 +2014 +17.527 +2019 +21.433 +Table 8.6 (source: https://data.worldbank.org) +Construct a histogram that represents these data. +Solution +In this case, the years are the labels, and the data we are interested in are the GDP numbers. Once you have the table +above (including the labels) entered into a spreadsheet, click and drag to select the full table. Then, in the “Insert” menu, +click “Chart.” The result may not be a bar chart; if it’s not, select “Column chart” in the drop-down menu “Chart type” in +the Chart Editor. If you want, you can edit things like the chart title in the “Customize” tab in the Chart Editor. +Figure 8.22 (data source: https://data.worldbank.org) +YOUR TURN 8.13 +1. The following table shows the world record times (as of February 2020) of the various 100m women’s swimming +events in international competition: +Event +Time +Name +Nationality +Freestyle +51.71 +Sarah Sjöström +Sweden +Backstroke +57.57 +Regan Smith +United States +Breaststroke +64.10 +Lilly King +United States +Butterfly +55.48 +Sarah Sjöström +Sweden +Table 8.7 (source: https://swimswam.com/records/womens- +world-records-lcm/) +Make a visualization of these times using the events as the labels. +8.2 • Visualizing Data +847 + +Misleading Graphs +Graphical representations of data can be manipulated in ways that intentionally mislead the reader. There are two +primary ways this can be done: by manipulating the scales on the axes and by manipulating or misrepresenting areas of +bars. Let’s look at some examples of these. +EXAMPLE 8.14 +Misleading Graphs +The table below shows the teams, and their payrolls, in the English Premier League, the top soccer organization in the +United Kingdom. +Team +Salary (£1,000,000s) +Team +Salary (£1,000,000s) +Manchester United F.C. +175.7 +Newcastle United F.C. +56.9 +Manchester City F.C. +136.5 +Aston Villa F.C. +52.3 +Chelsea F.C. +132.8 +Fulham F.C. +52.1 +Arsenal F.C. +130.7 +Southampton F.C. +49.6 +Tottenham Hotspur F.C. +129.2 +Wolverhampton Wanderers F.C. +49.5 +Liverpool F.C. +118.6 +Brighton & Hove Albion +43.7 +Crystal Palace +85.0 +Burnley F.C. +35.5 +Everton F.C. +82.5 +West Bromwich Albion F.C. +23.8 +Leicester City +73.7 +Leeds United F.C. +22.5 +West Ham United F.C. +69.2 +Sheffield United F.C. +19.7 +Table 8.8 (source: www.spotrac.com) +How might someone present this data in a misleading way? +Solution +Step 1: Let’s focus on the top five teams. Here’s a bar chart of their payrolls: +848 +8 • Statistics +Access for free at openstax.org + +Figure 8.23 (data source: www.spotrac.com) +Step 2: Now, here’s another bar chart visualizing exactly the same data: +Figure 8.24 (data source: www.spotrac.com) +Step 3: You should notice that despite using the same data, these two graphs look strikingly different. In the second +graph, the gap between Manchester United and the other four teams looks significantly larger than in the first graph. +The scale on the vertical axis has been manipulated here. The first graph's axis starts at zero, while the lowest value on +the second graph's axis is 120. This trick has a strong impact on the viewer’s perception of the data. +Beware of vertical axes that don’t start at zero! They overemphasize differences in heights. +Step 4: To further emphasize the difference this creates in our perception, let's look at that data again, but this time +using graphics instead of colored areas on our bar graph. +8.2 • Visualizing Data +849 + +Figure 8.25 (data source: www.spotrac.com) +This graph uses an image of a £10 banknote in place of the bars. Using an image that evokes the context of the data in +place of a standard, “boring” bar is a common tool that people use when creating infographics. However, this is generally +not a good practice because it distorts the data. Notice that our “bars” (the banknotes) are just as tall here as they were +in the previous figure. But, to maintain the right proportions, the widths had to be adjusted as well, which changes the +area (height × width) of each bar. A key point is that when looking at rectangles, the human eye tends to process areas +more easily than heights. +Beware of infographics! Areas overemphasize a difference that should be measured with a height! +Step 5: Now, let’s look at all 20 teams. This histogram indicates that the data are right-skewed, with the highest number +of teams having a payroll between £40 million and £80 million: +Figure 8.26 (data source: www.spotrac.com) +850 +8 • Statistics +Access for free at openstax.org + +Step 6: Now let's view this same data in another chart: +Figure 8.27 (data source: www.spotrac.com) +Step 7: Even though this chart uses the same data, the skew seems to be reversed. Why? Well, even though this graph +looks like a histogram, it isn’t. Look closely at the labels on the horizontal axis; they don't correspond to spots on the axis, +but instead provide a range, meaning this is a bar graph based on a binned frequency distribution. +When we review these ranges, we can see that the last range is misleading as it consists of all data “over 80.” If the bins +all had the same width, that last bin would run from 80 to 120. However, we can see from the histogram that the +maximum value for this data is between 160 and 200. If the last bin in this bar graph were labeled honestly, it would read +“80–200,” which would drive home the fact that the width of that bar is misleading. +Always check the horizontal axis on histograms! The widths of all the bars should be equal. +YOUR TURN 8.14 +1. Take a look again at the win totals for teams in Major League Baseball in 2019 : +Team +Wins +Team +Wins +HOU +107 +PHI +81 +LAD +106 +TEX +78 +NYY +103 +SFG +77 +MIN +101 +CIN +75 +ATL +97 +CHW +72 +Table 8.9 (source: +https://www.espn.com/mlb/ +standings/_/season/2019/view) +8.2 • Visualizing Data +851 + +Team +Wins +Team +Wins +OAK +97 +LAA +72 +TBR +96 +COL +71 +CLE +93 +SDP +70 +WSN +93 +PIT +69 +STL +91 +SEA +68 +MIL +89 +TOR +67 +NYM +86 +KCR +59 +ARI +85 +MIA +57 +BOS +84 +BAL +54 +CHC +84 +DET +47 +Table 8.9 (source: +https://www.espn.com/mlb/ +standings/_/season/2019/view) +Make one good and one misleading chart showing the number of wins by the top ten teams. Then, looking at all +the teams, make one good and one misleading histogram for the win totals. +VIDEO +How to Spot a Misleading Graph (https://openstax.org/r/MisleadingGraphs) +WHO KNEW? +Napoleon's Failed Invasion +One of the most famous data visualizations ever created is the cartographic depiction by Charles Joseph Minard of +Napoleon’s disastrous attempted invasion of Russia. +852 +8 • Statistics +Access for free at openstax.org + +Figure 8.28 Minard’s Napoleon Map (credit: Carte de Charles Minard/Wikimedia, public domain) +Minard’s chart is remarkable in that it shows not just how the size of Napoleon’s army shrank drastically over time, but +also the location on the map, the direction the army was traveling at the time, and the temperature during the +retreat. +Check Your Understanding +The medical office at a zoo tracks the animals it treats each week. The table shows the classifications for a particular +week: +Mammal +Mammal +Reptile +Bird +Mammal +Amphibian +Mammal +Mammal +Mammal +Reptile +Mammal +Bird +Mammal +Bird +Reptile +Reptile +Amphibian +Mammal +Bird +Mammal +Amphibian +Mammal +Mammal +Bird +7. Create a bar graph of the data without technology. +8. Create a pie chart of the data using technology. +Employees at a college help desk track the number of people who request assistance each week. The table gives a +sample of the results : +142 +153 +158 +156 +141 +143 +139 +158 +156 +146 +137 +153 +136 +127 +157 +148 +132 +139 +155 +167 +143 +168 +133 +157 +138 +156 +164 +130 +148 +136 +9. Make a stem-and-leaf plot of the data. +10. Create a histogram of the data. Use bins of width 5. +8.2 • Visualizing Data +853 + +The following are data on the admission rates of the different branch campuses in the University of California system, +along with the out-of-state tuition and fee cost: +Campus +Admission Rate +Cost ($) +Berkeley +0.1484 +43,176 +Davis +0.4107 +43,394 +Irvine +0.2876 +42,692 +Los Angeles +0.1404 +42,218 +Merced +0.6617 +42,530 +Riverside +0.5057 +42,819 +San Diego +0.3006 +43,159 +Santa Barbara +0.322 +43,383 +Santa Cruz +0.4737 +42,952 +(source: https://data.ed.gov) +11. Create a bar graph that illustrates the differences in admission rates among the different campuses. +12. Create two bar graphs for the out-of-state tuition. One should give an unbiased perception of the differences +among them, and the other should overemphasize those differences. +SECTION 8.2 EXERCISES +The table below shows the answers to the question, “Which social media platform, if any, do you use most frequently?” +None +Twitter +Snapchat +Snapchat +Twitter +Facebook +Instagram +Snapchat +Twitter +None +Snapchat +Instagram +Instagram +Facebook +None +Instagram +Snapchat +Twitter +Snapchat +Instagram +Instagram +Twitter +Snapchat +Twitter +Facebook +None +Instagram +Instagram +Twitter +Instagram +1. Make a bar chart to visualize these responses. +2. Make a pie chart to visualize these responses. +A sample of students at a large university were asked whether they were full-time students living on campus (Full-Time +Residential, FTR), full-time students who commuted (FTC), or part-time students (PT). The raw data are in the table +below: +FTR +FTR +FTC +PT +FTR +PT +FTR +FTC +FTR +FTC +FTC +FTR +FTR +PT +FTC +FTC +854 +8 • Statistics +Access for free at openstax.org + +FTC +PT +FTC +FTC +PT +FTR +FTC +PT +FTC +PT +FTR +PT +FTC +FTC +FTR +PT +3. Make a bar chart to visualize these responses. +4. Make a pie chart to visualize these responses. +Students in a statistics class were asked how many countries (besides their home countries) they had visited; the table +below gives the raw responses: +0 +2 +1 +1 +3 +2 +0 +2 +0 +1 +0 +2 +0 +1 +0 +1 +1 +0 +1 +0 +0 +0 +0 +2 +1 +1 +0 +1 +1 +0 +5. Create a bar graph visualizing these data (treating the responses as categorical). +6. Create a pie chart visualizing these data. +The purchasing department for a chain of bookstores wants to make sure they’re buying the right types of books to put +on the shelves, so they take a sample of 20 books that customers bought in the last five days and record the genres: +Nonfiction +Young Adult +Romance +Cooking +Young Adult +Young Adult +Thriller +Young Adult +Nonfiction +True Crime +Romance +Nonfiction +Thriller +True Crime +Romance +True Crime +Thriller +Romance +Young Adult +Young Adult +7. Create a bar graph to visualize these data. +8. Create a pie chart to visualize these data. +An elementary school class is administered a standardized test for which scores range from 0 to 100, as shown below: +60 +54 +71 +80 +63 +72 +70 +88 +88 +67 +74 +79 +50 +99 +64 +98 +55 +64 +86 +92 +72 +65 +88 +80 +65 +(source: +http://www.nwslsoccer.com) +9. Make a stem-and-leaf plot to visualize these results. +10. Make a histogram to visualize these results. Use bins of width 10. +The following table gives the final results for the 2021 National Women’s Soccer League season. The columns are +standings points (PTS; teams earn three points for a win and one point for a tie), wins (W), losses (L), ties (T), goals +scored by that team (GF), and goals scored against that team (GA). +8.2 • Visualizing Data +855 + +Team +PTS +W +L +T +GF +GA +Portland Thorns FC +44 +13 +6 +5 +33 +17 +OL Reign +42 +13 +8 +3 +37 +24 +Washington Spirit +39 +11 +7 +6 +29 +26 +Chicago Red Stars +38 +11 +8 +5 +28 +28 +NJ/NY Gotham FC +35 +8 +5 +11 +29 +21 +North Carolina Courage +33 +9 +9 +6 +28 +23 +Houston Dash +32 +9 +10 +5 +31 +31 +Orlando Pride +28 +7 +10 +7 +27 +32 +Racing Louisville FC +22 +5 +12 +7 +21 +40 +Kansas City Current +16 +3 +14 +7 +15 +36 +(source: http://www.nwslsoccer.com) +11. Make a stem-and-leaf plot for PTS. +12. Make a histogram for PTS, using bins of width 5. +13. Make a histogram for GF, using bins of width 5. +14. Make a histogram for GA, using bins of width 5. +For the following exercises, use the "CUNY" (https://openstax.org/r/Chapter8_Data-Sets) dataset–which gives the +location (borough) of each college in the City University of New York (CUNY) system, the highest degree offered, and +the proportions of total degrees awarded in a partial list of disciplines–to identify the right visualization to address each +question. Then, create those visualizations. +15. What is the highest degree offered in colleges across the CUNY system? +16. What is the distribution of the proportion of degrees awarded in Information Science across the CUNY system? +17. In which boroughs are the CUNY colleges located? +18. What are the proportions of degrees awarded across the listed humanities fields (Foreign Language, English, +Humanities, Philosophy & Religion, History) at City College? +19. What proportions of degrees are awarded in Social Service at the different institutions located in Manhattan? +For the following exercises, use the data found in the "Receivers" (https://openstax.org/r/Chapter8_Data-Sets) dataset +on the top 25 receivers (by number of receptions; data collected from pro-football-reference.com) in the NFL during the +2020 season. +20. Make a stem-and-leaf plot for the longest receptions (“Long”). +21. Make a stem-and-leaf plot for receptions. +22. Make a histogram for yards. +23. Make a histogram for yards per reception (“Yds/Rec”). +24. Make a histogram for the longest receptions (“Long”). +25. Make a histogram for receptions. +26. Make a histogram for age. +27. Describe the distribution of age as left-skewed, symmetric, or right-skewed. +28. Describe the distribution of receptions as left-skewed, symmetric, or right-skewed. +29. Describe the distribution of yards as left-skewed, symmetric, or right-skewed. +30. Describe the distribution of touchdowns (“TD”) as left-skewed, symmetric, or right-skewed. +31. Describe the distribution of longest receptions as left-skewed, symmetric, or right-skewed. +856 +8 • Statistics +Access for free at openstax.org + +8.3 Mean, Median and Mode +Figure 8.29 What does it mean to say someone has average height? (credit: modification of work “I’m the tallest” by Jenn +Durfey/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Calculate the mode of a dataset. +2. +Calculate the median of a dataset. +3. +Calculate the mean of a dataset. +4. +Contrast measures of central tendency to identify the most representative average. +5. +Solve application problems involving mean, median, and mode. +What exactly do we mean when we describe something as "average"? Is the height of an average person the height that +more people share than any other? What if we line up every person in the world, in order from shortest to tallest, and +find the person right in the middle: Is that person’s height the average? Or maybe it’s something more complicated. +Imagine a game where you and a friend are trying to guess the typical person’s height. Once the guesses are made, you +bring in every person and measure their height. You and your friend figure out how far off each of your guesses were +from the actual value, then square that number. The result is the number of points you earn for that person. After we +check every height and award points accordingly, the person with the lower score wins (because a lower score means +that person’s guess was, overall, closer to the actual values). Could we define the average height to be the number that +you should guess to give you the smallest possible score? +Each of these three methods of determining the “average” is commonly used. They are all methods of measuring +centrality (or central tendency). Centrality is just a word that describes the middle of a set of data. All give potentially +different results, and all are useful for different reasons. In this section, we’ll explore each of these methods of finding +the “average.” +The Mode +In our discussion of average heights, the first possible definition we offered was the height that more people share than +any other. This is the mode, or the value that appears most often. If there are two modes, the data are bimodal. +Let’s look at some examples. +EXAMPLE 8.15 +Finding the Mode Using a Stem-and-Leaf Plot +In Example 8.9, we looked at a stem-and-leaf plot of the sale prices (in dollars) of a particular collectible trading card: +8.3 • Mean, Median and Mode +857 + +0 +5 8 9 +1 +0 0 0 3 4 4 5 5 5 5 6 9 9 +2 +0 0 0 0 5 5 9 9 +3 +0 0 0 5 5 +4 +0 0 5 +5 +6 +0 +What is the mode price? +Solution +The mode is the price that appears most often. Both 15 and 20 appear 4 times, more than any other values. So, they are +the modes (and we can conclude that this set of data is bimodal). +YOUR TURN 8.15 +1. In Example 8.10, we constructed a stem-and-leaf plot for the number of times in one minute that different +crickets chirped: +8 +2 2 4 5 9 9 9 +9 +1 1 2 3 7 9 +10 +0 1 2 3 3 4 4 4 5 6 6 7 9 9 +11 +5 6 +12 +0 +What is the mode of the number of chirps in a minute? +When we have a complete list of the data or a stem-and-leaf plot, it’s pretty straightforward to find the mode; we just +need to find the number that appears most often. If we’re given a frequency distribution instead, the technique is +different (but just as straightforward): we’re looking for the number with the highest frequency. +EXAMPLE 8.16 +Finding the Mode Using a Frequency Distribution +In Example 8.3, we created a frequency distribution of the number of siblings of conflict resolution class attendees. +858 +8 • Statistics +Access for free at openstax.org + +Number of Siblings +Frequency +0 +5 +1 +13 +2 +6 +3 +3 +4 +2 +5 +1 +What is the mode of the number of siblings? +Solution +The mode is the value that appears the most often, which means it has the greatest frequency. Thirteen of the +respondents have one sibling, more than any other number. So, the mode is 1. +YOUR TURN 8.16 +1. In Your Turn 8.3, you found a frequency distribution for the number of people who shared a residence with +people in a sample. +Number of People in the Residence +Frequency +1 +12 +2 +13 +3 +8 +4 +6 +5 +1 +What’s the mode of the number of people in these residences? +What happens if there is no number in the data that appears more than once? In that case, by our definition, every data +value is a mode. But according to some other definitions, the data would have no mode. In practice, though, it doesn’t +really matter; if no data value appears more than once, then the mode is not helpful at all as a measure of centrality. +The Median +Let's revisit our example of trying to identify the height of the “average” person. If we lined everyone up in order by +height and found the person right in the middle, that person’s height is called the median, or the value that is greater +than no more than half and less than no more than half of the values. +Let’s look at a really simple example. Consider the following list of numbers: 11, 12, 13, 13, 14. Is the first number on the +list, 11, the median? There are no values less than 11 (that’s 0%), and there are four values greater than 11 (that’s 80%). +Since more than 50% of the data are greater than 11, the definition is violated; it’s not the median. Here’s a chart with the +rest of the data, with red shading to show where the definition is violated: +8.3 • Mean, Median and Mode +859 + +Data +Value +Number of Values +Below +Percentage of Values +Below +Number of Values +Above +Percentage of Values +Above +11 +0 +0% +4 +80% +12 +1 +20% +3 +60% +13 +2 +40% +1 +20% +14 +4 +80% +0 +0% +Table 8.10 +Only 13 has no violations, so it’s the median according to the definition. In practice, we find the median just like we +described in the average height example: by lining up all the data values in order from smallest to largest and picking +the value in the middle. For our easy example (with data values 11, 12, 13, 13, 14), that first 13 is right in the middle; +there are two values to the left and two values to the right. If there’s not one value right in the middle, we pick the two +closest, then choose the number exactly between them. For example, let’s say we have the data 41, 44, 46, 53. Since +there are an even number of data values in our list, we can’t pick the one right in the middle. The two closest to the +middle are 44 and 46, so we’ll choose the number halfway between those to be the median: 45. As this example shows, +the median (unlike the mode) doesn’t have to be a number in our original set of data. +In the examples we’ve looked at so far, it’s been pretty easy to identify which number is right in the middle. If we had a +very large dataset, though, it might be harder. Fortunately, we have some formulas to help us with that. +FORMULA +Suppose we have a set of data with +values, ordered from smallest to largest. If +is odd, then the median is the data +value at position +. If +is even, then we find the values at positions +and +. If those values are named +and +, then the median is defined to be +. +Let’s put those formulas to work in an example. +EXAMPLE 8.17 +Finding the Median Using a Stem-and-Leaf Plot +In Example 8.9, we looked at a stem-and-leaf plot that contained 33 sale prices (in dollars) of a particular collectible +trading card: +0 +5 8 9 +1 +0 0 0 3 4 4 5 5 5 5 6 9 9 +2 +0 0 0 0 5 5 9 9 +3 +0 0 0 5 5 +4 +0 0 5 +5 +6 +0 +860 +8 • Statistics +Access for free at openstax.org + +What is the median price? +Solution +Step 1: Since 33 is odd, the median is the data value at position +, where +is the number of values in the dataset. +There are 33 total values, so our formula becomes +. That means we want to look for the 17th number in the +dataset. +Step 2: We'll want to count from the lowest value to the 17th number. We can use our stem and leaf plot to do this. +1 2 3 +0 +5 8 9 +4 5 6 7 8 9 10 11 12 13 14 15 16 +1 +0 0 0 3 4 4 5 5 5 5 6 9 9 +17 +2 +0 0 0 0 5 5 9 9 +3 +0 0 0 5 5 +4 +0 0 5 +5 +6 +0 +The seventeenth number is 20, so the median is 20. +YOUR TURN 8.17 +1. Consider the data given in this stem-and-leaf plot (there are 17 data values): +12 +1 2 2 5 +13 +0 3 4 4 6 8 +14 +2 5 9 9 +15 +0 3 +16 +17 +0 +What is the median of this data? +Now, let’s tackle an example with an even number of values. +8.3 • Mean, Median and Mode +861 + +EXAMPLE 8.18 +Finding the Median +In Example 8.10, we looked at the number of times different crickets (of differing species, genders, etc.) chirped in a one- +minute span. That data is again provided below: +89 +97 +82 +102 +84 +99 +115 +105 +89 +109 +107 +89 +101 +109 +116 +103 +100 +91 +93 +103 +120 +91 +85 +104 +104 +82 +106 +92 +104 +106 +Find the median. +Solution +Step 1: In order to find the median, we first need to sort the data so that they’re in order, smallest to largest: +82 +82 +84 +85 +89 +89 +89 +91 +91 +92 +93 +97 +99 +100 +101 +102 +103 +103 +104 +104 +104 +105 +106 +106 +107 +109 +109 +115 +116 +120 +Step 2: Next, we figure out how many data values we have. Counting them up, we see there are 30, which is even. +Step 3: Since we have an even number of data values, we need to find the values in positions +and +. +These are 101 and 102. +Step 4: We use the formula to compute the median: +. +YOUR TURN 8.18 +1. This table gives the records of the Major League Baseball teams at the end of the 2019 season: +Team +Wins +Losses +Team +Wins +Losses +HOU +107 +55 +PHI +81 +81 +LAD +106 +56 +TEX +78 +84 +NYY +103 +59 +SFG +77 +85 +Table 8.11 (source: www.mlb.com +(http://www.mlb.com)) +862 +8 • Statistics +Access for free at openstax.org + +Team +Wins +Losses +Team +Wins +Losses +MIN +101 +61 +CIN +75 +87 +ATL +97 +65 +CHW +72 +89 +OAK +97 +65 +LAA +72 +90 +TBR +96 +66 +COL +71 +91 +CLE +93 +69 +SDP +70 +92 +WSN +93 +69 +PIT +69 +93 +STL +91 +71 +SEA +68 +94 +MIL +89 +73 +TOR +67 +95 +NYM +86 +76 +KCR +59 +103 +ARI +85 +77 +MIA +57 +105 +BOS +84 +78 +BAL +54 +108 +CHC +84 +78 +DET +47 +114 +Table 8.11 (source: www.mlb.com +(http://www.mlb.com)) +What is the median number of wins? +EXAMPLE 8.19 +Finding the Median Using a Frequency Distribution +In Example 8.3, we created a frequency distribution of the number of siblings of the people who attended a conflict +resolution class. Let's review that data again: +Number of Siblings +Frequency +0 +5 +1 +13 +2 +6 +3 +3 +4 +2 +5 +1 +8.3 • Mean, Median and Mode +863 + +What is the median of the number of siblings? +Solution +There are 30 data values total, so the median is between the 15th and 16th values in the ordered list. There are five 0s +and thirteen 1s according to the frequency distribution, so items one through five are all 0s and items six through +eighteen are all 1s. Since both items fifteen and sixteen are 1s, the median is 1. +YOUR TURN 8.19 +1. In Your Turn 8.3, you found a frequency distribution for the number of people who shared a residence with +people in a sample. Let's review that data again: +Number of People in the Residence +Frequency +1 +12 +2 +13 +3 +8 +4 +6 +5 +1 +What’s the median of the number of people in these residences? +The Mean +Recall our example of ways we could identify the “average” height of an individual. The last method we discussed was +also the most complicated. It involved a game where the player guesses a height, then figures out how far off that guess +is from every single person’s height. Those differences get squared and added together to get a score. Our next measure +of centrality gives the lowest possible score: No other guess would beat it in the game. Given a dataset containing n total +values, the mean of the dataset is the sum of all the data values, divided by n. +This is a computation you have likely done before. In many places, including spreadsheet programs like Microsoft Excel +and Google Sheets, this number is called the average. For statisticians, though, the word average has too many possible +meanings, so they prefer the one we’ll use: mean. +EXAMPLE 8.20 +Finding the Mean +Compute the mean of the numbers 12, 15, 17, 18, 18, and 19. +Solution +The mean is the sum of the values, divided by the number of values on the list. So, we get: +YOUR TURN 8.20 +1. Compute the mean of the numbers 5, 8, 11, 12, 12, 12, 15, 18, and 20. Round your answer to three decimal +places. +864 +8 • Statistics +Access for free at openstax.org + +EXAMPLE 8.21 +Finding the Mean Using a Frequency Distribution +Refer again to the frequency distribution of the number of siblings people who attended a conflict resolution class +reported: +Number of Siblings +Frequency +0 +5 +1 +13 +2 +6 +3 +3 +4 +2 +5 +1 +What is the mean of the number of siblings? +Solution +Step 1: We compute the mean by adding up all the data values and then dividing by the number of data values on the +list. +Step 2: Adding up the frequencies, we get +data values in our list. +Step 3: Now, to find the sum of all the data values, we could simply reconstruct the raw data and add up all the numbers +there. But, there’s an easier way: Remember that repeatedly adding a number to itself is the definition of multiplication. +So, for example, since there are six 2s in our data, the sum of all those 2s must be +. +Step 4: Let’s add a column to our distribution for these products: +Number of Siblings +Frequency +(Number of Siblings) +(Frequency) +0 +5 +0 +1 +13 +13 +2 +6 +12 +3 +3 +9 +4 +2 +8 +5 +1 +5 +Step 5: So, the sum of all our data values is +. The mean is +. +YOUR TURN 8.21 +1. Refer again to the frequency distribution of the number of people who shared a residence with people in a +sample: +8.3 • Mean, Median and Mode +865 + +Number of People in the Residence +Frequency +1 +12 +2 +13 +3 +8 +What’s the mean of the number of people in these residences? +As the number of data values we are considering grows, the computation for the mean gets more and more +complicated. That’s why people generally trust technology to perform that computation. +VIDEO +Compute Measures of Centrality Using Google Sheets (https://openstax.org/r/Google-sheet) +Note that a recent update to Google Sheets introduced a new function called “MODE.MULT,” which will find every mode +(not just the first one on the list). +EXAMPLE 8.22 +Using Google Sheets to Compute Measures of Centrality +The dataset "InState" (https://openstax.org/r/Chapter8_Data-Sets) contains the in-state tuitions of every college and +university in the country that reported that data to the Department of Education. Find the mode, median, and mean of +those in-state tuition values. +Solution +Step 1: To find the mode, we select an empty cell type “=MODE(”, click on the header of the column to insert a reference +to the column into our formula, and then close the parentheses. When we hit the enter key, our formula is replaced with +the mode: $13,380. +Step 2: We can find the median and mean using the same process, except using the functions “MEDIAN” and “AVERAGE” +in place of “MODE”. We find that the median is $11,207 and the mean is $15,476.79. +YOUR TURN 8.22 +1. The dataset "MLB2019" (https://openstax.org/r/Chapter8_Data-Sets) gives the number of wins for every Major +League Baseball team in the 2019 season. Use Google Sheets to find the mode(s), median, and mean. +Which Is Better: Mode, Median, or Mean? +If the mode, median, and mean all purport to measure the same thing (centrality), why do we need all three? The answer +is complicated, as each measure has its own strengths and weaknesses. The mode is simple to compute, but there may +be more than one. Further, if no data value appears more than once, the mode is entirely unhelpful. As for the mean and +median, the main difference between these two measures is how each is affected by extreme values. +Consider this example: the mean and median of 1, 2, 3, 4, 5 are both 3. But what if the dataset is instead 1, 2, 3, 4, 10? +The median is still 3, but the mean is now 4. What this example shows is that the mean is sensitive to extreme values, +while the median isn’t. This knowledge can help us decide which of the two is more relevant for a given dataset. If it is +important that the really high or really low values are reflected in the measure of centrality, then the mean is the better +option. If very high or low values are not important, however, then we should stick with the median. +The decision between mean and median only really matters if the data are skewed. If the data are symmetric, then the +mean and median are going to be approximately equal, and the distinction between them is irrelevant. If the data are +skewed, the mean gets pulled in the direction of the skew (i.e., if the data are right-skewed, then the mean will be bigger +866 +8 • Statistics +Access for free at openstax.org + +than the median; if the data are left-skewed, the opposite relation is true). +EXAMPLE 8.23 +Choosing Which Measure of Centrality to Use +For the following situations, decide which measure(s) of centrality would be best: +1. +You found a used car that you like, and you want to know if the price is too high or too low. You find a list of sale +prices for that make and model, and you see that the distribution of those prices is skewed to the right. Some of the +prices at the high end are close to the original sale price of the car, so you guess that those cars might have really +low mileage, or have other enhancements added on that increased the value (but which don’t apply to the car you +found). +2. +You are asked to analyze the responses to a survey. One of the questions asked, “How strongly do you agree with +the statement, ‘I believe my elected representatives have my best interests in mind when they vote’?” Responses are +a number between 1 and 5, with 1 representing “strongly disagree” and 5 representing “strongly agree.” +3. +You are asked to find the “average” household income for a zip code. Those values are skewed right. +4. +Thinking back to the situation at the beginning of this section: you want to find “average” height. The data you’ve +collected seem to be distributed symmetrically. +Solution +1. +In this situation, the high values are not comparable to the value of the car you found and we don’t want them to +affect the results. Also, we’re unlikely to find many repeated values, so the mode is probably not useful. Median is +best. +2. +Here, we want to know what a typical result is. The mean doesn’t really make sense; it involves adding the numbers +together, so it would treat two “strongly disagree” and two “strongly agree” responses (those add to +) as exactly the same as four “neutral” responses ( +). But those are really +different situations; the first shows a strong polarization in the responses, while the second represents strong +indifference. The mode is probably the best choice (because the data are actually categorical), but the median would +be good too. +3. +The mode isn’t going to be useful in this situation because it’s unlikely you will find many households that have +exactly the same income. The mean and median will be different because of the skew, so the choice comes down to +the extreme values. Remember that the data are skewed right, so high values are prevalent. Because these high +values are important for our analysis, we want them to be reflected in the results. Thus, the mean is best. That being +said, the median is also useful; it allows us to say something like “50% of the households surveyed make more than” +the median. +4. +Because we aren’t likely to find many people with exactly the same height, the mode won’t be useful. Since the data +are symmetric, the mean and the median will be about the same. So, it doesn’t really matter which of those two we +choose. +YOUR TURN 8.23 +For the following situations, decide which measure(s) of centrality would be best: +1. The data come from a survey question where the responses range from 1 (“not interested”) to 10 (“very +interested”). +2. The data are fuel efficiency measures (in miles per gallon) for various vehicles. The data are right-skewed. +3. The data are scores on an exam, and they are left-skewed. +Check Your Understanding +13. Use the given stem-and-leaf plot to determine the mode, median, and mean: +8 +8 9 +9 +0 0 7 +8.3 • Mean, Median and Mode +867 + +10 +2 5 6 7 8 +11 +1 2 2 2 4 5 7 9 9 +12 +0 0 3 5 7 +13 +0 1 1 4 +A survey of college students asked how many courses those students were currently taking. The results are +summarized in this frequency distribution: +Number of Classes +Frequency +1 +1 +2 +3 +3 +16 +4 +8 +5 +4 +14. Find the mode of the number of classes. +15. Find the median of the number of classes. +16. Find the mean of the number of classes. +Employees at a college help desk track the number of people who request assistance each week, as recorded below: +142 +153 +158 +156 +141 +143 +155 +167 +143 +168 +139 +158 +156 +146 +137 +153 +138 +156 +164 +130 +136 +127 +157 +148 +132 +139 +133 +157 +148 +136 +17. Compute the mode. +18. Compute the median. +19. Compute the mean. +The following table provides admission rates of the different branch campuses in the University of California system, +along with the out-of-state tuition and fee cost: +Campus +Admission Rate +Cost ($) +Berkeley +0.1484 +43,176 +Davis +0.4107 +43,394 +Irvine +0.2876 +42,692 +Los Angeles +0.1404 +42,218 +(source: https://data.ed.gov) +868 +8 • Statistics +Access for free at openstax.org + +Campus +Admission Rate +Cost ($) +Merced +0.6617 +42,530 +Riverside +0.5057 +42,819 +San Diego +0.3006 +43,159 +Santa Barbara +0.322 +43,383 +Santa Cruz +0.4737 +42,952 +(source: https://data.ed.gov) +20. Compute the mode, median, and mean admission rate. +21. Compute the mode, median, and mean cost. +22. Consider the data in “InState,” (https://openstax.org/r/Chapter8_Data-Sets), which shows in-state tuition and fees +for most institutions of higher learning in the United States. Compute the mode, median, and mean. +23. Workers for a particular company have complained that their wages are too low, while the management says +wages are plenty high. If the company’s salaries are right-skewed, which measure of centrality will the workers +choose to represent salary? Which will management choose? +SECTION 8.3 EXERCISES +Use the steam-and-leaf plot below to answer the questions. Round all answers to three decimal places. +0 +1 1 1 2 6 6 7 +1 +0 0 0 0 2 5 5 9 +2 +0 0 5 9 +3 +0 5 +4 +0 +1. Find the mode. +2. Find the median. +3. Find the mean. +Use the stem-and-leaf plot below to answer the following questions. Round all answers to three decimal places. +7 +8 +8 +0 6 +9 +1 1 4 8 +10 +0 0 3 5 9 +11 +0 0 1 1 1 2 6 8 +12 +0 1 5 +8.3 • Mean, Median and Mode +869 + +4. Find the mode. +5. Find the median. +6. Find the mean. +The following table shows the frequency distribution for a number of countries visited by students. Use this table to +answer the questions, rounding all answers to three decimal places. +Number of Countries +Frequency +0 +13 +1 +11 +2 +5 +3 +1 +7. Find the mode of the number of countries visited. +8. Find the median of the number of countries visited. +9. Find the mean of the number of countries visited. +The following table shows the frequency distribution for the number of fumbles by a selection of football players. Use +this table to answer the questions, rounding all answers to three decimal places. +Number of Fumbles +Frequency +0 +5 +1 +13 +2 +3 +3 +4 +10. Find the mode of the number of fumbles. +11. Find the median of the number of fumbles. +12. Find the mean of the number of fumbles. +A public opinion poll about an upcoming election asked respondents, “How many political advertisements do you recall +seeing on television in the last 24 hours?” Use this data to answer the questions, rounding all answers to three decimal +places. +6 +2 +5 +5 +2 +2 +4 +1 +3 +0 +1 +2 +1 +6 +2 +5 +2 +4 +8 +6 +3 +3 +4 +2 +5 +3 +4 +2 +2 +3 +13. Compute the mode. +14. Compute the median. +15. Compute the mean. +For the following exercises, use the data found in “TNSchools” (https://openstax.org/r/Chapter8_Data-Sets), which has +data on many institutions of higher education in the state of Tennessee. Here are what the columns represent: +870 +8 • Statistics +Access for free at openstax.org + +Column Name +Description +AdmRate +Proportion of applicants that are admitted +UGEnr +Number of undergraduate students +PTUG +Proportion of undergraduates who attend part-time +InState +Tuition and fees for in-state students +OutState +Tuition and fees for out-of-state students +FacSal +Mean monthly faculty salary +Pell +Proportion of students receiving Pell Grants +MedDebt +Median student loan debt at degree completion +StartAge +Mean age at the time of entry +Female +Proportion of students who identify as female +(source: https://data.ed.gov) +16. What is the mean undergraduate enrollment at these schools? +17. What is the median undergraduate enrollment at these schools? +18. Generate a histogram for undergraduate enrollment, and use it to describe the effect of the distribution on the +mean and the median as measures of centrality. +19. What is the mean proportion of part-time undergraduates at these schools? +20. What is the median proportion of part-time undergraduates at these schools? +21. Generate a histogram for proportion of part-time undergraduates, and use it to describe the effect of the +distribution on the mean and the median as measures of centrality. +22. What is the mean of the monthly faculty salaries at these schools? +23. What is the median of the monthly faculty salaries at these schools? +24. Generate a histogram for the monthly faculty salaries, and use it to describe the effect of the distribution on the +mean and the median as measures of centrality. +The dataset “AvgSAT” (https://openstax.org/r/Chapter8_Data-Sets) gives the average SAT score for incoming students +(the use of “average” here stands in for “mean” to avoid confusion in the following problems) for every institution in the +United States that reported the data (from data.ed.gov). Use that dataset to answer the following questions. +25. What is the mean of the average SAT scores? +26. What is the median of the average SAT scores? +27. Construct a histogram of average SAT scores, and describe how the distribution affects the mean and the +median. +For the following exercises, use the table, which gives the final results for the 2021 National Women’s Soccer League +season. The columns are standings points (PTS; teams earn three points for a win and one point for a tie), wins (W), +losses (L), ties (T), goals scored by that team (GF), and goals scored against that team (GA). +Team +Points +W +L +T +GF +GA +Portland Thorns FC +44 +13 +6 +5 +33 +17 +OL Reign +42 +13 +8 +3 +37 +24 +(source: http://www.nwslsoccer.com) +8.3 • Mean, Median and Mode +871 + +Team +Points +W +L +T +GF +GA +Washington Spirit +39 +11 +7 +6 +29 +26 +Chicago Red Stars +38 +11 +8 +5 +28 +28 +NJ/NY Gotham FC +35 +8 +5 +11 +29 +21 +North Carolina Courage +33 +9 +9 +6 +28 +23 +Houston Dash +32 +9 +10 +5 +31 +31 +Orlando Pride +28 +7 +10 +7 +27 +32 +Racing Louisville FC +22 +5 +12 +7 +21 +40 +Kansas City Current +16 +3 +14 +7 +15 +36 +(source: http://www.nwslsoccer.com) +28. Compute the mode(s), median, and mean of points. +29. Compute the mode(s), median, and mean of wins. +30. Compute the mode(s), median, and mean of losses. +31. Compute the mode(s), median, and mean of ties. +32. Compute the mode(s), median, and mean of goals scored (GF). +33. Compute the mode(s), median, and mean of goals against (GA). +For the following exercises, use the data in “CUNY” (https://openstax.org/r/Chapter8_Data-Sets) (taken from +data.ed.gov, which gives proportions of degrees awarded in various disciplines) to answer the given questions: +34. Compute the median and mean proportions of degrees awarded in Information Science. +35. Compute the median and mean proportions of degrees awarded in Education. +36. Compute the median and mean proportions of degrees awarded in Humanities. +37. Compute the median and mean proportions of degrees awarded in Biology. +38. Compute the median and mean proportions of degrees awarded in Physical Science. +39. Compute the median and mean proportions of degrees awarded in Social Science. +872 +8 • Statistics +Access for free at openstax.org + +8.4 Range and Standard Deviation +Figure 8.30 Measures of spread help us get a better understanding of test scores. (credit: "Standardized test exams form +with answers bubbled" by Marco Verch Professional Photographer/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Calculate the range of a dataset +2. +Calculate the standard deviation of a dataset +Measures of centrality like the mean can give us only part of the picture that a dataset paints. For example, let’s say +you’ve just gotten the results of a standardized test back, and your score was 138. The mean score on the test is 120. So, +your score is above average! But how good is it really? If all the scores were between 100 and 140, then you know your +score must be among the best. But if the scores ranged from 0 to 200, then maybe 140 is good, but not great (though +still above average). Knowing information about how the data are spread out can help us put a particular data value in +better context. In this section, we’ll look at two numbers that help us describe the spread in the data: the range and the +standard deviation. These numbers are called measures of dispersion. +The Range +Our first measure of dispersion is the range, or the difference between the maximum and minimum values in the set. +It’s the measure we used in the standardized test example above. +Let’s look at a couple of examples. +EXAMPLE 8.24 +Finding the Range +You survey some of your friends to find out how many hours they work each week. Their responses are: 5, 20, 8, 10, 35, +12. What is the range? +Solution +The maximum value in the set is 35 and the minimum is 5, so the range is +. +YOUR TURN 8.24 +1. On your morning commute, you decide to record how long you have to wait each time you get caught at a red +light. Here are the times in seconds: 12, 58, 35, 79, 21. What is the range? +For large datasets, finding the maximum and minimum values can be daunting. There are two ways to do it in a +spreadsheet. First, you can ask the spreadsheet program to sort the data from smallest to largest, then find the first and +last numbers on the sorted list. The second method uses built-in functions to find the minimum and maximum. +8.4 • Range and Standard Deviation +873 + +VIDEO +Find the Minimum and Maximum Using Google Sheets (https://openstax.org/r/min-max_Google-Sheet) +In either method, once you’ve found the maximum and minimum, all you have to do is subtract to find the range. +EXAMPLE 8.25 +Finding the Range with Google Sheets +The data in “AvgSAT” (https://openstax.org/r/Chapter8_Data-Sets) contains the average SAT score for students attending +every institution of higher learning in the US for which data is available. What is the range of these average SAT scores? +Solution +Step 1: To find the maximum, click on an empty cell in the spreadsheet, type “=MAX(”, and then click on the letter that +marks the top of the column containing the AvgSAT data. That inserts a reference to the column into our function. Then +we close the parentheses and hit the enter key. The formula is replaced with the maximum value in our data: 1566. +Step 2: Using the same process (but with “MIN” instead of “MAX”), we find the minimum value is 785. +Step 3: So, the range is +. +YOUR TURN 8.25 +1. The file “InState” (https://openstax.org/r/Chapter8_Data-Sets) contains in-state tuition costs (in dollars) for every +institution of higher learning in the US for which data is available. What is the range of these costs? +The range is very easy to compute, but it depends only on two of the data values in the entire set. If there happens to be +just one unusually high or low data value, then the range might give a distorted measure of dispersion. Our next +measure takes every single data value into account, making it more reliable. +The Standard Deviation +The standard deviation is a measure of dispersion that can be interpreted as approximately the average distance of +every data value from the mean. (This distance from the mean is the “deviation” in “standard deviation.”) +FORMULA +The standard deviation is computed as follows: +Here, +represents each data value, +is the mean of the data values, +is the number of data values, and the capital +sigma ( ) indicates that we take a sum. +To compute the standard deviation using the formula, we follow the steps below: +1. +Compute the mean of all the data values. +2. +Subtract the mean from each data value. +3. +Square those differences. +4. +Add up the results in step 3. +5. +Divide the result in step 4 by +6. +Take the square root of the result in step 5. +Let’s see that process in action. +874 +8 • Statistics +Access for free at openstax.org + +EXAMPLE 8.26 +Computing the Standard Deviation +You surveyed some of your friends to find out how many hours they work each week. Their responses were: 5, 20, 8, 10, +35, 12. What is the standard deviation? +Solution +Let’s follow the six steps mentioned previously to compute the standard deviation. +Step 1: Find the mean: +. +Step 2: Subtract the mean from each data value. To help keep track, let’s do this in a table. In the first row, we’ll list each +of our data values (and we’ll label the row +); in the second, we’ll subtract +from each data value. +5 +20 +8 +10 +35 +12 +−10 +5 +–7 +–5 +20 +–3 +Step 3: Square the differences. Let’s add a row to our table for those values: +5 +20 +8 +10 +35 +12 +−10 +5 +–7 +–5 +20 +–3 +100 +25 +49 +25 +400 +9 +Step 4: Add up those squares: +. +Step 5: Divide the sum by +. Since we have 6 data values, that gives us +. +Step 6: Take the square root of the result: +. +Thus, the standard deviation is +. +YOUR TURN 8.26 +1. On your morning commute, you decide to record how long you have to wait each time you get caught at a red +light. Here are the times in seconds: 12, 58, 35, 79, 21. +What is the standard deviation? +The computation for the standard deviation is complicated, even for just a small dataset. We’d never want to compute it +without technology for a large dataset! Luckily, technology makes this calculation easy. +VIDEO +Find the Standard Deviation Using Google Sheets (https://openstax.org/r/StanDev_Google-Sheet) +EXAMPLE 8.27 +Finding the Standard Deviation with Google Sheets +The data in “AvgSAT” (https://openstax.org/r/Chapter8_Data-Sets) contains the average SAT score for students attending +every institution of higher learning in the US for which data is available. What is the standard deviation of these average +SAT scores? +8.4 • Range and Standard Deviation +875 + +Solution +To find the standard deviation, we click in an empty cell in our spreadsheet and then type “=STDEV(”. Next, click on the +letter at the top of the column containing our data; this will put a reference to that column into our formula. Then close +the parentheses with and hit the enter key. The formula is replaced with the result: 125.517. +YOUR TURN 8.27 +1. The file “InState” (https://openstax.org/r/Chapter8_Data-Sets) contains in-state tuition costs (in dollars) for every +institution of higher learning in the US for which data is available. What is the standard deviation of these costs? +Check Your Understanding +24. Given the data 1, 4, 5, 5, and 10, find the range. +25. Given the data 1, 4, 5, 5, and 10, find the standard deviation using the process outlined in the definition. +Employees at a college help desk track the number of people who request assistance each week, as listed below: +142 +153 +158 +156 +141 +143 +139 +158 +156 +146 +137 +153 +136 +127 +157 +148 +132 +139 +155 +167 +143 +168 +133 +157 +138 +156 +164 +130 +148 +136 +26. Compute the range. +27. Compute the standard deviation. +The following are data on the admission rates of the different branch campuses in the University of California system, +along with the out-of-state tuition and fee cost. +Campus +Admission Rate +Cost ($) +Berkeley +0.1484 +43,176 +Davis +0.4107 +43,394 +Irvine +0.2876 +42,692 +Los Angeles +0.1404 +42,218 +Merced +0.6617 +42,530 +Riverside +0.5057 +42,819 +San Diego +0.3006 +43,159 +(source: https://data.ed.gov) +876 +8 • Statistics +Access for free at openstax.org + +Campus +Admission Rate +Cost ($) +Santa Barbara +0.322 +43,383 +Santa Cruz +0.4737 +42,952 +(source: https://data.ed.gov) +28. Compute the range of the admission rate. +29. Compute the standard deviation of the admission rate. +Using the data from Table 8.21, find the: +30. Range of the cost. +31. Standard deviation of the cost. +SECTION 8.4 EXERCISES +Given the data 81, 84, 89, 85, 86, 84: +1. Compute the range. +2. Compute the standard deviation using the formula. +Given the data 127, 167, 156, 158, 156, 157: +3. Compute the range. +4. Compute the standard deviation using the formula. +For the following exercises, use the data found in “TNSchools” (https://openstax.org/r/Chapter8_Data-Sets), which has +data on many institutions of higher education in the state of Tennessee. Here are what the columns represent: +Column Name +Description +AdmRate +Proportion of applicants that are admitted +UGEnr +Number of undergraduate students +PTUG +Proportion of undergraduates who attend part-time +InState +Tuition and fees for in-state students +OutState +Tuition and fees for out-of-state students +FacSal +Mean monthly faculty salary +Pell +Proportion of students receiving Pell Grants +MedDebt +Median student loan debt at degree completion +StartAge +Mean age at the time of entry +Female +Proportion of students who identify as female +(source: https://data.ed.gov) +5. Find the range of admissions rates. +6. Find the standard deviation of admissions rates. +7. Find the range of undergraduate enrollments. +8. Find the standard deviation of undergraduate enrollments. +9. Find the range of the proportions of undergraduates attending part-time. +8.4 • Range and Standard Deviation +877 + +10. Find the standard deviation of the proportions of undergraduates attending part-time. +11. Find the range of in-state tuition costs. +12. Find the standard deviation of in-state tuition costs. +13. Find the range of median debt. +14. Find the standard deviation of median debt. +For the following exercises, use the table below, which gives the final results for the 2021 National Women’s Soccer +League season. The columns are standings points (PTS; teams earn three points for a win and one point for a tie), wins +(W), losses (L), ties (T), goals scored by that team (GF), and goals scored against that team (GA). +Team +PTS +W +L +T +GF +GA +Portland Thorns FC +44 +13 +6 +5 +33 +17 +OL Reign +42 +13 +8 +3 +37 +24 +Washington Spirit +39 +11 +7 +6 +29 +26 +Chicago Red Stars +38 +11 +8 +5 +28 +28 +NJ/NY Gotham FC +35 +8 +5 +11 +29 +21 +North Carolina Courage +33 +9 +9 +6 +28 +23 +Houston Dash +32 +9 +10 +5 +31 +31 +Orlando Pride +28 +7 +10 +7 +27 +32 +Racing Louisville FC +22 +5 +12 +7 +21 +40 +Kansas City Current +16 +3 +14 +7 +15 +36 +15. Compute the standard deviation and range of points. +16. Compute the standard deviation and range of wins. +17. Compute the standard deviation and range of losses. +18. Compute the standard deviation and range of ties. +19. Compute the standard deviation and range of goals scored (GF). +20. Compute the standard deviation and range of goals against (GA). +878 +8 • Statistics +Access for free at openstax.org + +8.5 Percentiles +Figure 8.31 Two students graduating with the same class rank could be in different percentiles depending on the school +population. (credit: "graduation caps" by John Walker/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Compute percentiles. +2. +Solve application problems involving percentiles. +A college admissions officer is comparing two students. The first, Anna, finished 12th in her class of 235 people. The +second, Brian, finished 10th in his class of 170 people. Which of these outcomes is better? Certainly 10 is less than 12, +which favors Brian, but Anna’s class was much bigger. In fact, Anna beat out 223 of her classmates, which is +of her classmates, while Brian bested 160 out of 170 people, or 94%. Comparing the proportions of the data values that +are below a given number can help us evaluate differences between individuals in separate populations. These +proportions are called percentiles. If +of the values in a dataset are less than a number +, then we say that +is at the +th percentile. +Finding Percentiles +There are some other terms that are related to "percentile" with meanings you may infer from their roots. Remember +that the word percent means “per hundred.” This reflects that percentiles divide our data into 100 pieces. The word +quartile has a root that means “four.” So, if a data value is at the first quantile of a dataset, that means that if you break +the data into four parts (because of the quart-), this data value comes after the first of those four parts. In other words, +it’s greater than 25% of the data, placing it at the 25th percentile. Quintile has a root meaning “five,” so a data value at +the third quintile is greater than three-fifths of the data in the set. That would put it at the 60th percentile. The general +term for these is quantiles (the root quant– means “number”). +In Mean, Median, and Mode, we defined the median as a number that is greater than no more than half of the data in a +dataset and is less than no more than half of the data in the dataset. With our new term, we can more easily define it: +The median is the value at the 50th percentile (or second quartile). +Let’s look at some examples. +EXAMPLE 8.28 +Finding Percentiles +Consider the dataset 5, 8, 12, 1, 2, 16, 2, 15, 20, 22. +1. +At what percentile is the value 5? +8.5 • Percentiles +879 + +2. +What value is at the 60th percentile? +Solution +Before we can answer these two questions, we must put the data in increasing order: 1, 2, 2, 5, 8, 12, 15, 16, 20, 22. +1. +There are three values (1, 2, and 2) in the set that are less than 5, and there are ten values in the set. Thus, 5 is at the +percentile. +2. +To find the value at the 60th percentile, we note that there are ten data values, and 60% of ten is six. Thus, the +number we want is greater than exactly six of the data values. Thus, the 60th percentile is 15. +YOUR TURN 8.28 +Consider the dataset 2, 5, 8, 16, 12, 1, 8, 6, 15, 4. +1. What value is at the 80th percentile? +2. At what percentile is the value 12? +In each of the examples above, the computations were made easier by the fact that the we were looking for percentiles +that “came out evenly” with respect to the number of values in our dataset. Things don’t always work out so cleanly. +Further, different sources will define the term percentile in different ways. In fact, Google Sheets has three built-in +functions for finding percentiles, none of which uses our definition. Some of the definitions you’ll see differ in the +inequality that is used. Ours uses “less than or equal to,” while others use “less than” (these correspond roughly to +Google Sheets’ ‘PERCENTILE.INC’ and ‘PERCENTILE.EXC’). Some of them use different methods for interpolating values. +(This is what we did when we first computed medians without technology; if there were an even number of data values +in our dataset, found the mean of the two values in the middle. This is an example of interpolation. Most computerized +methods use this technique.) Other definitions don’t interpolate at all, but instead choose the closest actual data value to +the theoretical value. Fortunately, for large datasets, the differences among the different techniques become very small. +So, with all these different possible definitions in play, what will we use? For small datasets, if you’re asked to compute +something involving percentiles without technology , use the technique we used in the previous example. In all other +cases, we’ll keep things simple by using the built-in ‘PERCENTILE’ and ‘PERCENTRANK’ functions in Google Sheets (which +do the same thing as the ‘PERCENTILE.INC’ and ‘PERCENTRANK.INC’ functions; they’re “inclusive, interpolating” +definitions). +VIDEO +Using RANK, PERCENTRANK, and PERCENTILE in Google Sheets (https://openstax.org/r/Using-RANK-PERCENTRANK- +and-PERCENTILE) +EXAMPLE 8.29 +Using Google Sheets to Compute Percentiles: Average SAT Scores +The data in “AvgSAT” (https://openstax.org/r/Chapter8_Data-Sets) contains the average SAT score for students attending +every institution of higher learning in the US for which data is available. +1. +What score is at the 3rd quartile? +2. +What score is at the 40th percentile? +3. +At what percentile is Albion College in Michigan (average SAT: 1132)? +4. +At what percentile is Oregon State University (average SAT: 1205)? +Solution +1. +The 3rd quartile is the 75th percentile, so we’ll use the PERCENTILE function. Click on an empty cell, and type +“=PERCENTILE(“. Next, enter the data: click on the letter at the top of the column containing the average SAT scores. +Then, tell the function which percentile we want; it needs to be entered as a decimal. So, type a comma (to separate +the two pieces of information we’re giving this function), then type “0.75” and close the parentheses with “)”. The +result should look like this (assuming the data are in column C): “=PERCENTILE(C:C, 0.75)”. When you hit the enter +key, the formula will be replaced with the average SAT score at the 75th percentile: 1199. +2. +Using the PERCENTILE function, we find that an average SAT of 1100 is at the 40th percentile. +880 +8 • Statistics +Access for free at openstax.org + +3. +Since we want to know the percentile for a particular score, we’ll use the PERCENTRANK function. Like the +PERCENTILE function, we need to give PERCENTRANK two pieces of information: the data, and the value we care +about. So, click on an empty cell, type “=PERCENTRANK(“, and then click on the letter at the top of the column +containing the data. Next, type a comma and then the value we want to find the percentile for; in this case, we’ll +type “, 1132”. Finally, close the parentheses with “)” and hit the enter key. The formula will be replaced with the +information we want: an average SAT of 1132 is at the 54th percentile. +4. +Using the PERCENTRANK function, an average SAT of 1205 is at the 76.1th percentile. +YOUR TURN 8.29 +Looking again at the “AvgSAT” (https://openstax.org/r/Chapter8_Data-Sets) dataset: +1. What score is at the 15th percentile? +2. What score is at the 90th percentile? +3. At what percentile is the University of Missouri (Columbia campus), whose average SAT score is 1244? +4. At what percentile is Rice University in Texas, whose average SAT score is 1513? +EXAMPLE 8.30 +Using Google Sheets to Compute Percentiles: In-State Tuition +The dataset "InState" (https://openstax.org/r/Chapter8_Data-Sets) contains the in-state tuitions of every college and +university in the country that reported that data to the Department of Education. Use that dataset to answer these +questions. +1. +What tuition is at the second quintile? +2. +What tuition is at the 95th percentile? +3. +At what percentile is Walla Walla University in Washington (in-state tuition: $28,035)? +4. +At what percentile is the College of Saint Mary in Nebraska (in-state tuition: $20,350)? +Solution +1. +The second quintile is the 40th percentile; using PERCENTILE in Google Sheets, we get $8,400. +2. +Using PERCENTILE again, we get $44,866. +3. +Using PERCENTRANK, we get the 81.6th percentile. +4. +Using PERCENTRANK, we get the 74.8th percentile. +YOUR TURN 8.30 +Looking again at the "InState" (https://openstax.org/r/Chapter8_Data-Sets) dataset, answer these questions. +1. What tuition is at the 10th percentile? +2. What tuition is at the fourth quintile? +3. At what percentile is the main campus of New Mexico State University (in-state tuition: $6,686)? +4. At what percentile is Bowdoin College in Maine (in-state tuition: $53,922)? +Check Your Understanding +Given the data 10, 12, 14, 18, 21, 23, 24, 25, 29, and 30, compute the following without technology: +32. The value at the 30th percentile +33. The value at the first quintile +34. At what percentile 29 falls +35. At what percentile 24 falls +For the remainder of these problems, use the dataset "MLB2019," (https://openstax.org/r/Chapter8_Data-Sets) which +gives the number of wins for each Major League Baseball team in the 2019 season. Use Google Sheets to compute your +answers. +8.5 • Percentiles +881 + +36. How many wins is at the 30th percentile? +37. How many wins is at the 90th percentile? +38. How many wins is at the first quartile? +39. At what percentile are the Chicago Cubs (CHC, 84 wins)? +40. At what percentile are the Tampa Bay Rays (TBR, 96 wins)? +41. At what percentile are the Toronto Blue Jays (TOR, 67 wins)? +SECTION 8.5 EXERCISES +For the following exercises, use the following twenty data values to answer the questions without technology: 1, 4, 6, 7, +12, 15, 21, 25, 29, 30, 31, 33, 39, 43, 44, 45, 51, 55, 60, 63 +1. What data value is at the 10th percentile? +2. What data value is at the 55th percentile? +3. What data value is at the 90th percentile? +4. What data value is at the 30th percentile? +5. What data value is at the first quartile? +6. What data value is at the third quintile? +7. At what percentile is 29? +8. At what percentile is 55? +9. At what percentile is 4? +10. At what percentile is 51? +For the following exercises, use the data in "TNSchools" (https://openstax.org/r/Chapter8_Data-Sets), which has data on +many institutions of higher education in the state of Tennessee. Here are what the columns represent: +Column Name +Description +AdmRate +Proportion of applicants that are admitted +UGEnr +Number of undergraduate students +PTUG +Proportion of undergraduates who attend part-time +InState +Tuition and fees for in-state students +OutState +Tuition and fees for out-of-state students +FacSal +Mean monthly faculty salary +Pell +Proportion of students receiving Pell Grants +MedDebt +Median student loan debt at degree completion +StartAge +Mean age at the time of entry +Female +Proportion of students who identify as female +(source: https://data.ed.gov) +11. What admission rate is at the second quintile? +12. What admission rate is at the 80th percentile? +13. What admission rate is at the 90th percentile? +14. At what percentile is East Tennessee State University for admission rate? +15. At what percentile is Rhodes College for admission rate? +16. At what percentile is Freed-Hardeman University for admission rate? +17. What proportion of part-time undergraduate enrollment is at the third quartile? +18. What proportion of part-time undergraduate enrollment is at the 15th percentile? +19. What proportion of part-time undergraduate enrollment is at the 40th percentile? +882 +8 • Statistics +Access for free at openstax.org + +20. At what percentile is Lee University for proportion of part-time undergraduate enrollment? +21. At what percentile is Fisk University for proportion of part-time undergraduate enrollment? +22. At what percentile is Middle Tennessee State University for proportion of part-time undergraduate enrollment? +23. What median student loan debt is at the 10th percentile? +24. What median student loan debt is at the fourth quintile? +25. What median student loan debt is at the 85th percentile? +26. At what percentile is Carson-Newman College for median student loan debt? +27. At what percentile is Austin Peay State University for median student loan debt? +28. At what percentile is Belmont University for median student loan debt? +29. What mean starting age is at the first quartile? +30. What mean starting age is at the 67th percentile? +31. What mean starting age is at the 35th percentile? +32. At what percentile is the University of the South for mean starting age? +33. At what percentile is Lincoln Memorial University for mean starting age? +34. At what percentile is the University of Tennessee-Chattanooga for mean starting age? +35. What proportion of students who identify as female is at the third quintile? +36. What proportion of students who identify as female is at the 12th percentile? +37. What proportion of students who identify as female is at the 85th percentile? +38. At what percentile is Martin Methodist College for proportion of students who identify as female? +39. At what percentile is Tennessee Technological University for proportion of students who identify as female? +40. At what percentile is Maryville College for proportion of students who identify as female? +8.6 The Normal Distribution +Figure 8.32 Symmetric, bell-shaped distributions arise naturally in many different situations, including coin flips. +Learning Objectives +After completing this section, you should be able to: +1. +Describe the characteristics of the normal distribution. +2. +Apply the 68-95-99.7 percent groups to normal distribution datasets. +3. +Use the normal distribution to calculate a +-score. +4. +Find and interpret percentiles and quartiles. +Many datasets that result from natural phenomena tend to have histograms that are symmetric and bell-shaped. +Imagine finding yourself with a whole lot of time on your hands, and nothing to keep you entertained but a coin, a +pencil, and paper. You decide to flip that coin 100 times and record the number of heads. With nothing else to do, you +repeat the experiment ten times total. Using a computer to simulate this series of experiments, here’s a sample for the +number of heads in each trial: +54, 51, 40, 42, 53, 50, 52, 52, 47, 54 +It makes sense that we’d get somewhere around 50 heads when we flip the coin 100 times, and it makes sense that the +result won’t always be exactly 50 heads. In our results, we can see numbers that were generally near 50 and not always +50, like we thought. +8.6 • The Normal Distribution +883 + +Moving Toward Normality +Let’s take a look at a histogram for the dataset in our section opener: +Figure 8.33 +This is interesting, but the data seem pretty sparse. There were no trials where you saw between 43 and 47 heads, for +example. Those results don’t seem impossible; we just didn’t flip enough times to give them a chance to pop up. So, let’s +do it again, but this time we'll perform 100 coin flips 100 times. Rather than review all 100 results, which could be +overwhelming, let's instead visualize the resulting histogram. +Figure 8.34 +From the histogram, we see that most of the trials resulted in between, say, 44 and 56 heads. There were some more +unusual results: one trial resulted in 70 heads, which seems really unlikely (though still possible!). But we’re starting to +maybe get a sense of the distribution. More data would help, though. Let’s simulate another 900 trials and add them to +the histogram! +884 +8 • Statistics +Access for free at openstax.org + +Figure 8.35 +We can still see that 70 is a really unusual observation, though we came close in another trial (one that had 68 heads). +Now, the distribution is coming more into focus: It looks quite symmetric and bell-shaped. Let’s just go ahead and take +this thought experiment to an extreme conclusion: 10,000 trials. +Figure 8.36 +The distribution is pretty clear now. Distributions that are symmetric and bell-shaped like this pop up in all sorts of +natural phenomena, such as the heights of people in a population, the circumferences of eggs of a particular bird +species, and the numbers of leaves on mature trees of a particular species. All of these have bell-shaped distributions. +Additionally, the results of many types of repeated experiments generally follow this same pattern, as we saw with the +coin-flipping example; this fact is the basis for much of the work done by statisticians. It’s a fact that’s important enough +to have its own name: the Central Limit Theorem. +8.6 • The Normal Distribution +885 + +PEOPLE IN MATHEMATICS +John Kerrich +Having enough time on your hands to actually perform this coin-flipping experiment may sound far-fetched, but the +English mathematician John Kerrich found himself in just such a situation. While he was studying abroad in Denmark +in 1940, that country was invaded by the Germans. Kerrich was captured and placed in an internment camp, where he +remained for the duration of the war. Kerrich knew that he had all kinds of time on his hands. He also studied +statistics, so he knew what should happen theoretically if he flipped a coin many, many times. He also knew of nobody +who had ever tested that theory with an actual, large-scale experiment. So, he did it: While he was incarcerated, +Kerrich flipped a regular coin 10,000 times and recorded the results. Sure enough, the theory held up! +The Normal Distribution +In the coin flipping example above, the distribution of the number of heads for 10,000 trials was close to perfectly +symmetric and bell-shaped: +Figure 8.37 +Because distributions with this shape appear so often, we have a special name for them: normal distributions. Normal +distributions can be completely described using two numbers we’ve seen before: the mean of the data and the standard +deviation of the data. You may remember that we described the mean as a measure of centrality; for a normal +distribution, the mean tells us exactly where the center of the distribution falls. The peak of the distribution happens at +the mean (and, because the distribution is symmetric, it’s also the median). The standard deviation is a measure of +dispersion; for a normal distribution, it tells us how spread out the histogram looks. To illustrate these points, let’s look at +some examples. +EXAMPLE 8.31 +Identifying the Mean of a Normal Distribution +This graph shows three normal distributions. What are their means? +Figure 8.38 +Solution +Step 1: Take a look at the three curves on the graph. Since the mean of a normal distribution occurs at the peak, we +886 +8 • Statistics +Access for free at openstax.org + +should look for the highest point on each distribution. Let’s draw a line from each curve's peak down to the axis, so we +can see where these peaks occur: +Figure 8.39 +Step 2: The peak of the red (leftmost) distribution occurs over the number 1 on the horizontal axis. Thus, the mean of +the red distribution is 1. Similarly, the mean of the blue (middle) distribution is 2, and the mean of the yellow (rightmost) +distribution is 3. +YOUR TURN 8.31 +1. Identify the means of these three distributions: +EXAMPLE 8.32 +Identifying the Standard Deviation of a Normal Distribution +This graph shows three distributions, all with mean 2. What are their standard deviations? +Figure 8.40 +Solution +Step 1: Identifying the standard deviation from a graph can be a little bit tricky. Let’s focus in on the yellow (lowest +peaked) curve: +8.6 • The Normal Distribution +887 + +Figure 8.41 +Step 2: Notice that the graph curves downward in the middle, and curves upwards on the ends. Highlighted in red is the +part that curves downward and in green, the part that curves upward: +Figure 8.42 +Step 3: The places where the graph changes from curving up to curving down (or vice versa) are called inflection points. +Let’s identify where those occur by dropping a line straight down from each: +Figure 8.43 +Step 4: We can estimate that the inflection points occur at +and +; the mean is at +(as shown by the +middle dotted line). The difference between the mean and the location of either inflection point is the standard +deviation; since +, we conclude that the standard deviation of the green distribution is 3. +Step 5: Now, looking at the other two graphs, let's first identify the inflection points: +888 +8 • Statistics +Access for free at openstax.org + +Figure 8.44 +Step 6: The red (tallest peaked) distribution has inflection points at 1 and 3, and the mean is 2. Thus, the standard +deviation of the red distribution is +. The blue (lower peaked) distribution has inflection points at 0 and +4, and its mean is also 2. So, the standard deviation of the blue distribution is 2. +YOUR TURN 8.32 +1. Estimate the standard deviations of this normal distribution, centered at 5: +Let’s put it all together to identify a completely unknown normal distribution. +EXAMPLE 8.33 +Identifying the Mean and Standard Deviation of a Normal Distribution +Using the graph, identify the mean and standard deviation of the normal distribution. +Figure 8.45 +8.6 • The Normal Distribution +889 + +Solution +Step 1: Let’s start by putting dots on the graph at the peak and at the inflection points, then drop lines from those points +straight down to the axis: +Figure 8.46 +Step 2: From the red (middle) line, we can see that the mean of this distribution is 55. The blue (outermost) lines are each +3 units away from the mean (at 52 and 58), so the standard deviation is 3. +YOUR TURN 8.33 +1. Identify the mean and standard deviation of this distribution. Anything within 5 on the standard deviation is +acceptable. +Properties of Normal Distributions: The 68-95-99.7 Rule +The most important property of normal distributions is tied to its standard deviation. If a dataset is perfectly normally +distributed, then 68% of the data values will fall within one standard deviation of the mean. For example, suppose we +have a set of data that follows the normal distribution with mean 400 and standard deviation 100. This means 68% of the +data would fall between the values of 300 (one standard deviation below the mean: +) and 500 (one +standard deviation above the mean: +). Looking at the histogram below, the shaded area represents 68% +of the total area under the graph and above the axis: +890 +8 • Statistics +Access for free at openstax.org + +Figure 8.47 +Since 68% of the area is in the shaded region, this means that +of the area is found in the unshaded +regions. We know that the distribution is symmetric, so that 32% must be divided evenly into the two unshaded tails: +16% in each. +Of course, datasets in the real world are never perfect; when dealing with actual data that seem to follow a symmetric, +bell-shaped distribution, we’ll give ourselves a little bit of wiggle room and say that approximately 68% of the data fall +within one standard deviation of the mean. +The rule for one standard deviation can be extended to two standard deviations. Approximately 95% of a normally +distributed dataset will fall within 2 standard deviations of the mean. If the mean is 400 and the standard deviation is +100, that means 95% calculation describes the way we compute standardized scores. (two standard deviations below the +mean: +) and 600 (two standard deviations above the mean: +). We can visualize +this in the following histogram: +Figure 8.48 +As before, since 95% of the data are in the shaded area, that leaves 5% of the data to go into the unshaded tails. Since +the histogram is symmetric, half of the 5% (that’s 2.5%) is in each. +We can even take this one step further: 99.7% of normally distributed data fall within 3 standard deviations of the mean. +In this example, we’d see 99.7% of the data between 100 (calculated as +) and 700 (calculated as +). We can see this in the histogram below, although you may need to squint to find the unshaded +bits in the tails! +8.6 • The Normal Distribution +891 + +Figure 8.49 +This observation is formally known as the 68-95-99.7 Rule. +EXAMPLE 8.34 +Using the 68-95-99.7 Rule to Find Percentages +1. +If data are normally distributed with mean 8 and standard deviation 2, what percent of the data falls between 4 and +12? +2. +If data are normally distributed with mean 25 and standard deviation 5, what percent of the data falls between 20 +and 30? +3. +If data are normally distributed with mean 200 and standard deviation 15, what percent of the data falls between +155 and 245? +Solution +1. +Let’s look at a table that sets out the data values that are even multiples of the standard deviation (SD) above and +below the mean: +Mean +2 +4 +6 +8 +10 +12 +14 +Since 4 and 12 represent two standard deviations above and below the mean, we conclude that 95% of the data will +fall between them. +2. +Let’s build another table: +Mean +10 +15 +20 +25 +30 +35 +40 +We can see that 20 and 30 represent one standard deviation above and below the mean, so 68% of the data fall in +that range. +3. +Let’s make one more table: +Mean +155 +170 +185 +200 +215 +230 +245 +Since 155 and 245 are three standard deviations above and below the mean, we know that 99.7% of the data will fall +between them. +892 +8 • Statistics +Access for free at openstax.org + +YOUR TURN 8.34 +1. If data are distributed normally with mean 0 and standard deviation 3, what percent of the data fall between +–9 and 9? +2. If data are distributed normally with mean 50 and standard deviation 10, what percent of the data fall +between 30 and 70? +3. If data are distributed normally with mean 60 and standard deviation 5, what percent of the data fall between +55 and 65? +EXAMPLE 8.35 +Using the 68-95-99.7 Rule to Find Data Values +1. +If data are distributed normally with mean 100 and standard deviation 20, between what two values will 68% of the +data fall? +2. +If data are distributed normally with mean 0 and standard deviation 15, between what two values will 95% of the +data fall? +3. +If data are distributed normally with mean 14 and standard deviation 2, between what two values will 99.7% of the +data fall? +Solution +1. +The 68-95-99.7 Rule tells us that 68% of the data will fall within one standard deviation of the mean. So, to find the +values we seek, we’ll add and subtract one standard deviation from the mean: +and +. Thus, we know that 68% of the data fall between 80 and 120. +2. +Using the 68-95-99.7 Rule again, we know that 95% of the data will fall within 2 standard deviations of the mean. +Let’s add and subtract two standard deviations from that mean: +and +. So, 95% of +the data will fall between -30 and 30. +3. +Once again, the 68-95-99.7 Rule tells us that 99.7% of the data will fall within three standard deviations of the mean. +So, let’s add and subtract three standard deviations from the mean: and +. Thus, we conclude that +99.7% of the data will fall between 8 and 20. +YOUR TURN 8.35 +1. If data are distributed normally with mean 70 and standard deviation 5, between what two values will 68% of +the data fall? +2. If data are distributed normally with mean 40 and standard deviation 7, between what two values will 95% of +the data fall? +3. If data are distributed normally with mean 200 and standard deviation 30, between what two values will +99.7% of the data fall? +There are more problems we can solve using the 68-95-99.7 Rule. but first we must understand what the rule implies. +Remember, the rule says that 68% of the data falls within one standard deviation of the mean. Thus, with normally +distributed data with mean 100 and standard deviation 10, we have this distribution: +8.6 • The Normal Distribution +893 + +Figure 8.50 +Since we know that 68% of the data lie within one standard deviation of the mean, the implication is that 32% of the data +must fall beyond one standard deviation away from the mean. Since the histogram is symmetric, we can conclude that +half of the 32% (or 16%) is more than one standard deviation above the mean and the other half is more than one +standard deviation below the mean: +Figure 8.51 +Further, we know that the middle 68% can be split in half at the peak of the histogram, leaving 34% on either side: +Figure 8.52 +So, just the “68” part of the 68-95-99.7 Rule gives us four other proportions in addition to the 68% in the rule. Similarly, +the “95” and “99.7” parts each give us four more proportions: +894 +8 • Statistics +Access for free at openstax.org + +Figure 8.53 +Figure 8.54 +We can put all these together to find even more complicated proportions. For example, since the proportion between +100 and 120 is 47.5% and the proportion between 100 and 110 is 34%, we can subtract to find that the proportion +between 110 and 120 is +: +Figure 8.55 +EXAMPLE 8.36 +Finding Other Proportions Using the 68-95-99.7 Rule +Assume that we have data that are normally distributed with mean 80 and standard deviation 3. +1. +What proportion of the data will be greater than 86? +2. +What proportion of the data will be between 74 and 77? +3. +What proportion of the data will be between 74 and 83? +Solution +Before we can answer these questions, we must mark off sections that are multiples of the standard deviation away +from the mean: +8.6 • The Normal Distribution +895 + +Figure 8.56 +1. +To figure out what proportion of the data will be greater than 86, let's start by shading in the area of data that are +above 86 in our figure, or the data more than two standard deviations above the mean. +Figure 8.57 +We saw in Figure 8.47 that this proportion is 2.5%. +2. +To figure out what proportion of the data will be between 74 and 77, let's start by shading in that area of data. These +are data that are more than one but less than two standard deviations below the mean. +Figure 8.58 +From Figure 8.47, we know that the proportion of data less than two standard deviations below the mean is 47.5%. +And, from YOUR TURN 8.33, we know that 34% of the data is less than one standard deviation below the mean: +896 +8 • Statistics +Access for free at openstax.org + +Figure 8.59 +Subtracting, we see that the proportion of data between 74 and 77 is 13.5%. +3. +To figure out what proportion of the data will be between 74 and 83, let's start by shading in that area of data in our +figure. +Figure 8.60 +Next, we'll break this region into two pieces at the mean: +Figure 8.61 +From Figure 8.47, we know the blue (leftmost) region represents 47.5% of the data. And, using YOUR TURN 8.33, we +get that the red (rightmost) region covers 34% of the data. Adding those together, the proportion we want is 81.5%. +YOUR TURN 8.36 +Suppose we have data that are normally distributed with mean 500 and standard deviation 100. What proportions of +the data fall in these ranges? +8.6 • The Normal Distribution +897 + +1. 300 to 500 +2. 600 to 800 +3. 400 to 700 +Standardized Scores +When we want to apply the 68-95-99.7 Rule, we must first figure out how many standard deviations above or below the +mean our data fall. This calculation is common enough that it has its own name: the standardized score. Values above +the mean have positive standardized scores, while those below the mean have negative standardized scores. Since it's +common to use the letter +to represent a standard score, this value is also often referred to as a +-score. +So far, we’ve only really considered +-scores that are whole numbers, but in general they can be any number at all. For +example, if we have data that are normally distributed with mean 80 and standard deviation 6, the value 85 is five units +above the mean, which is less than one standard deviation. Dividing by the standard deviation, we get +. Since 85 is +of +one standard deviation above the mean, we’d say that the standardized score for 85 is +(which is positive, since +). This calculation describes the way we compute standardized scores. +FORMULA +If +is a member of a normally distributed dataset with mean +and standard deviation +, then the standardized score +for +is +If you know a +-score but not the original data value +, you can find it by solving the previous equation for +: +The symbols +and +are the Greek letters mu and sigma. They are the analogues of the English letters +and , +which stand for mean and standard deviation. +If you convert every data value in a dataset into its +-score, the resulting set of data will have mean 0 and standard +deviation 1. This is why we call these standardized scores: the normal distribution with mean 0 and standard +deviation 1 is often called the standard normal distribution. +EXAMPLE 8.37 +Standardizing Data +Suppose we have data that are normally distributed with mean 50 and standard deviation 6. Compute the standardized +scores (rounded to three decimal places) for these data values: +1. +52 +2. +40 +3. +68 +Solution +For each of these, we’ll plug the given values into the formula. Remember, the mean is +and the standard +deviation is +: +1. +2. +3. +898 +8 • Statistics +Access for free at openstax.org + +YOUR TURN 8.37 +1. Suppose we have data that are normally distributed with mean 75 and standard deviation 5. Compute +standardized scores for each of these data values: 66, 83, and 72. +EXAMPLE 8.38 +Converting Standardized Scores to Original Values +Suppose we have data that are normally distributed with mean 10 and standard deviation 2. Convert the following +standardized scores into data values. +1. +1.4 +2. +−0.9 +3. +3.5 +Solution +We’ll use the formula previously introduced to convert +-scores into +-values. In this case, the mean is +and the +standard deviation is +: +1. +2. +3. +YOUR TURN 8.38 +1. Suppose you have a normally distributed dataset with mean 2 and standard deviation 20. Convert these +standardized scores to data values: –2.3, 1.4, and 0.2. +Using Google Sheets to Find Normal Percentiles +The 68-95-99.7 Rule is great when we’re dealing with whole-number +-scores. However, if the +-score is not a whole +number, the Rule isn’t going to help us. Luckily, we can use technology to help us out. We’ll talk here about the built-in +functions in Google Sheets, but other tools work similarly. +Let’s say we’re working with normally distributed data with mean 40 and standard deviation 7, and we want to know at +what percentile a data value of 50 would fall. That corresponds to finding the proportion of the data that are less than +50. If we create our histogram and mark off whole-number multiples of the standard deviation like we did before, we’ll +see why the 68-95-99.7 Rule isn’t going to help: +Figure 8.62 +Since 50 doesn’t line up with one of our lines, the 68-95-99.7 Rule fails us. Looking back at Figure 8.47 and Figure 8.48, +the best we can say is that 50 is between the 84th and 99.5th percentiles, but that’s a pretty wide range. Google Sheets +has a function that can help; it’s called NORM.DIST. Here’s how to use it: +1. +Click in an empty cell in your worksheet. +8.6 • The Normal Distribution +899 + +2. +Type “=NORM.DIST(“ +3. +Inside the parentheses, we must enter a list of four things, separated by commas: the data value, the mean, the +standard deviation, and the word “TRUE”. These have to be entered in this order! +4. +Close the parentheses, and hit Enter. The result is then displayed in the cell; convert it to a percent to get the +percentile. +So, for our example, we should type “=NORM.DIST(50, 40, 7, TRUE)” into an empty cell, and hit Enter. The result is +0.9234362745; converting to a percent and rounding, we can conclude that 50 is at the 92nd percentile. Let’s walk +through a few more examples. +EXAMPLE 8.39 +Using Google Sheets to Find Percentiles +Suppose we have data that are normally distributed with mean 28 and standard deviation 4. At what percentile do each +of the following data values fall? +1. +30 +2. +23 +3. +35 +Solution +1. +By entering “=NORM.DIST(30, 28, 4, TRUE)” we find that 30 is at the 69th percentile. +2. +By entering “=NORM.DIST(23, 28, 4, TRUE)” we find that 23 is at the 11th percentile. +3. +By entering “=NORM.DIST(35, 28, 4, TRUE)” we find that 35 is at the 96th percentile. +YOUR TURN 8.39 +1. Suppose you have data that are normally distributed with mean 20 and standard deviation 6. Determine at what +percentiles these data values fall: 25, 12, and 31. +Google Sheets can also help us go the other direction: If we want to find the data value that corresponds to a given +percentile, we can use the NORM.INV function. For example, if we have normally distributed data with mean 150 and +standard deviation 25, we can find the data value at the 30th percentile as follows: +1. +Click on an empty cell in your worksheet. +2. +Type “=NORM.INV(“ +3. +Inside the parentheses, we’ll enter a list of three numbers, separated by commas: the percentile in question +expressed as a decimal, the mean, and the standard deviation. These must be entered in this order! +4. +Close the parentheses and hit Enter. The desired data value will be in the cell! +In our example, we want the 30th percentile; converting 30% to a decimal gives us 0.3. So, we’ll type “=NORM.INV(0.3, +150, 25)” to get 136.8899872; let’s round that off to 137. +EXAMPLE 8.40 +Using Google Sheets to Find the Data Value Corresponding to a Percentile +Suppose we have data that are normally distributed with mean 47 and standard deviation 9. Find the data values +(rounded to the nearest tenth) corresponding to these percentiles: +1. +75th (that’s the third quartile) +2. +12th +3. +90th +Solution +1. +By entering “=NORM.INV(0.75, 47, 9)” we find that 53.1 is at the 75th percentile. +2. +By entering “=NORM.INV(0.12, 47, 9)” we find that 36.4 is at the 12th percentile. +3. +By entering “=NORM.INV(0.9, 47, 9)” we find that 58.5 is at the 90th percentile. +900 +8 • Statistics +Access for free at openstax.org + +YOUR TURN 8.40 +1. Suppose we have data that are normally distributed with mean 5 and standard deviation 1.6. Identify which data +values (rounded to the nearest tenth) correspond to these percentiles: 25th (the first quartile), 80th, and 10th. +Check Your Understanding +For each of these problems, assume we’re working with normally distributed data with mean 100 and standard +deviation 12. +42. What percentage of the data falls between 76 and 124? Use the 68-95-99.7 Rule. +43. What percentage of the data falls between 100 and 112? Use the 68-95-99.7 Rule. +44. At what percentile does 112 fall? Use the 68-95-99.7 Rule. +45. What’s the +-score of the data value 107? Round to three decimal places. +46. What data value’s +-score is –2.4? +47. At what percentile does 107 fall? Use Google Sheets (or another technology). +48. What data value is at the 90th percentile? Use Google Sheets (or another technology), and round to the nearest +hundredth. +SECTION 8.6 EXERCISES +For the following exercises, explain how you can tell the histogram does NOT represent normally distributed data. +1. +2. +For the following exercises, use the 68-95-99.7 Rule to answer the given questions about normally distributed data with +mean 100 and standard deviation 5. +3. What proportion of the data fall between 95 and 105? +4. What proportion of the data fall between 90 and 110? +5. what proportion of the data fall between 85 and 100? +6. What proportion of the data fall between 110 and 120? +7. What proportion of the data are less than 90? +8. What proportion of the data are greater than 105? +9. What proportion of the data fall between 90 and 105? +10. What proportion of the data are between 95 and 115? +8.6 • The Normal Distribution +901 + +For the following exercises, use the 68-95-99.7 Rule to answer the given questions about normally distributed data with +mean 9 and standard deviation 1. +11. What proportion of the data are less than 7? +12. What proportion of the data are greater than 12? +13. What proportion of the data are between 6 and 12? +14. What proportion of the data are between 8 and 9? +15. What proportion of the data are between 6 and 8? +16. What proportion of the data are between 8 and 11? +17. What proportion of the data are less than 10? +18. What proportion of the data are greater than 6? +In the following exercises, convert the given data values to standardized scores. Assume the data are distributed +normally with mean 15 and standard deviation 3. Round to the nearest hundredth. +19. +20. +21. +22. +23. +24. +25. +26. +In the following exercises, convert the given +-scores to data values. Assume the data are distributed normally with +mean 15 and standard deviation 3. +27. +28. +29. +30. +31. +32. +33. +34. +For the following exercises, answer the questions about normally distributed data with mean 200 and standard +deviation 20. Round percentiles to the nearest whole number and round data values to the nearest tenth. +35. At what percentile is +? +36. At what percentile is +? +37. At what percentile is +? +38. At what percentile is +? +39. At what percentile is +? +40. At what percentile is +? +41. At what percentile is +? +42. At what percentile is +? +43. What data value is at the 40th percentile? +44. What data value is at the 10th percentile? +45. What data value is at the 55th percentile? +46. What data value is at the 95th percentile? +47. What data value is at the 33rd percentile? +48. What data value is at the third quartile? +49. What data value is at the 65th percentile? +50. What data value is at the 99th percentile? +902 +8 • Statistics +Access for free at openstax.org + +8.7 Applications of the Normal Distribution +Figure 8.63 Standardized test results generally adhere to the normal distribution. (credit: “Taking a Test” by Marco Verch +Professional Photographer/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Apply the normal distribution to real-world scenarios. +As we saw in The Normal Distribution, the word “standardized” is closely associated with the normal distribution. This is +why tests like college entrance exams, state achievement tests for K–12 students, and Advanced Placement tests are +often called “standardized tests”: scores are assigned in a way that forces them to follow a normal distribution, with a +mean and standard deviation that are consistent from year to year. Standardization also allows people like college +admissions officers to directly compare an applicant who took the ACT (a college entrance exam) to an applicant who +instead chose to take the SAT (a different college entrance exam). Standardization allows us to compare individuals from +different groups; this is among the most important applications of the normal distribution. We’ll explore this and other +real-world uses of the normal distribution in this section. +College Entrance Exams +There are two good ways to compare two data values from different groups: using +-scores and using percentiles. The +two methods will always give consistent results (meaning that we won’t find, for example, that the first value is better +using +-scores but the second value is better using percentiles), so use whichever method is more comfortable for you. +EXAMPLE 8.41 +Evaluating College Entrance Exam Scores +According to the Digest of Education Statistics (https://openstax.org/r/nces_ed_gov), composite scores on the SAT have +mean 1060 and standard deviation 195, while composite scores on the ACT have mean 21 and standard deviation 5. +1. +At what percentile would an SAT score of 990 fall? +2. +What is the z-score of an ACT score of 27? +3. +Which is better: a score of 1450 on the SAT or 29 on the ACT? +Solution +1. +Using Google Sheets, we can answer this question with the formula “=NORM.DIST(990, 1060, 195, TRUE)”. A score of +990 would fall at the 36th percentile. +2. +Using the formula +, we get +. +3. +Let’s compare the values using both percentiles and +-values: +Percentiles: Using “=NORM.DIST(1450, 1060, 195, TRUE)” we find that an SAT score of 1450 is at the 98th percentile. +Meanwhile, by entering “=NORM.DIST(29, 21, 5, TRUE)” we see that an ACT score of 29 is around the 95th percentile. +Since it’s at a higher percentile, we can conclude that an SAT score of 1450 is better than an ACT score of 29. +8.7 • Applications of the Normal Distribution +903 + +-scores: Using the formula, we see that the +-score for an SAT score of 1450 is +, while the +-score for an ACT score of 29 is +. Since it has a higher +-score, an SAT score of 1450 is better than +an ACT score of 29. +YOUR TURN 8.41 +According to the Graduate Management Admission Council (GMAC) (https://openstax.org/r/mba_com), the mean +score on the GMAT (an entrance exam for graduate schools in business management) is 565, with standard +deviation 116. For the LSAT (an entrance exam for law schools), Kaplan (https://openstax.org/r/kaptest_com) informs +us that the mean score is 150 with standard deviation 10. +1. What is the +-score for a GMAT score of 715? +2. At what percentile is an LSAT score of 166? +3. Which is better: a GMAT score of 650 or an LSAT score of 161? +Coin flipping +In the opening of The Normal Distribution, we saw that the number of heads we get when we flip a coin 100 times is +distributed normally. It can be shown that if +is the number of flips, then the mean of that distribution is +and the +standard deviation is +(as long as +). So, for 100 flips, the mean of the distribution is 50 and the standard +deviation is 5. In that opening example, one of our early runs gave us 70 heads in 100 flips, which we noted seemed +unusual. Using the normal distribution, we can identify exactly how unusual that really is. Using Google Sheets, the +formula “=NORM.DIST(70, 50, 5, TRUE)” gives us 0.999968, which is the 99.997th percentile! How is that useful? Suppose +you need to test whether a coin is fair, and so you flip it 100 times. While we might be suspicious if we get 70 heads out +of the 100 flips, we now have a numerical measure for how unusual that is: If the coin were fair, we would expect to see +70 heads (or more) only +of the time. That’s really unlikely! Analysis like this is related to +hypothesis testing, an important application of statistics in the sciences and social sciences. +EXAMPLE 8.42 +Flipping a Coin +Let’s say we flip a coin 64 times and count the number of heads. +1. +What would be the mean of the corresponding distribution? +2. +What would be the standard deviation of the corresponding distribution? +3. +Suppose we got 25 heads, which seems a little low. At what percentile would 25 heads fall? +Solution +1. +Since +, the mean is +. +2. +Again using +, we get a standard deviation of +. +3. +Using “=NORM.DIST(25, 32, 4, TRUE)”, we see that 25 heads is at the 4th percentile. Reading and Interpreting Scatter +Plots +YOUR TURN 8.42 +You flip a coin 144 times and count the number of heads. +1. What is the mean of the corresponding distribution? +2. What is the standard deviation of the corresponding distribution? +3. You flipped 81 heads. At what percentile does that fall? +Analyzing Data That Are Normally Distributed +Whenever we’re working with a dataset that has a distribution that looks symmetric and bell-shaped, we can use +904 +8 • Statistics +Access for free at openstax.org + +techniques associated with the normal distribution to analyze the data. +EXAMPLE 8.43 +Using Normal Techniques to Analyze Data +The data in “AvgSAT” (https://openstax.org/r/Chapter8_Data-Sets) contains the average SAT score for students attending +every institution of higher learning in the United States for which data is available. In Example 8.12, we created a +histogram for these data: +Figure 8.64 (data source: https://data.ed.gov) +This distribution is fairly symmetric (it’s just a little right-skewed) and bell-shaped, so we can use normal distribution +techniques to analyze the data. +1. +What is the mean of these average SAT scores? +2. +What is the standard deviation of these SAT scores? +3. +Using the answers to the previous two questions, use NORM.DIST in Google Sheets to estimate at what percentile +the University at Buffalo in New York (average SAT: 1250) falls. +4. +Use PERCENTRANK to find the actual percentile of the University at Buffalo, and see how close the estimate in the +previous question came. +Solution +1. +Using the AVERAGE function in Google Sheets, we find that the mean is 1141.174. +2. +Using the STDEV function, we get that the standard deviation is 125.517. +3. +Entering “=NORM.DIST(1250, 1141, 125.517, TRUE)” into Google Sheets, we estimate that the University at Buffalo is +at the 81st percentile. +4. +Using PERCENTRANK, we find that the actual percentile is the 84th. These are close! +YOUR TURN 8.43 +1. Again using the data in “AvgSAT” (https://openstax.org/r/Chapter8_Data-Sets), find the average SAT score of a +school at the 35th percentile in two ways: using NORM.INV and using PERCENTILE. +WHO KNEW? +Political Meddling Exposed +The normal distribution pops up in some unusual places. Recently, a team at Duke University has been using statistics +to help identify partisan gerrymandering, where electoral districts have been carefully drawn in a way that benefits +one political party over another. In their analysis, they found that hypothetical election results in randomly drawn +districts are normally distributed. By using techniques similar to the ones we used above, they can quantify precisely +8.7 • Applications of the Normal Distribution +905 + +how biased a particular electoral map is by finding the percentile rank of the actual election result on the normal +distribution of the hypothetical results. You can find out more about their work at the "Quantifying Gerrymandering" +site here. (https://openstax.org/r/quantifying-gerrymandering) +Check Your Understanding +For the following problems, recall that the SAT exam has mean 1060 and standard deviation 195, and that composite +scores on the ACT have mean 21 and standard deviation 5. +49. At what percentile would an SAT score of 940 fall? Round to the nearest whole number. +50. What score would be at the 67th percentile on the ACT? +51. Which is a better score: 1300 on the SAT or 27 on the ACT? +For the following problems, recall that if we flip a coin at least 20 times, the distribution of the number of heads is +approximately normal with mean equal to half the number of flips and standard deviation equal to half of the square +root of the number of flips. +52. Suppose we flip a coin 120 times. What are the mean and standard deviation of the corresponding distribution +of heads? +53. Let’s say you flip 70 heads in 120 flips. At what percentile would that fall? +54. How many heads would be at the 30th percentile? Round to the nearest whole number. +For the following problems, use the data in “World Tax,” (https://openstax.org/r/Chapter8_Data-Sets) which gives the +tax revenue of many countries of the world in 2017, expressed as a percentage of their gross domestic products. You +can assume that the distribution is approximately normal (but you can make a histogram to check, if you want). +55. What tax revenue percentage falls at the third quartile? Answer this question using Google Sheets in two ways: +using PERCENTILE and using NORM.INV. +56. What tax revenue percentage falls at the 20th percentile? Answer this question using Google Sheets in two +ways: using PERCENTILE and using NORM.INV. +57. At what percentile does the United Kingdom (25.62%, found on row 46 in the spreadsheet) fall? Answer this +question using Google Sheets in two ways: using PERCENTRANK and using NORM.DIST. +58. At what percentile does Kiribati (21.97%, found on row 62 in the spreadsheet) fall? Answer this question using +Google Sheets in two ways: using PERCENTRANK and using NORM.DIST. +SECTION 8.7 EXERCISES +For the following exercises, assume we’re looking at results from two different standardized tests. The first, called the +ABC, has mean 250 and standard deviation 50. The second, called the XYZ, has mean 80 and standard deviation 10. +1. What would be the +-score of a result of 322 on the ABC? +2. What would be the +-score of a result of 57 on the XYZ? +3. At what percentile would a result of 211 on the ABC fall? Round your answer to the nearest percentile. +4. At what percentile would a result of 94 on the XYZ fall? Round your answer to the nearest percentile. +5. What score would fall at the first quartile on the ABC? Round your answer to the nearest whole number. +6. What score would fall at the 60th percentile on the XYZ? Round your answer to the nearest whole number. +7. Which score is better: 202 on the ABC or 72 on the XYZ? +8. Which score is better: 324 on the ABC or 94 on the XYZ? +For the following exercises, recall that if we flip a coin at least 20 times, the distribution of the number of heads is +approximately normal with mean equal to half the number of flips and standard deviation equal to half of the square +root of the number of flips. +9. What would be the mean and standard deviation for the number of heads in 80 coin flips? +10. What would be the mean and standard deviation for the number of heads in 144 coin flips? +11. How many heads would be at the 80th percentile for 80 coin flips? Round your answer to the nearest whole +number. +12. How many heads would be at the 20th percentile for 144 coin flips? Round your answer to the nearest whole +number. +13. What is the +-score for 51 heads in 80 coin flips? Round your answer to the nearest hundredth. +14. What is the +-score for 87 heads in 144 coin flips? Round your answer to the nearest hundredth. +15. Which would be more unusual: flipping 51 heads in 80 tries or flipping 87 heads in 144 tries? How do you know? +For the following exercises, use the data in “Wheat,” (https://openstax.org/r/Chapter8_Data-Sets) which gives the yield +906 +8 • Statistics +Access for free at openstax.org + +of wheat (in bushels) produced over several equally sized plots. (These data were collected as part of an experiment +reported in “The experimental error of field trials” (https://openstax.org/r/sets-059) by W.B. Mercer and A.D. Hall). +16. Create a histogram of Yield to check that it’s approximately normally distributed. +17. Find the mean and standard deviation of Yield. Round answers to the nearest thousandth. +18. What proportion of yields does the normal distribution predict should fall below 4.56? Round to the nearest +thousandth. +19. At what percentile does the normal distribution predict a yield of 3.6 would fall? Round to the nearest whole +number. +20. What yield does the normal distribution estimate would fall at the 80th percentile? Round to the nearest +hundredth. +21. What yield does the normal distribution estimate would fall at the 95th percentile? Round to the nearest +hundredth. +22. If one of these plots yields 2.8 bushels, would that be a good or bad result? How do you know? +8.8 Scatter Plots, Correlation, and Regression Lines +Figure 8.65 A scatter plot is a visualization of the relationship between quantitative dataset. +Learning Objectives +After completing this section, you should be able to: +1. +Construct a scatter plot for a dataset. +2. +Interpret a scatter plot. +3. +Distinguish among positive, negative and no correlation. +4. +Compute the correlation coefficient. +5. +Estimate and interpret regression lines. +One of the most powerful tools statistics gives us is the ability to explore relationships between two datasets containing +quantitative values, and then use that relationship to make predictions. For example, a student who wants to know how +well they can expect to score on an upcoming final exam may consider reviewing the data on midterm and final exam +scores for students who have previously taken the class. It seems reasonable to expect that there is a relationship +between those two datasets: If a student did well on the midterm, they were probably more likely to do well on the final +than the average student. Similarly, if a student did poorly on the midterm, they probably also did poorly on the final +exam. +Of course, that relationship isn’t set in stone; a student’s performance on a midterm exam doesn’t cement their +performance on the final! A student might use a poor result on the midterm as motivation to study more for the final. A +student with a really good grade on the midterm might be overconfident going into the final, and as a result doesn’t +prepare adequately. +The statistical method of regression can find a formula that does the best job of predicting a score on the final exam +based on the student’s score on the midterm, as well as give a measure of the confidence of that prediction! In this +section, we’ll discover how to use regression to make these predictions. First, though, we need to lay some graphical +groundwork. +8.8 • Scatter Plots, Correlation, and Regression Lines +907 + +Relationships Between Quantitative Datasets +Before we can evaluate a relationship between two datasets, we must first decide if we feel that one might depend on +the other. In our exam example, it is appropriate to say that the score on the final depends on the score on the midterm, +rather than the other way around: if the midterm depended on the final, then we’d need to know the final score first, +which doesn’t make sense. +Here’s another example: if we collected data on home purchases in a certain area, and noted both the sale price of the +house and the annual household income of the purchaser, we might expect a relationship between those two. Which +depends on the other? In this case, sale price depends on income: people who have a higher income can afford a more +expensive house. If it were the other way around, people could buy a new, more expensive house and then expect a +raise! (This is very bad advice.) +It's worth noting that not every pair of related datasets has clear dependence. For example, consider the percent of a +country’s budget devoted to the military and the percent earmarked for public health. These datasets are generally +related: as one goes up, the other goes down. However, in this case, there’s not a preferred choice for dependence, as +each could be seen as depending on the other. When exploring the relationship between two datasets, if one set seems +to depend on the other, we’ll say that dataset contains values of the response variable (or dependent variable). The +dataset that the response variable depends on contains values of what we call the explanatory variable (or +independent variable). If no dependence relationship can be identified, then we can assign either dataset to either role. +EXAMPLE 8.44 +Identifying Explanatory and Response Variables +For each of the following pairs of related datasets, identify which (if any) should be assigned the role of response variable +and which should be assigned to be the explanatory variable. +1. +A person’s height and weight +2. +A professional basketball player’s salary and their average points scored per game (which is a measure of how good +they are at basketball) +3. +The length and width of leaves on a tree +Solution +1. +As people get taller, their weight tends to increase. But if a person goes on a diet and loses weight, we don’t expect +them to also get shorter. So, weight depends on height. That means we’ll say that the response variable is weight +and the explanatory variable is height. +2. +The more points a basketball player scores, the more money they should make. But if a basketball player gets a +raise, we wouldn’t expect them to get better at basketball as a result. So, the response is salary and the explanatory +is points per game. +3. +The way that the length and width of leaves are connected isn’t clear. It seems reasonable that as the width goes up, +so would the length. But the other direction is also plausible: as the length goes up, so does the width. Without a +clear dependence relationship, we’re free to declare either to be the response and the other to be the explanatory. +YOUR TURN 8.44 +Given these pairs of datasets, identify which (if either) would be the best choice for the response variable. +1. A person’s age and their annual income +2. A student’s GPA and their score on the SAT +3. A student’s GPA and the number of hours they spend studying per week +Once we’ve assigned roles to our two datasets, we can take the first step in visualizing the relationship between them: +creating a scatter plot. +Creating Scatter Plots +A scatter plot is a visualization of the relationship between two quantitative sets of data. The scatter plot is created by +turning the datasets into ordered pairs: the first coordinate contains data values from the explanatory dataset, and the +second coordinate contains the corresponding data values from the response dataset. These ordered pairs are then +plotted in the +-plane. Let's return to our exam example to put this into practice. +908 +8 • Statistics +Access for free at openstax.org + +EXAMPLE 8.45 +Creating Scatter Plots Without Technology +Students are exploring the relationship between scores on the midterm exam and final exam in their math course. Here +are some of the scores reported by their classmates: +Name +Midterm grade +Final grade +Student 1 +88 +84 +Student 2 +71 +80 +Student 3 +75 +77 +Student 4 +94 +95 +Student 5 +68 +73 +Create a scatter plot to visualize the data. +Solution +Step 1: Since it makes more sense to think of the final exam score as being dependent on the midterm exam score, we’ll +let the final grade be the response. So, let’s think of these two datasets as a set of ordered pairs, midterm first, final +second: +Name +(Midterm, Final) +Student 1 +(88, 84) +Student 2 +(71, 80) +Student 3 +(75, 77) +Student 4 +(94, 95) +Student 5 +(68, 73) +Step 2: Next, let’s make the axes. On the horizontal axis, make sure the range of values is sufficient to cover all of the +explanatory data. For our data, that’s 68 to 94. Similarly, the vertical axis should cover all of the response data (73 to 95): +Figure 8.66 +8.8 • Scatter Plots, Correlation, and Regression Lines +909 + +Step 3: Our first point is (88, 84). So, we’ll locate 88 on the horizontal axis, 84 on the vertical axis, and identify the point +that’s directly above the first location and horizontally level with the second: +Figure 8.67 +Step 4: Repeat this process to place the other four points on the graph: +Figure 8.68 +Step 5: Finally, label the axes: +Figure 8.69 +YOUR TURN 8.45 +1. Create a scatter plot to visualize the following data, showing the top five NFL receivers by number of receptions +for the 2019 season. Treat Yards as the response: +910 +8 • Statistics +Access for free at openstax.org + +Player +Receptions +Yards +Stefon Diggs +127 +1535 +Davante Adams +115 +1374 +DeAndre Hopkins +115 +1407 +Darren Waller +107 +1196 +Travis Kelce +105 +1416 +Table 8.12 (source: https://www.pro- +football-reference.com/years/2019/) +For large datasets, it’s impractical to create scatter plots manually. Luckily, Google Sheets automates this process for us. +VIDEO +Making Scatter Plots in Google Sheets (https://openstax.org/r/Scatter-Plots) +EXAMPLE 8.46 +Creating Scatter Plots in Google Sheets +The dataset “NHL19” (https://openstax.org/r/Chapter8_Data-Sets) gives the results of the 2018–2019 National Hockey +League season. The columns are team, wins (W), losses (L), overtime losses (OTL), total points (PTS), goals scored by the +team (GF), goals scored against the team (GA), and goal differential (the difference in GF and GA). Use Google Sheets to +create a scatter plot for GF vs. GA. +When we talk about plotting one set versus another, the first is the response and the second is explanatory. +Solution +Step 1: Open the dataset in Google Sheets, and click and drag to select the data we want to visualize (in this case, we +want the columns for GF and GA; make sure you include those labels in the selection). +Step 2: Next, click on the “Insert” menu, then click “Chart.” Sheets will automatically choose a chart format; if the result +isn’t a scatter plot, click on the drop-down menu under “Chart type” in the Chart Editor on the right side of the window +and select “Scatter chart.” +Step 3: Next, check that the correct choices were made for the horizontal and vertical axes. In this case, we want to see +GF on the vertical axis and GA on the horizontal axis. If Sheets made the wrong choice, we can fix it in the Chart Editor by +clicking on the name of the dataset under “X-axis” to open up a dropdown menu, then selecting the variable that should +go on the horizontal axis (GA in this case). +Step 4: Now, click on the variable under “Series” and select the one that should go on the vertical axis (GF). If you had to +make that change, the axis labels in the graph may also need changing; those labels can be fixed using the “Customize” +tab in the Chart Editor under “Chart & axis titles.” Your result should look like this: +8.8 • Scatter Plots, Correlation, and Regression Lines +911 + +Figure 8.70 (data source: www.nhl.com) +YOUR TURN 8.46 +1. With the data in "NHL19," (https://openstax.org/r/Chapter8_Data-Sets) use Google Sheets to create a scatter plot +for points (PTS) vs. wins (W). +Reading and Interpreting Scatter Plots +Scatter plots give us information about the existence and strength of a relationship between two datasets. To break that +information down, there are a series of questions we might ask to help us. First: Is there a curved pattern in the data? If +the answer is “yes,” then we can stop; none of the linear regression techniques from here to the end of this section are +appropriate. Figure 8.71 and Figure 8.74 show several examples of scatter plots that can help us identify these curved +patterns. +Figure 8.71 Curved pattern +912 +8 • Statistics +Access for free at openstax.org + +Figure 8.72 No curved pattern +Figure 8.73 Curved pattern +Figure 8.74 No curved pattern +Once we have confirmed that there is no curved pattern in our data, we can move to the next question: Is there a linear +relationship? To answer this, we must look at different values of the explanatory variable and determine whether the +corresponding response values are different, on average. It's important to look at the values “on average” because, in +general, our scatter plots won’t include just one corresponding response point for each value of the explanatory variable +(i.e., there may be multiple response values for each explanatory value). So, we try to look for the center of those points. +Let’s look again at Figure 8.74, but consider some different values for the explanatory variable. Let’s highlight the points +whose +-values are around 50 and those that are around 80: +8.8 • Scatter Plots, Correlation, and Regression Lines +913 + +Figure 8.75 +Now, we can estimate the middle of each group of points. Let's add our estimated averages to the plot as starred points: +Figure 8.76 +Since those two starred points occur at different heights, we can conclude that there’s likely a relationship worth +exploring. +Here’s another example using a different set of data: +Figure 8.77 +Let’s look again at the points near 50 and near 80, and estimate the middles of those clusters: +914 +8 • Statistics +Access for free at openstax.org + +Figure 8.78 +Notice that there’s not much vertical distance between our two starred points. This tells us that there’s not a strong +relationship between these two datasets. +Positive and Negative Linear Relationships +Another way to assess whether there is a relationship between two datasets in a scatter plot is to see if the points seem +to be clustered around a line (specifically, a line that’s not horizontal). The stronger the clustering around that line is, the +stronger the relationship. +Once we’ve established that there’s a relationship worth exploring, it’s time to start quantifying that relationship. Two +datasets have a positive linear relationship if the values of the response tend to increase, on average, as the values of +the explanatory variable increase. If the values of the response decrease with increasing values of the explanatory +variable, then there is a negative linear relationship between the two datasets. The strength of the relationship is +determined by how closely the scatter plot follows a single straight line: the closer the points are to that line, the +stronger the relationship. The scatter plots in Figure 8.74 to Figure 8.80 depict varying strengths and directions of linear +relationships. +Figure 8.79 Perfect negative relationship +8.8 • Scatter Plots, Correlation, and Regression Lines +915 + +Figure 8.80 Strong negative relationship +Figure 8.81 Weak negative relationship +Figure 8.82 No relationship +916 +8 • Statistics +Access for free at openstax.org + +Figure 8.83 Weak positive relationship +Figure 8.84 Strong positive relationship +Figure 8.85 Perfect positive relationship +The strength and direction (positive or negative) of a linear relationship can also be measured with a statistic called the +correlation coefficient (denoted ). Positive values of +indicate a positive relationship, while negative values of +indicate a negative relationship. Values of +close to 0 indicate a weak relationship, while values close to +correspond +to a very strong relationship. Looking again at Figure 8.74 to Figure 8.80, the correlation coefficients for each, in +sequential order, are: ‒1, ‒0.97, ‒0.55, ‒0.03, 0.61, 0.97, and 1. +There’s no firm rule that establishes a cutoff value of +to divide strong relationships from weak ones, but +is often +given as the dividing line (i.e., if +or +the relationship is strong, and if +the relationship is +weak). +8.8 • Scatter Plots, Correlation, and Regression Lines +917 + +The formula for computing +is very complicated; it’s almost never done without technology. Google Sheets will do the +computation for you using the CORREL function. The syntax works like this: if your explanatory values are in cells A2 to +A50 and the corresponding response values are in B2 to B50, then you can find the correlation coefficient by entering +“=CORREL(A2:A50, B2:B50)”. (Note that the order doesn’t matter for correlation coefficients; “=CORREL(B2:B50, A2:A50)” +will give the same result.) +Let’s put all of this together in an example. +EXAMPLE 8.47 +Interpreting Scatter Plots +Consider the four scatter plots below: +1. +Figure 8.86 +2. +Figure 8.87 +3. +Figure 8.88 +918 +8 • Statistics +Access for free at openstax.org + +4. +Figure 8.89 +For each of these, answer the following questions: +a. +Is there a curved pattern in the data? If yes, stop here. If no, continue to part b. +b. +Classify the strength and direction of the relationship. Make a guess at the value of . +Solution +1. +a. +Yes, there is a curved pattern. +2. +a. +No, there’s no curved pattern. +b. +Since the points tend upward as we move from left to right, this is a positive relationship. The points seem +pretty closely grouped around a line, so it’s fairly strong. Comparing this scatter plot to those in Figure 8.79 to +Figure 8.85, we can see that the relationship is stronger than the one in Figure 8.83 ( +) but not as strong +as the one in Figure 8.84 ( +). So, the value of the correlation coefficient is somewhere between the two. +We might guess that +. +3. +a. +No, there’s no curved pattern. +b. +Since the points tend downward as we move from left to right, this is a negative relationship. The points are not +tightly grouped around a line, but the pattern is clear. It looks like it has approximately the same strength as +the plot in Figure 8.83, just with the opposite sign. So, we might guess that +. +4. +a. +No, there’s no curved pattern. +b. +Since the points don’t really tend upward or downward as we move from left to right, there is no real +relationship here. Thus, +. +YOUR TURN 8.47 +For each of the plots below, answer the following questions: +a. +Is there a curved pattern in the data? If yes, stop here. If no, continue to part b. +b. +Classify the strength and direction of the relationship. Make a guess at the value of . +8.8 • Scatter Plots, Correlation, and Regression Lines +919 + +1. +2. +3. +920 +8 • Statistics +Access for free at openstax.org + +4. +EXAMPLE 8.48 +Finding the Correlation Coefficient +The data that were plotted in the previous example can be found in the dataset “correlationcoefficient1” +(https://openstax.org/r/Chapter8_Data-Sets). All of them share the same values for the explanatory variable +. The four +responses are labeled +through +. Compute the correlation coefficients for each, if appropriate, using Google Sheets. +Round to the nearest hundredth. +Solution +Step 1: There is a curved pattern in the data, so the correlation coefficient isn’t meaningful. +Step 2: Using “=CORREL(A2:A101, C2:C101)” we get +. +Step 3: Using “=CORREL(A2:A101, D2:D101)” we get +. +Step 4: Using “=CORREL(A2:A101, E2:E101)” we get +. +YOUR TURN 8.48 +1. The data that were plotted in Your Turn 8.47 can be found in the dataset “correlationcoefficient2”. All of them +share the same values for the explanatory variable +. The four responses are labeled +through +. Compute +the correlation coefficients for each, if appropriate, using Google Sheets. Round to the nearest hundredth. +WORK IT OUT +Winning with Statistics +Billy Beane, the former general manager of the Oakland A’s baseball team, famously took his low budget team to +unprecedented heights by using statistics to identify undervalued players; his story is recounted in the book +Moneyball (which was later made into a movie, with Brad Pitt playing Beane). You can do the same thing: Take a look +at team statistics in the sport of your choice (https://openstax.org/r/sports-reference) and try to identify a statistic +that’s most closely related to winning (meaning that it has the highest correlation coefficient with team wins). +Linear Regression +The final step in our analysis of the relationship between two datasets is to find and use the equation of the regression +line. For a given set of explanatory and response data, the regression line (also called the least-squares line or line of +best fit) is the line that does the best job of approximating the data. +What does it mean to say that a particular line does the “best job” of approximating the data? The way that statisticians +8.8 • Scatter Plots, Correlation, and Regression Lines +921 + +characterize this “best line” is rather technical, but we’ll include it for the sake of satisfying your curiosity (and backing up +the claim of "best"). Imagine drawing a line that looks like it does a pretty good job of approximating the data. Most of +the points in the scatter plot will probably not fall exactly on the line; the distance above or below the line a given point +falls is called that point’s residual. We could compute the residuals for every point in the scatter plot. If you take all those +residuals and square them, then add the results together, you get a statistic called the sum of squared errors for the line +(the name tells you what it is: “sum” because we’re adding, “squared” because we’re squaring, and “errors” is another +word for “residuals”). The line that we choose to be the “best” is the one that has the smallest possible sum of squared +errors. The implied minimization (“smallest”) is where the “least” in “least squares” comes from; the “squares” comes from +the fact that we’re minimizing the sum of squared errors. This is very similar to the process we outlined in the "game" +that we used to introduce the mean. Both the regression line and the mean are designed to minimize a sum of squared +errors. Here ends the super technical part. +Finding the Equation of the Regression Line +So, how do we find the equation of the regression line? Recall the point-slope form of the equation of a line: +FORMULA +If a line has slope +and passes through a point +, then the point-slope form of the equation of the line is: +The regression line has two properties that we can use to find its equation. First, it always passes through the point of +means. If +and +are the means of the explanatory and response datasets, respectively, then the point of means is +. We’ll use that as the point in the point-slope form of the equation. Second, if +and +are the standard +deviations of the explanatory and response datasets, respectively, and if +is the correlation coefficient, then the slope is +. Putting all that together with the point-slope formula gives us this: +FORMULA +Suppose +and +are explanatory and response datasets that have a linear relationship. If their means are +and +respectively, their standard deviations are +and +respectively, and their correlation coefficient is , then the +equation of the regression line is: +. +Let's walk through an example. +EXAMPLE 8.49 +Finding the Equation of the Regression Line from Statistics +Suppose you have datasets +and +with the following statistics: +has mean 21 and standard deviation 4, +has mean 8 +and standard deviation 2, and their correlation coefficient is −0.4. What’s the equation of the regression line? +Solution +Step 1: We’re given +, +, +, +, and +. Let's start with the formula for the equation of the +regression line: +Step 2: Plugging in our values gives us: +Step 3: Our final regression line equation is: +922 +8 • Statistics +Access for free at openstax.org + +YOUR TURN 8.49 +1. Suppose you have datasets +and +with the following statistics: +has mean 100 and standard deviation 5, +has +mean 200 and standard deviation 20, and their correlation coefficient is 0.75. What’s the equation of the +regression line? +As you can see, finding the equation of the regression line involves a lot of steps if you have to find all of the values of +the needed quantities yourself. But, as usual, technology comes to our rescue. This video (which you actually watched +earlier when learning how to create scatter plots) covers the regression line at around the 3:30 mark. Note that Google +Sheets calls it the "trendline." +Let's put this into practice. +EXAMPLE 8.50 +Finding the Equation of the Regression Line Using Google Sheets +In Example 8.46, we considered the relationship between goals scored (GF) and goals against (GA) using the dataset +“NHL19” (https://openstax.org/r/Chapter8_Data-Sets). Recreate the scatter plot in Google Sheets, and use it to find the +equation of the regression line. +Solution +Once we have recreated the scatter plot, we find the equation of the regression line by clicking the three dots at the top +right of the plot, selecting “Edit chart,” then clicking on “Customize” and “Series.” We add the regression line by checking +the box next to “Trendline,” and then we show the equation by selecting “Use Equation” in the drop-down menu under +“Label.” The equation of the tangent line is +. +YOUR TURN 8.50 +1. In Your Turn 8.46, you created a scatter plot for points (PTS) vs. wins (W) using the dataset “NHL19” +(https://openstax.org/r/Chapter8_Data-Sets). Recreate the scatter plot in Google Sheets, and use it to find the +equation of the regression line. +Using the Equation of the Regression Line +Once we’ve found the equation of the regression line, what do we do with it? We’ll look at two possible applications: +making predictions and interpreting the slope. +We can use the equation of the regression line to predict the response value +for a given explanatory value +. All we +have to do is plug that explanatory value into the formula and see what response value results. This is useful in two +ways: first, it can be used to make a guess about an unknown data value (like one that hasn’t been observed yet). +Second, it can be used to evaluate performance (meaning, we can predict an outcome given a particular event). In +Example 8.45, we created a scatter plot of final exam scores vs. midterm exam scores using this data: +Name +Midterm Grade +Final Grade +Allison +88 +84 +Benjamin +71 +80 +Carly +75 +77 +Daniel +94 +95 +Elmo +68 +73 +8.8 • Scatter Plots, Correlation, and Regression Lines +923 + +The equation of the regression line is +, where +is the final exam score and +is the midterm exam +score. If Frank scored 85 on the midterm, then our prediction for his final exam score is +. To +use the regression line to evaluate performance, we use a data value we’ve already observed. For example, Allison +scored 88 on the midterm. The regression line predicts that someone who scores an 88 on the midterm will get +on the final. Allison actually scored 84 on the final, meaning she underperformed +expectations by almost 4 points +. +The second application of the equation of the regression line is interpreting the slope of the line to describe the +relationship between the explanatory and response datasets. For the exam data in the previous paragraph, the slope of +the regression line is 0.687. Recall that the slope of a line can be computed by finding two points on the line and dividing +the difference in the +-values of those points by the difference in the +-values. Keeping that in mind, we can interpret our +slope as +. Multiplying both sides of that equation by the denominator of the fraction, +we get +. Thus, a one-point increase in the midterm score +would result in a predicted increase in the final score of 0.687 points. A ten-point drop in the midterm score would give +us a decrease in the predicted final score of 6.87 points. In general, the slope gives us the predicted change in the +response that corresponds to a one unit increase in the explanatory variable. +EXAMPLE 8.51 +Applying the Equation of the Regression Line +The data in “MLB2019Off” (https://openstax.org/r/Chapter8_Data-Sets) gives offensive team stats for the 2019 Major +League Baseball season. Use that dataset to answer the following questions: +1. +What is the equation of the regression line for runs (R) vs. hits (H)? +2. +How many runs would we expect a team to score if the team got 1500 hits in a season? +3. +Did the Kansas City Royals (KCR) overperform or underperform in terms of runs scored, based on their hit total? By +how much? +4. +Write a sentence to interpret the slope of the regression line. +Solution +1. +Using Google Sheets, we find that the regression line equation is +, where +is the number of runs +scored and +is the number of hits. +2. +Plugging 1500 into the equation of the regression line, we get +. We would predict that a +team with 1500 hits would score 870 runs. +3. +The Royals had 1356 hits, so we would predict their run total to be +. They actually scored +691 runs, so they underperformed expectations by 52 runs +. +4. +The slope gives us the predicted change in the response that corresponds to a one unit increase in the explanatory +variable. So, we expect one additional hit to result in 0.884 more runs. Since 0.884 runs doesn’t really make sense, +we can get a better interpretation by multiplying through by ten or one hundred: Ten additional hits will result in +almost nine additional runs, or a hundred additional hits will yield on average just over 88 additional runs. +YOUR TURN 8.51 +Using the “MLB2019Off” (https://openstax.org/r/Chapter8_Data-Sets) dataset, answer the following: +1. What is the equation of the regression line for the number of times a runner is caught stealing a base (CS) vs. +the number of successful stolen bases (SB)? +2. How many times would we expect a team to be caught stealing if the team steals 70 bases in a season? +3. Did the Philadelphia Phillies (PHI) overperform or underperform in terms of getting caught stealing, based on +their stolen base total? By how much? +4. Write a sentence to interpret the slope of the regression line. +924 +8 • Statistics +Access for free at openstax.org + +WHO KNEW? +Math and the Movies +Statistics and regression are used by Hollywood movie producers to decide what movies to make, and to predict how +much money they’ll earn at the box office. According to the American Statistical Association (https://openstax.org/r/ +American-Statistical-Association), not only do producers use statistics to identify the next potential blockbuster, but +they’ve also pinned down how much money awards add to the bottom line. (An Academy Award is worth about $3 +million!) In addition, studios use their streaming services to gather data about their customers and the types of +movies they watch; this data helps them learn what kinds of entertainment their customers want more of. +WORK IT OUT +Collecting and Analyzing Your Own Data +This section has demonstrated many pairs of related quantitative datasets. Think about some quantitative variables +that you can ask your classmates about, which might be related. Once you have some ideas, collect the data from +your classmates. Then analyze the data by creating a scatter plot, finding the equation of the regression line (if +appropriate), and interpreting it. +Extrapolation +A very common misuse of regression techniques involves extrapolation, which involves making a prediction about +something that doesn't belong in the dataset. +EXAMPLE 8.52 +More Applying the Equation of the Regression Line +The data in “WNBA2019” (https://openstax.org/r/Chapter8_Data-Sets) gives team statistics from the 2019 WNBA season. +Use that dataset to answer these questions about team wins (W) and the proportion of team field goals made (FG%, the +number of shots made divided by the number of shots attempted. Even though this column is labeled using a percent +sign, the values are not expressed as percentages): +1. +What is the equation of the regression line for wins vs. proportion of made field goals? +2. +How many wins would we expect for a team that makes 42% of its shots? +3. +Did the New York Liberty overperform or underperform in terms of wins, based on the team’s proportion of made +field goals? +4. +Write a sentence to interpret the slope of the regression line. +Solution +1. +Using Google Sheets, we get the equation +, where +is the proportion of field goals made +and +is the number of wins. +2. +Since 42% corresponds to a proportion of 0.42, we’ll plug 0.42 into the regression equation for +, which gives us +. We would predict that a team that makes 42% of its shots would win about 16 +games. +3. +The New York Liberty made 41.4% of their shots, so we expect they would have +wins. In fact, they had only 10 wins, so they underperformed expectations by over 5 wins. +4. +Step 1: The slope gives us the expected increase in the response that corresponds to a one unit increase in the +explanatory variable. If we simply go with that interpretation, we would get a sentence like “We expect an increase +in proportion of field goals made of 1 would result in an additional 178 wins.” However, that sentence doesn't make +much sense. Let's consider why. +Step 2: First, proportions must be between 0 and 1, the proportion of made field goals can’t be increased by 1 and +still make sense. Second, the total number of games played is only 34, so no team could get an additional 178 wins! +So, we’ll have to change the units. +Step 3: Since the proportions of made field goals are often expressed as a percentage, we could try to use that. If +we express the slope as a fraction with 1 in the denominator (remember, the denominator represents the +8.8 • Scatter Plots, Correlation, and Regression Lines +925 + +proportion of field goals made), then convert the denominator to a percentage and simplify, we get +. +Step 4: So, an increase in field goal percentage of 1% would result in an expected increase of 1.78 wins. +YOUR TURN 8.52 +Use the data in “WNBA2019” (https://openstax.org/r/Chapter8_Data-Sets) to answer these questions about the +relationship between the proportion of made field goals (FG%) and the proportion of made three-point field goals +(3P%): +1. What is the equation of the regression line for proportion of made three-point field goals vs. proportion of +made field goals? +2. What proportion of made three-point field goals would we expect for a team that makes 44% of its field +goals? +3. Did the Dallas Wings overperform or underperform in terms of proportion of made three-point field goals, +based on the team’s proportion of made field goals? +4. Write a sentence to interpret the slope of the regression line. +Correlation Does Not Imply Causation +One of the most common fallacies about statistics has to do with the relationship between two datasets. In the dataset +“Public” (https://openstax.org/r/Chapter8_Data-Sets), we find that the correlation coefficient between the 75th percentile +math SAT score and the 75th percentile verbal SAT score is 0.92, which is really strong. The slope of the regression line +that predicts the verbal score from the math score is 0.729, which we might interpret as follows: “If the 75th percentile +math SAT score goes up by 10 points, we’d expect the corresponding verbal SAT score to go up by just over 7 points.” +Does the increasing math score cause the increase in the verbal score? Probably not. What’s really going on is that +there’s a third variable that’s affecting them both: To raise the SAT math score by 10 points, a school will recruit students +who do better on the SAT in general; these students will also naturally have higher SAT verbal scores. This third variable +is sometimes called a lurking variable or a confounding variable. Unless all possible lurking variables are ruled out, we +cannot conclude that one thing causes another. +926 +8 • Statistics +Access for free at openstax.org + +PEOPLE IN MATHEMATICS +Dr. Talithia Williams +Figure 8.90 A photo of Dr. Williams (credit: Used by permission of Talithia Williams) +Dr. Talithia Williams is a statistician on the faculty of Harvey Mudd College, and the first Black woman to achieve +tenure at this university. She advocates for more women to become involved in the fields of engineering and science, +and is on the board of directors for the EDGE Foundation, an organization that helps women obtain advanced +degrees in mathematics (EDGE standing for Enhancing Diversity in Graduate Education). In 2018, Dr. Williams +published the book Power in Numbers: The Rebel Women of Mathematics, a retrospective look at historical female +figures who have contributed to the development of the field of mathematics. +Dr. Williams earned a Master’s degree in Mathematics from Howard University and a Master’s in Statistics from Rice +University, and also went on to earn her Ph.D. in Statistics from Rice. She has held research appointments at the Jet +Propulsion Laboratory, the National Security Agency, and NASA. Her research focuses on the environmental and +medical applications of statistics. In 2014, she gave a popular TED talk titled “Own Your Body’s Data” +(https://openstax.org/r/TED-Talk_talithia_williams) that discussed the potential insights to be gained from collecting +personal health data. She was even recently a host for the NOVA Wonders documentary series and a narrator for the +NOVA Universe series on PBS. +To stay up-to-date on Dr. Williams’s accomplishments, you can follow her on Twitter (https://openstax.org/r/ +twit_talithiaw) or her Facebook account (https://openstax.org/r/facebk_talithiaw). +WHO KNEW? +Statistics and Eugenics +Some of the brightest minds in the history of statistics unfortunately decided to use their considerable intellects to +further a pseudoscience known as eugenics. Eugenicists took Charles Darwin’s theories of evolution and ruthlessly +applied them to the human race. Francis Galton (1822–1911), a cousin of Darwin and also the mathematician who +invented the formula for standard deviation, claimed that people in the British upper classes possessed higher +8.8 • Scatter Plots, Correlation, and Regression Lines +927 + +intelligence due to their superior breeding. Karl Pearson (1857–1936), who derived the formula for the correlation +coefficient, argued in National Life from the Standpoint of Science, that, instead of providing social welfare programs, +nations could better improve the fortunes of the poor by waging “war with inferior races.” Ronald Fisher (1890–1962) +was possibly the most important statistician of the 20th century, having invented several new techniques (including +the ubiquitous analysis of variance), and yet he also founded the Cambridge University Undergraduates Eugenics +Society, whose self-prescribed goal was to evangelize “not by precept only, but by example, the doctrine of a new +natural ability of worth and blood.” +When eugenics took hold in the United States, it was used to justify terrible acts by the government, including the +forced sterilization of individuals with mental illness, epilepsy, a physical impairment (like blindness), or a criminal +history. The Nazi regime took these ideas to their ultimate, terrible conclusion: killing people who had mental or +physical disabilities, or who were born into an “inferior” race. Over six million people died in this Holocaust, one of the +darkest events in human history. To learn more, watch this video about Francis Galton and the legacy of eugenics +(https://openstax.org/r/Galton-and-eugenics) +Check Your Understanding +59. Make a scatter plot for the following data without technology: +20 +11 +8 +22 +25 +13 +15 +17 +13 +10 +For the following problems, answer these questions: +a. +Is there a curved pattern in the data? If yes, stop here. +b. +Classify the strength and direction of the relationship. Make a guess at the value of . +60. +928 +8 • Statistics +Access for free at openstax.org + +61. +Use the data in "MLB2019Off" (https://openstax.org/r/Chapter8_Data-Sets) to investigate the relationship between +slugging percentage (SLG, explanatory) and runs scored (R, response). +62. What’s the correlation coefficient? Round to the nearest hundredth. +63. What’s the equation of the regression line? Round the slope and intercept to the nearest whole number. +The regression equation used to predict average monthly faculty salary (FacSal) from out-of-state tuition (OutState) +using the data in “TNSchools” (https://openstax.org/r/Chapter8_Data-Sets) is +64. Predict the average monthly faculty salary for a school that charges $30,000 in out-of-state tuition. +65. Maryville College charges $34,880 for out-of-state students, and their average monthly faculty salary is $6,765. +Do they pay faculty more or less than expected? By how much? +66. Write a sentence to interpret the slope. +SECTION 8.8 EXERCISES +1. This table contains data for the first five schools (alphabetically) that fielded an NCAA Division I men’s basketball +team in the 2018–2019 season. It shows the total number of points each team scored (PF) and the total number of +points their opponents scored against them (PA). Create a scatter plot without technology of PA vs. PF. +School +PF +PA +Abilene Christian +2502 +2161 +Air Force +2179 +2294 +Akron +2271 +2107 +Alabama A&M +1938 +2285 +Alabama-Birmingham +2470 +2370 +(source: www.sports-reference.com) +2. This table contains data for the first five schools (alphabetically) that fielded an NCAA Division I men’s basketball +team in the 2018–2019 season. It shows the total number of field goals each team scored (FG) and the total +number of three-point field goals they scored (3P). Create a scatter plot without technology of 3P vs. FG. +8.8 • Scatter Plots, Correlation, and Regression Lines +929 + +School +FG +3P +Abilene Christian +897 +251 +Air Force +802 +234 +Akron +797 +297 +Alabama A&M +736 +182 +Alabama-Birmingham +906 +234 +(source: www.sports-reference.com) +For the following exercises, use the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets), on every school +that fielded an NCAA Division I men’s basketball team in the 2018–2019 season. +3. Use Google Sheets to create a scatter plot of points scored against a team (PA) vs. points scored by the team +(PF). +4. Use Google Sheets to create a scatter plot of number of three-point field goals made (3P) vs. total field goals +made (FG). +5. Use Google Sheets to create a scatter plot of number of fouls (Fouls) vs. number of blocks (BLK). +6. Use Google Sheets to create a scatter plot of points scored (PF) vs. percent of three-point shots made (3P%). +For the following exercises, answer the following questions: +a. +Is there a curved pattern in the data? If yes, stop here. If no, continue to part b. +b. +Classify the strength and direction of the relationship. Make a guess at the value of . +7. +8. +930 +8 • Statistics +Access for free at openstax.org + +9. +10. +11. +12. +8.8 • Scatter Plots, Correlation, and Regression Lines +931 + +13. +14. +For the following exercises, use the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets) on every school +that fielded an NCAA Division I men’sbasketball team in the 2018–2019 season. +15. What is the correlation coefficient for points scored against a team (PA) vs. points scored by the team (PF)? +Round to the nearest hundredth. +16. What is the equation of the regression line for PA vs. PF? +17. Predict the total number of points scored against a team that itself scores 2200 points. +18. Georgia Tech scored 2091 points, and had 2130 points scored against them. Is their PA higher or lower than +expected? By how much? +19. Write a sentence that interprets the slope of the regression line for PA vs. PF. +20. What is the correlation coefficient for three-point field goals made (3P) vs. total field goals made (FG)? Round to +the nearest hundredth. +21. What is the equation of the regression line for 3P vs. FG? +22. How many three-point field goals made would you expect for a team that made 1000 total field goals? +23. Seton Hall made 888 field goals; of those, 240 were three-point field goals. Did they make more or fewer three- +point field goals than expected? How many more or fewer? +24. Write a sentence to interpret the slope of the regression line for 3P vs. FG. +For the following exercises, use the datasets “Public” (https://openstax.org/r/Chapter8_Data-Sets) and “Private” +(https://openstax.org/r/Chapter8_Data-Sets), which give many institutions of higher learning in the United States +(public institutions in “Public” and private, non-profit institutions in “Private”), the schools’ 75th percentiles on the math +section of the SAT (SATM75), the verbal section of the SAT (SATV75), the math section of the ACT (ACTM75), and the +English section of the ACT (ACTE75). It also gives the schools’ admission rates (AdmRate) and total annual cost of +attendance (Cost). +25. It might seem reasonable to expect the cost to attend a school to go down as the proportion of applicants +admitted goes up. Create two scatter plots (one for private schools, one for public) to investigate that hunch. +Can we use linear regression to describe that relationship for these? Why or why not? +26. Find the correlation between the 75th percentiles of the two sections of the SAT at public schools and at private +schools. Which has a stronger relationship? +27. What score would we predict falls at the 75th percentile on the verbal section of the SAT at a public school +where the 75th percentile on the math section of the SAT is 500? +932 +8 • Statistics +Access for free at openstax.org + +28. What score would we predict falls at the 75th percentile on the verbal section of the SAT at a private school +where the 75th percentile on the math section of the SAT is 500? +29. Find the slope of the regression line that we would use to predict the 75th percentile SAT math score from the +75th percentile ACT English score at public schools, and write a sentence to interpret that slope. +30. Predict the cost of attendance at a public school whose 75th percentile on the SAT verbal section is 700. +31. The cost of attendance at DePauw University, a private school, is $62,567. The 75th percentile on the SAT math +section is 680. Is DePauw more or less expensive that we would predict based on the SAT math score? By how +much? +32. The cost of attendance at Coastal Carolina University, a public school, is $24,599. The 75th percentile of ACT +English scores at Coastal Carolina is 24. Is the cost higher or lower than we would expect based on the ACT +English score? By how much? +33. Find the equation of the regression line that we would use to predict the 75th percentile ACT English score from +the 75th percentile ACT math score at public institutions. +34. Find the equation of the regression line that we would use to predict cost of attendance at public schools using +the 75th percentile ACT math score. +35. Does the University of Hawai’i at Hilo have a higher or lower 75th percentile verbal SAT score (590) than we’d +expect based on its 75th percentile math SAT score (580)? By how much? +36. Find the slope of the regression line we would use to estimate cost from the 75th percentile SAT math scores at +public institutions. Write a sentence to interpret that slope. +37. Find the slope of the regression line we would use to estimate cost from the 75th percentile SAT math scores at +private institutions. Write a sentence to interpret that slope. +38. Look at the scatter plots that show the relationship between cost and the 75th percentiles of the various test +scores at private institutions. Which (if any) of the four exhibit a pattern that rules out analysis using linear +regression? +39. Look at the scatter plots that show the relationship between cost and the 75th percentiles of the various test +scores at public institutions. Which (if any) of the four exhibit a pattern that rules out analysis using linear +regression? +40. Looking at public institutions, rank the four test scores from highest to lowest in terms of the strength of their +relationships to cost. +8.8 • Scatter Plots, Correlation, and Regression Lines +933 + +Chapter Summary +Key Terms +8.1 Gathering and Organizing Data +• +sample +• +units +• +data +• +population +• +simple random sample +• +systematic random sample +• +stratified sample +• +cluster sample +• +quantitative data +• +categorical data +• +categorical frequency Distribution +• +binned frequency distribution +8.2 Visualizing Data +• +proportion +• +bar chart +• +pie chart +• +stem-and-leaf plot +• +distribution (of quantitative data) +• +histogram +• +bar chart for labeled data +8.3 Mean, Median and Mode +• +mode +• +bimodal +• +median +• +mean +8.4 Range and Standard Deviation +• +range +• +standard deviation +8.5 Percentiles +• +percentile +• +quartile +• +quintile +• +quantile +8.6 The Normal Distribution +• +normal distribution +• +68-95-99.7 Rule +• +standardized score ( -score) +8.8 Scatter Plots, Correlation, and Regression Lines +• +response variable (dependent variable) +• +explanatory variable (independent variable) +• +scatter plot +• +positive linear relationship +• +negative linear relationship +• +correlation coefficient +• +regression line (least-squares line, line of best fit) +934 +8 • Chapter Summary +Access for free at openstax.org + +Key Concepts +8.1 Gathering and Organizing Data +• +Categorical data places units into groups (categories), while quantitative data is a numerical measure of a property +of a unit. +• +The sampling method for a study depends on the way that randomization is used to select units for the sample. +• +Frequency distributions help to summarize data by counting the number of units that fall into a particular category +or range of quantitative values. +8.2 Visualizing Data +• +Categorical data can be visualized using pie charts or bar charts; quantitative data can be visualized using stem-and- +leaf plots or histograms. +• +Areas in pie charts and bar charts represent proportions of the data falling into a particular category, while areas in +histograms represent proportions of the data that fall into a given range of data values (or “bins”). Stem-and-leaf +plots are visual representations of entire datasets. +• +By manipulating the axes, changing widths of bars, or making bad choices for bins, we can create data +visualizations that misrepresent the distribution of data. +8.3 Mean, Median and Mode +• +The mode of a dataset is the value that appears the most frequently. The median is a value that is greater than or +equal to no more than 50% of the data and less than or equal to no more than 50% of the data. The mean is the +sum of all the data values, divided by the number of units in the dataset. +• +The median of a dataset is not affected by outliers, but the mean will be biased toward outliers. This distinction +might affect which measure of centrality is used to summarize a dataset. +8.4 Range and Standard Deviation +• +The range of a dataset is the difference between its largest and smallest values. The standard deviation is +approximately the mean difference (in absolute value) that individual units fall from the mean of the dataset. +8.5 Percentiles +• +The percentile rank of a data value is the percentage of all values in the dataset that are less than or equal to the +given value. +8.6 The Normal Distribution +• +Normally distributed data follow a bell-shaped, symmetrical distribution. +• +The mean of normally distributed data falls at the peak of the distribution. The standard deviation of normally- +distributed data is the distance from the peak to either of the inflection points. +• +Data that are normally distributed follow the 68-95-99.7 Rule, which says that approximately 68% of the data fall +within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three +standard deviations. +• +The +-score for a data value is the number of standard deviations that value falls above (or below, if the +-score is +negative) the mean. +• +We can use the normal distribution to estimate percentiles. +8.7 Applications of the Normal Distribution +• +We can use +-scores to compare data values from different datasets. +8.8 Scatter Plots, Correlation, and Regression Lines +• +If one variable affects the value of another variable, we say the first is an explanatory variable and the second is a +response variable. +• +Scatter plots place a point in the +-plane for each unit in the dataset. The +-value is the value of the explanatory +variable, and the +-value is the value of the response variable. +• +The correlation coefficient +gives us information about the strength and direction of the relationship between two +variables. If +is positive, the relationship is positive: an increase in the value of the explanatory variable tends to +correspond to an increase in the value of the response variable. If +is negative, the relationship is negative: an +increase in the value of the explanatory variable tends to correspond to a decrease in the value of the response +variable. Values of +that are close to 0 indicate weak relationships, while values close to –1 or indicate strong +8 • Chapter Summary +935 + +relationships. +• +The regression line for a relationship between two variables is the line that best represents the data. It can be used +to predict values of the response variable for a given value of the explanatory variable. +Videos +8.2 Visualizing Data +• +Make a Simple Bar Graph in Google Sheets (https://openstax.org/r/Bar-Graphs) +• +Create Pie Charts Using Google Sheets (https://openstax.org/r/Pie-Charts) +• +Make a Histogram Using Google Sheets (https://openstax.org/r/Histograms) +• +How to Spot a Misleading Graph (https://openstax.org/r/Misleading-Graphs) +8.3 Mean, Median and Mode +• +Compute Measures of Centrality Using Google Sheets (https://openstax.org/r/Computing-Measures-of-Centrality) +8.4 Range and Standard Deviation +• +Find the Minimum and Maximum Using Google Sheets (https://openstax.org/r/Finding-Minimum-and-Maximum) +• +Find the Standard Deviation Using Google Sheets (https://openstax.org/r/Finding-Standard-Deviation) +8.5 Percentiles +• +Using RANK, PERCENTRANK, and PERCENTILE in Google Sheets (https://openstax.org/r/Using-RANK-PERCENTRANK- +and-PERCENTILE) +8.8 Scatter Plots, Correlation, and Regression Lines +• +Making Scatter Plots in Google Sheets (https://openstax.org/r/Scatter-Plots) +Formula Review +8.3 Mean, Median and Mode +Suppose we have a set of data with +values, ordered from smallest to largest. If +is odd, then the median is the data +value at position +. If +is even, then we find the values at positions +and +. If those values are named +and +, +then the median is defined to be +. +8.4 Range and Standard Deviation +Here, s is the standard deviation, +represents each data value, +is the mean of the data values, +is the number of data +values, and the capital sigma ( ) indicates that we take a sum. +8.6 The Normal Distribution +If +is a member of a normally distributed dataset with mean +and standard deviation +, then the standardized score for +is +If you know a +-score but not the original data value +, you can find it by solving the previous equation for +: +8.8 Scatter Plots, Correlation, and Regression Lines +If a line has slope +and passes through a point +, then the point-slope form of the equation of the line is: +Suppose +and +are explanatory and response datasets that have a linear relationship. If their means are +and +respectively, their standard deviations are +and +respectively, and their correlation coefficient is , then the equation +of the regression line is: +936 +8 • Chapter Summary +Access for free at openstax.org + +Projects +1. +Browse through some news websites to find five stories that report on data and include data visualizations. Can you +tell from the report how the data were collected? Was randomization used? Are the visualizations appropriate for +the data? Are the visualizations presented in a way that might bias the reader? +2. +We discussed three measures of centrality in this chapter: the mode, the median, and the mean. In a broader +context, the mean as we discussed it is more properly called the arithmetic mean, to distinguish it from other types +of means. Examples of these include the geometric mean, harmonic mean, truncated mean, and weighted mean. +How are these computed? How do they compare to the arithmetic mean? In what situations would each of these be +preferred to the arithmetic mean? +3. +Simpson’s Paradox is a statistical phenomenon that can sometimes appear when we observe a relationship within +several subgroups of a population, but when the data for all thegroups are analyzed all together, the opposite +relationship appears. Find some examples of Simpson’s Paradox in real-world situations, and write a paragraph or +two that would explain the concept to someone who had never studied statistics before. +8 • Chapter Summary +937 + +Chapter Review +Gathering and Organizing Data +Decide whether randomization is being used in the selection of these samples. If it is, identify the type of random +sample (simple, systematic, cluster, or stratified). +1. College students want to gauge interest in a new club they're thinking about starting. They choose three +residence halls at random and slide a survey form under the door to every room. +2. The managers of a campus dining hall want feedback on a new dish that is being served, so they ask the first 10 +people who choose that dish what they think of it. +3. The administration of a large university system with five campuses wants to administer a survey to students in +the system. The administration chooses 50 students at random from each campus; the 250 students selected +form thesample. +4. A sample of college students were asked how many of their meals were provided by a campus dining hall on the +previous day. Here are the results: +0 +1 +0 +3 +2 +3 +3 +2 +2 +0 +2 +1 +1 +0 +0 +1 +2 +0 +3 +0 +1 +3 +2 +3 +0 +1 +3 +0 +Create a frequency distribution for these data. +Visualizing Data +5. Create a bar chart to visualize the data in question 4. +6. Use the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets) on every school that fielded an NCAA +Division I men’s basketball team in the 2018–2019 season, to create a histogram of the number of free throws +made by each team (FT). +Mean, Median and Mode +7. The data below show the top ten scorers (by points per game) in the National Basketball Association for the +2018–2019 season. Find the mode, median, and mean age of these players without technology. +Rank +Player +Age +1 +James Harden +29 +2 +Paul George +28 +3 +Giannis Antetokounmpo +24 +4 +Joel Embiid +24 +5 +Stephen Curry +30 +6 +Devin Booker +22 +7 +Kawhi Leonard +27 +Table 8.13 (source: https://www.basketball- +reference.com/leagues/ +NBA_2019_leaders.html) +938 +8 • Chapter Summary +Access for free at openstax.org + +Rank +Player +Age +8 +Kevin Durant +30 +9 +Damian Lillard +28 +10 +Bradley Beal +25 +Table 8.13 (source: https://www.basketball- +reference.com/leagues/ +NBA_2019_leaders.html) +8. Using the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets), find the mean, median, and mode of the +number of free throws made by each team (FT). +Range and Standard Deviation +9. Using the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets), find the range and standard deviation of +the number of free throws made by each team (FT). +Percentiles +Use the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets), to answer the following: +10. What number of free throws (FT) is at the 40th percentile? +11. At what percentile is Syracuse University, which made 480 free throws? +The Normal Distribution +12. Identify the means and standard deviations of these normal distributions: +Answer the following about data that are distributed normally with mean 200 and standard deviation 20: +13. What proportion of the data are between 160 and 240? +14. What’s the standardized score of the data value 235? +15. At what percentile would the data value 187 fall? +16. What data value would be at the 90th percentile? +Applications of the Normal Distribution +Use the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets) to answer the following: +17. What is the standardized score for Purdue Universitiy’s number of free throws (FT = 461)? +18. At what percentile would we estimate Purdue’s value of FT to fall, using the normal distribution? +19. At what percentile does Purdue’s value of FT actually fall? +Scatter Plots, Correlation, and Regression Lines +For each of the following scatter plots, decide whether linear regression would be appropriate. If it is, classify the +strength and direction of the relationship. +8 • Chapter Summary +939 + +20. +21. +22. +Use the data in “NBA2019” (https://openstax.org/r/Chapter8_Data-Sets) to answer the following: +23. What is the correlation coefficient between the number of field goals attempted per game (FGA) and number of +points scored per game (PTS)? +24. What is the equation of the regression line we would use to predict PTS from FGA? +25. Write a sentence to interpret the slope of the regression line. +26. Predict the number of points per game for a player who attempts 16 field goals per game. +27. Danilo Gallinari attempted 13 field goals per game, and averaged 19.8 points per game. Did he score more or +fewer points per game than expected? By how much? +940 +8 • Chapter Summary +Access for free at openstax.org + +Chapter Test +Decide whether randomization is being used in the selection of these samples. If it is, identify the type of random +sample (simple, systematic, cluster, or stratified). +1. A vinyl record dealer is trying to price a large collection she’s thinking of buying. She looks at every tenth record +on the shelf and notes the value. +2. A court clerk is charged with identifying one hundred people for a jury pool for upcoming legal hearings. He +has an alphabetized list of registered voters in his jurisdiction, so he uses a random number generator to pick +one hundred names from the list. +3. A state agricultural officer is worried about the spread of a parasitic disease among cattle. He chooses 30 cattle +farms at random and tests each cow on those 30 farms for the disease. +4. A sample of donations to a blood bank contained these blood types: +A +O +A +O +A +A +AB +O +O +A +AB +A +O +O +A +A +O +O +B +O +B +O +O +O +B +A +O +A +O +O +A +A +Create a frequency distribution for these data. +5. Create a bar chart without technology to visualize the data in question 4. +6. Use the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets) to create a histogram of the number of +total rebounds by each team (TRB). +7. The table below shows the number of players on the rosters of each English Premier League team. Find the mode, +median, and mean without technology. +Team +Active Players +Team +Active Players +Manchester United F.C. +27 +Newcastle United F.C. +28 +Manchester City F.C. +24 +Aston Villa F.C. +24 +Chelsea F.C. +25 +Fulham F.C. +27 +Arsenal F.C. +28 +Southampton F.C. +24 +Tottenham Hotspur F.C. +27 +Wolverhampton Wanderers F.C. +22 +Liverpool F.C. +30 +Brighton & Hove Albion +31 +Crystal Palace +29 +Burnley F.C. +22 +Everton F.C. +23 +West Bromwich Albion F.C. +26 +Leicester City +25 +Leeds United F.C. +25 +West Ham United F.C. +24 +Sheffield United F.C. +27 +Table 8.14 (source: https://fbref.com/en/comps/9/Premier-League- +Stats#all_stats_squads_standard) +8 • Chapter Summary +941 + +8. Using the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets), find the mean, median, and mode of the +total number of rebounds made by each team (TRB). +9. Using the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets), find the range and standard deviation of +the total number of rebounds made by each team (TRB). +Use the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets) to answer the following: +10. What number of total rebounds (TRB) is at the 3rd quartile? +11. At what percentile is the University of Evansville, which had 1095 total rebounds? +Answer the following about data that are distributed normally with mean 50 and standard deviation 5: +12. What proportion of the data are between 35 and 65? +13. What’s the standardized score of the data value 37? +14. At what percentile would the data value 58 fall? +15. What data value would be at the 40th percentile? +Use the data in “MBB2019” (https://openstax.org/r/Chapter8_Data-Sets) to answer the following: +16. What is the standardized score for the number of total rebounds (TRB) recorded by Mississippi State University +(TRB = 1215)? +17. At what percentile would we estimate Mississippi State’s TRB value fall, using the normal distribution? +18. At what percentile does Mississippi State’s TRB value actually fall? +Use the data in “NBA2019” (https://openstax.org/r/Chapter8_Data-Sets) to answer the following: +19. What is the correlation coefficient between the number of two-point field goals made per game (2P) and +number of points scored per game (PTS)? +20. What is the equation of the regression line we would use to predict PTS from 2P? +21. Write a sentence to interpret the slope of the regression line. +22. Predict the number of points per game for a player who makes 6 two-point field goals per game. +23. J.J. Redick made 2.8 field goals per game, and averaged 18.1 points per game. Did he score more or fewer +points per game than expected? By how much? +942 +8 • Chapter Summary +Access for free at openstax.org + +Figure 9.1 A road sign in Finland, a country that uses the metric system. (credit: modification of work by Anna Järvenpää/ +Flickr, Public Domain) +Chapter Outline +9.1 The Metric System +9.2 Measuring Area +9.3 Measuring Volume +9.4 Measuring Weight +9.5 Measuring Temperature +Introduction +You are planning a road trip from your home state to the sunny beaches of Mexico and need to prepare a budget. While +in the United States, gasoline is sold in gallons and distances are measured in miles, but in almost any other country you +will find that gasoline is sold in liters and distance is measured in kilometers. +Whether you’re traveling, baking, watching an international sporting event, working with machine tools, or using +scientific equipment, it's important to understand the metric system, or the International System of Units (SI). The +metric system is a decimal measuring system that uses meters, liters, and grams to quantify length, capacity, and mass. +It is used in all but three countries in the world, including the United States. +9 +METRIC MEASUREMENT +9 • Introduction +943 + +9.1 The Metric System +Figure 9.2 A scale that measures weight in both metric and customary units. (credit: “Weighing the homemade cheddar” +by Ruth Hartnup/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify units of measurement in the metric system and their uses. +2. +Order the six common prefixes of the metric system. +3. +Convert between like unit values. +Even if you don’t travel outside of the United States, many specialty grocery stores utilize the metric system. For +example, if you want to make authentic tamales you might visit the nearest Hispanic grocery store. While shopping, you +discover that there are two brands of masa for the same price, but one bag is marked 1,200 g and the other 1 kg. Which +one is the better deal? Understanding the metric system allows you to understand that 1,200 grams is equivalent to 1.2 +kg, so the 1,200 g bag is the better deal. +Units of Measurement in the Metric System +Units of measurement provide common standards so that regardless of where or when an object or substance is +measured, the results are consistent. When measuring distance, the units of measure might be feet, meters, or miles. +Weight might be expressed in terms of pounds or grams. Volume or capacity might be measured in gallons, or liters. +Understanding how metric units of measure relate to each other is essential to understanding the metric system: +• +The metric unit for distance is the meter (m). A person’s height might be written as 1.8 meters (1.8 m). A meter +slightly longer than a yard (3 feet), while a centimeter is slightly less than half an inch. +• +The metric unit for area is the square meter (m2). The area of a professional soccer field is 7,140 square meters +(7,140 m2). +• +The metric base unit for volume is the cubic meter (m3). However, the liter (L), which is a metric unit of capacity, is +used to describe the volume of liquids. Soda is often sold in 2-liter (2 L) bottles. +• +The gram (g) is a metric unit of mass but is commonly used to express weight. The weight of a paper clip is +approximately 1 gram (1 g). +• +The metric unit for temperature is degrees Celsius (°C). The temperature on a warm summer day might be 24 °C. +While the U.S. Customary System of Measurement uses ounces and pounds to distinguish between weight units of +different sizes, in the metric system a base unit is combined with a prefix, such as kilo– in kilogram, to identify the +relationship between smaller or larger units. +When using abbreviations to represent metric measures, always separate the quantity and the units with a space, +with no spaces between the letters or symbols in the units. For example, 7 millimeters is written as 7 mm, not 7mm. +944 +9 • Metric Measurement +Access for free at openstax.org + +TECH CHECK +It is important to be able to convert between the U.S. Customar System of Measurement and the metric system. +However, in this chapter we’ll focus on converting units within the metric system. Why? Typing “200 centimeters in +inches” into any browser search bar will instantly convert those measures for you (Figure 9.3). You’ll have an +opportunity in the Projects to work between measurement systems. +Figure 9.3 (credit: Screenshot/Google) +Let’s be honest. Most of us use computers or smartphones to perform many of the calculations and conversions we +were taught in math class. But there is value in understanding the metric system since it exists all around us, and +most importantly, knowing how the different metric units relate to each other allows you to compare prices, find the +right tool in a workshop, or acclimate when in another country. No matter the circumstance, you cannot avoid the +metric system. +EXAMPLE 9.1 +Determining the Correct Base Unit +Which base unit would be used to express the following? +1. +amount of water in a swimming pool +2. +length of an electrical wire +3. +weight of one serving of peanuts +Solution +1. +Liquid volume is generally expressed in units of liters (L). +2. +Length is measured in units of meters (m). +3. +Weight is commonly expressed in units of grams (g). +YOUR TURN 9.1 +Determine the correct base measurement for each of the following. +1. weight of a laptop +2. width of a table +3. amount of soda in a pitcher +While there are other base units in the metric system, our discussions in this chapter will be limited to units used to +express length, area, volume, weight, and temperature. +9.1 • The Metric System +945 + +Metric Prefixes +Unlike the U.S. Customary System of Measurement in which 12 inches is equal to 1 foot and 3 feet are equal to 1 yard, +the metric system is structured so that the units within the system get larger or smaller by a power of 10. For example, a +centimeter is +, or 100 times smaller than a meter, while the kilometer is 103, or 1,000 times larger than a meter. +The metric system combines base units and unit prefixes reasonable to the size of a measured object or substance. The +most used prefixes are shown in Table 9.1. An easy way to remember the order of the prefixes, from largest to smallest, +is the mnemonic King Henry Died From Drinking Chocolate Milk. +Prefix +kilo– +hecto– +deca– +base unit +deci– +cent– +milli– +Abbreviation +Magnitude +Table 9.1 Metric Prefixes +EXAMPLE 9.2 +Ordering the Magnitude of Units +Order the measures from smallest unit to largest unit. +centimeter, millimeter, decimeter +Solution +Looking at the metric prefixes, we can see that the prefix order from smallest unit to largest unit is milli-, centi-, deci-, so +the order of the units from smallest to largest is millimeter, centimeter, decimeter. +YOUR TURN 9.2 +1. Order the measures from largest unit to smallest unit. +hectogram, decagram, kilogram +EXAMPLE 9.3 +Determining Reasonable Values for Length +What is a reasonable value for the length of a person’s thumb: 5 meters, 5 centimeters, or 5 millimeters? +Solution +Given that a meter is slightly longer than a yard, 5 meters is not a reasonable value for the length of a person’s thumb. +Since a millimeter is 10 times smaller than a centimeter, which is approximately +inch, 5 millimeters is not a reasonable +estimate for the length of a person’s thumb. The correct answer is 5 centimeters. +YOUR TURN 9.3 +1. What is a reasonable estimate for the length of a hallway: 2.5 kilometers, 2.5 meters, or 2.5 centimeters? +Converting Metric Units of Measure +Imagine you order a textbook online and the shipping detail indicates the weight of the book is 1 kg. By attaching the +letter “k” to the base unit of gram (g), the unit used to express the measure is +or 1,000 times greater than a gram. +One kilogram is equivalent to 1,000 grams. +The tip of a highlighter measures approximately 1 cm. The letter “c” attached to the base unit of meter (m) means the +946 +9 • Metric Measurement +Access for free at openstax.org + +unit used to express the measure is +of a meter. One meter is equivalent to 100 centimeters. +A conversion factor is used to convert from smaller metric units to bigger metric units and vice versa. It is a number +that when used with multiplication or division converts from one metric unit to another, both having the same base unit. +In the metric system, these conversion factors are directly related to the powers of 10. The most common used +conversion factors are shown in Figure 9.4. +Figure 9.4 Common Metric Conversion Factors for (a) Meters, (b) Liters, and (c) Grams +EXAMPLE 9.4 +Converting Metric Distances Using Multiplication +The firehouse is 13.45 km from the library. How many meters is it from the firehouse to the library? +Solution +When converting from a larger unit to a smaller unit, use multiplication. The conversion factor from kilometer to the +base unit of meters is 1,000. +So, the firehouse is 13,450 meters away from the firehouse. +YOUR TURN 9.4 +1. The record for the men’s high jump is 2.45 m. What is the record when expressed in centimeters? +EXAMPLE 9.5 +Converting Metric Capacity Using Division +How many liters is 3,565 milliliters? +Solution +When converting from a smaller unit to a larger unit, use division. The conversion factor from milliliter to the base unit of +liters is 1,000. +So, 3,565 milliliters is 3.565 liters. +9.1 • The Metric System +947 + +YOUR TURN 9.5 +1. A bottle of cleaning solution measures 7.6 liters. How many decaliters is that? +EXAMPLE 9.6 +Converting Metric Units of Mass to Solve Problems +Caroline and Aiyana are working on a chemistry experiment together and must perform calculations using +measurements taken during the experiment. Due to miscommunication, Caroline took measurements in centigrams and +Aiyana used milligrams. Convert Caroline’s measurement of 125 centigrams to milligrams. +Solution +When converting from a larger unit to a smaller unit, use multiplication. The conversion factor from centigrams to the +milligrams is 10. +So, the 125 centigrams are 1,250 milligrams. +YOUR TURN 9.6 +1. Convert Aiyana’s measurement of 1,457 mg to centigrams. +EXAMPLE 9.7 +Converting Metric Units of Volume to Solve Problems +A bottle contains 500 mL of juice. If the juice is packaged in 24-bottle cases, how many liters of juice does the case +contain? +Solution +Step 1: Multiply the amount of juice in each bottle by the number of bottles. +Step 2: Divide by 1,000 to convert from milliliters to liters. +So, there are 12 liters of juice in each case. +YOUR TURN 9.7 +1. A hospital orders 250 doses of liquid amoxicillin. Each dose is 5 mL. How many liters of amoxicillin did the +hospital order? +PEOPLE IN MATHEMATICS +Valerie Antoine +In the 1970s, people were told that they must learn the metric system because the United States was soon going to +948 +9 • Metric Measurement +Access for free at openstax.org + +convert to using metric measurements. Children and young adults probably watched educational cartoons about the +metric system on Saturday mornings. +In 1975, President Gerald Ford signed the Metric Conversion Act and created a board of 17 people commissioned to +coordinate the voluntary switch to the metric system in the United States. Among those 17 people was Valerie +Antoine, an engineer who made it her life’s work to push for this change. Despite President Ronald Reagan dissolving +the board in 1982, effectively killing the move to the metric system at the time, Antoine continued the movement out +of her own home as the executive director of the U.S. Metric Association. Reagan’s decision followed intensive +lobbying by American businesses whose factories used machinery designed to use customary measurements by +workers trained in customary measurements. There was also intense public pressure from American citizens who +didn’t want to go through the time consuming and expensive process of changing the country’s entire infrastructure. +Fueled by a Congressional mandate in 1992 that required all federal agencies make the switch to the metric system, +Antoine never gave up hope that the metric system would trickle down from the government and find its way into +American schools, homes, and everyday life. +VIDEO +U.S. Office of Education: Metric Education (https://openstax.org/r/U.S._Office_Education) +EXAMPLE 9.8 +Converting Grams to Solve Problems +The nutrition label on a jar of spaghetti sauce indicates that one serving contains 410 mg of sodium. You have poured +two servings over your favorite pasta before recalling your doctor’s advice about keeping your sodium consumption +below 1 g per meal. Have you followed your doctor’s recommendation? +Solution +Step 1: Multiply the number of servings by the amount of sodium in each serving. +Step 2: Divide by 1,000 to convert from milligrams to grams. +You have followed doctor’s recommendation because 0.82 g is less than 1 gram. +YOUR TURN 9.8 +1. The FDA recommends that you consume less than 0.5 g of caffeine daily. A cup of coffee contains 95 mg of +caffeine and a can of soda contains 54 mg. If you drink 2 cups of coffee and 3 cans of soda, have you kept your +day’s caffeine consumption to the FDA recommendation? Explain. +EXAMPLE 9.9 +Comparing Different Units +A student carefully measured 0.52 cg of copper for a science experiment, but their lab partner said they need 6 mg of +copper total. How many more centigrams of copper does the student need to add? +Solution +Step 1: Convert these two measurements into a common unit. Since the question asks for the number of centigrams, +9.1 • The Metric System +949 + +convert 6 mg to centigrams, which is 0.6 cg. +Step 2: Find the difference by subtracting +which is 0.08 cg. This means the student must add another 0.08 cg +of copper. +YOUR TURN 9.9 +1. Kyrie boasted he jumped out of an airplane at an altitude of 3,810 meters on his latest skydive trip. His friend +said they beat Kyrie because their jump was at an altitude of 3.2 km. Whose skydive was at a greater altitude? +WHO KNEW? +The United States and the Metric System +Did you know that the metric system pervades daily life in the United States already? While Americans still may +purchase gallons of milk and measure house sizes in square feet, there are many instances of the metric system. +Photographers buy 35 mm film and use 50 mm lenses. When you have a headache, you might take 600 mg of +ibuprofen. And if you are eating a low-carb diet you probably restrict your carb intake to fewer than 20 g of carbs +daily. Did you know even the dollar is metric? In the video, Neil DeGrasse Tyson and comedian co-host Chuck Nice +provide an amusing perspective on the metric system. +The International System of Units (SI) is the current international standard metric system and is the most widely used +system around the world. In most English-speaking countries SI units such as meter, liter, and metric ton are spelled +metre, litre, and tonne. +VIDEO +Neil deGrasse Tyson Explains the Metric System (https://openstax.org/r/Tyson_Explains_Metric_System) +WORK IT OUT +Get to Know the Metric System +Just how much is the metric system a part of your life now? Probably more than you think. For the next 24 hours, take +notice as you move through your daily activities. When you are shopping, are the package sizes provided in metric +units? Change the weather app on your phone to display the temperature in degrees Celsius. Are you able to tell what +kind of day it will be now? While the United States is not officially using the metric system, you will still find the metric +system all around you. +Check Your Understanding +1. Which metric base unit would be used to measure the height of a door? +2. Which metric base unit would be used to measure your weight? +3. Which is greater: 12 hectoliters or 12 centiliters? +4. Convert 1,520 cm to meters (m). +5. Convert 1.34 km to decameters (dam). +6. Convert 12,700 cg to hectograms (hg). +7. Convert 750 km to millimeters (mm). +8. Which is the larger measurement: 0.04 dam or 40 cm? +950 +9 • Metric Measurement +Access for free at openstax.org + +SECTION 9.1 EXERCISES +For the following exercises, determine the base unit of the metric system described. Choose from liter, gram, or meter. +1. amount of soda in a bottle +2. weight of a book you are mailing +3. amount of gasoline needed to fill a car’s tank +4. height of a computer desk +5. weight of a dog at the veterinarian’s office +6. dimensions of the newest HD TV +7. distance a student athlete ran on the treadmill during their latest workout +8. amount of water to add to bleach for mopping floors +9. Write the order of the metric prefixes from greatest to least. +For the following exercises, choose the smaller of the two units. +10. decagrams or decigrams +11. centimeters or millimeters +12. liters or kiloliters +13. decigrams or centigrams +14. decameters or hectometers +15. milliliters or deciliters +16. Convert 158 hectometers to meters (m). +17. Convert 12.3 cg to grams (g). +18. Convert 160 dam to kilometer (km). +19. You purchase 10 kg of candy. You divide the candy into 25 bags. How many grams of candy are in each bag? +20. An outdoor track is 400 m long. If you run 10 laps around the track, how many kilometers have you run? +21. An aspirin tablet is 650 mg. If you take 2 aspirin twice in one day, how many grams of aspirin have you taken? +22. A juice box contains 450 mL of juice. If there are 6 juice boxes in a package, how many liters of juice are in the +package? +23. Carlos consumed 4 cans of his favorite energy drink. If each can contains 111 mg of caffeine, how many grams of +caffeine did he consume? +24. Celeste ran a total distance of 72,548 m in 1 week while training for a 5K fun run. How many kilometers did she +run? +25. During week one of their diet, Dakota consumed 413 g of carbs. After speaking to their doctor, they only consumed +210 g of carbs the second week. How many fewer milligrams of carbs did they consume the second week? +26. Kaylea gained 2.3 kg of muscle weight in 9 months of working out. Cho gained 250 decagrams of muscle during +that same time. Who gained more muscle weight? How many grams more? +9.1 • The Metric System +951 + +9.2 Measuring Area +Figure 9.5 A painter uses an extension roller to paint a wall. (credit: "Paint Rollers are effective" by WILLPOWER STUDIOS/ +Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify reasonable values for area applications. +2. +Convert units of measures of area. +3. +Solve application problems involving area. +Area is the size of a surface. It could be a piece of land, a rug, a wall, or any other two-dimensional surface with +attributes that can be measured in the metric unit for distance-meters. Determining the area of a surface is important to +many everyday activities. For example, when purchasing paint, you’ll need to know how many square units of surface +area need to be painted to determine how much paint to buy. +Square units indicate that two measures in the same units have been multiplied together. For example, to find the area +of a rectangle, multiply the length units and the width units to determine the area in square units. +Note that to accurately calculate area, each of the measures being multiplied must be of the same units. For example, to +find an area in square centimeters, both length measures (length and width) must be in centimeters. +FORMULA +The formula used to determine area depends on the shape of that surface. Here we will limit our discussions to the +area of rectangular-shaped objects like the one in Figure 9.6. Given this limitation, the basic formula for area is: +Figure 9.6 Rectangle with Length +and Width +Labeled +952 +9 • Metric Measurement +Access for free at openstax.org + +Reasonable Values for Area +Because area is determined by multiplying two lengths, the magnitude of difference between different square units is +exponential. In other words, while a meter is 100 times greater in length than a centimeter, a square meter +is +times greater in area than a square centimeter +. The relationships between benchmark +metric area units are shown in the following table. +Units +Relationship +Conversion Rate +to +to +to +An essential understanding of metric area is to identify reasonable values for area. When testing for reasonableness you +should assess both the unit and the unit value. Only by examining both can you determine whether the given area is +reasonable for the situation. +EXAMPLE 9.10 +Determining Reasonable Units for Area +Which unit of measure is most reasonable to describe the area of a sheet of paper: +, +, or +? +Solution +In the U.S. Customary System of Measurement, the length and width of paper is usually measured in inches. In the +metric system centimeters are used for measures usually expressed in inches. Thus, the most reasonable unit of +measure to describe the area of a sheet of paper is square centimeters. Square kilometers is too large a unit and square +millimeters is too small a unit. +YOUR TURN 9.10 +1. Which unit of measure is most reasonable to describe the area of a forest: km2, cm2, or mm2? +EXAMPLE 9.11 +Determining Reasonable Values for Area +You want to paint your bedroom walls. Which represents a reasonable value for the area of the walls: 100 cm2, 100 m2, +or 100 km2? +Solution +An area of 100 cm2 is equivalent to a surface of +which is much too small for the walls of a bedroom. An +area of 100 km2 is equivalent to a surface of +which is much too large for the walls of a bedroom. So, a +reasonable value for the area of the walls is 100 m2. +9.2 • Measuring Area +953 + +YOUR TURN 9.11 +1. Which represents a reasonable value for the area of the top of a kitchen table: 1,800 mm2, 1,800 cm2, or +? +EXAMPLE 9.12 +Explaining Reasonable Values for Area +A landscaper is hired to resod a school’s football field. After measuring the length and width of the field they determine +that the area of the football field is 5,350 km2. Does their calculation make sense? Explain your answer. +Solution +No, kilometers are used to determine longer distances, such as the distance between two points when driving. A football +field is less than 1 kilometer long, so a more reasonable unit of value would be m2. An area of 5,350 km2 can be +calculated using the dimensions 53.5 by 100, which are reasonable dimensions for the length and width of a football +field. So, a more reasonable value for the area of the football field is 5,350 m2. +YOUR TURN 9.12 +1. A crafter uses four letter-size sheets of paper to create a paper mosaic in the shape of a square. They decide they +want to frame the paper mosaic, so they measure and determine that the area of the paper mosaic is +. Does their calculation make sense? Explain your answer. +Converting Units of Measures for Area +Just like converting units of measure for distance, you can convert units of measure for area. However, the conversion +factor, or the number used to multiply or divide to convert from one area unit to another, is not the same as the +conversion factor for metric distance units. Recall that the conversion factor for area is exponentially relative to the +conversion factor for distance. The most frequently used conversion factors are shown in Figure 9.7. +Figure 9.7 Common Conversion Factors for Metric Area Units +VIDEO +Converting Metric Units of Area (https://openstax.org/r/Metric_Units_of_Area) +EXAMPLE 9.13 +Converting Units of Measure for Area Using Division +A plot of land has an area of 237,500,000 m2. What is the area in square kilometers? +Solution +Use division to convert from a smaller metric area unit to a larger metric area unit. To convert from m2 to km2, divide the +value of the area by 1,000,000. +954 +9 • Metric Measurement +Access for free at openstax.org + +The plot of land has an area of 237.5 km2. +YOUR TURN 9.13 +1. A roll of butcher paper has an area of 1,532,900 cm2. What is the area of the butcher paper in square meters? +EXAMPLE 9.14 +Converting Units of Measure for Area Using Multiplication +A plot of land has an area of 0.004046 km2. What is the area of the land in square meters? +Solution +Use multiplication to convert from a larger metric area unit to a smaller metric area unit. To convert from km2 to m2, +multiply the value of the area by 1,000,000. +The plot of land has an area of 4,046 m2. +YOUR TURN 9.14 +1. A bolt of fabric has an area of 136.5 m2. What is the area of the bolt of fabric in square centimeters? +EXAMPLE 9.15 +Determining Area by Converting Units of Measure for Length First +A computer chip measures 10 mm by 15 mm. How many square centimeters is the computer chip? +Solution +Step 1: Convert the measures of the computer chip into centimeters +Step 2: Use the area formula to determine the area of the chip. +The computer chip has an area of +. +YOUR TURN 9.15 +1. A piece of fabric measures 100 cm by 106 cm. What is the area of the fabric in square meters? +Solving Application Problems Involving Area +While it may seem that solving area problems is as simple as multiplying two numbers, often determining area requires +more complex calculations. For example, when measuring the area of surfaces, you may need to account for portions of +the surface that are not relevant to your calculation. +9.2 • Measuring Area +955 + +EXAMPLE 9.16 +Solving for the Area of Complex Surfaces +One side of a commercial building is 12 meters long by 9 meters high. There is a rolling door on this side of the building +that is 4 meters wide by 3 meters high. You want to refinish the side of the building, but not the door, with aluminum +siding. How many square meters of aluminum siding are required to cover this side of the building? +Solution +Step 1: Determine the area of the side of the building. +Step 2: Determine the area of the door. +Step 3: Subtract the area of the door from the area of the side of the building. +So, you need to purchase +of aluminum siding. +YOUR TURN 9.16 +1. You want to cover a garden with topsoil. The garden is 5 meters by 8 meters. There is a path in the middle of the +garden that is 8 meters long and 0.75 meters wide. What is the area of the garden you need to cover with +topsoil? +When calculating area, you must ensure that both distance measurements are expressed in terms of the same distance +units. Sometimes you must convert one measurement before using the area formula. +EXAMPLE 9.17 +Solving for Area with Distance Measurements of Different Units +A national park has a land area in the shape of a rectangle. The park measures 2.2 kilometers long by 1,250 meters wide. +What is the area of the park in square kilometers? +Solution +Step 1: Use a conversion fraction to convert the information given in meters to kilometers. +Step 2: Multiply to find the area. +The park has an area of 2.75 km2. +YOUR TURN 9.17 +1. An Olympic pool measures 50 meters by 2,500 centimeters. What is the surface area of the pool in square +956 +9 • Metric Measurement +Access for free at openstax.org + +meters? +When calculating area, you may need to use multiple steps, such as converting units and subtracting areas that are not +relevant. +EXAMPLE 9.18 +Solving for Area Using Multiple Steps +A kitchen floor has an area of 15 m2. The floor in the kitchen pantry is 100 cm by 200 cm. You want to tile the kitchen and +pantry floors using the same tile. How many square meters of tile do you need to buy? +Solution +Step 1: Determine the area of the pantry floor in square centimeters. +Step 2: Divide the area in cm2 by the conversion factor to determine the area in m2 since the other measurement for the +kitchen floor is in m2. +The area of the kitchen pantry floor is 2 m2. +Step 3: Add the two areas of the pantry and the kitchen floors together. +So, you need to buy 17 m2 of tile. +YOUR TURN 9.18 +1. Your bedroom floor has an area of 25 m2. The living room floor measures 600 cm by 750 cm. How many square +meters of carpet do you need to buy to carpet the floors in both rooms? +WHO KNEW? +The Origin of the Metric System +The metric system is the official measurement system for every country in the world except the United States, Liberia, +and Myanmar. But did you know it originated in France during the French Revolution in the late 18th century? At the +time there were over 250,000 different units of weights and measures in use, often determined by local customs and +economies. For example, land was often measured in days, referring to the amount of land a person could work in a +day. +VIDEO +Why the Metric System Matters (https://openstax.org/r/Metric_System_Matters) +Check Your Understanding +For the following exercises, determine the most reasonable value for each area. +9. bedroom wall: 12 km2, 12 m2, 12 cm2, or 12 mm2 +10. city park: 1,200 km2, 1,200 m2, 1,200 cm2, or 1,200 mm2 +9.2 • Measuring Area +957 + +11. kitchen table: 2.5 km2, 2.5 m2, 2.5 cm2, or 2.5 mm2 +For the following exercises, convert the given area to the units shown. +12. 20,000 cm2 = __________ m2 +13. 5.7 m2 = __________ cm2 +14. 217 cm2 = __________ mm2 +15. A wall measures 4 m by 2 m. A doorway in the wall measures 0.5 m by 1.6 m. What is the area of the wall not taken +by the door in square meters? +SECTION 9.2 EXERCISES +For the following exercises, determine the most reasonable value for each area. +1. kitchen floor: 16 km2, 16 m2, 16 cm2, or 16 mm2 +2. national Park: 1,000 km2, 1,000 m2, 1,000 cm2, or 1,000 mm2 +3. classroom table: 5 km2, 5 m2, 5 cm2, or 5 mm2 +4. window: 9,000 km2, 9,000 m2, 9,000 cm2, or 9,000 mm2 +5. paper napkin: 10,000 km2, 10,000 m2, 10,000 cm2, or 10,000 mm2 +6. parking lot: 45,000 km2, 45,000 m2, 45,000 cm2, or 45,000 mm2 +For the following exercises, convert the given area to the units shown. +7. +8. +9. +10. +11. +12. +13. +14. +15. +16. +17. +18. +For the following exercises, determine the area. +19. +cm, +m +20. +mm, +m +21. +km, +m +22. +cm, +m +23. +cm, +mm +24. +m, +km +25. +cm, +mm +26. +mm, +m +27. +km, +m +28. A notebook is 200 mm by 300 mm. A sticker on the notebook cover measures 2 cm by 2 cm. How many square +centimeters of the notebook cover is still visible? +29. A bedroom wall is 4 m by 2.5 m. A window on the wall is 1 m by 2 m. How much wallpaper is needed to cover the +wall? +30. A wall is 5 m by 2.5 m. A picture hangs on the wall that is 600 mm by 300 mm. How much of the wall, in m2, is not +958 +9 • Metric Measurement +Access for free at openstax.org + +covered by the picture? +31. What is the area of the shape that is shown? +32. A quilter made a design using a small square, a medium square, and a large square. What is the area of the +shaded parts of the design that is shown? +33. A landscaper makes a plan for a walkway in a backyard, as shown. How many square meters of patio brick does +the landscaper need to cover the walkway? +34. A painter completed a portrait. The height of the portrait is 36 cm. The width is half as long as the height. What is +the area of the portrait in square meters? +35. A room has an area of 137.5 m2. The length of the room is 11 m. What is the width of the room? +36. A wall is 4.5 m long by 3 m high. A can of paint will cover an area of 10 m2. How many cans of paint are needed if +each wall needs 2 coats of paint? +37. A soccer field is 110 meters long and 75 meters wide. If the cost of artificial turf is $30 per square meter, what is +the cost of covering the soccer field with artificial turf? +38. A window is 150 centimeters wide and 90 centimeters high. If three times the area of the window is needed for +curtain material, how much curtain material is needed in square meters? +39. A rectangular flower garden is 5.5 meters wide and 8.4 meters long. A path with a width of 1 meter is laid around +the garden. What is the area of the path? +40. A dining room is 8 meters wide by 6 meters long. Wood flooring costs $12.50 per square meter. How much will it +costs to install wood flooring in the dining room? +9.2 • Measuring Area +959 + +9.3 Measuring Volume +Figure 9.8 Packing cartons sit on a loading dock ready to be filled. (credit: “boxing day” by Erich Ferdinand/Flickr, CC BY +2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify reasonable values for volume applications. +2. +Convert between like units of measures of volume. +3. +Convert between different unit values. +4. +Solve application problems involving volume. +Volume is a measure of the space contained within or occupied by three-dimensional objects. It could be a box, a pool, a +storage unit, or any other three-dimensional object with attributes that can be measured in the metric unit for +distance–meters. For example, when purchasing an SUV, you may want to compare how many cubic units of cargo the +SUV can hold. +Cubic units indicate that three measures in the same units have been multiplied together. For example, to find the +volume of a rectangular prism, you would multiply the length units by the width units and the height units to determine +the volume in square units: +Note that to accurately calculate volume, each of the measures being multiplied must be of the same units. For example, +to find a volume in cubic centimeters, each of the measures must be in centimeters. +FORMULA +The formula used to determine volume depends on the shape of the three-dimensional object. Here we will limit our +discussions to the area to rectangular prisms like the one in Figure 9.9 Given this limitation, the basic formula for +volume is: +960 +9 • Metric Measurement +Access for free at openstax.org + +Figure 9.9 Rectangular Prism with Height +, Length +, and Width +Labeled. +Reasonable Values for Volume +Because volume is determined by multiplying three lengths, the magnitude of difference between different cubic units is +exponential. In other words, while a meter is 100 times greater in length than a centimeter, a cubic meter, m3, is +times greater in area than a cubic centimeter, cm3. This relationship between benchmark +metric volume units is shown in the following table. +Units +Relationship +Conversion Rate +km3 to m3 +1 km3 = 1,000,000,000 m3 +m3 to dm3 +1 m3 = 1,000 dm3 +dm3 to cm3 +1 cm3 = 1,000 dm3 +cm3 to mm3 +1 cm3 = 1,000 mm3 +To have an essential understanding of metric volume, you must be able to identify reasonable values for volume. When +testing for reasonableness you should assess both the unit and the unit value. Only by examining both can you +determine whether the given volume is reasonable for the situation. +EXAMPLE 9.19 +Determining Reasonable Values for Volume +A grandparent wants to send cookies to their grandchild away at college. Which represents a reasonable value for the +volume of a box to ship the cookies: +• +3,375 km3, +• +3,375 m3, or +• +3,375 cm3? +Solution +A volume of 3,375 km3 is equivalent to a rectangular prism with dimensions of +which is far too +large for a shipping box. An area of 3,375 m3 is equivalent to a surface of +which is also too large. A +reasonable value for the volume of the box is 3,375 cm3. +9.3 • Measuring Volume +961 + +YOUR TURN 9.19 +1. Which represents a reasonable value for the volume of a storage area: +8 m3, 8 cm3, or 8 mm3? +EXAMPLE 9.20 +Identifying Reasonable Values for Volume +A food manufacturer is prototyping new packaging for one of its most popular products. Which represents a reasonable +value for the volume of the box: +• +2 dm3, +• +2 cm3, or +• +2 mm3? +Solution +A decimeter is equal to 10 centimeters. A box with a volume of 2 dm3 might have the dimensions +, +or +, which is reasonable. A box with a volume of 2 cm3 or 2 mm3 would be too small. +YOUR TURN 9.20 +1. Which represents a reasonable value for the volume of a fish tank: +40,000 mm3, 40,000 cm3, or 40,000 m3? +EXAMPLE 9.21 +Explaining Reasonable Values for Volume +A farmer has a hay loft. They calculate the volume of the hayloft as 64 cm3. Does the calculation make sense? Explain +your answer. +Solution +No. Centimeters are used to determine smaller distances, such as the length of a pencil. A hayloft is more than 64 +centimeters long, so a more reasonable unit of value would be m2. A volume of 64 m3 can be calculated using the +dimensions 4 meters by 4 meters by 4 meters, which are reasonable dimensions for a hayloft. So, a more reasonable +value for the volume of the hayloft is 64 m3. +YOUR TURN 9.21 +1. An artist creates a glass paperweight. They decide they want to box and ship the paperweight, so they measure +and determine that the volume of the cubic box is 125,000 mm3. Does their calculation make sense? Explain your +answer. +Converting Like Units of Measures for Volume +Just like converting units of measure for distance, you can convert units of measure for volume. However, the conversion +factor, the number used to multiply or divide to convert from one volume unit to another, is different from the +conversion factor for metric distance units. Recall that the conversion factor for volume is exponentially relative to the +conversion factor for distance. The most frequently used conversion factors are illustrated in Figure 9.10. +962 +9 • Metric Measurement +Access for free at openstax.org + +Figure 9.10 Common Conversion Factors for Metric Volume Units +EXAMPLE 9.22 +Converting Like Units of Measure for Volume Using Multiplication +A pencil case has a volume of 1,700 cm3. What is the volume in cubic millimeters? +Solution +Use multiplication to convert from a larger metric volume unit to a smaller metric volume unit. To convert from cm3 to +mm3, multiply the value of the volume by 1,000. +The pencil case has a volume of 1,700,000 mm3. +YOUR TURN 9.22 +1. A jewelry box has a volume of 8 cm3. What is the volume of the jewelry box in cubic millimeters? +VIDEO +How to Convert Cubic Centimeters to Cubic Meters (https://openstax.org/r/Convert_CC_to_CM) +EXAMPLE 9.23 +Converting Like Units of Measure for Volume Using Multi-Step Multiplication +A shipping container has a volume of 33.2 m3. What is the volume in cubic centimeters? +Solution +Use multiplication to convert a larger metric volume unit to a smaller metric volume unit. To convert from m3 to cm3, +first multiply the value of the volume by 1,000 to convert from m3 to dm3, and then multiply again by 1,000 to convert +from dm3 to cm3. +The shipping container has a volume of 32,200,000 cm3. +YOUR TURN 9.23 +1. A gasoline storage tank has a volume of 37.854 m3. What is the volume of the storage tank in cubic centimeters? +EXAMPLE 9.24 +Converting Like Units of Measure for Volume Using Multi-Step Division +A holding tank at the local aquarium has a volume of 22,712,000,000 cm3. What is the volume in cubic meters? +9.3 • Measuring Volume +963 + +Solution +Figure 9.10 indicates that when converting from a smaller metric volume unit to a larger metric volume unit you divide +using the given conversion factor. To convert from cm3 to m3, divide the value of the volume by 1,000 to first convert +from cm3 to dm3, then divide again to convert from dm3 to m3. +The holding tank has a volume of 22,712 m3. +YOUR TURN 9.24 +1. A warehouse has a volume of 465,000,000 cm3. What is the volume of the warehouse in cubic meters? +Understanding Other Metric Units of Volume +When was the last time you purchased a bottle of soda? Was the volume of the bottle expressed in cubic centimeters or +liters? The liter (L) is a metric unit of capacity often used to express the volume of liquids. A liter is equal in volume to 1 +cubic decimeter. A milliliter is equal in volume to 1 cubic centimeter. So, when a doctor orders 10 cc (cubic centimeters) of +saline to be administered to a patient, they are referring to 10 mL of saline. +The most frequently used factors for converting from cubic meters to liters are listed in Table 9.2. +m3 to L +m3 to mL +Table 9.2 Relationships Between Metric +Volume and Metric Capacity Units +VIDEO +Converting Metric Units of Volume (https://openstax.org/r/Converting_Metric_Units_of_Volume) +EXAMPLE 9.25 +Converting Different Units of Measure for Volume +A holding tank at the local aquarium has a volume of 22,712,000,000 cm3? What is the capacity of the holding tank in +liters? +Solution +Use division to convert from cubic centimeters to liters. To determine the equivalent volume in liters, convert from cm3 +to L by dividing the value of the volume in cm3 by 1,000. +The holding tank holds 22,712,000 L of water. +964 +9 • Metric Measurement +Access for free at openstax.org + +YOUR TURN 9.25 +1. A gas can has a volume of 19,000 cm3. How much gas, in liters, does the gas can hold? +EXAMPLE 9.26 +Converting Different Units of Measure for Volume Using Multiplication +An airplane used 150 m3 of fuel to fly from New York to Hawaii. How many liters of fuel did the airplane use? +Solution +Because 1 liter is equivalent to 1 cubic decimeter, use multiplication to convert from m3 to dm3. Multiply the value of the +volume by 1,000 to convert from m3 to dm3. Because +, the resulting value is equivalent to the number of +liters used. +The airplane used 150,000 liters of fuel. +YOUR TURN 9.26 +1. A gasoline storage tank has a volume of 37.854 m3. What is the volume of the storage tank in liters? +EXAMPLE 9.27 +Converting Different Units of Measure for Volume Using Multi-Step Division +How many liters can a pitcher with a volume of 8,000,000 mm3 hold? +Solution +Use division to convert from a smaller metric volume unit to a larger metric volume unit. To convert from mm3 to dm3, +Step 1: Divide by 1,000 to convert from mm3 to cm3. +Step 2: Divide again by 1,000 to convert from cm3 to dm3. +Step 3: Use the unit value to express the volume in terms of liters. +The pitcher can hold 8 liters of liquid. +YOUR TURN 9.27 +1. A glass jar has a volume of 800,000 mm3. How many mL of liquid can the glass jar hold? +Solving Application Problems Involving Volume +Knowing the volume of an object lets you know just how much that object can hold. When making a bowl of punch you +might want to know the total amount of liquid a punch bowl can hold. Knowing how many liters of gasoline a car’s tank +can hold helps determine how many miles a car can drive on a full tank. Regardless of the application, understanding +volume is essential to many every day and professional tasks. +9.3 • Measuring Volume +965 + +EXAMPLE 9.28 +Using Volume to Solve Problems +A cubic shipping carton’s dimensions measure +. A company wants to fill the carton with smaller cubic +boxes that measure +. How many of the smaller boxes will fit in each shipping carton? +Solution +Step 1: Determine the volume of the shipping carton. +Step 2: Use the appropriate conversion factor to convert the volume of the shipping carton from m3 to cm3. +Step 3: Determine the volume of the smaller boxes. +Step 4: Divide the volume of the shipping carton, in cm3, by the volume of the smaller box, in cm3. +The shipping carton will hold 8,000 smaller boxes. +YOUR TURN 9.28 +1. A factory can mill 300 cubic meters of flour each day. They package the flour in boxes that measure +. How many boxes of flour does the factory produce each day? +EXAMPLE 9.29 +Solving Volume Problems with Different Units +A carton of juice measures 6 cm long, 6 cm wide and 20 cm tall. A factory produces 28,800 liters of orange juice each day. +How many cartons of orange juice are produced each day? +Solution +Step 1: Find the volume of the carton in cubic centimeters. +Step 2: Convert the volume in cm3 to liters. +Step 3: Divide the number of liters of orange juice produced each day by the volume of each carton. +966 +9 • Metric Measurement +Access for free at openstax.org + +The factory produces 40,000 cartons of orange juice each day. +YOUR TURN 9.29 +1. An ice cream maker boxes frozen yogurt mix in boxes that measure 25 cm long, 8 cm wide and 35 cm tall. They +produce 42,000 liters of frozen yogurt mix each day. How many boxes of frozen yogurt mix are produced each +day? +EXAMPLE 9.30 +Solving Complex Volume Problems +A fish tank measures 60 cm long, 15 cm wide and 34 cm tall (Figure 9.11). The tank is 25 percent full. How many liters of +water are needed to completely fill the tank? +Figure 9.11 +Solution +Step 1: Determine the volume of the fish tank in cubic centimeters. +Step 2: Convert the volume in cm3 to volume in liters. +Step 3: Since the tank is 25 percent full, the tank is 75 percent empty. Convert 75 percent to its decimal equivalent. +Multiply the total volume by 75 percent expressed in decimal form to determine how many liters of water are required to +fill the tank. +So, 22.95 liters of water are needed to fill the tank. +9.3 • Measuring Volume +967 + +YOUR TURN 9.30 +1. A fish tank measures 75 cm long, 20 cm wide and 25 cm tall. The tank is 50 percent full. How many liters of water +are needed to completely fill the tank? +WORK IT OUT +How Does Shape Affect Volume? +Take two large sheets of card stock. Roll one piece to tape the longer edges together to make a cylinder. Tape the +cylinder to the other piece of card stock which serves as the base of the cylinder. Fill the cylinder to the top with +cereal. Pour the cereal from the cylinder into a plastic storage or shopping bag. Remove the cylinder from the base +and the tape from the cylinder. Re-roll the cylinder along the shorter edges a tape together. Attach the new cylinder to +the base. Pour the cereal from the plastic bag into the cylinder. What do you observe? How does the shape of a +container affect its volume? +Check Your Understanding +For the following exercises, determine the most reasonable value for each volume. +16. Terrarium: 50,000 km3, 50,000 m3, 50,000 cm3, or 50,000 mm3 +17. Milk carton: 236,000 L, 236 L, 236,000 mL, or 236 mL +18. Box of crackers: 1,500 km3, 1,500 m3, 1,500 cm3, or 1,500 mm3 +For the following exercises, Convert the given volume to the units shown. +19. 42,500 mm3 = __________ cm3 +20. 1.5 dm3 = __________ mL +21. 6.75 cm3 = __________ mm3 +For the following exercises, determine the volume of objects with the dimensions shown. +22. +________ L +23. +________ mL +24. +________ m3 +SECTION 9.3 EXERCISES +For the following exercises, determine the most reasonable value for each volume. +1. Fish tank: +71,120 km3, 71,120 m3, 71,120 cm3, or 71,120 mm3 +2. Juice box: +125,000 L, 125 L, 125,000 mL, or 125 mL +3. Box of cereal: +2,700 km3, 2,700 m3, 2,700 cm3, or 2,700 mm3 +4. Water bottle: +5 L, 0.5 L, 5 mL, or 0.5 mL +5. Shoe box: +968 +9 • Metric Measurement +Access for free at openstax.org + +3,600 km3, 3.6 m3, 3,600 cm3, or 3,600 mm3 +6. Swimming pool: +45 L, 45,000 L, 45 mL, or 45,000 mL +For the following exercises, convert the given volume to the units shown. +7. 38,861 mm3 = __________ cm3 +8. 13 dm3 = __________ mL +9. 874 cm3 = __________ mm3 +10. 4 m3 = __________ cm3 +11. 0.00003 m3 = _________ mm3 +12. 57,500 mm3 = _______ L +13. 0.007 m3 = __________ L +14. 8,600 cm3 = _________ m3 +15. 45.65 m3 = _______ cm3 +16. 0.06 m3 = __________ dm3 +17. 0.081 m3 = _________ mL +18. 3,884,000 mm3 = _______ m3 +For the following exercises, determine the volume of objects with the dimensions shown. +19. +________ L +20. +________ mL +21. +________ m3 +22. +________ cm3 +23. +________ m3 +24. +________ L +25. +________ mL +26. +________ cm3 +27. +________ m3 +28. A box has dimensions of +. The box currently holds 1,250 cm3 of rice. How many cubic +centimeters of rice are needed to completely fill the box? +29. The dimensions of a medium storage unit are +. What is the volume of a small storage area with +dimensions half the size of the medium unit? +30. How much liquid, in liters, can a container with dimensions of +hold? +31. What is the volume of the rectangular prism that is shown? +9.3 • Measuring Volume +969 + +32. A box is 15 centimeters long and 5 centimeters wide. The volume of the box is 225 cm3. What is the height of the +box? +33. Kareem mixed two cartons of orange juice, three 2-liter bottles of soda water and six cans of cocktail fruits to make +a fruit punch for a party. The cartons of orange juice and cans of cocktail fruits each have a volume of 500 cm3. +How much punch, in liters, did Kareem make? +34. A holding tank has dimensions of +. If the tank is half-full, how more liters of liquid can the tank +hold? +35. A large plastic storage bin has dimensions of +. A medium bin’s dimensions are half the size +of the large bin. A small bin’s dimensions are the size of the medium bin. If the storage bins come in a set of +3—small, medium, and large—what is the total volume of the storage bin set in cubic centimeters? +36. A soft serve ice cream machine holds a 19.2 liter bag of ice cream mix. If the average serving size of an ice cream +cone is 120 mL, how many cones can be made from each bag of mix? +37. A shipping carton has dimensions of +. How many boxes with dimensions of +will fit in the shipping carton? +38. A recipe for chili makes 3.5 liters of chili. If a restaurant serves chili in 250 mL bowls, how many bowls of chili can +they serve? +39. A contractor is building an in-ground pool. They excavate a pit that measures +. The dirt is being +taken away in a truck that holds 30 m3. How many trips will the truck have to make to cart away all of the dirt? +40. A juice dispenser measures +. How many 375 mL servings will a full dispenser serve? +9.4 Measuring Weight +Figure 9.12 Weight scale at the local Antigua market (credit: “Weight scale at the local Antigua market” by Lucía García +González/Flickr, CC0 1.0 Public Domain Dedication) +Learning Objectives +After completing this section, you should be able to: +1. +Identify reasonable values for weight applications. +2. +Convert units of measures of weight. +3. +Solve application problems involving weight. +In the metric system, weight is expressed in terms of grams or kilograms, with a kilogram being equal to 1,000 grams. A +paper clip weighs about 1 gram. A liter of water weighs about 1 kilogram. In fact, in the same way that 1 liter is equal in +volume to 1 cubic decimeter, the kilogram was originally defined as the mass of 1 liter of water. In some cases, +particularly in scientific or medical settings where small amounts of materials are used, the milligram is used to express +weight. At the other end of the scale is the metric ton (mt), which is equivalent to 1,000 kilograms. The average car +weighs about 2 metric tons. +Any discussion about metric weight must also include a conversation about mass. Scientifically, mass is the amount of +970 +9 • Metric Measurement +Access for free at openstax.org + +matter in an object whereas weight is the force exerted on an object by gravity. The amount of mass of an object +remains constant no matter where the object is. Identical objects located on Earth and on the moon will have the same +mass, but the weight of the objects will differ because the moon has a weaker gravitational force than Earth. So, objects +with the same mass will weigh less on the moon than on Earth. +Since there is no easy way to measure mass, and since gravity is just about the same no matter where on Earth you go, +people in countries that use the metric system often use the words mass and weight interchangeably. While scientifically +the kilogram is only a unit of mass, in everyday life it is often used as a unit of weight as well. +Reasonable Values for Weight +To have an essential understanding of metric weight, you must be able to identify reasonable values for weight. When +testing for reasonableness, you should assess both the unit and the unit value. Only by examining both can you +determine whether the given weight is reasonable for the situation. +VIDEO +Metric System: Units of Weight (https://openstax.org/r/Metric_System:_units_of_weight) +EXAMPLE 9.31 +Identifying Reasonable Units for Weight +Which is the more reasonable value for the weight of a newborn baby: +• +3.5 kg or +• +3.5 g? +Solution +Using our reference weights, a baby weighs more than 3.5 paperclips, so 3.5 kilograms is a more reasonable value for +the weight of a newborn baby. +YOUR TURN 9.31 +1. Which represents a reasonable value for the weight of a penny: +2.5 g or 2.5 kg +EXAMPLE 9.32 +Determining Reasonable Values for Weight +Which of the following represents a reasonable value for the weight of three lemons? +• +250 g, +• +2,500 g, or +• +250 kg? +Solution +Because a kilogram is about 2.2 pounds, we can eliminate 250 kg as it is way too heavy. 2,500 grams is equivalent to 2.5 +kilograms, or about five pounds, which is again, too heavy. So, a reasonable value for the weight of three lemons would +be 250 grams. +YOUR TURN 9.32 +1. Which represents a reasonable value for the weight of a car: +1,300 g, 130 kg, or 1,300 kg? +9.4 • Measuring Weight +971 + +WHO KNEW? +How Do You Measure the Weight of a Whale? +It is impossible to weigh a living whale. Fredrik Christiansen from the Aarhus Institute of Advanced Studies in +Denmark developed an innovative way to measure the weight of whales. Using images taken from a drone and +computer modeling, the weight of a whale can be estimated with great accuracy. +VIDEO +Using Drones to Weigh Whales? (https://openstax.org/r/Using_drones_to_weigh_whales?) +EXAMPLE 9.33 +Explaining Reasonable Values for Weight +The blue whale is the largest living mammal on Earth. Which of the following is a reasonable value for the weight of a +blue whale: 149 g, 149 kg, or 149 mt? Explain your answer. +Solution +A reasonable value for the weight of a blue whale is 149 metric tons. Both 149 g and 149 kg are much too small a value +for the largest living mammal on Earth. +YOUR TURN 9.33 +1. The Etruscan shrew is one of the world’s smallest mammals. It has a huge appetite, eating almost twice its +weight in food each day. Its heart beats at a rate of 25 beats per second! Which of the following is a reasonable +value for the weight of an Etruscan shrew: 2 g, 2 kg, or 2 mt? Explain your answer. +Converting Like Units of Measures for Weight +Just like converting units of measure for distance, you can convert units of measure for weight. The most frequently used +conversion factors for metric weight are illustrated in Figure 9.13. +Figure 9.13 Common Conversion Factors for Metric Weight Units +EXAMPLE 9.34 +Converting Metric Units of Weight Using Multistep Division +How many kilograms are in 24,300,000 milligrams? +Solution +Use division to convert from a smaller metric weight unit to a larger metric weight unit. To convert from milligrams to +kilograms, +Step 1: Divide the value of the weight in milligrams by 1,000 to first convert from milligrams to grams. +972 +9 • Metric Measurement +Access for free at openstax.org + +Step 2: Divide by 1,000 again to convert from grams to kilograms. +So, 24,300,000 milligrams are equivalent to 24.3 kilograms. +YOUR TURN 9.34 +1. How many kilograms are in 175,000 milligrams? +EXAMPLE 9.35 +Converting Metric Units of Weight Using Multiplication +The average ostrich weighs approximately 127 kilograms. How many grams does an ostrich weigh? +Solution +Use multiplication to convert from a larger metric weight unit to a smaller metric weight unit. To convert from kilograms +to grams, multiply the value of the weight by 1,000. +The average ostrich weighs 127,000 grams. +YOUR TURN 9.35 +1. The world’s heaviest tomato weighed 4.869 kg when measured on July 15, 2020. How much did the tomato weigh +in grams? +EXAMPLE 9.36 +Converting Metric Units of Weight Using Multistep Multiplication +How many milligrams are there in 0.025 kilograms? +Solution +Use multiplication to convert from a larger metric weight unit to a smaller metric weight unit. To convert from kilograms +to grams, +Step 1: Multiply the value of the weight by 1,000. +Step 2: Multiply the result by 1,000 to convert from grams to milligrams. +So, 0.025 kilograms is equivalent to 25,000 milligrams. +9.4 • Measuring Weight +973 + +YOUR TURN 9.36 +1. How many milligrams are there in 1.23 kilograms? +VIDEO +Metric Units of Mass: Convert mg, g, and kg (https://openstax.org/r/Convert_mg-g-kg) +Solving Application Problems Involving Weight +From children’s safety to properly cooking a pie, knowing how to solve problems involving weight is vital to everyday life. +Let’s review some ways that knowing how to work with metric weight can facilitate important decisions and delicious +eating. +EXAMPLE 9.37 +Comparing Weights to Solve Problems +The maximum weight for a child to safely use a car seat is 29 kilograms. If a child weighs 23,700 grams, can the child +safely use the car seat? +Solution +Step 1: Convert the child’s weight in grams to kilograms. +Step 2: Compare the two weights. +Yes, the child can safely use the car seat. +YOUR TURN 9.37 +1. The dosage recommendations for a popular brand of acetaminophen are listed in table below. What is the +recommended dosage for a child who weighs 17,683 grams? +Weight +Dosage +11 kg to 15 kg +5 mL +16 kg to 21 kg +7.5 mL +22 kg to 27 kg +10 mL +EXAMPLE 9.38 +Solving Multistep Weight Problems +A recipe for scones calls for 350 grams of flour. How many kilograms of flour are required to make 4 batches of scones? +Solution +Step 1: Multiply the grams of flour need by 4 to determine the total amount of flour needed. +974 +9 • Metric Measurement +Access for free at openstax.org + +Step 2: Convert from grams to kilograms. +So, 1.4 kilograms of flour are needed to make four batches of scones. +YOUR TURN 9.38 +1. A croissant recipe calls for 500 g of flour. How many kilograms of flour are required to make 10 batches of +croissants? +EXAMPLE 9.39 +Solving Complex Weight Problems +The average tomato weighs 140 grams. A farmer needs to order boxes to pack and ship their tomatoes to local grocery +stores. They estimate that this year’s harvest will yield 125,000 tomatoes. A box can hold 12 kilograms of tomatoes. How +many boxes does the farmer need? +Solution +Step 1: Determine the total estimated weight of the harvested tomatoes. +Step 2: Convert the total weight from grams to kilograms. +Step 3: Divide the weight of the tomatoes by the weight each box can hold. +So, the farmer will need to order 1,458 boxes. +YOUR TURN 9.39 +1. The average potato weighs 225 grams. A grocery chain orders 5,000 bags of potatoes. Each bag weighs 5 kg. +Approximately how many potatoes did they order? +Check Your Understanding +For the following exercises, determine the most reasonable value for each weight. +25. Candy bar: +50 kg, 50 g, or 50 mg +26. Lion: +180 kg, 180 g, or 180 mg +27. Basketball: +624 kg, 624 g, or 624 mg +For the following exercises, convert the given weight to the units shown. +28. 8,900 g = __________ kg +9.4 • Measuring Weight +975 + +29. 17 g = __________ mg +30. 0.07 kg = __________ g +For the following exercises, determine the total weight in the units shown. +31. three 48 g granola bars ________ kg +32. seven 28 g cheese slices ________ mg +33. six 15 mg tea bags ________ g +SECTION 9.4 EXERCISES +For the following exercises, determine the most reasonable value for each weight. +1. Aspirin tablet: +300 kg, 300 g, or 300 mg +2. Elephant: +5,000 kg, 5,000 g, or 5,000 mg +3. Baseball: +145 kg, 145 g, or 145 mg +4. Orange: +115 kg, 115 g, or 115 mg +5. Pencil: +6 kg, 6 g, or 6 mg +6. Automobile: +1,300 kg, 1,300 g, or 1,300 mg +For the following exercises, convert the given weight to the units shown. +7. 3,500 g = __________ kg +8. 53 g = __________ mg +9. 0.02 kg = __________ g +10. 200 mg = __________ g +11. 2.3 g = _________ mg +12. 20 kg = _______ mg +13. 2,300 kg = __________ g +14. 8,700 mg = _________ g +15. 9,730 mg = _______ kg +16. 0.0078 kg = __________ g +17. 2.34 g = _________ mg +18. 234.5 mg = _______ g +For the following exercises, determine the total weight in the units shown. +19. Three 350 mg tablets ________ g +20. Seven 115 g soap bars ________ kg +21. Six 24 g batteries ________ mg +22. Fifty 3.56 g pennies ________ mg +23. Eight 2.25 kg bags of potatoes ________ g +24. Four 23 kg sacks of flour ________ g +25. Ten 2.5 kg laptops ________ g +26. Seven 1,150 g chickens ________ kg +27. Ninety 4,500 mg marbles ________ kg +28. There are 26 bags of flour. Each bag weighs 5 kg. What is the total weight of the flour? +29. The average female hippopotamus weighs 1,496 kg. The average male hippopotamus weighs 1,814 kg. How much +heavier, in grams, is the male hippopotamus than the female hippopotamus? +30. Twelve pieces of cardboard weigh 72 grams. What is the weight of one piece of cardboard? +31. Miguel’s backpack weighs 2.4 kg and Shanayl’s backpack weighs 2,535 grams. Whose backpack is heavier and by +how much? +32. A souvenir chocolate bar weighs 1.815 kg. If you share the candy bar equally with two friends, how many grams of +chocolate does each person get? +976 +9 • Metric Measurement +Access for free at openstax.org + +33. You purchase 10 bananas that weigh 50 grams each. If bananas cost $5.50 per kilogram, how much did you pay? +34. A family-size package of ground meat costs $15.75. The package weighs 4.5 kg. What is the cost per gram of the +meat? +35. A box containing 6 identical books weighs 7.2 kg. The box weighs 600 g. What is the weight of each book in grams? +36. A 2.316 kg bag of candy is equally divided into 12 party bags. What is the weight of the candy, in grams, in each +party bag? +37. A store has 450 kg of flour at the beginning of the day. At the end of the day the store has 341 kg of flour. If flour +costs $0.35 per kilogram, how much flour, in dollars, did the store sell that day? +38. A local restaurant offers lobster for $110 per kilogram. What is the price for a lobster that weighs 450 grams? +39. A student’s backpack weighs 575 grams. Their books weigh 3.5 kg. If the student’s weight while wearing their +backpack is 58.25 kg, how much does the student weigh in kilograms? +40. The weight of a lamb is 41 kg 340 g. What is the total weight, in kilograms, of four lambs of the same weight? +9.5 Measuring Temperature +Figure 9.14 A thermometer that measures temperature in both customary and metric units. (credit: “Thermometer” by +Jeff Djevdet/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Convert between Fahrenheit and Celsius. +2. +Identify reasonable values for temperature applications. +3. +Solve application problems involving temperature. +When you touch something and it feels warm or cold, what is that really telling you about that substance? Temperature +is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice +cube. The faster those atoms and molecules move, the higher the temperature. +In the metric system, temperature is measured using the Celsius (°C) scale. Because temperature is a condition of the +physical properties of a substance, the Celsius scale was created with 100 degrees separating the point at which water +freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules +change from one state of matter to another—from solid (ice) to liquid (water) to gas (water vapor). +When reading temperatures, it’s important to look beyond the degree symbol to determine which temperature scale +the units express. For example, 13 °C reads “13 degrees Celsius,” indicating that the temperature is expressed using +the Celsius scale, while 13 °F reads “13 degrees Fahrenheit,” indicating that the temperature is expressed using the +9.5 • Measuring Temperature +977 + +Fahrenheit scale. +VIDEO +Misconceptions About Temperature (https://openstax.org/r/Misconceptions_About_Temperature) +WHO KNEW? +How Many Temperature Scales Are There? +Did you know that in addition to Fahrenheit and Celsius, there is a third temperature scale widely used throughout +the world? The Kelvin scale starts at absolute zero, the lowest possible temperature at which there is no heat energy +present at all. It is primarily used by scientists to measure very high or very low temperatures when water is not +involved. +Converting Between Fahrenheit and Celsius Temperatures +Understanding how to convert between Fahrenheit and Celsius temperatures is an essential skill in understanding +metric temperatures. You likely know that below 32 °F means freezing temperatures and perhaps that the same holds +true for 0 °C. While it may be difficult to recall that water boils at 212 °F, knowing that it boils at 100 °C is a fairly easy +thing to remember. +But what about all the temperatures in between? What is the temperature in degrees Celsisus on a scorching summer +day? What about a cool autumn afternoon? If a recipe instructs you to preheat the oven to 350 °F, what Celsius +temperature do you set the oven at? +Figure 9.15 lists common temperatures on both scales, because we don’t use Celsius temperatures daily it’s difficult to +remember them. Fortunately, we don’t have to. Instead, we can convert temperatures from Fahrenheit to Celsius and +from Celsius to Fahrenheit using a simple algebraic expression. +978 +9 • Metric Measurement +Access for free at openstax.org + +Figure 9.15 Common Temperatures +9.5 • Measuring Temperature +979 + +FORMULA +The formulas used to convert temperatures from Fahrenheit to Celsius or from Celsius to Fahrenheit are outlined in +Table 9.3. +Fahrenheit to Celsius +Celsius to Fahrenheit +Table 9.3 Temperature Conversion Formulas +EXAMPLE 9.40 +Converting Temperatures from Fahrenheit to Celsius +A recipe calls for the oven to be set to 392 °F. What is the temperature in Celsius? +Solution +Use the formula in Table 9.3 to convert from Fahrenheit to Celsius. +So, 392 °F is equivalent to 200 °C. +YOUR TURN 9.40 +1. What is 482 °F in Celsius? +EXAMPLE 9.41 +Converting Temperatures from Celsius to Fahrenheit +On a sunny afternoon in May, the temperature in London was 20 °C. What was the temperature in degrees Fahrenheit? +Solution +Use the formula in Table 9.3 to convert from Celsius to Fahrenheit. +The temperature was 68 °F. +YOUR TURN 9.41 +1. What is 75 °C in Fahrenheit? +980 +9 • Metric Measurement +Access for free at openstax.org + +EXAMPLE 9.42 +Comparing Temperatures in Celsius and Fahrenheit +A manufacturer requires a vaccine to be stored in a refrigerator at temperatures between 36 °F and 46 °F. The +refrigerator in the local pharmacy cools to 3 °C. Can the vaccine be stored safely in the pharmacy’s refrigerator? +Solution +Use the formula in Table 9.3 to convert from Celsius to Fahrenheit. +Then, compare the temperatures. +Yes. 37.4 °F falls within the acceptable range to store the vaccine, so it can be stored safely in the pharmacy’s refrigerator. +YOUR TURN 9.42 +1. In July, the average temperature in Madrid is 16.7 °C. The average temperature in Toronto in July is 57.4 °F. Which +city has the higher temperature, and by how many degrees Celsius? +Reasonable Values for Temperature +While knowing the exact temperature is important in most cases, sometimes an approximation will do. When trying to +assess the reasonableness of values for temperature, there is a quicker way to convert temperatures for an +approximation using mental math. These simpler formulas are listed in Table 9.4. +FORMULA +The formulas used to estimate temperatures from Fahrenheit to Celsius or from Celsius to Fahrenheit are outlined in +Table 9.4. +Fahrenheit to Celsius +Celsius to Fahrenheit +Table 9.4 Estimate Temperature Conversion +VIDEO +Temperature Conversion Trick (http://openstax.org/r/Temperature_Conversion_Trick) +EXAMPLE 9.43 +Using Benchmark Temperatures to Determine Reasonable Values for Temperatures +Which is the more reasonable value for the temperature of a freezer? +• +5 °C or +• +–5 °C? +9.5 • Measuring Temperature +981 + +Solution +We know that water freezes at 0 °C. So, the more reasonable value for the temperature of a freezer is −5 °C, which is +below 0 °C. At temperature of 5 °C is above freezing. +YOUR TURN 9.43 +1. To make candy apples, you must boil the sugar mixture for about 20 minutes. Which is the more reasonable +value for the temperature of the mixture when boiling: +• +48 °C or +• +148 °C? +EXAMPLE 9.44 +Using Estimation to Determine Reasonable Values for Temperatures +The average body temperature is generally accepted as 98.6 °F. What is a reasonable value for the average body +temperature in degrees Celsius: +• +98.6 °C, +• +64.3 °C, or +• +34.3 °C? +Solution +To estimate the average body temperature in degrees Celsius, subtract 30 from the temperature in degrees Fahrenheit, +and divide the result by 2. +A reasonable value for average body temperature is 34.3 °C. +YOUR TURN 9.44 +1. Which represents a reasonable value for temperature of a hot summer day: +• +102 °C, +• +37 °C, or +• +20 °C? +EXAMPLE 9.45 +Using Conversion to Determine Reasonable Values for Temperatures +Which is a reasonable temperature for storing chocolate: +• +28 °C, +• +18 °C, or +• +2 °C? +Solution +Use the formula in Table 9.3 to determine the temperature in degrees Fahrenheit. +982 +9 • Metric Measurement +Access for free at openstax.org + +A temperature of 82.4 °F would be too hot, causing the chocolate to melt. A temperature of 35.6 °F is very close to +freezing, which would affect the look and feel of the chocolate. So, a reasonable temperature for storing chocolate is 18 +°C, or 64.4 °F. +YOUR TURN 9.45 +1. Which represents a reasonable temperature for cooking chili: +• +240 °C, +• +60 °C, or +• +6 °C? +Solving Application Problems Involving Temperature +Whether traveling abroad or working in a clinical laboratory, knowing how to solve problems involving temperature is an +important skill to have. Many food labels express sizes in both ounces and grams. Most rulers and tape measures are +two-sided with one side marked in inches and feet and the other in centimeters and meters. And while many +thermometers have both Fahrenheit and Celsius scales, it really isn’t practical to pull out a thermometer when cooking a +recipe that uses metric units. Let’s review at few instances where knowing how to fluently use the Celsius scale helps +solve problems. +EXAMPLE 9.46 +Using Subtraction to Solve Temperature Problems +The temperature in the refrigerator is 4 °C. The temperature in the freezer is 21 °C lower. What is the temperature in the +freezer? +Solution +Use subtraction to find the difference. +So, the temperature in the freezer is −17 °C. +YOUR TURN 9.46 +1. At 6 PM, the temperature was 4 °C. By 6 AM the temperature had fallen by 6 °C. What was the temperature at 6 +AM? +EXAMPLE 9.47 +Using Addition to Solve Temperature Problems +A scientist was using a liquid that was 35 °C. They needed to heat the liquid to raise the temperature by 6 °C. What was +the temperature after the scientist heated it? +Solution +Use addition to find the new temperature. +The temperature of the liquid was 41 °C after the scientist heated it. +YOUR TURN 9.47 +1. A hot dog was cooked in a microwave oven. The hot dog was 4 °C when it was put in the microwave and the +9.5 • Measuring Temperature +983 + +temperature increased by 59 °C when it was taken out. What was the temperature of the cooked hot dog? +EXAMPLE 9.48 +Solving Complex Temperature Problems +The optimum temperature for a chemical compound to develop its unique properties is 392 °F. When the heating +process begins, the temperature of the compound is 20 °C. For safety purposes the compound can only be heated 9 °C +every 15 minutes. How long until the compound reaches its optimum temperature? +Solution +Step 1: Determine the optimum temperature in degrees Celsius using the formula in Table 9.3. +Step 2: Subtract the starting temperature. +Step 3: Determine the number of 15-minute cycles needed to heat the compound to its optimum temperature. +Step 4: Multiply the number of cycles needed by 15 minutes and convert the product to hours and minutes. +So, it will take 2 hours and 15 minutes for the compound to reach its optimum temperature. +YOUR TURN 9.48 +1. After reaching a temperature of 302 °F a chemical compound cools at the rate of 5 °C every 6 minutes. How long +will it take until the compound has reached a temperature of zero degree Celsius. +VIDEO +Learn the Metric System in 5 Minutes (https://openstax.org/r/Metric_System_in_5_Minutes) +Check Your Understanding +For the following exercises, determine the most reasonable value for each temperature. +34. Popsicle: +28.5 °C or +28.5 °F +35. Room temperature: +20 °F or +20 °C +36. Hot coffee: +71.1 °C or +984 +9 • Metric Measurement +Access for free at openstax.org + +71.1 °F +For the following exercises, use mental math to approximate the temperature. +37. 450 °F = __________ °C +38. 35 °C = __________ °F +39. 100 °F = __________ °C +For the following exercises, convert the temperatures to the nearest degree. +40. 225 °F = __________ °C +41. 27 °C = __________ °F +42. 750 °F = __________ °C +SECTION 9.5 EXERCISES +For the following exercises, determine the most reasonable value for each temperature. +1. Human body: +37 °C or +37 °F +2. An ice cube: +32 °C or +0 °C +3. Boiling water: +212 °C +or 212 °F +4. Summer day: +8 °C, +23 °C, or +75 °C +5. Winter day: +33 °C, +30 °C, or +3 °C +6. Lava: +700 °C, +70 °C, or +7 °C +For the following exercises, use mental math to approximate the temperature. +7. 500 °F = __________ °C +8. 25 °C = __________ °F +9. 350 °F = __________ °C +10. 150 °C = __________ °F +11. ��4 °F = __________ °C +12. 73 °C = __________ °F +13. 72 °F = __________ °C +14. 10 °C = __________ °F +15. 1,020 °F = __________ °C +For the following exercises, convert the temperatures to the nearest degree. +16. 450 °F = __________ °C +17. 35 °C = __________ °F +18. 525 °F = __________ °C +19. 140 °C = __________ °F +20. –40 °F = __________ °C +21. 67 °C = __________ °F +22. 85 °F = __________ °C +23. 15 °C = __________ °F +24. 1,200 °F = __________ °C +25. 112 °F = __________ °C +26. 105 °C = __________ °F +9.5 • Measuring Temperature +985 + +27. 125 °F = __________ °C +28. A cup of tea is 30 °C. After adding ice, the temperature of the tea decreased 3 °C. What is the temperature of the +tea now? +29. A liquid with a temperature of 15 °C is placed on a stovetop. The liquid is heated at the rate of 1.5 °C per minute. +What is the temperature of the liquid after 10 minutes? +30. The temperature inside a store is 22 °C. Outside the temperature is 37 °C. How much cooler is it in the store than +outside? +31. A boiling pan of water is 212 °F. As the water cools the temperature drops 6 °C every 2 minutes. How many +minutes until the temperature of the water reaches 40 °C? +32. On May 1 the temperature was 79 °F. On June 1 the temperature was 25 °C. Which day was warmer? +33. For each log placed on a fire, the temperature increases 45 °C. How many logs are needed for the campfire to +increase 315 °C? +34. The temperature of a Bunsen burner flame increases 572 °F each minute. About how many minutes does it take +for the flame to increase 1,200 °C? +35. The instructions to cook a pizza say to set the oven at 425 °F. To the nearest degree, what is the temperature in +degrees Celsius? +36. In the evening the temperature was 3 °C. By morning, the temperature had fallen 4 °C. What is the temperature +now? +37. An 8,000 BTU air conditioner can cool a room 1 °C every 5 minutes. If the temperature in the room is 24 °C, how +long will it take the air conditioner to cool the room to 20 °C? +38. What is 95 °F in degrees Celsius? +39. What is 50 °C in degrees Fahrenheit? +40. Which is a reasonable value for the temperature of a room: +• +60 °C, +• +40 °C, or +• +20 °C? +986 +9 • Metric Measurement +Access for free at openstax.org + +Chapter Summary +Key Terms +9.1 The Metric System +• +metric system +• +meter (m) +• +gram (g) +• +liter (L) +• +square meter (m2) +• +cubic meter (m3) +• +degrees Celsius (°C) +• +conversion factor +9.2 Measuring Area +• +area +• +square units +9.3 Measuring Volume +• +volume +• +cubic units +9.4 Measuring Weight +• +mass +9.5 Measuring Temperature +• +temperature +Key Concepts +9.1 The Metric System +• +The metric system is decimal system of weights and measures based on base units of meter, liter, and gram. The +system was first proposed in 1670 and has since been adopted as the International System of Units and used in +nearly every country in the world. +• +Each successive unit on the metric scale is 10 times larger than the previous one. To convert between units with the +same base unit, you must either multiply or divide by a power of 10. +• +The most used prefixes are listed in Table 9.1. An easy way to remember the order of the prefixes, from largest to +smallest, is the mnemonic King Henry Died from Drinking Chocolate Milk. +9.2 Measuring Area +• +Area describes the size of a two-dimensional surface. It is the amount of space contained within the lines of a two- +dimensional space. +• +Area is measured in square meters units; in the metric system the base unit for area is square meters ( +). +9.3 Measuring Volume +• +Volume is a measure of the space contained within or occupied by three-dimensional objects. +• +Volume is measured in cubic units; the base unit for volume in the metric system is cubic meters (m3) +• +The liter (L) is a metric unit of capacity but is often used to express the volume of liquids. One liter is equivalent to +one cubic decimeter, which is the volume of a cube measuring +. +9.4 Measuring Weight +• +Mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The mass of +an object never changes; the weight of an object changes depending on the force of gravity. An object with the +same mass would weigh less on the moon than on Earth because the moon’s gravity is less than that of Earth. +• +In the metric system, weight and mass are often used interchangeably and are expressed in terms in grams or +kilograms. +9 • Chapter Summary +987 + +9.5 Measuring Temperature +• +Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a +stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature. +• +In the metric system, temperature is measured using the Celsius (°C) scale. +• +The Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at +which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of +matter to another—from solid (ice) to liquid (water) to gas (water vapor). +Videos +9.1 The Metric System +• +U.S. Office of Education: Metric Education (https://openstax.org/r/U.S._Office_Education) +• +Neil deGrasse Tyson Explains the Metric System (https://openstax.org/r/Tyson_Explains_Metric_System) +9.2 Measuring Area +• +Converting Metric Units of Area (https://openstax.org/r/Metric_Units_of_Area) +• +Why the Metric System Matters (https://openstax.org/r/Metric_System_Matters) +9.3 Measuring Volume +• +How to Convert Cubic Centimeters to Cubic Meters (https://openstax.org/r/Convert_CC_to_CM) +• +Converting Metric Units of Volume (https://openstax.org/r/Converting_Metric_Units_of_Volume) +9.4 Measuring Weight +• +Metric System: Units of Weight (https://openstax.org/r/Metric_System:_units_of_weight) +• +Using Drones to Weigh Whales? (https://openstax.org/r/Using_drones_to_weigh_whales?) +• +Metric Units of Mass: Convert mg, g, and kg (https://openstax.org/r/Convert_mg-g-kg) +9.5 Measuring Temperature +• +Misconceptions About Temperature (https://openstax.org/r/Misconceptions_About_Temperature) +• +Temperature Conversion Trick (http://openstax.org/r/Temperature_Conversion_Trick) +• +Learn the Metric System in 5 Minutes (https://openstax.org/r/Metric_System_in_5_Minutes) +Formula Review +9.1 The Metric System +You can convert between unit sizes with the same base unit using the conversion factors shown in Figure 9.4. +9.2 Measuring Area +To determine the area of rectangular-shaped objects: +Figure 9.16 Rectangle with Length +and Width +Labeled +You can convert between metric area units using the conversion factors shown in Figure 9.7. +9.3 Measuring Volume +To determine the volume of a rectangular prism: +988 +9 • Chapter Summary +Access for free at openstax.org + +Figure 9.17 Rectangular Prism with Height +, Length +, and Width +Labeled +You can convert between metric volume units and metric capacity units using the relationships shown in Table 9.2. +9.4 Measuring Weight +You can convert between metric weight units using the conversion factors shown in Figure 9.13. +9.5 Measuring Temperature +To convert temperature from Fahrenheit to Celsius: +To convert temperature from Celsius to Fahrenheit: +To estimate temperature from Fahrenheit to Celsius: +To estimate temperature from Celsius to Fahrenheit: +Projects +Cooking +1. +Take a favorite recipe that uses customary measures and convert the measures and cooking temperature to the +metric system. +2. +Find a recipe that uses metric measures and convert the measures and cooking temperature to the U.S. Customary +System of Measurement, using cups, tablespoons, or teaspoons as required. +3. +What did you observe? Was it easier to convert from one system to another? Which system allows for more precise +measurements? What kitchen tools would you need in your kitchen if you used the metric system? +Shopping +1. +Compare the average gas price in California to the average gas price in Puerto Rico. +2. +What conversions did you need to make to do the comparison? +3. +Do you think that the price of the gasoline is affected by the units in which it is sold? +Sports +1. +What system of measurement is used for track and field events? Why do you think this system is used? +2. +What system of measurement is used for football? Why do you think this system is used? +3. +Research various sports records. Which units of measurement is used? What do you think influenced the unit of +measure used? +9 • Chapter Summary +989 + +Chapter Review +The Metric System +For the following exercises, determine the base unit of the metric system described: liter, gram, or meter. +1. amount of juice in a carton +2. weight of a puppy +3. height of a wall +For the following exercises, choose the smaller of the two units. +4. hectogram or centigram +5. decameter or kilometer +6. milliliter or deciliter +7. Convert 236 cm to meters (m). +8. Convert 137.5 mg to grams (g). +9. You purchase 15 kg of birdseed. You divide the birdseed into 10 bags. How many grams of birdseed are in each +bag? +10. A track is 650 m long. If you run 8 laps around the track, how many kilometers did you run? +Measuring Area +For the following exercises, determine the most reasonable value for each area. +11. bedroom floor +12 km2 +12 m2 +12 cm2 +12 mm2 +12. nature preserve +800 km2 +800 m2 +800 cm2 +800 mm2 +13. desk surface +0.125 km2 +0.125 m2 +0.125 cm2 +0.125 mm2 +For the following exercises, convert the given area to the units shown. +14. 650,000 cm2 = __________ m2 +15. 17 m2 = __________ cm2 +16. 759 cm2 = __________ mm2 +17. A laptop is 30 cm by 20 cm. A sticker on the laptop cover measures 15 mm by 15 mm. How many square +centimeters of the laptop cover is still visible? +18. A dining room wall is 2.75 m by 2.25 m. A window on the wall is 0.75 m by 1.5 m. How many square meters is the +surface of the wall? +19. A floor is 6 m by 4 m. A rug on the floor measures 200 cm by 100 cm. How much of the floor, in m2, is not covered +by the rug? +20. A ceiling is 4 m by 2.5 m. A skylight in the ceiling measures 75 cm by 80 cm. How much of the ceiling, in m2, would +need to be painted if you decide to change the color of the ceiling? +Measuring Volume +For the following exercises, determine the most reasonable value for each volume. +21. aquarium +65,750 km3 +65,750 m3 +65,750 cm3 +65,750 mm3 +990 +9 • Chapter Summary +Access for free at openstax.org + +22. carton of soup +94,000 L +94 L +940,000 mL +940 mL +23. gift box +6,250 km3 +6,250 m3 +6,250 cm3 +6,250 mm3 +For the following exercises, convert the given volume to the units shown. +24. 59,837 mm3 = __________ cm3 +25. 12.5 dm3 = __________ mL +26. 368 cm3 = __________ mm3 +27. A box has dimensions of +. The box currently holds 1,750 cm3 of black beans. How many +cubic centimeters of black beans are needed to completely fill the box? +28. The dimensions of a small storage unit are +. What is the volume of a medium storage area with +dimensions twice the size of the small unit? +29. How much liquid, in liters, can a container with dimensions of +hold? +30. A box is 20 cm long and 15 cm wide. The volume of the box is 2,100 cm3. What is the height of the box? +Measuring Weight +31. What is the most reasonable estimate for the weight of a small stack of coins? +120 mg +12 g +1.2 kg +0.12 mt +For the following exercises, convert the given weight to the units shown. +32. 7,900 g = __________ kg +33. 81 g = __________ mg +34. 0.12 kg = __________ g +For the following exercises, determine the total weight in the units shown. +35. four 450 mg tablets ________ g +36. six 45 g candy bars ________ kg +37. seven 1.5 g buttons ________ mg +38. A box of chocolate weighs 2.75 kg. If you share the candy bar equally with 5 friends, how many grams of chocolate +does each person get? +39. You purchase 15 tomatoes that weigh 100 grams each. If tomatoes cost $1.18 per kilogram, how much did you +pay? +40. A value-size package of ground chicken costs $23.00. The package weighs 2.5 kg. What is the cost per gram of the +chicken? +Measuring Temperature +For the following exercises, convert the temperatures to the nearest degree. +41. 365 °F = __________ °C +42. 42 °C = __________ °F +43. 575 °F = __________ °C +44. 135 °C = __________ °F +45. –20 °F = __________ °C +46. 73 °C = __________ °F +47. The temperature of a welder’s torch increases 1,112 ºF every 3 minutes. About how many minutes does it take for +the torch to increase 1,200 ºC? +48. A substance is heated to 302 ºF. As the substance cools the temperature drops 4 ºC every 3 minutes. How many +9 • Chapter Summary +991 + +minutes until the temperature of the water reaches 130 ºC? +49. On November 1 the temperature was 40 ºF. On December 1 the temperature was 7 ºC. Which day was cooler? +50. The instructions to cook lasagna say to set the oven at 475 ºF. To the nearest degree, what is the temperature in +degrees Celsius? +Chapter Test +1. A student consumed 10 cans of their favorite soda last week. If each can contained 39 g of sugar, how many +milligrams of sugar did they consume? +2. An athlete ran a total distance of 154 km in one month while training for a 5K fun run. How many meters did they +run? +3. A doctor recommends their patient drinks 5 liters of water on days of vigorous exercise. If the patient drinks 3,400 +mL, how much more do they need to drink? +4. A garden walkway has the dimensions shown. How many square meters of brick are needed to cover the walkway? +5. A studio apartment has a length of 1,200 cm and a width that one half of the length. What is the area of the +apartment in meters? +6. A living room is 7 m long and 8 m wide. Carpet costs $9.50 per square meter. How much will it cost to carpet the +room? +7. Arif mixed two cartons of orange juice that each contain 600 cL, two bottles of cranberry juice that each contain +1,250 mL, and one 2-liter bottle of soda water to make punch for a party. How much punch, in liters, did Arif make? +8. A fish tank has the dimensions +. What is the capacity of the fish tank in liters? +9. A recipe for chili makes 5.4 L of chili. If a restaurant serves chili in 300 mL bowls, how many bowls of chili can they +serve? +10. If a person weighs 76 kg but loses 3 kilograms, how much does the person weigh in grams? +11. A car weighs approximately 2 metric tons (mt). If a used car dealer has 222,000 kg of cars in their lot, how many +cars are in the lot? +12. A restaurant has 126 kg of rice. If each serving contains 125 g, how many servings will they make? +13. A boiling pan of water is 212 °F. As the water cools the temperature drops 12 °C every 3 minutes. How many +minutes until the temperature of the water reaches 28 °C? +14. On May 1 the temperature was 75 °F. On June 1 the temperature was 28 °C. Which day was warmer? +15. For each log placed on a fire, the temperature increases 45 °C. How many logs are needed for the campfire to +increase 315 °C? +992 +9 • Chapter Summary +Access for free at openstax.org + +Figure 10.1 The School of Athens by the Renaissance artist Raphael, painted between 1509 and 1511, depicts some of +the greatest minds of ancient times. (credit: modification of work “School of Athens” by Raphael (1483–1520), Vatican +Museums/Wikimedia, Public Domain) +Chapter Outline +10.1 Points, Lines, and Planes +10.2 Angles +10.3 Triangles +10.4 Polygons, Perimeter, and Circumference +10.5 Tessellations +10.6 Area +10.7 Volume and Surface Area +10.8 Right Triangle Trigonometry +Introduction +The painting The School of Athens presents great figures in history such as Plato, Aristotle, Socrates, Euclid, Archimedes, +and Pythagoras. Other scientists are also represented in the painting. +To the ancient Greeks, the study of mathematics meant the study of geometry above all other subjects. The Greeks +looked for the beauty in geometry and did not allow their geometrical constructions to be “polluted” by the use of +anything as practical as a ruler. They permitted the use of only two tools—a compass for drawing circles and arcs, and an +unmarked straightedge to draw line segments. They would mark off units as needed. However, they never could be sure +of what the units meant. For instance, how long is an inch? These mathematicians defined many concepts. +The Greeks absorbed much from the Egyptians and the Babylonians (around 3000 BCE), including knowledge about +congruence and similarity, area and volume, angles and triangles, and made it their task to introduce proofs for +everything they learned. All of this historical wisdom culminated with Euclid in 300 BCE. +Euclid (325–265 BC) is known as the father of geometry, and his most famous work is the 13-volume collection known as +The Elements, which are said to be “the most studied books apart from the Bible.” Euclid brought together everything +offered by the Babylonians, the Egyptians, and the more refined contributions by the Greeks, and set out, successfully, to +organize and prove these concepts as he methodically developed formal theorems. +This chapter begins with a discussion of the most basic geometric tools: the point, the line, and the plane. All other +topics flow from there. Throughout the eight sections, we will talk about how to determine angle measurement and +learn how to recognize properties of special angles, such as right angles and supplementary angles. We will look at the +relationship of angles formed by a transversal, a line running through a set of parallel lines. We will explore the concepts +of area and perimeter, surface area and volume, and transformational geometry as used in the patterns and rigid +10 +GEOMETRY +10 • Introduction +993 + +motions of tessellations. Finally, we will introduce right-angle trigonometry and explore the Pythagorean Theorem. +10.1 Points, Lines, and Planes +Figure 10.2 The lower right-hand corner of The School of Athens depicts a figure representing Euclid illustrating to +students how to use a compass on a small chalkboard. (credit: modification of work “School of Athens” by Raphael +(1483–1520), Vatican Museums/Wikimedia, Public Domain) +Learning Objectives +After completing this section, you should be able to: +1. +Identify and describe points, lines, and planes. +2. +Express points and lines using proper notation. +3. +Determine union and intersection of sets. +In this section, we will begin our exploration of geometry by looking at the basic definitions as defined by Euclid. These +definitions form the foundation of the geometric theories that are applied in everyday life. +In The Elements, Euclid summarized the geometric principles discovered earlier and created an axiomatic system, a +system composed of postulates. A postulate is another term for axiom, which is a statement that is accepted as truth +without the need for proof or verification. There were no formal geometric definitions before Euclid, and when terms +could not be defined, they could be described. In order to write his postulates, Euclid had to describe the terms he +needed and he called the descriptions “definitions.” Ultimately, we will work with theorems, which are statements that +have been proved and can be proved. +Points and Lines +The first definition Euclid wrote was that of a point. He defined a point as “that which has no part.” It was later expanded +to “an indivisible location which has no width, length, or breadth.” Here are the first two of the five postulates, as they are +applicable to this first topic: +1. +Postulate 1: A straight line segment can be drawn joining any two points. +2. +Postulate 2: Any straight line segment can be extended indefinitely in a straight line. +Before we go further, we will define some of the symbols used in geometry in Figure 10.3: +994 +10 • Geometry +Access for free at openstax.org + +Figure 10.3 Basic Geometric Symbols for Points and Lines +From Figure 10.3, we see the variations in lines, such as line segments, rays, or half-lines. What is consistent is that two +collinear points (points that lie on the same line) are required to form a line. Notice that a line segment is defined by its +two endpoints showing that there is a definite beginning and end to a line segment. A ray is defined by two points on the +line; the first point is where the ray begins, and the second point gives the line direction. A half-line is defined by two +points, one where the line starts and the other to give direction, but an open circle at the starting point indicates that the +starting point is not part of the half-line. A regular line is defined by any two points on the line and extends infinitely in +both directions. Regular lines are typically drawn with arrows on each end. +EXAMPLE 10.1 +Defining Lines +For the following exercises, use this line (Figure 10.4). +Figure 10.4 +1. +Define +. +2. +Define +. +3. +Define +. +4. +Define +. +Solution +1. +The symbol +, two letters with a straight line above, refers to the line segment that starts at point +and ends at +point +. +2. +The letter +alone refers to point +. +3. +The symbol +, two letters with a line above containing arrows on both ends, refers to the line that extends +infinitely in both directions and contains the points +and +. +4. +The symbol +, two letters with a straight line above, refers to the line segment that starts at point +and ends at +point +. +YOUR TURN 10.1 +For the following exercises, use this line. +1. Define +2. Define +3. +4. +10.1 • Points, Lines, and Planes +995 + +There are numerous applications of line segments in daily life. For example, airlines working out routes between cities, +where each city’s airport is a point, and the points are connected by line segments. Another example is a city map. Think +about the intersection of roads, such that the center of each intersection is a point, and the points are connected by line +segments representing the roads. See Figure 10.5. +Figure 10.5 Air Line Routes +EXAMPLE 10.2 +Determining the Best Route +View the street map (Figure 10.6) as a series of line segments from point to point. For example, we have vertical line +segments +, +and +on the right. On the left side of the map, we have vertical line segments +, +The +horizontal line segments are +, +, +, +, +, +and +There are two diagonal line segments, +and +Assume that each location is on a corner and that you live next door to the library. +Figure 10.6 Street Map +1. +Let’s say that you want to stop at the grocery store on your way home from school. Come up with three routes you +might take to do your errand and then go home. In other words, name the three ways by the line segments in the +order you would walk, and which way do you think would be the most efficient route? +2. +How about stopping at the library after school? Name four ways you might travel to the library and which way do +you think is the most efficient? +996 +10 • Geometry +Access for free at openstax.org + +3. +Suppose you need to go to the post office and the dry cleaners on your way home from school. Name three ways +you might walk to do your errands and end up at home. Which way do you think is the most efficient way to walk, +get your errands done, and go home? +Solution +1. +From school to the grocery store and home: first way +; second way +; +third way +. It seems that the third way is the most efficient way. +2. +From school to the library: first way +; second way +; third way +; fourth +way +. The first way should be the most efficient way. +3. +From school to the post office or dry cleaners to home: first way +; second way +; third way +. The third way would be the most efficient way. +YOUR TURN 10.2 +1. Using the street map in Figure 10.6, find two ways you would stop at the dry cleaners and the grocery store after +school on your way home. +Parallel Lines +Parallel lines are lines that lie in the same plane and move in the same direction, but never intersect. To indicate that the +line +and the line +are parallel we often use the symbol +The distance +between parallel lines remains +constant as the lines extend infinitely in both directions. See Figure 10.7. +Figure 10.7 Parallel Lines +Perpendicular Lines +Two lines that intersect at a +angle are perpendicular lines and are symbolized by +. If +and +are perpendicular, +we write +When two lines form a right angle, a +angle, we symbolize it with a little square +See Figure 10.8. +Figure 10.8 Perpendicular Lines +EXAMPLE 10.3 +Identifying Parallel and Perpendicular Lines +Identify the sets of parallel and perpendicular lines in Figure 10.9. +10.1 • Points, Lines, and Planes +997 + +Figure 10.9 +Solution +Drawing these lines on a grid is the best way to distinguish which pairs of lines are parallel and which are perpendicular. +Because they are on a grid, we assume all lines are equally spaced across the grid horizontally and vertically. The grid +also tells us that the vertical lines are parallel and the horizontal lines are parallel. Additionally, all intersections form a +angle. Therefore, we can safely say the following: +, the line containing the points +and +is parallel to the line containing the points +and +. +, the line containing the points +and +is parallel to the line containing the points +and +. +, the line containing the points +and +is perpendicular to the line containing the points +and +. We know +this because both lines trace grid lines, and intersecting grid lines are perpendicular. +We can also state that +; the line containing the points +and +is perpendicular to the line containing the +points +and +because both lines trace grid lines, which are perpendicular by definition. +We also have +; the line containing the points +and +is perpendicular to the line containing the points +and +because both lines trace grid lines, which are perpendicular by definition. +Finally, we see that +; the line containing the points +and +is perpendicular to the line containing the points +and +because both lines trace grid lines, which are perpendicular by definition. +YOUR TURN 10.3 +1. Identify the sets of parallel and perpendicular lines in the given figure. +998 +10 • Geometry +Access for free at openstax.org + +Defining Union and Intersection of Sets +Union and intersection of sets is a topic from set theory that is often associated with points and lines. So, it seems +appropriate to introduce a mini-version of set theory here. First, a set is a collection of objects joined by some common +criteria. We usually name sets with capital letters. For example, the set of odd integers between 0 and 10 looks like this: +When it involves sets of lines, line segments, or points, we are usually referring to the union or +intersection of set. +The union of two or more sets contains all the elements in either one of the sets or elements in all the sets referenced, +and is written by placing this symbol +in between each of the sets. For example, let set +and let set +Then, the union of sets A and B is +The intersection of two or more sets contains only the elements that are common to each set, and we place this symbol +in between each of the sets referenced. For example, let’s say that set +and let set +Then, +the intersection of sets +and +is +EXAMPLE 10.4 +Defining Union and Intersection of Sets +Use the line (Figure 10.10) for the following exercises. Draw each answer over the main drawing. +Figure 10.10 +1. +Find +. +2. +Find +. +3. +Find +. +4. +Find +. +5. +Find +. +Solution +1. +Find +. This is the intersection of the ray +and the ray +Intersection includes only the elements that +are common to both lines. For this intersection, only the line segment +is common to both rays. Thus, +Figure 10.11 +2. +Find +. The problem is asking for the union of two line segments, +and +Union includes all elements +10.1 • Points, Lines, and Planes +999 + +in the first line and all elements in the second line. Since +is part of +, +Figure 10.12 +3. +Find +. This is the union of the line +with the line segment +. As the line segment +is included on +the line +then the union of these two lines equals the line +. +Figure 10.13 +4. +Find +. The intersection of the line +with line segment +is the set of elements common to both lines. +In this case, the only element in common is the line segment +Figure 10.14 +5. +Find +. This is the intersection of the ray +and the line segment +Intersection includes elements +common to both lines. There are no elements in common. The intersection yields the empty set, as shown in Figure +10.15. Therefore, +. +Figure 10.15 +YOUR TURN 10.4 +For the following exercises, use the line shown to identify and draw the union or intersection of sets. +1. +2. +3. +Planes +A plane, as defined by Euclid, is a “surface which lies evenly with the straight lines on itself.” A plane is a two-dimensional +surface with infinite length and width, and no thickness. We also identify a plane by three noncollinear points, or points +that do not lie on the same line. Think of a piece of paper, but one that has infinite length, infinite width, and no +thickness. However, not all planes must extend infinitely. Sometimes a plane has a limited area. +We usually label planes with a single capital letter, such as Plane +, as shown in Figure 10.16, or by all points that +determine the edges of a plane. In the following figure, Plane +contains points +and +, which are on the same line, +and point +, which is not on that line. By definition, +is a plane. We can move laterally in any direction on a plane. +1000 +10 • Geometry +Access for free at openstax.org + +Figure 10.16 Plane +One way to think of a plane is the Cartesian coordinate system with the +-axis marked off in horizontal units, and +-axis +marked off in vertical units. In the Cartesian plane, we can identify the different types of lines as they are positioned in +the system, as well as their locations. See Figure 10.17. +Figure 10.17 Cartesian Coordinate Plane +This plane contains points +, +, and +. Points +and +are colinear and form a line segment. Point +is not on that line +segment. Therefore, this represents a plane. +To give the location of a point on the Cartesian plane, remember that the first number in the ordered pair is the +horizontal move and the second number is the vertical move. Point +is located at +point +is located at +and point +is located at +We can also identify the line segment as +Two other concepts to note: Parallel planes do not intersect and the intersection of two planes is a straight line. The +equation of that line of intersection is left to a study of three-dimensional space. See Figure 10.18. +Figure 10.18 Parallel and Intersecting Planes +To summarize, some of the properties of planes include: +• +Three points including at least one noncollinear point determine a plane. +10.1 • Points, Lines, and Planes +1001 + +• +A line and a point not on the line determine a plane. +• +The intersection of two distinct planes is a straight line. +EXAMPLE 10.5 +Identifying a Plane +For the following exercises, refer to Figure 10.19 +Figure 10.19 +1. +Identify the location of points +, +, +, and +. +2. +Describe the line from point +to point +. +3. +Describe the line from point +containing point +. +4. +Does this figure represent a plane? +Solution +1. +Point +is located at +point +is located at +point +is located at +point +is located at +2. +The line from point +to point +is a line segment +3. +The line from point +containing point +is a ray +starting at point +in the direction of +. +4. +Yes, this figure represents a plane because it contains at least three points, points +and +form a line segment, +and neither point +nor point +is on that line segment. +YOUR TURN 10.5 +For the following exercises, refer to the given figure. +1002 +10 • Geometry +Access for free at openstax.org + +1. Identify the location of points +, +, +, +, +, and +. +2. Describe the line that includes point +and point +. +3. Describe the line from +to point +. +4. Describe the line from +to point +. +5. Does this figure represent a plane? +EXAMPLE 10.6 +Intersecting Planes +Name two pairs of intersecting planes on the shower enclosure illustration (Figure 10.20). +Figure 10.20 +Solution +The plane +intersects plane +, and plane +intersects plane +. +YOUR TURN 10.6 +1. Name two pairs of intersecting planes on the shower enclosure shown. +10.1 • Points, Lines, and Planes +1003 + +PEOPLE IN MATHEMATICS +Plato +Part of a remarkable chain of Greek mathematicians, Plato (427–347 BC) is known as the teacher. He was responsible +for shaping the development of Western thought perhaps more powerfully than anyone of his time. One of his +greatest achievements was the founding of the Academy in Athens where he emphasized the study of geometry. +Geometry was considered by the Greeks to be the “ultimate human endeavor.” Above the doorway to the Academy, an +inscription read, “Let no one ignorant of geometry enter here.” +The curriculum of the Academy was a 15-year program. The students studied the exact sciences for the first 10 years. +Plato believed that this was the necessary foundation for preparing students’ minds to study relationships that +require abstract thinking. The next 5 years were devoted to the study of the “dialectic.” The dialectic is the art of +question and answer. In Plato’s view, this skill was critical to the investigation and demonstration of innate +mathematical truths. By training young students how to prove propositions and test hypotheses, he created a culture +in which the systematic process was guaranteed. The Academy was essentially the world’s first university and held the +reputation as the ultimate center of learning for more than 900 years. +Check Your Understanding +Use the graph for the following exercises. +1. Identify the type of line containing point +and point +. +1004 +10 • Geometry +Access for free at openstax.org + +2. Identify the type of line containing points +and +. +3. Identify the type of line that references point +and contains point +. +Use the line for the following exercises. +4. Determine +. +5. Determine +. +6. Determine +. +7. How can you determine whether two lines are parallel? +8. How can you determine whether two lines are perpendicular? +9. Determine if the illustration represents a plane. +SECTION 10.1 EXERCISES +Use this graph for the following exercises. +1. Identify the kind of line passing through the points +and +. +2. Identify the kind of line passing through the points +and +. +3. Identify all sets of parallel lines. +4. Identify all sets of perpendicular lines. +Use the line for the following exercises. +10.1 • Points, Lines, and Planes +1005 + +5. Find +. +6. Find +. +7. Find +. +8. Find +9. Find +10. Find +For the following exercises, reference this line. +11. Name the points in the set +. +12. Name the points contained in the set +. +13. Name the points in the set +. +14. Name the points in the set +. +15. Name the points in the set +. +16. Name the points in the set +. +17. Name the points in the set +. +18. Name the points in the set +. +Use the given figure for the following exercises. +19. Describe the lighter plane (right) in the drawing. +20. Describe the darker plane (left) in the drawing. +1006 +10 • Geometry +Access for free at openstax.org + +10.2 Angles +Figure 10.21 This modern architectural design emphasizes sharp reflective angles as part of the aesthetic through the +use of glass walls. (credit: “Société Générale @ La Défense @ Paris�� by Images Guilhem Vellut/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify and express angles using proper notation. +2. +Classify angles by their measurement. +3. +Solve application problems involving angles. +4. +Compute angles formed by transversals to parallel lines. +5. +Solve application problems involving angles formed by parallel lines. +Unusual perspectives on architecture can reveal some extremely creative images. For example, aerial views of cities +reveal some exciting and unexpected angles. Add reflections on glass or steel, lighting, and impressive textures, and the +structure is a work of art. Understanding angles is critical to many fields, including engineering, architecture, +landscaping, space planning, and so on. This is the topic of this section. +We begin our study of angles with a description of how angles are formed and how they are classified. An angle is the +joining of two rays, which sweep out as the sides of the angle, with a common endpoint. The common endpoint is called +the vertex. We will often need to refer to more than one vertex, so you will want to know the plural of vertex, which is +vertices. +In Figure 10.22, let the ray +stay put. Rotate the second ray +in a counterclockwise direction to the size of the angle +you want. The angle is formed by the amount of rotation of the second ray. When the ray +continues to rotate in a +counterclockwise direction back to its original position coinciding with ray +the ray will have swept out +We call +the rays the “sides” of the angle. +Figure 10.22 Vertex and Sides of an Angle +Classifying Angles +Angles are measured in radians or degrees. For example, an angle that measures +radians, or 3.14159 radians, is equal +to the angle measuring +An angle measuring +radians, or 1.570796 radians, measures +To translate degrees to +10.2 • Angles +1007 + +radians, we multiply the angle measure in degrees by +For example, to write +in radians, we have +To translate radians to degrees, we multiply by +For example, to write +radians in degrees, we have +Another example of translating radians to degrees and degrees to radians is +To write in degrees, we have +To write +in radians, we have +. However, we will use degrees throughout this chapter. +FORMULA +To translate an angle measured in degrees to radians, multiply by +To translate an angle measured in radians to degrees, multiply by +Several angles are referred to so often that they have been given special names. A straight angle measures +; a +right angle measures +an acute angle is any angle whose measure is less than +and an obtuse angle is any +angle whose measure is between +and +See Figure 10.23. +Figure 10.23 Classifying and Naming Angles +An easy way to measure angles is with a protractor (Figure 10.24). A protractor is a very handy little tool, usually made of +transparent plastic, like the one shown here. +Figure 10.24 Protractor (credit: modification of work “School drawing tools” by Marco Verch/Flickr, CC BY 2.0) +With a protractor, you line up the straight bottom with the horizontal straight line of the angle. Be sure to have the +center hole lined up with the vertex of the angle. Then, look for the mark on the protractor where the second ray lines +up. As you can see from the image, the degrees are marked off. Where the second ray lines up is the measurement of +the angle. +Make sure you correctly match the center mark of the protractor with the vertex of the angle to be measured. +Otherwise, you will not get the correct measurement. Also, keep the protractor in a vertical position. +Notation +Naming angles can be done in couple of ways. We can name the angle by three points, one point on each of the sides +1008 +10 • Geometry +Access for free at openstax.org + +and the vertex point in the middle, or we can name it by the vertex point alone. Also, we can use the symbols +or +before the points. When we are referring to the measure of the angle, we use the symbol +. See Figure 10.25. +Figure 10.25 Naming an Angle +We can name this angle +, or +, or +EXAMPLE 10.7 +Classifying Angles +Determine which angles are acute, right, obtuse, or straight on the graph (Figure 10.26). You may want to use a +protractor for this one. +Figure 10.26 +Solution +Acute angles measure less +than +Obtuse angles measure between +and +Right angles +measure +Straight angles +measure +Most angles can be classified visually or by description. However, if you are unsure, use a protractor. +10.2 • Angles +1009 + +YOUR TURN 10.7 +1. Determine which angles are acute, obtuse, right, and straight in the graph. +Adjacent Angles +Two angles with the same starting point or vertex and one common side are called adjacent angles. In Figure 10.27, +angle +is adjacent to +. Notice that the way we designate an angle is with a point on each of its two sides +and the vertex in the middle. +Figure 10.27 Adjacent Angles +Supplementary Angles +Two angles are supplementary if the sum of their measures equals +In Figure 10.28, we are given that +so what is +These are supplementary angles. Therefore, because +, and as +we have +Figure 10.28 Supplementary Angles +EXAMPLE 10.8 +Solving for Angle Measurements and Supplementary Angles +Solve for the angle measurements in Figure 10.29. +Figure 10.29 +1010 +10 • Geometry +Access for free at openstax.org + +Solution +Step 1: These are supplementary angles. We can see this because the two angles are part of a horizontal line, and a +horizontal line represents +Therefore, the sum of the two angles equals +Step 2: +Step 3: Find the measure of each angle: +Step 4: We check: +YOUR TURN 10.8 +1. Solve for the angle measurements in the figure shown. +Complementary Angles +Two angles are complementary if the sum of their measures equals +In Figure 10.30, we have +and +What is the +These are complementary angles. Therefore, because +the +Figure 10.30 Complementary Angles +EXAMPLE 10.9 +Solving for Angle Measurements and Complementary Angles +Solve for the angle measurements in Figure 10.31. +10.2 • Angles +1011 + +Figure 10.31 +Solution +We have that +Then, +, +, and +YOUR TURN 10.9 +1. Find the measure of each angle in the illustration. +Vertical Angles +When two lines intersect, the opposite angles are called vertical angles, and vertical angles have equal measure. For +example, Figure 10.32 shows two straight lines intersecting each other. One set of opposite angles shows angle markers; +those angles have the same measure. The other two opposite angles have the same measure as well. +Figure 10.32 Vertical Angles +EXAMPLE 10.10 +Calculating Vertical Angles +In Figure 10.33, one angle measures +Find the measures of the remaining angles. +Figure 10.33 +Solution +The 40-degree angle and +are vertical angles. Therefore, +Notice that +and +are supplementary angles, meaning that the sum of +and +equals +Therefore, +. +Since +and +are vertical angles, then +equals +1012 +10 • Geometry +Access for free at openstax.org + +YOUR TURN 10.10 +1. Given the two intersecting lines in the figure shown and +∡ +find the measure of the remaining angles. +Transversals +When two parallel lines are crossed by a straight line or transversal, eight angles are formed, including alternate +interior angles, alternate exterior angles, corresponding angles, vertical angles, and supplementary angles. See Figure +10.34. Angles 1, 2, 7, and 8 are called exterior angles, and angles 3, 4, 5, and 6 are called interior angles. +Figure 10.34 Transversal +Alternate Interior Angles +Alternate interior angles are the interior angles on opposite sides of the transversal. These two angles have the same +measure. For example, +and +are alternate interior angles and have equal measure; +and +are alternate +interior angles and have equal measure as well. See Figure 10.35. +Figure 10.35 Alternate Interior Angles +Alternate Exterior Angles +Alternate exterior angles are exterior angles on opposite sides of the transversal and have the same measure. For +example, in Figure 10.36, +and +are alternate exterior angles and have equal measures; +and +are alternate +exterior angles and have equal measures as well. +Figure 10.36 Alternate Exterior Angles +10.2 • Angles +1013 + +Corresponding Angles +Corresponding angles refer to one exterior angle and one interior angle on the same side as the transversal, which have +equal measures. In Figure 10.37, +and +are corresponding angles and have equal measures; +and +are +corresponding angles and have equal measures; +and +are corresponding angles and have equal measures; +and +are corresponding angles and have equal measures as well. +Figure 10.37 Corresponding Angles +EXAMPLE 10.11 +Evaluating Space +You live on the corner of First Avenue and Linton Street. You want to plant a garden in the far corner of your property +(Figure 10.38) and fence off the area. However, the corner of your property does not form the traditional right angle. You +learned from the city that the streets cross at an angle equal to +What is the measure of the angle that will border +your garden? +Figure 10.38 +Solution +As the angle between Linton Street and First Avenue is +the supplementary angle is +Therefore, the garden will +form a +angle at the corner of your property. +YOUR TURN 10.11 +1. Suppose you have a similar property to the one in Figure 10.53, but the angle that corresponds to the garden +corner is +. What is the measure between the two cross streets? +EXAMPLE 10.12 +Determining Angles Formed by a Transversal +In Figure 10.39 given that angle 3 measures +find the measures of the remaining angles and give a reason for your +solution. +1014 +10 • Geometry +Access for free at openstax.org + +Figure 10.39 +Solution +by vertical angles. +by corresponding angles. +by vertical angles. +by supplementary angles. +by vertical angles. +by alternate exterior angles. +by vertical angles. +YOUR TURN 10.12 +1. In the given figure if +∡ +, find the +∡ , +∡ , and +∡ . +EXAMPLE 10.13 +Measuring Angles Formed by a Transversal +In Figure 10.40 given that angle 2 measures +find the measure of the remaining angles and state the reason for your +solution. +Figure 10.40 +Solution +by vertical angles, because +and +are the opposite angles formed by two intersecting lines. +by supplementary angles to +or +We see that +and +form a straight angle as does +and +A straight angle measures +so +by vertical angles, because +and +are the two opposite angles formed by two intersecting lines. +by corresponding angles because they are the same angle formed by the transversal crossing two +parallel lines, one exterior and one interior. +10.2 • Angles +1015 + +by vertical angles because +and +are the two opposite angles formed by two intersecting lines. +by alternate exterior angles because, like vertical angles, these angles are the opposite angles +formed by the transversal intersecting two parallel lines. +by vertical angles because these are the opposite angles formed by two intersecting lines. +YOUR TURN 10.13 +1. In the provided figure given that the +∡ +, find +∡ , and +∡ +EXAMPLE 10.14 +Finding Missing Angles +Find the measures of the angles 1, 2, 4, 11, 12, and 14 in Figure 10.41 and the reason for your answer given that +and +are parallel. +Figure 10.41 +Solution +, supplementary angles +, vertical angles +, vertical angles +, corresponding angles +, vertical angles +, supplementary angles +YOUR TURN 10.14 +1. Using Figure 10.58, find the measures of angles 5, 6, 7, 8, and 9. +1016 +10 • Geometry +Access for free at openstax.org + +WHO KNEW? +The Number 360 +Did you ever wonder why there are +in a circle? Why not +or +The number 360 was chosen by +Babylonian astronomers before the ancient Greeks as the number to represent how many degrees in one complete +rotation around a circle. It is said that they chose 360 for a couple of reasons: It is close to the number of days in a +year, and 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, … +Check Your Understanding +Classify the following angles as acute, right, obtuse, or straight. +10. +∡ +11. +∡ +12. +∡ +13. +∡ +For the following exercises, determine the measure of the angles in the given figure. +14. Find the measure of ∡ and state the reason for your solution. +15. Find the measure of ∡ and state the reason for your solution. +16. Find the measure of ∡ and state the reason for your solution. +SECTION 10.2 EXERCISES +Classify the angles in the following exercises as acute, obtuse, right, or straight. +1. +∡ +2. +∡ +3. +∡ +4. +∡ +5. +∡ +6. +∡ +7. Use the given figure to solve for the angle measurements. +8. Use the given figure to solve for the angle measurements. +10.2 • Angles +1017 + +9. Give the measure of the supplement to +Use the given figure for the following exercises. Let angle 2 measure +. +10. Find the measure of angle 1 and state the reason for your solution. +11. Find the measure of angle 3 and state the reason for your solution. +12. Find the measure of angle 4 and state the reason for your solution. +13. Find the measure of angle 5 and state the reason for your solution. +14. Find the measure of angle 6 and state the reason and state the reason for your solution. +15. Find the measure of angle 7 and state the reason for your solution. +16. Find the measure of angle 8 and state the reason for your solution. +17. Use the given figure to solve for the angle measurements. +Use the given figure for the following exercises. +18. Find the measure of angle 3 and explain the reason for your solution. +19. Find the measure of angle 8 and explain the reason for your solution. +1018 +10 • Geometry +Access for free at openstax.org + +10.3 Triangles +Figure 10.42 The appearance of triangles in buildings is part of modern-day architectural design. (credit: "Inside +Hallgrímskirkja church, Reykjavik, Iceland" by O Palsson/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify triangles by their sides. +2. +Identify triangles by their angles. +3. +Determine if triangles are congruent. +4. +Determine if triangles are similar. +5. +Find the missing side of similar triangles. +How were the ancient Greeks able to calculate the radius of Earth? How did soldiers gauge their target? How was it +possible centuries ago to estimate the height of a sail at sea? Triangles have always played a significant role in how we +find heights of objects too high to measure or distances between objects too far away to calculate. In particular, the +concept of similar triangles has countless applications in the real world, and we shall explore some of those applications +in this section. +Technology has given us instruments that allow us to find measurements of distant objects with little effort. However, it +is all based on the properties of triangles discovered centuries ago. In this section, we will explore the various types of +triangles and their special properties, as well as how to measure interior and exterior angles. We will also explore +congruence theorems and similarity. +Identifying Triangles +Joining any three noncollinear points with line segments produces a triangle. For example, given points +, +, and +, +connected by the line segments +and +we have a triangle, as shown in Figure 10.43. +Figure 10.43 Triangle +Triangles are classified by their angles and their sides. All angles in an acute triangle measure +One of the angles in +a right triangle measures +symbolized by □. One angle in an obtuse triangle measures between +and +Sides +that have equal length are indicated by the same hash marks. Figure 10.44 illustrates the shapes of the basic triangles, +10.3 • Triangles +1019 + +their names, and their properties. +A few other facts to remember as we move forward: +• +The points where the line segments meet are called the vertices (plural for vertex). +• +We often refer to sides of a triangle by the angle they are opposite. In other words, side +is opposite angle +, side +is opposite angle +, and side +is opposite angle +. +Figure 10.44 Types of Triangles +We want to add a special note about right triangles here, as they are referred to more than any other triangle. The side +opposite the right angle is its longest side and is called the hypotenuse, and the sides adjacent to the right angle are +called the legs. +One of the most important properties of triangles is that the sum of the interior angles equals +Euclid discovered +and proved this property using parallel lines. The completed sketch is shown in Figure 10.45. +Figure 10.45 Sum of Interior Angles +This is how the proof goes: +Step 1: Start with a straight line +and a point +not on the line. +Step 2: Draw a line through point +parallel to the line +. +Step 3: Construct two transversals (a line crossing the parallel lines), one angled to the right and one angled to the left, +to intersect the parallel lines. +Step 4: Because of the property that alternate interior angles inside parallel lines are equal, we have that +1020 +10 • Geometry +Access for free at openstax.org + +Step 5: Notice that +by the straight angle property. +Step 6: Therefore, by substitution, +and +we have that +Therefore, the sum of the interior angles of a +EXAMPLE 10.15 +Finding Measures of Angles Inside a Triangle +Find the measure of each angle in the triangle shown (Figure 10.46). We know that the sum of the angles must equal +Figure 10.46 +Solution +Step 1: As the sum of the interior angles equals +we can use algebra to find the measures: +Step 2: Now that we have the value of +, we can substitute 45 into the other two expressions to find the measure of +those angles: +Step 3: Then, +, +, and +YOUR TURN 10.15 +1. Find the measures of each angle in the triangle shown. +10.3 • Triangles +1021 + +EXAMPLE 10.16 +Finding Angle Measures +Find the measure of angles numbered 1–5 in Figure 10.47. +Figure 10.47 +Solution +The +because it is supplementary with the unknown angle of the adjacent triangle. The unknown angle +measures +The +because of vertical angles. The +because the angle that is supplementary to +the +measures +, and angle 5 is the unknown angle in that triangle. The +by vertical angles. Finally, +as it is the third angle in the triangle with angles measuring +and +YOUR TURN 10.16 +1. Find the measure of angles 1, 2, and 3 in the figure shown. +Congruence +If two triangles have equal angles and their sides lengths are equal, the triangles are congruent. In other words, if you +can pick up one triangle and place it on top of the other triangle and they coincide, even if you have to rotate one, they +are congruent. +EXAMPLE 10.17 +Determining If Triangles Are Congruent +In Figure 10.48, is the triangle +congruent to triangle +? +Figure 10.48 +Solution +Triangle +is congruent to triangle +. Angles +and +are congruent to angles +and +, which implies that angle +is congruent to angle +. Side +is congruent to side +, and side +is congruent to side +, which implies that +side +is congruent to side +. +1022 +10 • Geometry +Access for free at openstax.org + +YOUR TURN 10.17 +1. In the figure shown, is triangle +congruent to triangle +? +The Congruence Theorems +The following theorems are tools you can use to prove that two triangles are congruent. We use the symbol +to define +congruence. For example, +. +Side-Side-Side (SSS). If three sides of one triangle are equal to the corresponding sides of the second triangle, then the +triangles are congruent. See Figure 10.49. +Figure 10.49 Side-Side-Side (SSS) +We have that +, +and +then +Side-Angle-Side (SAS). If two sides of a triangle and the angle between them are equal to the corresponding two sides +and included angle of the second triangle, then the triangles are congruent. See Figure 10.50. We see that +and +, +, then +. +Figure 10.50 Side-Angle-Side (SAS) +Angle-Side-Angle (ASA). If two angles and the side between them in one triangle are congruent to the two +corresponding angles and the side between them in a second triangle, then the two triangles are congruent. See Figure +10.51. Notice that +, and +, +, then +Figure 10.51 Angle-Side-Angle (ASA) +10.3 • Triangles +1023 + +Angle-Angle-Side (AAS). If two angles and a nonincluded side of one triangle are congruent to two angles and the +nonincluded corresponding side of a second triangle, then the triangles are congruent. +See Figure 10.52. We see that +, +, and +, then +. +Figure 10.52 Angle-Angle-Side (AAS) +EXAMPLE 10.18 +Identifying Congruence Theorems +What congruence theorem is illustrated in Figure 10.53? +Figure 10.53 +Solution +AAS: Two angles and a non-included side in one triangle are congruent to the corresponding angles and side in the +second triangle. +YOUR TURN 10.18 +1. Identify the congruence theorem being illustrated in the figure shown. +EXAMPLE 10.19 +Determining the Congruence Theorem +What congruence theorem is illustrated in Figure 10.54? +1024 +10 • Geometry +Access for free at openstax.org + +Figure 10.54 +Solution +The SSS theorem. +YOUR TURN 10.19 +1. What congruence theorem is being illustrated in the figure shown? +Similarity +If two triangles have the same angle measurements and are the same shape but differ in size, the two triangles are +similar. The lengths of the sides of one triangle will be proportional to the corresponding sides of the second triangle. +Note that a single fraction +is called a ratio, but two fractions equal to each other is called a proportion, such as +This rule of similarity applies to all shapes as well as triangles. Another way to view similarity is by applying a scaling +factor, which is the ratio of corresponding measurements between an object or representation of the object, to an +image that produces the second, similar image. +For example, why are the two images in Figure 10.55 are similar? These two images have the same proportions between +elements. Therefore, they are similar. +Figure 10.55 Similarity +10.3 • Triangles +1025 + +EXAMPLE 10.20 +Determining If Triangles Are Similar +Are the two triangles shown in Figure 10.56 similar? +Figure 10.56 +Solution +Step 1: We will look at the proportions within each triangle. In triangle +(alpha), the side opposite the +angle +measures 7, and the side opposite the +angle measures 4. Then, the measures of the corresponding sides in triangle +(beta) measures 3.5 and 2, respectively. We have +This is the proportion +. The scaling factor is 0.5714. +Step 2: Let’s try another correspondence. In triangle +, the hypotenuse measures 8.06 and the side opposite the +angle measures 7. In triangle +, the hypotenuse measure 4.03 and the side opposite the +angle measures 3.5. We +have +Step 3: Now, let’s look at the proportions between triangle +and triangle +The side measuring 2 in triangle +corresponds to the side measuring 4 in triangle +, the side measuring 3.5 in triangle +corresponds to the side +measuring 7 in triangle +and the hypotenuse in triangle +corresponds to the hypotenuse in triangle +We have +Thus, the corresponding angles are equal and the proportions between each pair of corresponding sides equals 0.5. In +other words, the scaling factor is 0.5. Therefore, the triangles are similar. +YOUR TURN 10.20 +1. Is triangle +similar to triangle +in the figure shown? +1026 +10 • Geometry +Access for free at openstax.org + +EXAMPLE 10.21 +Proving Similarity +In Figure 10.57, is triangle +(delta) similar to triangle +(epsilon)? Find the lengths of sides +and +as part of your answer. +Figure 10.57 +Solution +We can see that all three angles in triangle +are equal to the corresponding angles in triangle +. That is enough to +determine similarity. However, we want to find the values of +and +to prove similarity. +Step 1: We have to do is set up the proportions between the corresponding sides. We have the side that measures 2.375 +in triangle +corresponding to the side measuring 1.069 in triangle +. We have the hypotenuse/side in triangle +measuring 6 corresponding to the hypotenuse/side labeled +in triangle +. And, finally, the side labeled +in triangle +corresponds to the side measuring 2.475 in triangle +Each proportion should be equal. We start with the proportion of the shorter sides. Thus +Step 2: We solve for +using the first proportion. Set the two ratios equal to each other, cross-multiply, and solve for +. +We have: +So, +. +Step 3: Checking that length in the proportion factor of 0.45, we have: +Step 4: Solving for +, we will use the same proportion we used to solve for +. We have: +10.3 • Triangles +1027 + +Step 5: We test the proportions. We have the following: +The proportions are all equal. Therefore, we have proven the property of similarity between triangle +and triangle +YOUR TURN 10.21 +1. Are these triangles similar? Find the lengths of sides +and +to prove your answer. +EXAMPLE 10.22 +Applying Similar Triangles +A person who is 5 feet tall is standing 50 feet away from the base of a tree (Figure 10.58). The tree casts a 57-foot +shadow. The person casts a 7-foot shadow. What is the height of the tree? +Figure 10.58 +Solution +The bigger triangle includes a tree at side +and the smaller triangle includes the person at the side labeled 5 ft. These +two triangles are similar because the smaller triangle fits inside the larger triangle at the smallest angle. It would fit +inside the larger triangle at either of the other two angles as well. That all angles are equal is one of the criteria for +similar triangles, so we can solve using proportions: +1028 +10 • Geometry +Access for free at openstax.org + +The tree is 40.7 feet tall. +YOUR TURN 10.22 +1. A person who is 6 feet tall is standing 100 feet away from the base of a tree. The tree casts a shadow 107.5-foot +shadow. The person’s shadow is 7.5 feet long. How tall is the tree? +EXAMPLE 10.23 +Finding Missing Lengths +At a certain time of day, a radio tower casts a shadow 180 feet long (Figure 10.59). At the same time, a 9-foot truck casts +a shadow 15 feet long. What is the height of the tower? +Figure 10.59 +Solution +These are similar triangles and the problem can be solved by using proportions: +The height of the tower is 108 ft. +YOUR TURN 10.23 +1. A tree casts a shadow of 180 feet early in the morning. A 10-foot high garage casts a shadow of 30 feet at the +10.3 • Triangles +1029 + +same time in the morning. What is the height of the tree? +PEOPLE IN MATHEMATICS +Thales of Miletus +Thales of Miletus, sixth century BC, is considered one of the greatest mathematicians and philosophers of all time. +Thales is credited with being the first to discover that the two angles at the base of an isosceles triangle are equal, +and that the two angles formed by intersecting lines are equal—that is, vertical or opposite angles, are equal. Thales +is also known for devising a method for measuring the height of the pyramids by similar right triangles. Figure 10.60 +shows his method. He measured the length of the shadow cast by the pyramid at the precise time when his own +shadow ended at the same place. +Figure 10.60 Thales and Similarity +He equated the vertical height of the pyramid with his own height; the horizontal distance from the pyramid to the tip +of its shadow with the distance from himself and the tip of his own shadow; and finally, the length of the shadow cast +off the top of the pyramid with length of his own shadow cast off the top of his head. Using proportions, as shown in +Figure 10.60, he essentially discovered the properties of similarity for right triangles. That is, +is similar to +Note that to be similar, all corresponding angles between the two triangles must be equal, and the +proportions from one side to another side within each triangle, as well as the proportions of the corresponding sides +between the two triangles must be equal. +Thales is also credited with discovering a method of determining the distance of a ship from the shoreline. Here is +how he did it, as illustrated in Figure 10.61. +1030 +10 • Geometry +Access for free at openstax.org + +Figure 10.61 Thales and Similar Triangles +Thales walked along the shoreline pointing a stick at the ship until it formed a +angle to the shore. Then he walked +along the shot and placed the stick in the ground at point +. He continued walking until he reached point +. Then, he +turned and walked away from the shore at a +angle until the stick he placed in the ground at point +lined up with +the ship, point +. This is how he created similar triangles and estimated the distance of the ship to the shore by using +proportions. +Check Your Understanding +17. Find the measure of the missing angle in the given figure. +18. Find the measure of the missing angle in the given figure. +19. In the isosceles triangle shown, find the missing angles. +20. In the figure shown given +is parallel to +, find +and +. +10.3 • Triangles +1031 + +21. Find +and +in the given figure. +SECTION 10.3 EXERCISES +For the following exercises, classify the triangle with the listed angle measurements as acute, right, or obtuse. +1. +∡ +, +∡ +, +∡ +2. +∡ +, +∡ +, +∡ +3. +∡ +, +∡ +, +∡ +4. +∡ +, +∡ +, +∡ +5. Find the missing angles in the given figure. +6. What are the measurements of each angle in the given figure? +7. Find the angle measurements in the given figure. +8. Find the angle measurements in the given figure. +1032 +10 • Geometry +Access for free at openstax.org + +For the following exercises, determine which congruence theorem is used to show that the two triangles are +congruent. +9. +10. +11. +12. Are these triangles similar? If so, what is the common proportion or the scaling factor? +13. Are the images in the given figure similar? If so, what is the common proportion or the scaling factor? +14. In the figure shown, given +is parallel to +find +and +. +10.3 • Triangles +1033 + +15. Find +and +in the given figure. +16. In the given figure, find the proportions and solve for +and +. +17. In the given figure, find the proportions and solve for . +18. A triangular plot of land has a perimeter of 2,400 ft. The longest side is 200 ft less than twice the shortest side. The +middle side is 200 ft less than the longest side. Find the lengths of the three sides. +19. The Orange Tree Hotel in Charleston, North Carolina has a fountain in the shape of a cylinder with a circular +foundation. The circumference of the foundation is 6 times the radius increased by 12.88 ft. Find the radius of the +circular foundation. (Use 3.14 as an approximation for +.) +1034 +10 • Geometry +Access for free at openstax.org + +10.4 Polygons, Perimeter, and Circumference +Figure 10.62 Geometric patterns are often used in fabrics due to the interest the shapes create. (credit: "Triangles" by +Brett Jordan/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify polygons by their sides. +2. +Identify polygons by their characteristics. +3. +Calculate the perimeter of a polygon. +4. +Calculate the sum of the measures of a polygon’s interior angles. +5. +Calculate the sum of the measures of a polygon’s exterior angles. +6. +Calculate the circumference of a circle. +7. +Solve application problems involving perimeter and circumference. +In our homes, on the road, everywhere we go, polygonal shapes are so common that we cannot count the many uses. +Traffic signs, furniture, lighting, clocks, books, computers, phones, and so on, the list is endless. Many applications of +polygonal shapes are for practical use, because the shapes chosen are the best for the purpose. +Modern geometric patterns in fabric design have become more popular with time, and they are used for the beauty they +lend to the material, the window coverings, the dresses, or the upholstery. This art is not done for any practical reason, +but only for the interest these shapes can create, for the pure aesthetics of design. +When designing fabrics, one has to consider the perimeter of the shapes, the triangles, the hexagons, and all polygons +used in the pattern, including the circumference of any circular shapes. Additionally, it is the relationship of one object to +another and experimenting with different shapes, changing perimeters, or changing angle measurements that we find +the best overall design for the intended use of the fabric. In this section, we will explore these properties of polygons, +the perimeter, the calculation of interior and exterior angles of polygons, and the circumference of a circle. +Identifying Polygons +A polygon is a closed, two-dimensional shape classified by the number of straight-line sides. See Figure 10.63 for some +examples. We show only up to eight-sided polygons, but there are many, many more. +10.4 • Polygons, Perimeter, and Circumference +1035 + +Figure 10.63 Types of Polygons +If all the sides of a polygon have equal lengths and all the angles are equal, they are called regular polygons. However, +any shape with sides that are line segments can classify as a polygon. For example, the first two shapes, shown in Figure +10.64 and Figure 10.64, are both pentagons because they each have five sides and five vertices. The third shape Figure +10.64 is a hexagon because it has six sides and six vertices. We should note here that the hexagon in Figure 10.64 is a +concave hexagon, as opposed to the first two shapes, which are convex pentagons. Technically, what makes a polygon +concave is having an interior angle that measures greater than +. They are hollowed out, or cave in, so to speak. +Convex refers to the opposite effect where the shape is rounded out or pushed out. +Figure 10.64 Polygons +While there are variations of all polygons, quadrilaterals contain an additional set of figures classified by angles and +whether there are one or more pairs of parallel sides. See Figure 10.65. +1036 +10 • Geometry +Access for free at openstax.org + +Figure 10.65 Types of Quadrilaterals +EXAMPLE 10.24 +Identifying Polygons +Identify each polygon. +1. +2. +3. +4. +10.4 • Polygons, Perimeter, and Circumference +1037 + +5. +6. +Solution +1. +This shape has six sides. Therefore, it is a hexagon. +2. +This shape has four sides, so it is a quadrilateral. It has two pairs of parallel sides making it a parallelogram. +3. +This shape has eight sides making it an octagon. +4. +This is an equilateral triangle, as all three sides are equal. +5. +This is a rhombus; all four sides are equal. +6. +This is a regular octagon, eight sides of equal length and equal angles. +YOUR TURN 10.24 +Identify the shape. +1. +2. +3. +4. +EXAMPLE 10.25 +Determining Multiple Polygons +What polygons make up Figure 10.66? +1038 +10 • Geometry +Access for free at openstax.org + +Figure 10.66 +Solution +Shapes 1 and 5 are hexagons; shapes 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15, and 16 are triangles; shapes 8 and 17 are +parallelograms; and shape 11 is a trapezoid. +YOUR TURN 10.25 +1. What polygons make up the figure shown? +Perimeter +Perimeter refers to the outside measurements of some area or region given in linear units. For example, to find out how +much fencing you would need to enclose your backyard, you will need the perimeter. The general definition of +perimeter is the sum of the lengths of the sides of an enclosed region. For some geometric shapes, such as rectangles +and circles, we have formulas. For other shapes, it is a matter of just adding up the side lengths. +A rectangle is defined as part of the group known as quadrilaterals, or shapes with four sides. A rectangle has two sets +of parallel sides with four angles. To find the perimeter of a rectangle, we use the following formula: +FORMULA +The formula for the perimeter +of a rectangle is +, twice the length +plus twice the width +. +For example, to find the length of a rectangle that has a perimeter of 24 inches and a width of 4 inches, we use the +formula. Thus, +The length is 8 units. +The perimeter of a regular polygon with +sides is given as +. For example, the perimeter of an equilateral +triangle, a triangle with three equal sides, and a side length of 7 cm is +. +10.4 • Polygons, Perimeter, and Circumference +1039 + +EXAMPLE 10.26 +Finding the Perimeter of a Pentagon +Find the perimeter of a regular pentagon with a side length of 7 cm (Figure 10.67). +Figure 10.67 +Solution +A regular pentagon has five equal sides. Therefore, the perimeter is equal to +. +YOUR TURN 10.26 +1. Find the perimeter of a square table that measures 30 inches across from one side to its opposite side. +EXAMPLE 10.27 +Finding the Perimeter of an Octagon +Find the perimeter of a regular octagon with a side length of 14 cm (Figure 10.68). +Figure 10.68 +Solution +A regular octagon has eight sides of equal length. Therefore, the perimeter of a regular octagon with a side length of 14 +cm is +. +YOUR TURN 10.27 +1. Find the perimeter of a regular heptagon with a side length of 3.2 as shown in the figure. +Sum of Interior and Exterior Angles +To find the sum of the measurements of interior angles of a regular polygon, we have the following formula. +FORMULA +The sum of the interior angles of a polygon with +sides is given by +1040 +10 • Geometry +Access for free at openstax.org + +For example, if we want to find the sum of the interior angles in a parallelogram, we have +Similarly, to find the sum of the interior angles inside a regular heptagon, we have +To find the measure of each interior angle of a regular polygon with +sides, we have the following formula. +FORMULA +The measure of each interior angle of a regular polygon with +sides is given by +For example, find the measure of an interior angle of a regular heptagon, as shown in Figure 10.69. We have +Figure 10.69 Interior Angles +EXAMPLE 10.28 +Calculating the Sum of Interior Angles +Find the measure of an interior angle in a regular octagon using the formula, and then find the sum of all the interior +angles using the sum formula. +Solution +An octagon has eight sides, so +. +Step 1: Using the formula +: +So, the measure of each interior angle in a regular octagon is +. +Step 2: The sum of the angles inside an octagon, so using the formula: +Step 3: We can test this, as we already know the measure of each angle is +. Thus, +. +10.4 • Polygons, Perimeter, and Circumference +1041 + +YOUR TURN 10.28 +1. Find the measure of each interior angle of a regular pentagon and then find the sum of the interior angles. +EXAMPLE 10.29 +Calculating Interior Angles +Use algebra to calculate the measure of each interior angle of the five-sided polygon (Figure 10.70). +Figure 10.70 +Solution +Step 1: Let us find out what the total of the sum of the interior angles should be. Use the formula for the sum of the +angles in a polygon with +sides: +. So, +. +Step 2: We add up all the angles and solve for +: +Step 3: We can then find the measure of each interior angle: +YOUR TURN 10.29 +1. Find the sum of the measures of the interior angles and then find the measure of each interior angle in the +figure shown. +An exterior angle of a regular polygon is an angle formed by extending a side length beyond the closed figure. The +1042 +10 • Geometry +Access for free at openstax.org + +measure of an exterior angle of a regular polygon with +sides is found using the following formula: +FORMULA +To find the measure of an exterior angle of a regular polygon with +sides we use the formula +In Figure 10.71, we have a regular hexagon +. By extending the lines of each side, an angle is formed on the +exterior of the hexagon at each vertex. The measure of each exterior angle is found using the formula, +. +Figure 10.71 Exterior Angles +Now, an important point is that the sum of the exterior angles of a regular polygon with +sides equals +This implies +that when we multiply the measure of one exterior angle by the number of sides of the regular polygon, we should get +For the example in Figure 10.71, we multiply the measure of each exterior angle, +, by the number of sides, six. +Thus, the sum of the exterior angles is +EXAMPLE 10.30 +Calculating the Sum of Exterior Angles +Find the sum of the measure of the exterior angles of the pentagon (Figure 10.72). +Figure 10.72 +Solution +Each individual angle measures +Then, the sum of the exterior angles is +YOUR TURN 10.30 +1. Find the sum of the measures of the exterior angles in the figure shown. +10.4 • Polygons, Perimeter, and Circumference +1043 + +Circles and Circumference +The perimeter of a circle is called the circumference. To find the circumference, we use the formula +where +is +the diameter, the distance across the center, or +where +is the radius. +FORMULA +The circumference of a circle is found using the formula +where +is the diameter of the circle, or +where +is the radius. +The radius is ½ of the diameter of a circle. The symbol +is the ratio of the circumference to the +diameter. Because this ratio is constant, our formula is accurate for any size circle. See Figure 10.73. +Figure 10.73 Circle Diameter and Radius +Let the radius be equal to 3.5 inches. Then, the circumference is +EXAMPLE 10.31 +Finding Circumference with Diameter +Find the circumference of a circle with diameter 10 cm. +Solution +If the diameter is 10 cm, the circumference is +YOUR TURN 10.31 +1. Find the circumference of a circle with a radius of 2.25 cm. +1044 +10 • Geometry +Access for free at openstax.org + +EXAMPLE 10.32 +Finding Circumference with Radius +Find the radius of a circle with a circumference of 12 in. +Solution +If the circumference is 12 in, then the radius is +YOUR TURN 10.32 +1. Find the radius of a circle with a circumference of +EXAMPLE 10.33 +Calculating Circumference for the Real World +You decide to make a trim for the window in Figure 10.74. How many feet of trim do you need to buy? +Figure 10.74 +Solution +The trim will cover the 6 feet along the bottom and the two 12-ft sides plus the half circle on top. The circumference of a +semicircle is ½ the circumference of a circle. The diameter of the semicircle is 6 ft. Then, the circumference of the +semicircle would be +Therefore, the total perimeter of the window is +You need to buy 39.4 ft of trim. +YOUR TURN 10.33 +1. To make a trim for the window in the figure shown, you need the perimeter. How many feet of trim do you need +to buy? +10.4 • Polygons, Perimeter, and Circumference +1045 + +PEOPLE IN MATHEMATICS +Archimedes +Figure 10.75 Archimedes (credit: “Archimedes” Engraving from the book Les vrais pourtraits et vies des hommes +illustres grecz, latins et payens (1586)/Wikimedia Commons, Public Domain) +The overwhelming consensus is that Archimedes (287–212 BCE) was the greatest mathematician of classical antiquity, +if not of all time. A Greek scientist, inventor, philosopher, astronomer, physicist, and mathematician, Archimedes +flourished in Syracuse, Sicily. He is credited with the invention of various types of pulley systems and screw pumps +based on the center of gravity. He advanced numerous mathematical concepts, including theorems for finding +surface area and volume. Archimedes anticipated modern calculus and developed the idea of the “infinitely small” and +the method of exhaustion. The method of exhaustion is a technique for finding the area of a shape inscribed within a +sequence of polygons. The areas of the polygons converge to the area of the inscribed shape. This technique evolved +to the concept of limits, which we use today. +1046 +10 • Geometry +Access for free at openstax.org + +One of the more interesting achievements of Archimedes is the way he estimated the number pi, the ratio of the +circumference of a circle to its diameter. He was the first to find a valid approximation. He started with a circle having +a diameter of 1 inch. His method involved drawing a polygon inscribed inside this circle and a polygon circumscribed +around this circle. He knew that the perimeter of the inscribed polygon was smaller than the circumference of the +circle, and the perimeter of the circumscribed polygon was larger than the circumference of the circle. This is shown +in the drawing of an eight-sided polygon. He increased the number of sides of the polygon each time as he got closer +to the real value of pi. The following table is an example of how he did this. +Sides +Inscribed Perimeter +Circumscribed Perimeter +4 +2.8284 +4.00 +8 +3.0615 +3.3137 +16 +3.1214 +3.1826 +32 +3.1365 +3.1517 +64 +3.1403 +3.1441 +Archimedes settled on an approximation of +after an iteration of 96 sides. Because pi is an irrational +number, it cannot be written exactly. However, the capability of the supercomputer can compute pi to billions of +decimal digits. As of 2002, the most precise approximation of pi includes 1.2 trillion decimal digits. +WHO KNEW? +The Platonic Solids +The Platonic solids (Figure 10.76) have been known since antiquity. A polyhedron is a three-dimensional object +constructed with congruent regular polygonal faces. Named for the philosopher, Plato believed that each one of the +solids is associated with one of the four elements: Fire is associated with the tetrahedron or pyramid, earth with the +cube, air with the octahedron, and water with the icosahedron. Of the fifth Platonic solid, the dodecahedron, Plato +said, “… God used it for arranging the constellations on the whole heaven.” +10.4 • Polygons, Perimeter, and Circumference +1047 + +Figure 10.76 Platonic Solids +Plato believed that the combination of these five polyhedra formed all matter in the universe. Later, Euclid proved that +exactly five regular polyhedra exist and devoted the last book of the Elements to this theory. These ideas were +resuscitated by Johannes Kepler about 2,000 years later. Kepler used the solids to explain the geometry of the +universe. The beauty and symmetry of the Platonic solids have inspired architects and artists from antiquity to the +present. +Check Your Understanding +In the following exercises, identify the regular polygons. +22. +23. +24. +25. Find the perimeter of the regular pentagon with side length of 6 cm. +26. Find the sum of the interior angles of a regular hexagon. +27. Find the measure of each interior angle of a regular hexagon. +28. Find the measurements of each angle in the given figure. +29. Find the circumference of the circle with radius equal to 3 cm. +SECTION 10.4 EXERCISES +In the following exercises, identify the polygons. +1. +1048 +10 • Geometry +Access for free at openstax.org + +2. +3. +4. Find the perimeter of a regular hexagon with side length equal to 12 cm. +5. A regular quadrilateral has a perimeter equal to 72 in. Find the length of each side. +6. The perimeter of an equilateral triangle is 72 cm. Find the length of each side. +7. Find the dimensions of a rectangular region with perimeter of 34 m, where the shorter side is 13 less than twice +the longer side. Let +. Then, the shorter side is +8. Find the perimeter of the figure shown. +9. Find the perimeter of a fenced-in area where the length is 22 m, and the width is ½ of the length plus 3. +10. You have 140 ft of fencing to enclose a rectangular region that borders a river. You do not have to fence in the side +that borders the river. The width is equal to +, and the length is equal to six times the width less 10. Find the +dimensions of the region to be fenced. +11. What is the measure of each interior angle of a regular hexagon? +12. What is the sum of the interior angles of a triangle? +13. Use algebra to find the measure of each angle of the quadrilateral shown. +14. Find the missing sides and angles of the parallelogram shown. +10.4 • Polygons, Perimeter, and Circumference +1049 + +15. Find the sum of the interior and exterior angles of the regular pentagon shown. +16. What is the measure of each exterior angle of a regular hexagon? +17. What is the sum of the exterior angles of a triangle? +18. What is the sum of the exterior angles of an octagon? +19. Find the circumference of a circle with radius 5. +20. Find the circumference of a circle with a diameter of 7. +21. Find the perimeter of the window in the figure shown. +22. Find the circumference of a circle with a radius of 1.25 cm. +23. The hands of a clock vary in length. If the hour hand is 5 inches long and the minute hand is 7 inches long, how far +does each hand travel in 12 hours? +1050 +10 • Geometry +Access for free at openstax.org + +24. A bicycle tire has a radius of 12 in. How many revolutions will the tire make if the bicycle travels approximately 377 +ft? +25. When Chicago installed its Centennial Ferris wheel at Navy Pier in 1995, there was much discussion about how big +it could be. The city wanted it to be as high as possible because the incredible views of the city would surely draw +the tourists. It was decided that the height of the wheel could safely reach 150 ft. If the wheel begins 10 ft above +ground level, what is the circumference of the Ferris wheel? +26. Find the perimeter of an equilateral triangle with side length equal to 21 cm. +27. What is the diameter of a circle whose circumference is 39.6 cm? +10.4 • Polygons, Perimeter, and Circumference +1051 + +10.5 Tessellations +Figure 10.77 Penrose tiling represents one type of tessellation. (credit: "Penrose Tiling" by Inductiveload/Wikimedia +Commons, Public Domain) +Learning Objectives +After completing this section, you should be able to: +1. +Apply translations, rotations, and reflections. +2. +Determine if a shape tessellates. +The illustration shown above (Figure 10.77) is an unusual pattern called a Penrose tiling. Notice that there are two types +of shapes used throughout the pattern: smaller green parallelograms and larger blue parallelograms. What's interesting +about this design is that although it uses only two shapes over and over, there is no repeating pattern. +In this section, we will focus on patterns that do repeat. Repeated patterns are found in architecture, fabric, floor tiles, +wall patterns, rug patterns, and many unexpected places as well. It may be a simple hexagon-shaped floor tile, or a +complex pattern composed of several different motifs. These two-dimensional designs are called regular (or periodic) +tessellations. There are countless designs that may be classified as regular tessellations, and they all have one thing in +common—their patterns repeat and cover the plane. +We will explore how tessellations are created and experiment with making some of our own as well. The topic of +tessellations belongs to a field in mathematics called transformational geometry, which is a study of the ways objects +can be moved while retaining the same shape and size. These movements are termed rigid motions and symmetries. +WHO KNEW? +M. C. Escher +A good place to start the study of tessellations is with the work of M. C. Escher. The Dutch graphic artist was famous +for the dimensional illusions he created in his woodcuts and lithographs, and that theme is carried out in many of his +tessellations as well. Escher became obsessed with the idea of the “regular division of the plane.” He sought ways to +divide the plane with shapes that would fit snugly next to each other with no gaps or overlaps, represent beautiful +patterns, and could be repeated infinitely to fill the plane. He experimented with practically every geometric shape +imaginable and found the ones that would produce a regular division of the plane. The idea is similar to dividing a +number by one of its factors. When a number divides another number evenly, there are no remainders, like there are +no gaps when a shape divides or fills the plane. +Escher went far beyond geometric shapes, beyond triangles and polygons, beyond irregular polygons, and used +other shapes like figures, faces, animals, fish, and practically any type of object to achieve his goal; and he did achieve +1052 +10 • Geometry +Access for free at openstax.org + +it, beautifully, and left it for the ages to appreciate. +VIDEO +M.C. Escher: How to Create a Tessellation (https://openstax.org/r/create_tessellation) +The Mathematical Art of M.C. Escher (https://openstax.org/r/mathematical_art) +Tessellation Properties and Transformations +A regular tessellation means that the pattern is made up of congruent regular polygons, same size and shape, including +some type of movement; that is, some type of transformation or symmetry. Here we consider the rigid motions of +translations, rotations, reflections, or glide reflections. A plane of tessellations has the following properties: +• +Patterns are repeated and fill the plane. +• +There are no gaps or overlaps. Shapes must fit together perfectly. (It was Escher who determined that a proper +tessellation could have no gaps and no overlaps.) +• +Shapes are combined using a transformation. +• +All the shapes are joined at a vertex. In other words, if you were to draw a circle around a vertex, it would include a +corner of each shape touching at that vertex. +• +For a tessellation of regular congruent polygons, the sum of the measures of the interior angles that meet at a +vertex equals +In Figure 10.78, the tessellation is made up of squares. There are four squares meeting at a vertex. An interior angle of a +square is +and the sum of four interior angles is +In Figure 10.79, the tessellation is made up of regular +hexagons. There are three hexagons meeting at each vertex. The interior angle of a hexagon is +and the sum of +three interior angles is +Both tessellations will fill the plane, there are no gaps, the sum of the interior angle +meeting at the vertex is +and both are achieved by translation transformations. These tessellations work because all +the properties of a tessellation are present. +Figure 10.78 Tessellation – Squares +Figure 10.79 Tessellation – Hexagons +The movements or rigid motions of the shapes that define tessellations are classified as translations, rotations, +10.5 • Tessellations +1053 + +reflections, or glide reflections. Let’s first define these movements and then look at some examples showing how these +transformations are revealed. +Translation +A translation is a movement that shifts the shape vertically, horizontally, or on the diagonal. Consider the trapezoid +in Figure 10.80. We have translated it 3 units to the right and 3 units up. That means every corner is moved by +the number of units and in the direction specified. Mathematicians will indicate this movement with a vector, an arrow +that is drawn to illustrate the criteria and the magnitude of the translation. The location of the translated trapezoid is +marked with the vertices, +but it is still the exact same shape and size as the original trapezoid +. +Figure 10.80 Translation +EXAMPLE 10.34 +Creating a Translation +Suppose you have a hexagon on a grid as in Figure 10.81. Translate the hexagon 5 units to the right and 3 units up. +Figure 10.81 +Solution +The best way to do this is by translating the individual points +. Once translated, the points become +1054 +10 • Geometry +Access for free at openstax.org + +Figure 10.82 +YOUR TURN 10.34 +1. Translate the hexagon with points +6 units down. +Rotation +The rotation transformation occurs when you rotate a shape about a point and at a predetermined angle. In Figure +10.83, the triangle is rotated around the rotation point by +and then translated 7 units up and 4 units over to the +right. That means that each corner is translated to the new location by the same number of units and in the same +direction. +10.5 • Tessellations +1055 + +Figure 10.83 Rotation +We can see that +is mapped to +by a rotation of +up and to the right. If rotated again by +, the triangle would +be upside down. +EXAMPLE 10.35 +Applying a Rotation +Figure 10.84 illustrates a tessellation begun with an equilateral triangle. Explain how this pattern is produced. +Figure 10.84 +Solution +A rotation to the right or to the left around the vertex by +six times, produces the hexagonal shape. The sixth rotation +brings the triangle back to its original position. Then, a reflection up and another one on the diagonal will reproduce the +pattern. When a shape returns to its original position by a rotation, we say that it has rotational symmetry. +YOUR TURN 10.35 +1. Starting with the triangle in the figure shown, explain how the pattern on the right was achieved. +1056 +10 • Geometry +Access for free at openstax.org + +Reflection +A reflection is the third transformation. A shape is reflected about a line and the new shape becomes a mirror image. +You can reflect the shape vertically, horizontally, or on the diagonal. There are two shapes in Figure 10.85. The +quadrilateral is reflected horizontally; the arrow shape is reflected vertically. +Figure 10.85 Reflection +Glide Reflection +The glide reflection is the fourth transformation. It is a combination of a reflection and a translation. This can occur by +first reflecting the shape and then gliding or translating it to its new location, or by translating first and then reflecting. +The example in Figure 10.86 shows a trapezoid, which is reflected over the dashed line, so it appears upside down. Then, +we shifted the shape horizontally by 6 units to the right. Whether we use the glide first or the reflection first, the end +result is the same in most cases. However, the tessellation shown in the next example can only be achieved by a +reflection first and then a translation. +10.5 • Tessellations +1057 + +Figure 10.86 Glide Reflection +EXAMPLE 10.36 +Applying the Glide Reflection +An obtuse triangle is reflected about the dashed line, and the two shapes are joined together. How does the tessellation +shown in Figure 10.87 materialize? +Figure 10.87 +Solution +The new shape is reflected horizontally and joined with the original shape. It is then translated vertically and horizontally +to make up the tessellation. Notice the blank spaces next to the vertical pattern. These areas are made up of the exact +original shape rotated +but with no line up the center. These rotated shapes are translated horizontally and +vertically, and thus, the plane is tessellated with no gaps. This is an example of a glide reflection where the order of the +transformations matters. +YOUR TURN 10.36 +1. Explain how this tessellation of equilateral triangles could be produced. +EXAMPLE 10.37 +Applying More Than One Tessellation +Show how this tessellation (Figure 10.88) can be achieved. +1058 +10 • Geometry +Access for free at openstax.org + +Figure 10.88 +Solution +This is a tessellation that has one color on the front of the trapezoid and a different color on the back. There is a +translation on the diagonal, and a reflection vertically. These are two separate transformations resulting in two new +placements of the trapezoid. We can call this a combination of two transformations or a glide reflection. +YOUR TURN 10.37 +1. How does this tessellation of the squares come about? +Interior Angles +The sum of the interior angles of a tessellation is +. In Figure 10.89, the tessellation is made of six triangles formed +into the shape of a hexagon. Each angle inside a triangle equals +, and the six vertices meet the sum of those interior +angles, +. +10.5 • Tessellations +1059 + +Figure 10.89 Interior Angles at the Vertex of Triangles +In Figure 10.90, the tessellation is made up of trapezoids, such that two of the interior angles of each trapezoid equals +and the other two angles equal +. Thus, the sum of the interior angles where the vertices of four trapezoids meet +equals +. +Figure 10.90 Interior Angles at the Vertex of Trapezoids +These tessellations illustrate the property that the shapes meet at a vertex where the interior angles sum to +. +Tessellating Shapes +We might think that all regular polygons will tessellate the plane by themselves. We have seen that squares do and +hexagons do. The pattern of squares in Figure 10.91 is a translation of the shape horizontally and vertically. The +hexagonal pattern in Figure 10.92, is translated horizontally, and then on the diagonal, either to the right or to the left. +This particular pattern can also be formed by rotations. Both tessellations are made up of congruent shapes and each +shape fits in perfectly as the pattern repeats. +Figure 10.91 Translation Horizontally and Vertically +Figure 10.92 Translation Horizontally and Slide Diagonally +1060 +10 • Geometry +Access for free at openstax.org + +We have also seen that equilateral triangles will tessellate the plane without gaps or overlaps, as shown in Figure 10.93. +The pattern is made by a reflection and a translation. The darker side is the face of the triangle and the lighter side is the +back of the triangle, shown by the reflection. Each triangle is reflected and then translated on the diagonal. +Figure 10.93 Reflection and Glide Translation +Escher experimented with all regular polygons and found that only the ones mentioned, the equilateral triangle, the +square, and the hexagon, will tessellate the plane by themselves. Let’s try a few other regular polygons to observe what +Escher found. +EXAMPLE 10.38 +Tessellating the Plane +Do regular pentagons tessellate the plane by themselves (Figure 10.94)? +Figure 10.94 +Solution +We can see that regular pentagons do not tessellate the plane by themselves. There is a gap, a gap in the shape of a +parallelogram. We conclude that regular pentagons will not tessellate the plane by themselves. +YOUR TURN 10.38 +1. Do regular heptagons tessellate the plane by themselves? +EXAMPLE 10.39 +Tessellating Octagons +Do regular octagons tessellate the plane by themselves (Figure 10.95)? +10.5 • Tessellations +1061 + +Figure 10.95 +Solution +Again, we see that regular octagons do not tessellate the plane by themselves. The gaps, however, are squares. So, two +regular polygons, an octagon and a square, do tessellate the plane. +YOUR TURN 10.39 +1. Do regular dodecagons (12-sided regular polygons) tessellate the plane by themselves? +Just because regular pentagons do not tessellate the plane by themselves does not mean that there are no pentagons +that tessellate the plane, as we see in Figure 10.96. +Figure 10.96 Tessellation of Pentagons +Another example of an irregular polygon that tessellates the plane is by using the obtuse irregular triangle from a +previous example. What transformations should be performed to produce the tessellation shown in Figure 10.97? +1062 +10 • Geometry +Access for free at openstax.org + +Figure 10.97 Tessellating with Obtuse Irregular Triangles +First, the triangle is reflected over the tip at point +, and then translated to the right and joined with the original triangle +to form a parallelogram. The parallelogram is then translated on the diagonal and to the right and to the left. +Naming +A tessellation of squares is named by choosing a vertex and then counting the number of sides of each shape touching +the vertex. Each square in the tessellation shown in Figure 10.98 has four sides, so starting with square +, the first +number is 4, moving around counterclockwise to the next square meeting the vertex, square +, we have another 4, +square +adds another 4, and finally square +adds a fourth 4. So, we would name this tessellation a 4.4.4.4. +The hexagon tessellation, shown in Figure 10.99 has six sides to the shape and three hexagons meet at the vertex. Thus, +we would name this a 6.6.6. The triangle tessellation, shown in Figure 10.100 has six triangles meeting the vertex. Each +triangle has three sides. Thus, we name this a 3.3.3.3.3.3. +Figure 10.98 4.4.4.4 +Figure 10.99 6.6.6 +Figure 10.100 3.3.3.3.3.3 +EXAMPLE 10.40 +Creating Your Own Tessellation +Create a tessellation using two colors and two shapes. +10.5 • Tessellations +1063 + +Solution +We used a parallelogram and an isosceles triangle. The parallelogram is reflected vertically and horizontally so that only +every other corner touches. The triangles are reflected vertically and horizontally and then translated over the +parallelogram. The result is alternating vertical columns of parallelograms and then triangles (Figure 10.101). +Figure 10.101 +YOUR TURN 10.40 +1. Create a tessellation using polygons, regular or irregular. +Check Your Understanding +30. What are the properties of repeated patterns that let them be classified as tessellations? +31. Explain how the using the transformation of a translation is applied to the movement of this shape starting with +point +. +32. Starting with the triangle with vertex +, describe how the transformation in this drawing is achieved. +1064 +10 • Geometry +Access for free at openstax.org + +33. Starting with a triangle with a darker face and a lighter back, describe how this pattern came about. +34. Name the tessellation in the figure shown. +SECTION 10.5 EXERCISES +1. What type of movements are used to change the orientation and placement of a shape? +2. What is the name of the motion that renders a shape upside down? +3. What do we call the motion that moves a shape to the right or left or on the diagonal? +4. If you are going to tessellate the plane with a regular polygon, what is the sum of the interior angles that surround +a vertex? +5. Does a regular heptagon tesselate the plane by itself? +6. What are the only regular polygons that will tessellate the plane by themselves? +7. What is the transformation called that revolves a shape about a point to a new position? +8. Transformational geometry is a study of what? +9. Describe how to achieve a rotation transformation. +10. Construct a +rotation of the triangle shown. +10.5 • Tessellations +1065 + +11. Shapes can be rotated around a point of rotation or a ____________. +12. What is the name of the transformation that involves a reflection and a translation? +13. What can a tessellation not have between shapes? +14. Describe the transformation shown. +15. What do we call a transformation that produces a mirror image? +16. Sketch the reflection of the shape about the dashed line. +17. Sketch the reflection of the shape about the dashed line. +18. Sketch the translation of the shape 3 units to the right and 3 units vertically. +1066 +10 • Geometry +Access for free at openstax.org + +19. Rotate the shape +about the rotation point using point +as your guide. +20. Do regular pentagons tessellate the plain by themselves? +21. What do regular tessellations have in common? +22. How would we name a tessellation of squares as shown in the figure? +23. How do we name a tessellation of octagons and squares as shown in the figure? +24. How would we name a tessellation of trapezoids as shown in the figure? +10.5 • Tessellations +1067 + +10.6 Area +Figure 10.102 The area of a regulation baseball diamond must adhere to specific measurements to be legal. (credit: +"Diagram of Regulation Diamond" by Erica Fischer from "Baseball" The World Book, 1920/Flickr, Public Domain, CC BY +2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Calculate the area of triangles. +2. +Calculate the area of quadrilaterals. +3. +Calculate the area of other polygons. +4. +Calculate the area of circles. +Some areas carry more importance than other areas. Did you know that in a baseball game, when the player hits the ball +and runs to first base that he must run within a 6-foot wide path? If he veers off slightly to the right, he is out. In other +words, a few inches can be the difference in winning or losing a game. Another example is real estate. On Manhattan +Island, one square foot of real estate is worth far more than real estate in practically any other area of the country. In +other words, we place a value on area. As the context changes, so does the value. +Area refers to a region measured in square units, like a square mile or a square foot. For example, to purchase tile for a +kitchen floor, you would need to know how many square feet are needed because tile is sold by the square foot. +Carpeting is sold by the square yard. As opposed to linear measurements like perimeter, which in in linear units. For +example, fencing is sold in linear units, a linear foot or yard. Linear dimensions refer to an outline or a boundary. Square +units refer to the area within that boundary. Different items may have different units, but either way, you must know the +linear dimensions to calculate the area. +Many geometric shapes have formulas for calculating areas, such as triangles, regular polygons, and circles. To calculate +areas for many irregular curves or shapes, we need calculus. However, in this section, we will only look at geometric +shapes that have known area formulas. The notation for area, as mentioned, is in square units and we write sq in or sq +cm, or use an exponent, such as +or +Note that linear measurements have no exponent above the units or we can +say that the exponent is 1. +Area of Triangles +The formula for the area of a triangle is given as follows. +1068 +10 • Geometry +Access for free at openstax.org + +FORMULA +The area of a triangle is given as +where +represents the base and +represents the height. +For example, consider the triangle in Figure 10.103. +Figure 10.103 Triangle 1 +The base measures 4 cm and the height measures 5 cm. Using the formula, we can calculate the area: +In Figure 10.104, the triangle has a base equal to 7 cm and a height equal to 3.5 cm. Notice that we can only find the +height by dropping a perpendicular to the base. The area is then +Figure 10.104 Triangle 2 +EXAMPLE 10.41 +Finding the Area of a Triangle +Find the area of this triangle that has a base of 4 cm and the height is 6 cm (Figure 10.105). +Figure 10.105 +Solution +Using the formula, we have +10.6 • Area +1069 + +YOUR TURN 10.41 +1. Find the area of the triangle with a base equal to 4 cm and the height equal to 4 cm. +Area of Quadrilaterals +A quadrilateral is a four-sided polygon with four vertices. Some quadrilaterals have either one or two sets of parallel +sides. The set of quadrilaterals include the square, the rectangle, the parallelogram, the trapezoid, and the rhombus. The +most common quadrilaterals are the square and the rectangle. +Square +In Figure 10.106, a +grid is represented with twelve +squares across each row, and twelve +squares down each column. If you count the little squares, the sum equals 144 squares. Of course, you do not +have to count little squares to find area—we have a formula. Thus, the formula for the area of a square, where +, is +. The area of the square in Figure 10.106 is +Figure 10.106 Area of a Square +FORMULA +The formula for the area of a square is +or +Rectangle +Similarly, the area for a rectangle is found by multiplying length times width. The rectangle in Figure 10.107 has width +equal to 5 in and length equal to 12 in. The area is +1070 +10 • Geometry +Access for free at openstax.org + +Figure 10.107 Area of a Rectangle +FORMULA +The area of a rectangle is given as +Many everyday applications require the use of the perimeter and area formulas. Suppose you are remodeling your home +and you want to replace all the flooring. You need to know how to calculate the area of the floor to purchase the correct +amount of tile, or hardwood, or carpet. If you want to paint the rooms, you need to calculate the area of the walls and +the ceiling to know how much paint to buy, and the list goes on. Can you think of other situations where you might need +to calculate area? +EXAMPLE 10.42 +Finding the Area of a Rectangle +You have a garden with an area of 196 square feet. How wide can the garden be if the length is 28 feet? +Solution +The area of a rectangular region is +Letting the width equal +: +YOUR TURN 10.42 +1. Find the area of a rectangular region with a length of 18 feet and a width 1/3 of the length. +EXAMPLE 10.43 +Determining the Cost of Floor Tile +Jennifer is planning to install vinyl floor tile in her rectangular-shaped basement, which measures 29 ft by 16 ft. How +many boxes of floor tile should she buy and how much will it cost if one box costs $50 and covers +Solution +The area of the basement floor is +We will divide this area by +Thus, +Therefore, +Jennifer will have to buy 24 boxes of tile at a cost $1,200. +YOUR TURN 10.43 +1. You remodel your kitchen and decide to change out the tile floor. The floor measures 30 ft by 15 ft. One box of +tile costs $45 and covers 10 ft2. How many boxes of tile should you buy, and what will it cost? +Parallelogram +The area of a parallelogram can be found using the formula for the area of a triangle. Notice in Figure 10.108, if we cut a +diagonal across the parallelogram from one vertex to the opposite vertex, we have two triangles. If we multiply the area +10.6 • Area +1071 + +of a triangle by 2, we have the area of a parallelogram: +Figure 10.108 Area of a Parallelogram +FORMULA +The area of a parallelogram is +For example, if we have a parallelogram with the base be equal to 10 inches and the height equal to 5 inches, the area +will be +EXAMPLE 10.44 +Finding the Area of a Parallelogram +In the parallelogram (Figure 10.109), if +find the exact area of the parallelogram. +Figure 10.109 +Solution +Using the formula of +we have +YOUR TURN 10.44 +1. Find the area of the parallelogram. +1072 +10 • Geometry +Access for free at openstax.org + +EXAMPLE 10.45 +Finding the Area of a Parallelogram Park +The boundaries of a city park form a parallelogram (Figure 10.110). The park takes up one city block, which is contained +by two sets of parallel streets. Each street measures 55 yd long. The perpendicular distance between streets is 39 yd. +How much sod, sold by the square foot, should the city purchase to cover the entire park and how much will it cost? The +sod is sold for $0.50 per square foot, installation is $1.50 per square foot, and the cost of the equipment for the day is +$100. +Figure 10.110 +Solution +Step 1: As sod is sold by the square foot, the first thing we have to do is translate the measurements of the park from +yards to feet. There are 3 ft to a yard, so 55 yd is equal to 165 ft, and 39 yd is equal to 117 ft. +Step 2: The park has the shape of a parallelogram, and the formula for the area is +: +Step 3: The city needs to purchase +of sod. The cost will be $0.50 per square foot for the sod and $1.50 per +square foot for installation, plus $100 for equipment: +YOUR TURN 10.45 +1. Suppose your city has a park just like the one in Example 10.45. The length of each street is 49 yd, and the +perpendicular distance between streets is 31 yd. How much sod, sold by the square foot at $0.45 per square foot +plus $1.00 per square foot for installation, and a flat $50 fee for equipment, should be purchased and what will +the cost be? +10.6 • Area +1073 + +Trapezoid +Another quadrilateral is the trapezoid. A trapezoid has one set of parallel sides or bases. The formula for the area of a +trapezoid with parallel bases +and +and height +is given here. +FORMULA +The formula for the area of a trapezoid is given as +For example, find the area of the trapezoid in Figure 10.111 that has base +equal to 8 cm, base +equal to 6 cm, and +height equal to 6 cm. +Figure 10.111 Area of a Trapezoid +The area is +. +EXAMPLE 10.46 +Finding the Area of a Trapezoid +(Figure 10.112) is a regular trapezoid with +Find the exact perimeter of +, and then find the area. +Figure 10.112 +Solution +The perimeter is the measure of the boundary of the shape, so we just add up the lengths of the sides. We have +Then, the area of the trapezoid using the formula is +. +YOUR TURN 10.46 +1. Find the area of the trapezoid shown. +1074 +10 • Geometry +Access for free at openstax.org + +Rhombus +The rhombus has two sets of parallel sides. To find the area of a rhombus, there are two formulas we can use. One +involves determining the measurement of the diagonals. +FORMULA +The area of a rhombus is found using one of these formulas: +• +where +and +are the diagonals. +• +where +is the base and +is the height. +For our purposes here, we will use the formula that uses diagonals. For example, if the area of a rhombus is +and the measure of +find the measure of +To solve this problem, we input the known values into the formula +and solve for the unknown. See Figure 10.113. +Figure 10.113 Area of a Rhombus +We have that +EXAMPLE 10.47 +Finding the Area of a Rhombus +Find the measurement of the diagonal +if the area of the rhombus is +and the measure of +Solution +Use the formula with the known values: +10.6 • Area +1075 + +YOUR TURN 10.47 +1. A rhombus has an area of +, the measure of +. Find the measure of +. +EXAMPLE 10.48 +Finding the Area of a Rhombus +You notice a child flying a rhombus-shaped kite on the beach. When it falls to the ground, it falls on a beach towel +measuring +by +You notice that one of the diagonals of the kite is the same length as the +width of the +towel. The second diagonal appears to be 2 in longer. What is the area of the kite (Figure 10.114)? +Figure 10.114 +Solution +Using the formula, we have: +YOUR TURN 10.48 +1. You have a kite that measures +If one of the diagonals measures 25 in, what is the length of the other +diagonal? +Area of Polygons +To find the area of a regular polygon, we need to learn about a few more elements. First, the apothem +of a regular +polygon is a line segment that starts at the center and is perpendicular to a side. The radius +of a regular polygon is also +a line segment that starts at the center but extends to a vertex. See Figure 10.115. +1076 +10 • Geometry +Access for free at openstax.org + +Figure 10.115 Apothem and Radius of a Polygon +FORMULA +The area of a regular polygon is found with the formula +where +is the apothem and +is the perimeter. +For example, consider the regular hexagon shown in Figure 10.116 with a side length of 4 cm, and the apothem +measures +Figure 10.116 Area of a Hexagon +We have the perimeter, +We have the apothem as +Then, the area is: +EXAMPLE 10.49 +Finding the Area of a Regular Octagon +Find the area of a regular octagon with the apothem equal to 18 cm and a side length equal to 13 cm (Figure 10.117). +Figure 10.117 +Solution +Using the formula, we have the perimeter +Then, the area is +YOUR TURN 10.49 +1. Find the area of the regular pentagon with the apothem equal to 5.5 cm and the side length equal to 7 cm. +10.6 • Area +1077 + +Changing Units +Often, we have the need to change the units of one or more items in a problem to find a solution. For example, suppose +you are purchasing new carpet for a room measured in feet, but carpeting is sold in terms of yards. You will have to +convert feet to yards to purchase the correct amount of carpeting. Or, you may need to convert centimeters to inches, or +feet to meters. In each case, it is essential to use the correct equivalency. +EXAMPLE 10.50 +Changing Units +Carpeting comes in units of square yards. Your living room measures 21 ft wide by 24 ft long. How much carpeting do +you buy? +Solution +We must convert feet to yards. As there are 3 ft in 1 yd, we have +and +Then, +YOUR TURN 10.50 +1. You want to carpet your bedroom, which measures 15 ft wide by 18 ft long. Carpeting is sold by the square yard, +so you must convert your measurements to yards. How much carpeting do you buy? +Area of Circles +Just as the circumference of a circle includes the number +so does the formula for the area of a circle. Recall that +is a +non-terminating, non-repeating decimal number: +. It represents the ratio of the circumference to the +diameter, so it is a critical number in the calculation of circumference and area. +FORMULA +The area of a circle is given as +where +is the radius. +For example, to find the of the circle with radius equal to 3 cm, as shown in Figure 10.118, is found using the formula +Figure 10.118 Circle with Radius 3 +We have +1078 +10 • Geometry +Access for free at openstax.org + +EXAMPLE 10.51 +Finding the Area of a Circle +Find the area of a circle with diameter of 16 cm. +Solution +The formula for the area of a circle is given in terms of the radius, so we cut the diameter in half. Then, the area is +YOUR TURN 10.51 +1. Find the area of the circle with a radius of 3 cm. +EXAMPLE 10.52 +Determining the Better Value for Pizza +You decide to order a pizza to share with your friend for dinner. The price for an 8-inch diameter pizza is $7.99. The price +for 16-inch diameter pizza is $13.99. Which one do you think is the better value? +Solution +The area of the 8-inch diameter pizza is +The area of the 16-inch diameter pizza is +Next, we divide the cost of each pizza by its area in square inches. Thus, +per +square inch and +per square inch. So clearly, the 16-inch pizza is the better value. +YOUR TURN 10.52 +1. You can buy a 9-inch diameter pizza for $10.99, or a 15-inch diameter pizza for $14.99. Which pizza is the better +value? +EXAMPLE 10.53 +Applying Area to the Real World +You want to purchase a tinted film, sold by the square foot, for the window in Figure 10.119. (This problem should look +familiar as we saw it earlier when calculating circumference.) The bottom part of the window is a rectangle, and the top +part is a semicircle. Find the area and calculate the amount of film to purchase. +10.6 • Area +1079 + +Figure 10.119 +Solution +First, the rectangular portion has +. For the top part, we have a semicircle with a diameter of 5 ft, so +the radius is 2.5 ft. We want one half of the area of a circle with radius 2.5 ft, so the area of the top semicircle part is +Add the area of the rectangle to the area of the semicircle. Then, the total area to be covered +with the window film is +YOUR TURN 10.53 +1. You decide to install a new front door with a semicircle top as shown in the figure. How much area will the new +door occupy? +Area within Area +Suppose you want to install a round hot tub on your backyard patio. How would you calculate the space needed for the +hot tub? Or, let’s say that you want to purchase a new dining room table, but you are not sure if you have enough space +for it. These are common issues people face every day. So, let’s take a look at how we solve these problems. +1080 +10 • Geometry +Access for free at openstax.org + +EXAMPLE 10.54 +Finding the Area within an Area +The patio in your backyard measures 20 ft by 10 ft (Figure 10.120). On one-half of the patio, you have a 4-foot diameter +table with six chairs taking up an area of approximately 36 sq feet. On the other half of the patio, you want to install a +hot tub measuring 6 ft in diameter. How much room will the table with six chairs and the hot tub take up? How much +area is left over? +Figure 10.120 +Solution +The hot tub has a radius of 3 ft. That area is then +The total square feet taken up with the +table and chairs and the hot tub is +The area left over is equal to the total area of the patio, +minus the area for the table and chairs and the hot tub. Thus, the area left over is +YOUR TURN 10.54 +1. Find the area of the shaded region in the given figure. +EXAMPLE 10.55 +Finding the Cost of Fertilizing an Area +A sod farmer wants to fertilize a rectangular plot of land 150 ft by 240 ft. A bag of fertilizer covers +and costs +$200. How much will it cost to fertilize the entire plot of land? +Solution +The plot of land is +It will take 7.2 bags of fertilizer to cover the land area. Therefore, the farmer will have to +purchase 8 bags of fertilizer at $200 a bag, which comes to $1,600. +10.6 • Area +1081 + +YOUR TURN 10.55 +1. You want to install sod on one-half of your parallelogram-shaped backyard as shown. The patio covers the other +half. Sod costs $50 a bag and covers +How much will it cost to buy the sod? +PEOPLE IN MATHEMATICS +Heron of Alexandria +Figure 10.121 Heron of Alexandria (credit: "Heron of Alexandria" from 1688 German translation of Hero's Pneumatics/ +Wikimedia Commons, Public Domain) +Heron of Alexandria, born around 20 A.D., was an inventor, a scientist, and an engineer. He is credited with the +invention of the Aeolipile, one of the first steam engines centuries before the industrial revolution. Heron was the +father of the vending machine. He talked about the idea of inserting a coin into a machine for it to perform a specific +action in his book, Mechanics and Optics. His contribution to the field of mathematics was enormous. Metrica, a +series of three books, was devoted to methods of finding the area and volume of three-dimensional objects like +pyramids, cylinders, cones, and prisms. He also discovered and developed the procedures for finding square roots +and cubic roots. However, he is probably best known for Heron’s formula, which is used for finding the area of a +triangle based on the lengths of its sides. Given a triangle +(Figure 10.122), +Figure 10.122 +Heron’s formula is +where +is the semi-perimeter calculated as +1082 +10 • Geometry +Access for free at openstax.org + +Check Your Understanding +35. Find the area of the triangle with base equal to 3 cm and height equal to 5 cm. +36. The area of the sail in the sailboat is +The shortest length is 10 ft. What is the height of the sail? +37. Find the area of this parallelogram. +38. Find the area of a regular hexagon with side length of 5 and apothem equal to 4.3. +39. Find the area of circle with a diameter of 16 in. +40. Find the area of the shaded region in the given figure. +41. A round tray sits on top of the dining room table. The radius of the tray is 15 in. What is the area taken up by the +tray? +SECTION 10.6 EXERCISES +For the following exercises, find the area of the figure with the given measurements. +1. Area of a triangle with base equal to 10 cm, and height equal to 15 cm. +2. Area of right triangle with base 54 cm, and height equal to 72 cm. +3. A triangle has an area of +If the base equals 5 cm, find the height. +4. Find the area of the triangle with base equal to 2.5 in, height equal to 0.7 in as shown. +10.6 • Area +1083 + +5. Find the area of the trapezoid with +, and +as shown. +6. Find the area of the trapezoid shown. +7. Find the area of a parallelogram with base equal to 50 cm and height equal to 35 cm as shown. +8. Find the area of a parallelogram with base equal to 20 in, and height equal to 22 in. +9. Find the area of the rhombus shown. +10. Find the area of the regular pentagon with the apothem equal to 5.5 in and the side length equal to 6 in as shown. +11. Find the area of the regular octagon with apothem equal to 7 cm and the side length is 3.5 cm as shown. +1084 +10 • Geometry +Access for free at openstax.org + +12. Find the area of a regular pentagon with apothem equal 1.5 in, and the sides are equal to 3 in as shown. +13. Find the area of a circle with radius of 3 cm as shown. +14. You are installing a countertop in the shape of a trapezoid with a round sink as shown in the figure. After the sink +is installed, how much area is left on the countertop? +15. Find the area of the shaded region in the given figure. +16. You have a structural post in the corner of your kitchen. The adjacent room is the family room and the two rooms +are separated by a transparent glass wall. To camouflage the post, you decide to have five shelves built that will fill +the corner, as shown in the figure. The rectangular shelves are 26 inches wide and 24 inches deep. The diameter of +the post is 6 inches. How much shelf area will be available after the shelves are built around the post? The laminate +material for the shelves runs $25 per square foot. How much will the laminate shelves cost? +10.6 • Area +1085 + +17. Find the area of the shaded region in the given figure. +18. Find the area of the shaded region in the given figure. +19. Find the area of the shaded region in the given figure. All measurements in centimeters. +20. Your property measures 2 miles wide by 2.5 miles long. You want to landscape it but the landscaper charges by the +square foot. How many square feet need to be landscaped? +21. Find the area of the window shown. +1086 +10 • Geometry +Access for free at openstax.org + +22. What is the area of a major league baseball diamond enclosed by the baselines if it forms a square with 90ft +between bases? +10.7 Volume and Surface Area +Figure 10.123 Volume is illustrated in this 3-dimensional view of an interior space. This gives a buyer a more realistic +interpretation of space. (credit: "beam render 10 with sun and cat tree" by monkeywing/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Calculate the surface area of right prisms and cylinders. +2. +Calculate the volume of right prisms and cylinders. +3. +Solve application problems involving surface area and volume. +Volume and surface area are two measurements that are part of our daily lives. We use volume every day, even though +we do not focus on it. When you purchase groceries, volume is the key to pricing. Judging how much paint to buy or how +many square feet of siding to purchase is based on surface area. The list goes on. An example is a three-dimensional +rendering of a floor plan. These types of drawings make building layouts far easier to understand for the client. It allows +the viewer a realistic idea of the product at completion; you can see the natural space, the volume of the rooms. This +section gives you practical information you will use consistently. You may not remember every formula, but you will +remember the concepts, and you will know where to go should you want to calculate volume or surface area in the +future. +We will concentrate on a few particular types of three-dimensional objects: right prisms and right cylinders. The adjective +“right” refers to objects such that the sides form a right angle with the base. We will look at right rectangular prisms, +10.7 • Volume and Surface Area +1087 + +right triangular prisms, right hexagonal prisms, right octagonal prisms, and right cylinders. Although, the principles +learned here apply to all right prisms. +Three-Dimensional Objects +In geometry, three-dimensional objects are called geometric solids. Surface area refers to the flat surfaces that +surround the solid and is measured in square units. Volume refers to the space inside the solid and is measured in cubic +units. Imagine that you have a square flat surface with width and length. Adding the third dimension adds depth or +height, depending on your viewpoint, and now you have a box. One way to view this concept is in the Cartesian +coordinate three-dimensional space. The +-axis and the +-axis are, as you would expect, two dimensions and suitable for +plotting two-dimensional graphs and shapes. Adding the +-axis, which shoots through the origin perpendicular to the +-plane, and we have a third dimension. See Figure 10.124. +Figure 10.124 Three-Dimensional Space +Here is another view taking the two-dimensional square to a third dimension. See Figure 10.125. +Figure 10.125 Going from Two Dimensions to Three Dimensions +To study objects in three dimensions, we need to consider the formulas for surface area and volume. For example, +suppose you have a box (Figure 10.126) with a hinged lid that you want to use for keeping photos, and you want to cover +the box with a decorative paper. You would need to find the surface area to calculate how much paper is needed. +Suppose you need to know how much water will fill your swimming pool. In that case, you would need to calculate the +volume of the pool. These are just a couple of examples of why these concepts should be understood, and familiarity +with the formulas will allow you to make use of these ideas as related to right prisms and right cylinders. +Figure 10.126 +Right Prisms +A right prism is a particular type of three-dimensional object. It has a polygon-shaped base and a congruent, regular +polygon-shaped top, which are connected by the height of its lateral sides, as shown in Figure 10.127. The lateral sides +form a right angle with the base and the top. There are rectangular prisms, hexagonal prisms, octagonal prisms, +triangular prisms, and so on. +1088 +10 • Geometry +Access for free at openstax.org + +Figure 10.127 Pentagonal Prism +Generally, to calculate surface area, we find the area of each side of the object and add the areas together. To calculate +volume, we calculate the space inside the solid bounded by its sides. +FORMULA +The formula for the surface area of a right prism is equal to twice the area of the base plus the perimeter of the base +times the height, +where +is equal to the area of the base and top, +is the perimeter of the base, and +is the height. +FORMULA +The formula for the volume of a rectangular prism, given in cubic units, is equal to the area of the base times the +height, +where +is the area of the base and +is the height. +EXAMPLE 10.56 +Calculating Surface Area and Volume of a Rectangular Prism +Find the surface area and volume of the rectangular prism that has a width of 10 cm, a length of 5 cm, and a height of 3 +cm (Figure 10.128). +Figure 10.128 +Solution +The surface area is +The volume is +YOUR TURN 10.56 +1. A rectangular solid has a width of 6 cm, length of 15 cm, and height or depth of 6 cm. Find the surface area and +the volume. +In Figure 10.129, we have three views of a right hexagonal prism. The regular hexagon is the base and top, and the +lateral faces are the rectangular regions perpendicular to the base. We call it a right prism because the angle formed by +10.7 • Volume and Surface Area +1089 + +the lateral sides to the base is +See Figure 10.127. +Figure 10.129 Right Hexagonal Prism +The first image is a view of the figure straight on with no rotation in any direction. The middle figure is the base or the +top. The last figure shows you the solid in three dimensions. To calculate the surface area of the right prism shown in +Figure 10.129, we first determine the area of the hexagonal base and multiply that by 2, and then add the perimeter of +the base times the height. Recall the area of a regular polygon is given as +where +is the apothem and +is the +perimeter. We have that +Then, the surface area of the hexagonal prism is +To find the volume of the right hexagonal prism, we multiply the area of the base by the height using the formula +The base is +and the height is 20 +. Thus, +EXAMPLE 10.57 +Calculating the Surface Area of a Right Triangular Prism +Find the surface area of the triangular prism (Figure 10.130). +Figure 10.130 +Solution +The area of the triangular base is +. The perimeter of the base is +Then, the surface area of the triangular prism is +YOUR TURN 10.57 +1. Find the surface area of the triangular prism shown. +1090 +10 • Geometry +Access for free at openstax.org + +EXAMPLE 10.58 +Finding the Surface Area and Volume +Find the surface area and the volume of the right triangular prism with an equilateral triangle as the base and height +(Figure 10.131). +Figure 10.131 +Solution +The area of the triangular base is +. Then, the surface area is +The volume formula is found by multiplying the area of the base by the height. We have that +YOUR TURN 10.58 +1. Find the surface area and the volume of the octagonal figure shown. +EXAMPLE 10.59 +Determining Surface Area Application +Katherine and Romano built a greenhouse made of glass with a metal roof (Figure 10.132). In order to determine the +heating and cooling requirements, the surface area must be known. Calculate the total surface area of the greenhouse. +10.7 • Volume and Surface Area +1091 + +Figure 10.132 +Solution +The area of the long side measures +Multiplying by 2 gives +The front (minus the +triangular area) measures +Multiplying by 2 gives +The floor measures +Each triangular region measures +Multiplying by 2 gives +Finally, +one side of the roof measures +Multiplying by 2 gives +Add them up and we have +YOUR TURN 10.59 +1. Calculate the surface area of a greenhouse with a flat roof measuring 12 ft wide, 25 ft long, and 8 ft high. +Right Cylinders +There are similarities between a prism and a cylinder. While a prism has parallel congruent polygons as the top and the +base, a right cylinder is a three-dimensional object with congruent circles as the top and the base. The lateral sides of a +right prism make a +angle with the polygonal base, and the side of a cylinder, which unwraps as a rectangle, makes a +angle with the circular base. +Right cylinders are very common in everyday life. Think about soup cans, juice cans, soft drink cans, pipes, air hoses, and +the list goes on. +In Figure 10.133, imagine that the cylinder is cut down the 12-inch side and rolled out. We can see that the cylinder side +when flat forms a rectangle. The +formula includes the area of the circular base, the circular top, and the area of the +rectangular side. The length of the rectangular side is the circumference of the circular base. Thus, we have the formula +for total surface area of a right cylinder. +Figure 10.133 Right Cylinder +FORMULA +The surface area of a right cylinder is given as +To find the volume of the cylinder, we multiply the area of the base with the height. +FORMULA +The volume of a right cylinder is given as +1092 +10 • Geometry +Access for free at openstax.org + +EXAMPLE 10.60 +Finding the Surface Area and Volume of a Cylinder +Given the cylinder in Figure 10.133, which has a radius of 5 inches and a height of 12 inches, find the surface area and +the volume. +Solution +Step 1: We begin with the areas of the base and the top. The area of the circular base is +Step 2: The base and the top are congruent parallel planes. Therefore, the area for the base and the top is +Step 3: The area of the rectangular side is equal to the circumference of the base times the height: +Step 4: We add the area of the side to the areas of the base and the top to get the total surface area. We have +. +Step 5: The volume is equal to the area of the base times the height. Then, +YOUR TURN 10.60 +1. Find the surface area and volume of the cylinder with a radius of 7cm and a height of 5 cm. +Applications of Surface Area and Volume +The following are just a small handful of the types of applications in which surface area and volume are critical factors. +Give this a little thought and you will realize many more practical uses for these procedures. +EXAMPLE 10.61 +Applying a Calculation of Volume +A can of apple pie filling has a radius of 4 cm and a height of 10 cm. How many cans are needed to fill a pie pan (Figure +10.134) measuring 22 cm in diameter and 3 cm deep? +10.7 • Volume and Surface Area +1093 + +Figure 10.134 +Solution +The volume of the can of apple pie filling is +The volume of the pan is +To find the number of cans of apple pie filling, we divide the volume of the pan by the +volume of a can of apple pie filling. Thus, +We will need 2.3 cans of apple pie filling to fill the pan. +YOUR TURN 10.61 +1. You are making a casserole that includes vegetable soup and pasta. The size of your cylindrical casserole dish +has a diameter of 10 inches and is 4 inches high. The pasta will consume the bottom portion of the casserole +dish about 1 inch high. The soup can has a diameter of 3 inches and is 4 inches high. After the pasta is added, +how many cans of soup can you add? +Optimization +Problems that involve optimization are ones that look for the best solution to a situation under some given conditions. +Generally, one looks to calculus to solve these problems. However, many geometric applications can be solved with the +tools learned in this section. Suppose you want to make some throw pillows for your sofa, but you have a limited amount +of fabric. You want to make the largest pillows you can from the fabric you have, so you would need to figure out the +dimensions of the pillows that will fit these criteria. Another situation might be that you want to fence off an area in your +backyard for a garden. You have a limited amount of fencing available, but you would like the garden to be as large as +possible. How would you determine the shape and size of the garden? Perhaps you are looking for maximum volume or +minimum surface area. Minimum cost is also a popular application of optimization. Let’s explore a few examples. +EXAMPLE 10.62 +Maximizing Area +Suppose you have 150 meters of fencing that you plan to use for the enclosure of a corral on a ranch. What shape would +give the greatest possible area? +Solution +So, how would we start? Let’s look at this on a smaller scale. Say that you have 30 inches of string and you experiment +with different shapes. The rectangle in Figure 10.135 measures 12 in long by 3 in wide. We have a perimeter of +and the area calculates as +The rectangle in Figure 10.135, measures 8 in +long and 7 in wide. The perimeter is +and the area is +In Figure 10.135, the +square measures 7.5 in on each side. The perimeter is then +and the area is +If you want a circular corral as in Figure 10.135, we would consider a circumference of +which gives a radius of +and an area of +1094 +10 • Geometry +Access for free at openstax.org + +Figure 10.135 +We see that the circular shape gives the maximum area relative to a circumference of 30 in. So, a circular corral with a +circumference of 150 meters and a radius of 23.87 meters gives a maximum area of +YOUR TURN 10.62 +1. You have 25 ft of rope to section off a rectangular-shaped garden. What dimensions give the maximum area that +can be roped off? +EXAMPLE 10.63 +Designing for Cost +Suppose you want to design a crate built out of wood in the shape of a rectangular prism (Figure 10.136). It must have a +volume of 3 cubic meters. The cost of wood is $15 per square meter. What dimensions give the most economical design +while providing the necessary volume? +Figure 10.136 +Solution +To choose the optimal shape for this container, you can start by experimenting with different sizes of boxes that will hold +3 cubic meters. It turns out that, similar to the maximum rectangular area example where a square gives the maximum +area, a cube gives the maximum volume and the minimum surface area. +As all six sides are the same, we can use a simplified volume formula: +where +is the length of a side. Then, to find the length of a side, we take the cube root of the volume. +We have +The surface area is equal to the sum of the areas of the six sides. The area of one side is +So, +the surface area of the crate is +. At $15 a square meter, the cost comes to +Checking the volume, we have +. +10.7 • Volume and Surface Area +1095 + +YOUR TURN 10.63 +1. Suppose you want to a build a container to hold 2 cubic feet of fabric swatches. You want to cover the container +in laminate costing $10 per square foot. What are the dimensions of the container that is the most economical? +What is the cost? +Check Your Understanding +42. Find the surface area of the equilateral triangular prism shown. +43. Find the surface area of the octagonal prism shown. +44. Find the volume of the octagonal prism shown with the apothem equal to +the side length equal to +and +the height equal to +45. Determine the surface area of the right cylinder where the radius of the base is +, and the height is +. +46. Find the volume of the cylinder where the radius of the base is +, and the height is +. +47. As an artist, you want to design a cylindrical container for your colored art pencils and another rectangular +container for some other tools. The cylindrical container will be 8 inches high with a diameter of 6 inches. The +rectangular container measures 10 inches wide by 8 inches deep by 4 inches high and has a lid. You found some +beautiful patterned paper to use to wrap both pieces. How much paper will you need? +SECTION 10.7 EXERCISES +1. Find the volume of the right triangular prism with the two side legs of the base equal to 10 m, the hypotenuse +equal to +, and the height or the length, depending on your viewpoint, is equal to 15 m. +1096 +10 • Geometry +Access for free at openstax.org + +2. Find the surface area of the right triangular prism in the Exercise 1 with the two legs of the base equal to 10 m, and +the height equal to 15 m. +3. Find the surface area of the right trapezoidal prism with side +, side +, the height is 10 cm, the +slant length is 12 cm, and the length is 24 cm. +4. Find the volume of the trapezoidal prism in the exercise above where the base and top have the following +measurements: side +, side +, the slant lengths are each +, and the height of the trapezoidal +base = +. The height or length of the three-dimensional solid is +. +5. Find the surface area of the octagonal prism. The base and top are regular octagons with the apothem equal to +10 m, a side length equal to 12 m, and a height of 30 m. +6. Find the volume for the right octagonal prism, with the apothem equal to 10 m, the side length of the base is +equal to 12 m, and the height equal to 30 m. +7. You decide to paint the living room. You will need the surface area of the 4 walls and the ceiling. The room +measures 20 ft long and 14 ft wide, and the ceiling is 8 ft high. +For the following exercises, find the surface area of each right cylinder. +8. +9. +10. +11. +12. +13. +For the following exercises, find the volume of each right cylinder to the nearest tenth. +14. +15. +16. +17. +18. +19. You have remodeled your kitchen and the exhaust pipe above the stove must pass through an overhead cabinet as +10.7 • Volume and Surface Area +1097 + +shown in the figure. Find the volume of the remaining space in the cabinet. +10.8 Right Triangle Trigonometry +Figure 10.137 In the lower left corner of the fresco The School of Athens by Raphael, the figure in white writing in the +book represents Pythagoras. Alongside him, to the right, the figure with the long, light-brown hair is said to depict +Archimedes. (credit: modification of work “School of Athens” by Raphael (1483–1520), Vatican Museums/Wikimedia, +Public Domain) +Learning Objectives +After completing this section, you should be able to: +1. +Apply the Pythagorean Theorem to find the missing sides of a right triangle. +2. +Apply the +and +right triangle relationships to find the missing sides of a triangle. +3. +Apply trigonometric ratios to find missing parts of a right triangle. +4. +Solve application problems involving trigonometric ratios. +This is another excerpt from Raphael’s The School of Athens. The man writing in the book represents Pythagoras, the +namesake of one of the most widely used formulas in geometry, engineering, architecture, and many other fields, the +Pythagorean Theorem. However, there is evidence that the theorem was known as early as 1900–1100 BC by the +Babylonians. The Pythagorean Theorem is a formula used for finding the lengths of the sides of right triangles. +Born in Greece, Pythagoras lived from 569–500 BC. He initiated a cult-like group called the Pythagoreans, which was a +secret society composed of mathematicians, philosophers, and musicians. Pythagoras believed that everything in the +world could be explained through numbers. Besides the Pythagorean Theorem, Pythagoras and his followers are +credited with the discovery of irrational numbers, the musical scale, the relationship between music and mathematics, +and many other concepts that left an immeasurable influence on future mathematicians and scientists. +The focus of this section is on right triangles. We will look at how the Pythagorean Theorem is used to find the +unknown sides of a right triangle, and we will also study the special triangles, those with set ratios between the lengths +of sides. By ratios we mean the relationship of one side to another side. When you think about ratios, you should think +about fractions. A fraction is a ratio, the ratio of the numerator to the denominator. Finally, we will preview trigonometry. +We will learn about the basic trigonometric functions, sine, cosine and tangent, and how they are used to find not only +unknown sides but unknown angles, as well, with little information. +1098 +10 • Geometry +Access for free at openstax.org + +Pythagorean Theorem +The Pythagorean Theorem is used to find unknown sides of right triangles. The theorem states that the sum of the +squares of the two legs of a right triangle equals the square of the hypotenuse (the longest side of the right triangle). +FORMULA +The Pythagorean Theorem states +where +and +are two sides (legs) of a right triangle and +is the hypotenuse, as shown in Figure 10.138. +Figure 10.138 Pythagorean Right Triangle +For example, given that side +and side +we can find the measure of side +using the Pythagorean Theorem. +Thus, +EXAMPLE 10.64 +Using the Pythagorean Theorem +Find the length of the missing side of the triangle (Figure 10.139). +Figure 10.139 +Solution +Using the Pythagorean Theorem, we have +10.8 • Right Triangle Trigonometry +1099 + +When we take the square root of a number, the answer is usually both the positive and negative root. However, lengths +cannot be negative, which is why we only consider the positive root. +YOUR TURN 10.64 +1. Use the Pythagorean Theorem to find the missing side of the right triangle shown. +Distance +The applications of the Pythagorean Theorem are countless, but one especially useful application is that of distance. In +fact, the distance formula stems directly from the theorem. It works like this: +In Figure 10.140, the problem is to find the distance between the points +and +We call the length from +point +to point +side +, and the length from point +to point +side +. To find side , we use the +distance formula and we will explain it relative to the Pythagorean Theorem. The distance formula is +such that +is a substitute for +in the Pythagorean Theorem and is equal to +and +is a substitute for +in the Pythagorean Theorem and is equal to +When we +plug in these numbers to the distance formula, we have +Thus, +, the hypotenuse, in the Pythagorean Theorem. +1100 +10 • Geometry +Access for free at openstax.org + +Figure 10.140 Distance +EXAMPLE 10.65 +Calculating Distance Using the Distance Formula +You live on the corner of First Street and Maple Avenue, and work at Star Enterprises on Tenth Street and Elm Drive +(Figure 10.141). You want to calculate how far you walk to work every day and how it compares to the actual distance (as +the crow flies). Each block measures 200 ft by 200 ft. +Figure 10.141 +Solution +You travel 7 blocks south and 9 blocks west. If each block measures 200 ft by 200 ft, then +. +As the crow flies, use the distance formula. We have +YOUR TURN 10.65 +1. How far is it to your workplace (as the crow flies) if the blocks in the previous example measure 100 ft by 100 ft? +10.8 • Right Triangle Trigonometry +1101 + +EXAMPLE 10.66 +Calculating Distance with the Pythagorean Theorem +The city has specific building codes for wheelchair ramps. Every vertical rise of 1 in requires that the horizontal length be +12 inches. You are constructing a ramp at your business. The plan is to make the ramp 130 inches in horizontal length +and the slanted distance will measure approximately 132.4 inches (Figure 10.142). What should the vertical height be? +Figure 10.142 +Solution +The Pythagorean Theorem states that the horizontal length of the base of the ramp, side a, is 130 in. The length of c, or +the length of the hypotenuse, is 132.4 in. The length of the height of the triangle is side b. +Then, by the Pythagorean Theorem, we have: +If you construct the ramp with a 25 in vertical rise, will it fulfill the building code? If not, what will have to change? +The building code states 12 in of horizontal length for each 1 in of vertical rise. The vertical rise is 25 in, which means +that the horizontal length has to be +So, no, this will not pass the code. If you must keep the vertical rise +at 25 in, what will the other dimensions have to be? Since we need a minimum of 300 in for the horizontal length: +The new ramp will look like Figure 10.143. +Figure 10.143 +YOUR TURN 10.66 +1. If 10 in is the maximum possible vertical rise as shown in the figure, how long would the ramp have to be to pass +the building code rule of 12 horizontal inches to 1 vertical inch? +Triangles +In geometry, as in all fields of mathematics, there are always special rules for special circumstances. An example is the +perfect square rule in algebra. When expanding an expression like +we do not have to expand it the long way: +1102 +10 • Geometry +Access for free at openstax.org + +If we know the perfect square formula, given as +we can skip the middle step and just start writing down the answer. This may seem trivial with problems like +However, what if you have a problem like +That is a different story. Nevertheless, we use the same +perfect square formula. The same idea applies in geometry. There are special formulas and procedures to apply in +certain types of problems. What is needed is to remember the formula and remember the kind of problems that fit. +Sometimes we believe that because a formula is labeled special, we will rarely have use for it. That assumption is +incorrect. So, let us identify the +triangle and find out why it is special. See Figure 10.144. +Figure 10.144 The +We see that the shortest side is opposite the smallest angle, and the longest side, the hypotenuse, will always be +opposite the right angle. There is a set ratio of one side to another side for the +triangle given as +or +Thus, you only need to know the length of one side to find the other two sides in a +triangle. +EXAMPLE 10.67 +Finding Missing Lengths in a +Triangle +Find the measures of the missing lengths of the triangle (Figure 10.145). +Figure 10.145 +Solution +We can see that this is a +triangle because we have a right angle and a +angle. The remaining angle, +therefore, must equal +Because this is a special triangle, we have the ratios of the sides to help us identify the +missing lengths. Side +is the shortest side, as it is opposite the smallest angle +and we can substitute +The +ratios are +We have the hypotenuse equaling 10, which corresponds to side , and side +is equal to 2 . +Now, we must solve for +: +Side +is equal to +or +The lengths are +10.8 • Right Triangle Trigonometry +1103 + +YOUR TURN 10.67 +1. Find the lengths of the missing sides in the given figure. +EXAMPLE 10.68 +Applying +Triangle to the Real World +A city worker leans a 40-foot ladder up against a building at a +angle to the ground (Figure 10.146). How far up the +building does the ladder reach? +Figure 10.146 +Solution +We have a +triangle, and the hypotenuse is 40 ft. This length is equal to 2 , where +is the shortest side. If +, then +. The ladder is leaning on the wall 20 ft up from the ground. +YOUR TURN 10.68 +1. You want to repair a window on the second floor of your home. If you place the ladder at a +angle to the +ground, the ladder just about reaches the window. How far from the wall should you place the ladder? How far +up will the ladder reach? Make a sketch as an aid. +1104 +10 • Geometry +Access for free at openstax.org + +Triangles +The +triangle is another special triangle such that with the measure of one side we can find the measures of +all the sides. The two angles adjacent to the +angle are equal, and each measures +If two angles are equal, so are +their opposite sides. The ratio among sides is +or +as shown in Figure 10.147. +Figure 10.147 +Triangles +EXAMPLE 10.69 +Finding Missing Lengths of a +Triangle +Find the measures of the unknown sides in the triangle (Figure 10.148). +Figure 10.148 +Solution +Because we have a +triangle, we know that the two legs are equal in length and the hypotenuse is a product +of one of the legs and +One leg measures 3, so the other leg, +, measures 3. Remember the ratio of +Then, the hypotenuse, , equals +YOUR TURN 10.69 +1. Find the measures of the unknown sides in the given figure. +Trigonometry Functions +Trigonometry developed around 200 BC from a need to determine distances and to calculate the measures of angles in +the fields of astronomy and surveying. Trigonometry is about the relationships (or ratios) of angle measurements to side +lengths in primarily right triangles. However, trigonometry is useful in calculating missing side lengths and angles in +other triangles and many applications. +NOTE: You will need either a scientific calculator or a graphing calculator for this section. It must have the capability +to calculate trigonometric functions and express angles in degrees. +Trigonometry is based on three functions. We title these functions using the following abbreviations: +10.8 • Right Triangle Trigonometry +1105 + +• +• +• +Letting +which is the hypotenuse of a right triangle, we have Table 10.1. The functions are given in terms +of +, +, and , and in terms of sides relative to the angle, like opposite, adjacent, and the hypotenuse. +Table 10.1 Trigonometric Ratios +We will be applying the sine function, cosine function, and tangent function to find side lengths and angle +measurements for triangles we cannot solve using any of the techniques we have studied to this point. In Figure 10.149, +we have an illustration mainly to identify +and the sides labeled +and +. +Figure 10.149 Angle +An angle +sweeps out in a counterclockwise direction from the positive +-axis and stops when the angle reaches the +desired measurement. That ray extending from the origin that marks +is called the terminal side because that is where +the angle terminates. Regardless of the information given in the triangle, we can find all missing sides and angles using +the trigonometric functions. For example, in Figure 10.150, we will solve for the missing sides. +Figure 10.150 Solving for Missing Sides +Let’s use the trigonometric functions to find the sides +and +. As long as your calculator mode is set to degrees, you do +not have to enter the degree symbol. First, let’s solve for +. +We have +and +Then, +Next, let’s find +. This is the cosine function. We have +Then, +1106 +10 • Geometry +Access for free at openstax.org + +Now we have all sides, +You can also check the sides using the +ratio of +Table 10.2 is a list +of common angles, which you should find helpful. +Table 10.2 Common Angles +EXAMPLE 10.70 +Using Trigonometric Functions +Find the lengths of the missing sides for the triangle (Figure 10.151). +Figure 10.151 +Solution +We have a +angle, and the length of the triangle on the +-axis is 6 units. +Step 1: To find the length of , we can use the cosine function, as +We manipulate this equation a bit to solve +for : +Step 2: We can use the Pythagorean Theorem to find the length of +. Prove that your answers are correct by using other +trigonometric ratios: +Step 3: Now that we have +, we can use the sine function to prove that +is correct. We have +10.8 • Right Triangle Trigonometry +1107 + +YOUR TURN 10.70 +1. Find the lengths of the missing sides in the given figure. +To find angle measurements when we have two side measurements, we use the inverse trigonometric functions +symbolized as +or +The –1 looks like an exponent, but it means inverse. For example, in the previous +example, we had +and +To find what angle has these values, enter the values for the inverse cosine +function +in your calculator: +You can also use the inverse sine function and enter the values of +in your calculator given +and +We have +Finally, we can also use the inverse tangent function. Recall +We have +EXAMPLE 10.71 +Solving for Lengths in a Right Triangle +Solve for the lengths of a right triangle in which +and +(Figure 10.152). +1108 +10 • Geometry +Access for free at openstax.org + +Figure 10.152 +Solution +Step 1: To find side +, we use the sine function: +Step 2: To find +, we use the cosine function: +Step 3: Since this is a +triangle and side +should equal +if we input 3 for +, we have +Put this in +your calculator and you will get +YOUR TURN 10.71 +1. Find the missing side and angles in the figure shown. +EXAMPLE 10.72 +Finding Altitude +A small plane takes off from an airport at an angle of +to the ground. About two-thirds of a mile (3,520 ft) from the +airport is an 1,100-ft peak in the flight path of the plane (Figure 10.153). If the plane continues that angle of ascent, find +its altitude when it is above the peak, and how far it will be above the peak. +10.8 • Right Triangle Trigonometry +1109 + +Figure 10.153 +Solution +To solve this problem, we use the tangent function: +The plane’s altitude when passing over the peak is 2,140 ft, and it is 1,040 ft above the peak. +YOUR TURN 10.72 +1. Suppose that the plane takes off at a +angle, and 1 mile from the airport is a 1,500-foot peak. At what altitude +will the plane pass over the peak? +EXAMPLE 10.73 +Finding Unknown Sides and Angles +Suppose you have two known sides, but do not know the measure of any angles except for the right angle (Figure +10.154). Find the measure of the unknown angles and the third side. +Figure 10.154 +Solution +Step 1: We can find the third side using the Pythagorean Theorem: +Now, we have all three sides. +Step 2: To find +we will first find +The angle +is the angle whose sine is +Step 3: To find +, we use the inverse sine function: +1110 +10 • Geometry +Access for free at openstax.org + +Step 4: To find the last angle, we just subtract: +. +YOUR TURN 10.73 +1. You know the lengths of two sides and the right angle as shown in the figure. Find the length of the third side +and the other angles. +Angle of Elevation and Angle of Depression +Other problems that involve trigonometric functions include calculating the angle of elevation and the angle of +depression. These are very common applications in everyday life. The angle of elevation is the angle formed by a +horizontal line and the line of sight from an observer to some object at a higher level. The angle of depression is the +angle formed by a horizontal line and the line of sight from an observer to an object at a lower level. +EXAMPLE 10.74 +Finding the Angle of Elevation +A guy wire of length 110 meters runs from the top of an antenna to the ground (Figure 10.155). If the angle of elevation +of an observer to the top of the antenna is +how high is the antenna? +Figure 10.155 +Solution +We are looking for the height of the tower. This corresponds to the +-value, so we will use the sine function: +The tower is 75 m high. +10.8 • Right Triangle Trigonometry +1111 + +YOUR TURN 10.74 +1. You travel to Chicago and visit the observation deck at Willis Tower, 1,450 ft above ground. You can see the +Magnificent Mile to the northeast 6,864 ft away. What is the angle of depression from the observation deck to +the Magnificent Mile? +EXAMPLE 10.75 +Finding Angle of Elevation +You are sitting on the grass flying a kite on a 50-foot string (Figure 10.156). The angle of elevation is +How high above +the ground is the kite? +Figure 10.156 +Solution +We can solve this using the sine function, +YOUR TURN 10.75 +1. You are flying a kite on a 60-foot string. The angle of elevation from the ground to the kite is +. How high above +the ground is the kite? +1112 +10 • Geometry +Access for free at openstax.org + +PEOPLE IN MATHEMATICS +Pythagoras and the Pythagoreans +The Pythagorean Theorem is so widely used that most people assume that Pythagoras (570–490 BC) discovered it. +The philosopher and mathematician uncovered evidence of the right triangle concepts in the teachings of the +Babylonians dating around 1900 BC. However, it was Pythagoras who found countless applications of the theorem +leading to advances in geometry, architecture, astronomy, and engineering. +Among his accolades, Pythagoras founded a school for the study of mathematics and music. Students were called the +Pythagoreans, and the school’s teachings could be classified as a religious indoctrination just as much as an academic +experience. Pythagoras believed that spirituality and science coexist, that the intellectual mind is superior to the +senses, and that intuition should be honored over observation. +Pythagoras was convinced that the universe could be defined by numbers, and that the natural world was based on +mathematics. His primary belief was All is Number. He even attributed certain qualities to certain numbers, such as +the number 8 represented justice and the number 7 represented wisdom. There was a quasi-mythology that +surrounded Pythagoras. His followers thought that he was more of a spiritual being, a sort of mystic that was all- +knowing and could travel through time and space. Some believed that Pythagoras had mystical powers, although +these beliefs were never substantiated. +Pythagoras and his followers contributed more ideas to the field of mathematics, music, and astronomy besides the +Pythagorean Theorem. The Pythagoreans are credited with the discovery of irrational numbers and of proving that +the morning star was the planet Venus and not a star at all. They are also credited with the discovery of the musical +scale and that different strings made different sounds based on their length. Some other concepts attributed to the +Pythagoreans include the properties relating to triangles other than the right triangle, one of which is that the sum of +the interior angles of a triangle equals +These geometric principles, proposed by the Pythagoreans, were proven +200 years later by Euclid. +WHO KNEW? +A Visualization of the Pythagorean Theorem +In Figure 10.157, which is one of the more popular visualizations of the Pythagorean Theorem, we see that square +is +attached to side +; square +is attached to side +; and the largest square, square , is attached to side . Side +measures 3 cm in length, side +measures 4 cm in length, and side +measures 5 cm in length. By definition, the area +of square +measures 9 square units, the area of square +measures 16 square units, and the area of square +measures 25 square units. Substitute the values given for the areas of the three squares into the Pythagorean +Theorem and we have +Thus, the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse, as stated in +the Pythagorean Theorem. +10.8 • Right Triangle Trigonometry +1113 + +Figure 10.157 +Check Your Understanding +48. Find the lengths of the unknown sides of the +triangle shown.. +49. Find the missing lengths of the +triangle shown. +50. Use the Pythagorean theorem to find the missing length in the triangle shown. +51. The sun casts a shadow over the roof of a house that ends 105 ft from the front door as shown in the figure. How +high is the house to the tip of the roof? +1114 +10 • Geometry +Access for free at openstax.org + +52. Find the measure of side +in the given figure. +53. Find the measure of side +in the given figure. +SECTION 10.8 EXERCISES +Use the Pythagorean theorem to answer the following exercises. Let +and +represent two legs of a right triangle, and +let +represent the hypotenuse. Find the lengths of the missing sides. +1. +2. +3. If +and +, find +. +4. If +and +, find +. +5. If +and +, find +. +Refer to the +triangle shown for the following exercises. +6. If +, find +and +7. If +, find +and . +8. If +, find +and +9. If +, find +and +. +10. If +, find +and +Refer to the +triangle shown in the following exercises. +10.8 • Right Triangle Trigonometry +1115 + +11. If +, find +. +12. If +, find +. +13. If +, find +. +14. If +, find +. +15. If +, find +. +16. In the provided figure given +, find the unknown sides and angles. +Evaluate the expressions in the following exercises. +17. +18. +19. +20. Use the figure shown to solve for all missing sides and angles given +. +21. Use the figure shown to solve for all missing sides and angles given +. +1116 +10 • Geometry +Access for free at openstax.org + +Chapter Summary +Key Terms +10.1 Points, Lines, and Planes +• +line segment +• +plane +• +union +• +intersection +• +parallel +• +perpendicular +10.2 Angles +• +vertex +• +right angle +• +acute angle +• +obtuse angle +• +straight angle +• +complementary +• +supplementary +10.3 Triangles +• +acute +• +obtuse +• +isosceles +• +equilateral +• +hypotenuse +• +congruence +• +similarity +• +scaling factor +10.4 Polygons, Perimeter, and Circumference +• +perimeter +• +polygon +• +pentagon +• +hexagon +• +heptagon +• +octagon +• +quadrilateral +• +trapezoid +• +parallelogram +• +circumference +10.5 Tessellations +• +tessellation +• +translation +• +reflection +• +rotation +• +glide reflection +10.6 Area +• +triangle +• +square +• +rectangle +• +rhombus +• +apothem +• +radius +• +circle +10 • Chapter Summary +1117 + +10.7 Volume and Surface Area +• +surface area +• +volume +• +right prism +• +right cylinder +10.8 Right Triangle Trigonometry +• +right triangle +• +sine +• +cosine +• +tangent +Key Concepts +10.1 Points, Lines, and Planes +• +Modern-day geometry began in approximately 300 BCE with Euclid’s Elements, where he defined the principles +associated with the line, the point, and the plane. +• +Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other. +• +The union of two sets, +and +, contains all points that are in both sets and is symbolized as +• +The intersection of two sets +and +includes only the points common to both sets and is symbolized as +10.2 Angles +• +Angles are classified as acute if they measure less than +obtuse if they measure greater than +and less than +right if they measure exactly +and straight if they measure exactly +• +If the sum of angles equals +, they are complimentary angles. If the sum of angles equals +, they are +supplementary. +• +A transversal crossing two parallel lines form a series of equal angles: alternate interior angles, alternate exterior +angles, vertical angles, and corresponding angles +10.3 Triangles +• +The sum of the interior angles of a triangle equals +• +Two triangles are congruent when the corresponding angles have the same measure and the corresponding side +lengths are equal. +• +The congruence theorems include the following: SAS, two sides and the included angle of one triangle are +congruent to the same in a second triangle; ASA, two angles and the included side of one triangle are congruent to +the same in a second triangle; SSS, all three side lengths of one triangle are congruent to the same in a second +triangle; AAS, two angles and the non-included side of one triangle are congruent to the same in a second triangle. +• +Two shapes are similar when the proportions between corresponding angles, sides or features of two shapes are +equal, regardless of size. +10.4 Polygons, Perimeter, and Circumference +• +Regular polygons are closed, two-dimensional figures that have equal side lengths. They are named for the number +of their sides. +• +The perimeter of a polygon is the measure of the outline of the shape. We determine a shape’s perimeter by +calculating the sum of the lengths of its sides. +• +The sum of the interior angles of a regular polygon with +sides is found using the formula +The +measure of a single interior angle of a regular polygon with +sides is determined using the formula +• +The sum of the exterior angles of a regular polygon is +The measure of a single exterior angle of a regular +polygon with +sides is found using the formula +• +The circumference of a circle is +where +is the radius, or +and +is the diameter. +10.5 Tessellations +• +A tessellation is a particular pattern composed of shapes, usually polygons, that repeat and cover the plane with no +gaps or overlaps. +• +Properties of tessellations include rigid motions of the shapes called transformations. Transformations refer to +translations, rotations, reflections, and glide reflections. Shapes are transformed in such a way to create a pattern. +1118 +10 • Chapter Summary +Access for free at openstax.org + +10.6 Area +• +The area +of a triangle is found with the formula +where +is the base and +is the height. +• +The area of a parallelogram is found using the formula +where +is the base and +is the height. +• +The area of a rectangle is found using the formula +where +is the length and +is the width. +• +The area of a trapezoid is found using the formula +where +is the height, +is the length of one +base, and +is the length of the other base. +• +The area of a rhombus is found using the formula +where +is the length of one diagonal and +is the +length of the other diagonal. +• +The area of a regular polygon is found using the formula +where +is the apothem and +is the perimeter. +• +The area of a circle is found using the formula +where +is the radius. +10.7 Volume and Surface Area +• +A right prism is a three-dimensional object that has a regular polygonal face and congruent base such that that +lateral sides form a +angle with the base and top. The surface area +of a right prism is found using the +formula +where +is the area of the base, +is the perimeter of the base, and +is the height. The +volume +of a right prism is found using the formula +where +is the area of the base and +is the height. +• +A right cylinder is a three-dimensional object with a circle as the top and a congruent circle is the base, and the side +forms a +angle to the base and top. The surface area of a right cylinder is found using the formula +where +is the radius and +is the height. The volume is found using the formula +where +is the radius and +is the height. +10.8 Right Triangle Trigonometry +• +The Pythagorean Theorem is applied to right triangles and is used to find the measure of the legs and the +hypotenuse according the formula +where c is the hypotenuse. +• +To find the measure of the sides of a special angle, such as a +triangle, use the ratio +where +each of the three sides is associated with the opposite angle and 2 +is associated with the hypotenuse, opposite the +angle. +• +To find the measure of the sides of the second special triangle, the +triangle, use the ratio +where each of the three sides is associated with the opposite angle and +is associated with the hypotenuse, +opposite the +angle. +• +The primary trigonometric functions are +and +• +Trigonometric functions can be used to find either the length of a side or the measure of an angle in a right triangle, +and in applications such as the angle of elevation or the angle of depression formed using right triangles. +Videos +10.5 Tessellations +• +M.C. Escher: How to Create a Tessellation (https://openstax.org/r/create_tessellation) +• +The Mathematical Art of M.C. Escher (https://openstax.org/r/mathematical_art) +Formula Review +10.2 Angles +To translate an angle measured in degrees to radians, multiply by +To translate an angle measured in radians to degrees, multiply by +10.4 Polygons, Perimeter, and Circumference +The formula for the perimeter +of a rectangle is +, twice the length +plus twice the width +. +The sum of the interior angles of a polygon with +sides is given by +The measure of each interior angle of a regular polygon with +sides is given by +10 • Chapter Summary +1119 + +To find the measure of an exterior angle of a regular polygon with +sides we use the formula +The circumference of a circle is found using the formula +where +is the diameter of the circle, or +where +is the radius. +10.6 Area +The area of a triangle is given as +where +represents the base and +represents the height. +The formula for the area of a square is +or +The area of a rectangle is given as +The area of a parallelogram is +The formula for the area of a trapezoid is given as +The area of a rhombus is found using one of these formulas: +• +where +and +are the diagonals. +• +where +is the base and +is the height. +The area of a regular polygon is found with the formula +where +is the apothem and +is the perimeter. +The area of a circle is given as +where +is the radius. +10.7 Volume and Surface Area +The formula for the surface area of a right prism is equal to twice the area of the base plus the perimeter of the base +times the height, +where +is equal to the area of the base and top, +is the perimeter of the base, and +is the height. +The formula for the volume of a rectangular prism, given in cubic units, is equal to the area of the base times the height, +where +is the area of the base and +is the height. +The surface area of a right cylinder is given as +The volume of a right cylinder is given as +10.8 Right Triangle Trigonometry +The Pythagorean Theorem states +where +and +are two sides (legs) of a right triangle and +is the hypotenuse. +Projects +1. +One of the reasons so many formulas in geometry were discovered was because of the importance in finding +measurements of lengths, areas, perimeter, and angles. Find at least five examples of how geometry can be used in +practical applications today. +2. +Who were the Pythagoreans? Why did this society exist? Explore what they did and discuss some of their beliefs. +1120 +10 • Chapter Summary +Access for free at openstax.org + +Chapter Review +Points, Lines, and Planes +Use the figure shown for the following exercises. +1. Find +. +2. Find +. +3. Name the points in the set +. +Angles +4. Given that +and +are parallel lines, solve the angle measurements for all the angles in the figure shown. +5. Find the measure of the vertical angles in the figure shown. +6. Classify the angle as acute, right, or obtuse: +. +7. Use the given figure to solve for the angles. +8. Use the given figure to solve for the angle measurements. +Triangles +9. Use the given figure to find the measure of the unknown angles. +10. Use algebra to find the measure of the angles in the figure shown. +10 • Chapter Summary +1121 + +11. The two triangles shown are congruent by what theorem? +12. The two triangles shown are congruent by what theorem? +13. The two triangles shown are congruent by what theorem? +14. Are the two figures shown similar? +15. Are the two triangles shown similar? +16. Find the scaling factor of the two similar triangles in the given figure. +17. Find the length of +in the figure shown. +Polygons, Perimeter, and Circumference +Identify the polygons. +1122 +10 • Chapter Summary +Access for free at openstax.org + +18. +19. +20. +21. Find the perimeter of a regular hexagon with side length 5 cm. +22. The perimeter of a triangle is 18 in the given figure. Find the length of the sides. +23. What is the measure of an interior angle of a regular heptagon? +24. What is the measure of an interior angle of a regular octagon? +25. Calculate the measure of each interior angle in the figure shown. +26. Find the measure of an exterior angle of a regular pentagon. +27. What is the sum of the measures of the exterior angles of a regular heptagon? +28. Find the circumference of a circle with radius 3.2 cm. +29. Find the diameter of a circle with a circumference of 35.6 in. +Tessellations +30. In what field of mathematics does the topic of tessellations belong? +31. Tessellations can have no _______or _________. +32. What type of transformation moves an object over horizontally by some number of units? +33. What type of transformation results in a mirror image of the shape? +34. Which regular polygons will tessellate the plane by themselves? +35. The sum of the interior angles of the shapes meeting at a vertex is equal to how many degrees? +36. How would you name this tessellation? +10 • Chapter Summary +1123 + +Area +37. What is the area of a triangle with a base equal to 5 cm and a height equal to 12 cm? +38. If the area of a triangle equals +and the base equals 8, what is the height? +39. Find the area of the parallelogram shown. +40. If the area of a trapezoid equals +, +, and the height equals +, find the length of +. +41. Find the area of an octagon if the apothem equals 10 cm and the side length is 12 cm. +42. The area of a circle is +. What is the radius? +43. You want to install a tinted protective shield on this window. How many square feet do you order? +44. Find the area of the shaded region in the given figure. +Volume and Surface Area +45. Find the surface area of a hexagonal prism with side length 5 cm, apothem 3.1 cm, and height 15 cm. +A triangular prism has an equilateral base and side lengths of 15 in, triangle height of 10 in, and prism length of 25 in. +46. Find the surface area of the triangular prism. +47. Find the volume of the triangular prism. +48. Find the volume of a right cylinder with radius equal to 1.5 cm and height equal to 5 cm. +A right cylinder has a radius of 6 cm and a height of 10 cm. +49. Find the surface area of the right cylinder. +50. Find the volume of the right cylinder. +Right Triangle Trigonometry +51. Find the lengths of the missing sides in the figure shown. +1124 +10 • Chapter Summary +Access for free at openstax.org + +52. Find the measure of the unknown side and angle. +Chapter Test +Use the given figure for the following exercises. +1. Find +. +2. Find +. +3. Use the given figure to find the angle measurements. +4. Given that +and +are parallel lines, solve the angle measurements for all the angles in the given figure. +5. Find the angle measurements in the given figure. +10 • Chapter Summary +1125 + +6. The two triangles shown are congruent by what theorem? +7. Find the sum of the interior angles of a regular heptagon. +8. Determine the scaling factor between these two similar triangles. +9. Find the measure of the missing angles in the figure shown. +10. Calculate the perimeter of a regular octagon with a side length of 5 cm. +11. Find the measurement of an interior angle of a regular heptagon. +12. Find the sum of the interior angles of a regular pentagon. +13. Find the measure of an exterior angle of a regular pentagon. +14. Find the circumference of the circle with a radius of 3.5 cm. +15. What are the four transformations that are used to produce tessellations? +16. Find the surface area of the triangular prism shown. +1126 +10 • Chapter Summary +Access for free at openstax.org + +17. Find the volume of the right cylinder in the given figure. +18. Find the missing length in the given figure. +19. Find the missing length in the given figure. +20. Find the length of side +in the given figure. +21. Find the length of side +in the given figure. +22. Find the measure of +in the given figure. +10 • Chapter Summary +1127 + +1128 +10 • Chapter Summary +Access for free at openstax.org + +Figure 11.1 Voters cast their ballots in one of the world’s many democracies. (credit: “Governor Votes Early” by Maryland +GovPics/Flickr, CC BY 2.0) +Chapter Outline +11.1 Voting Methods +11.2 Fairness in Voting Methods +11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem +11.4 Apportionment Methods +11.5 Fairness in Apportionment Methods +Introduction +Suppose a friend asked you, “When did you last vote?” What would your answer be? Maybe you would tell your friend +that the last time you voted was during the last presidential election, or perhaps you would tell your friend that you +prefer not to vote. When thinking about voting, presidential campaigns or advertisements for reelections may come to +mind, but you can cast your vote in many ways. Have you liked a post, followed a creator, friended a stranger, or clicked a +heart online today? In the digital age, it's possible to vote several times a day. Voting systems are not only the machines +that drive every democracy on Earth, but they are also the engines driving social media and many other aspects of life. A +deeper understanding of these voting systems will enhance your ability to successfully engage with the world in which +we live. +In this chapter, you will become one of the founders of the new democratic country of Imaginaria. You have a great +responsibility to the people of this fledgling democracy. You have been tasked with writing the portion of the +constitution that lays out voting procedures. In preparation for this important task, you will explore the various types of +voting systems, from school board elections to Twitter wars. You will see how these types are alike, how they differ, and +how they might be applied in Imaginaria. Most importantly, you will learn about the mathematically inherent advantages +and disadvantages of various voting systems so that you can make informed choices to better the lives of the +Imaginarians. +11 +VOTING AND APPORTIONMENT +11 • Introduction +1129 + +11.1 Voting Methods +Figure 11.2 President Barack Obama votes in the 2012 election. (credit: Pete Souza/White House, Public Domain) +Learning Objectives +After completing this section, you should be able to: +1. +Apply plurality voting to determine a winner. +2. +Apply runoff voting to determine a winner. +3. +Apply ranked-choice voting to determine a winner. +4. +Apply Borda count voting to determine a winner. +5. +Apply pairwise comparison and Condorcet voting to determine a winner. +6. +Apply approval voting to determine a winner. +7. +Compare and contrast voting methods to identify flaws. +Today is the day that you begin your quest to collaborate on the constitution of Imaginaria! Let’s begin by thinking about +the selection of a leader who can serve as president. It seems straightforward; if the majority of citizens prefer a +particular candidate, that candidate should win. But not all votes are decided by a simple majority. Why not? What are +the options? +Majority versus Plurality Voting +When an election involves only two options, a simple majority is a reasonable way to determine a winner. A majority is a +number equaling more than half, or greater than 50 percent of the total. +Let’s take a look at the outcomes of U.S. presidential elections to understand more. Table 11.1 displays the results of the +2000 U.S. presidential election. Like most presidential elections, this election involved more than two options. If that is +the case, is it possible that no single candidate will receive more than half of the votes cast? +Candidate (Party Label) +Popular Vote Total +Al Gore (Democrat) +50,999,897 +George W. Bush (Republican) +50,456,002 +Ralph Nader (Green) +2,882,955 +Patrick J. Buchanan (Reform/Independent) +448,895 +Table 11.1 Results of the Popular Vote for the 2000 U.S. Presidential +Election (source: https://www.fec.gov/introduction-campaign- +finance/election-and-voting-information/federal-elections-2000/ +president2000/) +1130 +11 • Voting and Apportionment +Access for free at openstax.org + +Candidate (Party Label) +Popular Vote Total +Harry Browne (Libertarian) +384,431 +Howard Phillips (Constitution) +98,020 +Other +134,900 +Total: +105,405,100 +Table 11.1 Results of the Popular Vote for the 2000 U.S. Presidential +Election (source: https://www.fec.gov/introduction-campaign- +finance/election-and-voting-information/federal-elections-2000/ +president2000/) +EXAMPLE 11.1 +Majority of Popular Vote in the 2000 U.S. Presidential Election +Refer back to Table 11.1. Did any single candidate secure the majority of popular votes? +Solution +Step 1: Calculate 50 percent of 105,405,100 by multiplying the decimal form of 50 percent, which is 0.50, by 158,394,605: +Step 2: Determine the minimum number of votes needed to have a majority. The minimum number of votes required is +the lowest counting number that is larger than 50 percent of the votes. To have a majority, an individual candidate must +have more than 52,702,550; so, a majority candidate must have 52,702,551 votes or more. +Step 3: Compare the number of votes each candidate received to 52,702,551. According to the data in Table 11.1, none +of the candidates secured a majority. +YOUR TURN 11.1 +1. According to the Cook Political Report, a total of 158,394,605 votes were cast in the 2020 U.S. presidential +election. Of those, 81,281,502 were cast for Joe Biden, 74,222,593 for Donald Trump, and 2,890,510 for other +candidates. Although U.S. presidential elections are not determined by popular vote, did either Joe Biden or +Donald Trump secure a majority of the votes? +Unlike in the 2000 U.S. presidential election, a candidate won the majority of votes in the 2020 election (see Table 11.1). It +is a common occurrence for no single candidate to receive a majority of the votes in an election with more than two +candidates. When this occurs, the candidate with the largest portion of the votes is said to have a plurality. +EXAMPLE 11.2 +Plurality of Popular Vote in the 2000 U.S. Presidential Election +Refer again to Table 11.1. In the 2000 U.S. presidential election, which candidate had a plurality of popular votes? +Solution +Al Gore secured 50,999,897 votes which was more than any other single candidate. Therefore, he had a plurality of the +popular votes. +Your plans for Imaginarian elections will likely include primary elections, or preliminary elections to select candidates for +a principal or general election. Table 11.2 displays the results of the 2018 U.S. Senate Republican primary for Maryland. +11.1 • Voting Methods +1131 + +Top Four Republican Candidates +Votes +Percentage of Party Votes +Cambell, Tony +51,426 +29.22% +Chaffee, Chris +42,328 +24.05% +Grigorian, Christina J. +30,756 +17.48% +Graziani, John R. +15,435 +8.77% +Total Votes +175,981 +100% +Table 11.2 (source: https://ballotpedia.org/ +United_States_Senate_election_in_Maryland_(June_26,_2018_Republican_primary)) +YOUR TURN 11.2 +1. Consider the results of the 2018 U.S. Senate primary for Maryland. Determine which candidate won the primary +for the Republicans (R). Did the candidate win a majority or a plurality of Republican votes? +Consider how election by plurality, not majority, is the most common method of selecting candidates for public office. +WHO KNEW? +The U.S. Electorial College: Winning the Presidency without a Plurality +In the 2000 U.S. presidential election, Al Gore had a plurality of the popular votes, but he did not win the election. +Why? This occurred because the U.S. president and vice president are elected by electors rather than a direct vote by +the citizens. The electors are part of the Electoral College, a body of people representing the states. Why was the +Electoral College created? The Electoral College was created as a compromise between those authors of the U.S. +Constitution who believed Congress should elect the president, and those who believed the citizens should vote +directly. The popular vote was not recorded until the presidential election of 1824. Since then, only five presidents +have been elected without winning a plurality of the popular vote: John Quincy Adams in 1824, Rutherford B. Hayes in +1876, Grover Cleveland in 1888, George Bush in 2000, and Donald Trump in 2016. +Runoff Voting +Has your family ever debated what to have for dinner? Suppose your family is deciding on a restaurant and exactly half +of you want to have pizza but the other half want hamburgers. How do you decide when the result is a tie? You need a +tiebreaker! +Will the new democracy of Imaginaria need tiebreakers? When no candidate satisfies the requirements to win the +election, a runoff election, or second election, is held to determine a winner. +How would runoff voting work in Imaginaria? There are many types of runoff voting systems, which are voting systems +that utilize a runoff election when the first round does not result in a winner. The method for implementing a runoff +election can vary widely, particularly in the criteria used to determine whether a candidate will be on the ballot in the +second election. For example, a two-round system is a runoff voting system in which only the top candidates advance to +the runoff election. In some two-round systems, only the top two candidates are on the second ballot, or it may be any +candidate who secures a certain percentage of the vote will advance. The Hare Method is another runoff voting system +in which only the candidate(s) with the very least votes are eliminated. This can potentially result in several rounds of +runoff elections. +1132 +11 • Voting and Apportionment +Access for free at openstax.org + +EXAMPLE 11.3 +Runoff Election for Condominium Association President +A condominium association elects a new president every two years by a two-round system of voting. If none of the +candidates receive a majority, the association charter states that the top two candidates will be eligible to participate in a +runoff election. In a particular year, five residents were nominated. The results of the first round are given in the table +below. +Candidate +Votes in First Round +Abou +18 +Baiocchi +10 +Campana +5 +Dali +11 +Eugene +4 +1. +Is there a winner based on the first round? Why or why not? +2. +If there is a winner, who won? If there should be a runoff, who will advance to the second round? +Solution +1. +A majority of 48 total votes is required to win. Begin by finding 50 percent of 48, which is calculated as follows: +. A majority is 25 or more. No candidate has a majority, so there is no winner based on the first round. +2. +Abou and Dali advance to the second round. +YOUR TURN 11.3 +The student government bylaws of a particular college require that a new president is elected annually by plurality +voting. In the event of a tie, the bylaws require the candidate(s) with the fewest votes to be removed from the ballot +and a runoff election to be held with the remaining candidates. This process is repeated until a single candidate +receives a plurality and wins the election. The results of two voting rounds are given in the table below. +Candidate +Votes in First Round +Votes in Second Round +Ferguson +158 +168 +Garcia +103 +104 +Hearn +157 +180 +Isaac +123 +123 +Jackman +58 +Eliminated +Kelly +72 +74 +Lim +158 +180 +1. Which two candidates tied in the second round? Does this mean there will be a third election? +11.1 • Voting Methods +1133 + +2. If there was a winner, who won? If there should be a runoff, who must be removed from the ballot? +Steps to Determine Winner by Plurality or Majority Election with Runoff +To determine the winner by plurality or when a majority election with runoff occurs, we take these three steps: +Step 1: If a majority is required to win the election, determine the number of votes needed to achieve a majority. This is +the least whole number greater than 50 percent of the total votes. If a majority is not required, move to Step 2. +Step 2: Count the number of votes for each candidate in the current round of voting. If a single candidate has enough +votes to win a plurality, or a majority as appropriate, then you are done! Otherwise, eliminate a predetermined number +of candidates based on the rules of the election. Elimination conditions may vary. For example, the rules may state that +the candidate(s) with the fewest votes will be eliminated (as in the Hare method), or that only the candidates meeting a +certain threshold will move on (as in a two-round system). Once the appropriate candidates are eliminated, move on to +Step 3. +Step 3: Hold a runoff election. If the runoff is simulated using a list of voter’s preferences, renumber the preferences to +reflect the remaining number of options in such a way that the original order of preference is retained. Then repeat Step +2. +Note: The second and third steps may be repeated as many times as necessary for voting procedures that allow multiple +runoffs. +EXAMPLE 11.4 +Family Dinner Night +The five members of the Chionilis family—Annette, Rene, Seema, Titus, and Galen—have decided to get takeout for +dinner. They are trying to decide on a restaurant. The options are Rainbow China, Dough Boys Pizza, Taco City, or +Caribbean Flavor. They will use majority election with runoffs where the restaurant with the fewest votes is eliminated in +each round. The preferences of each family member are listed by first initial in the table below. An entry of 1 represents +the person’s first choice; 2, their second; and so on. For example, Annette’s second choice is Dough Boys Pizza. +Options +A +R +S +T +G +Rainbow China +1 +3 +3 +1 +3 +Dough Boys Pizza +2 +2 +1 +2 +1 +Taco City +3 +4 +2 +4 +2 +Caribbean Flavor +4 +1 +4 +3 +4 +Use the information in the table to answer the following questions. +1. +Which common type of runoff voting method is this? +2. +List the results of each round of voting based on this information and determine which restaurant was ultimately +chosen. +Solution +1. +the Hare Method +2. +Step 1: Determine the number of votes necessary to have a majority. There are five family members, so 50 percent +of 5 is +. A majority is three or more votes. +Step 2: Count the number of votes for each restaurant in the first election. In a list of voter preferences, the 1s +represent the top choice of each voter, which corresponds to their vote in the first round. +Results of Round 1: +◦ +Rainbow China — 2 votes +◦ +Dough Boys — 2 votes +◦ +Taco City — 0 votes +1134 +11 • Voting and Apportionment +Access for free at openstax.org + +◦ +Caribbean Flavor — 1 vote +No restaurant received a majority. Eliminate Taco City, which has the fewest first place votes: +Options +A +R +S +T +G +Rainbow China +1 +3 +3 +1 +3 +Dough Boys Pizza +2 +2 +1 +2 +1 +Caribbean Flavor +4 +1 +4 +3 +4 +Step 3: Hold a runoff election. In other words, hold a second round. Since we have a list of the voters’ preferences +with the eliminated option removed, we will renumber the preferences as first, second, and third so that we keep +the original order of preference. The result is that we will count the second-place vote of any voter whose first +choice was eliminated. +Options +A +R +S +T +G +Rainbow China +1 +3 +2 +1 +2 +Dough Boys Pizza +2 +2 +1 +2 +1 +Caribbean Flavor +3 +1 +3 +3 +3 +Step 4: Repeat the process from Step 2. Count the number votes for each restaurant in the first-round election. +Since we are using a list of preferences, we need to count the number of 1s received by each restaurant. +Results of Round 2: +◦ +Rainbow China — 2 votes +◦ +Dough Boys — 2 votes +◦ +Caribbean Flavor — 1 vote +No single restaurant has three votes. Eliminate Caribbean Flavor, which has the fewest first place votes. +OPTIONS +A +R +S +T +G +Rainbow China +1 +3 +2 +1 +2 +Dough Boys Pizza +2 +2 +1 +2 +1 +Step 5: Repeat the process from Step 3. Hold another runoff election. This will be Round 3. Renumber the voters' +preferences as first and second this time. +Options +A +R +S +T +G +Rainbow China +1 +2 +2 +1 +2 +Dough Boys Pizza +2 +1 +1 +2 +1 +11.1 • Voting Methods +1135 + +Step 6: Repeat the process from Step 2 one last time. Count the number of first place votes for each remaining +restaurant. +Results of Round 3: +◦ +Rainbow China — 2 votes +◦ +Dough Boys — 3 votes +Determine whether any one choice has a majority. Yes! Dough Boys has three votes, so it is the winner! +YOUR TURN 11.4 +1. There are six members on the board of a Parent Teacher Association (PTA) at a local elementary school: the +president (P), the vice president (V), the recording secretary (R), the liaison to the administration (L), the treasurer +(T), and the chief fundraiser (C). The board must decide which equipment to purchase for the classrooms with +moneys from their annual fundraisers. The preferences of the board members are shown in the table below. +P +V +R +L +T +C +Option A +3 +4 +1 +3 +4 +1 +Option B +1 +3 +2 +1 +3 +2 +Option C +2 +2 +3 +2 +1 +4 +Option D +4 +1 +4 +4 +2 +3 +The board uses plurality method and a runoff in the event of a tie, such that the option(s) with the least votes will +be eliminated in each round. Which option will be chosen? +Ranked-Choice Voting +In Example 11.4 and Your Turn 11.4, you were given a list that ordered each voter’s preferences. This ordering is called a +preference ranking. A ballot in which a voter is required to give an ordering of their preferences is a ranked ballot, and +any voting system in which a voter uses a ranked ballot is referred to as ranked voting. +The vote for the Academy Awards uses a ranked ballot. The table below provides an example of a ranked ballot for the +2020 Academy Award nominees for Best Director. +Candidate for Best Director +Rank top choice as 1, next choice as 2, and so on. +Martin Scorsese, The Irishman +1 +2 +3 +4 +5 +Todd Phillips, Joker +1 +2 +3 +4 +5 +Sam Mendes, 1917 +1 +2 +3 +4 +5 +Quentin Tarantino, Once Upon a Time in Hollywood +1 +2 +3 +4 +5 +Bong Joon-ho, Parasite +1 +2 +3 +4 +5 +As you decide on the voting methods that will be used in your new democracy, budget must be a consideration. You +1136 +11 • Voting and Apportionment +Access for free at openstax.org + +might consider a particular type of ranked voting called ranked-choice voting (RCV), which simulates a series of runoff +elections without the usual time and expense involved when voters must repeatedly return to the polls, like we did in +Example 11.4. +The method of ranked-choice voting (RCV), also called instant runoff voting (IRV), is a version of the Hare Method, +using preference ranking so that, if no single candidate receives a majority, the least popular selections can be +eliminated and the results can be recounted, without the need for more elections. +Ranked voting can be confused with ranked-choice voting, but ranked voting is a more general category which +includes ranked-choice voting and several other voting methods. +As we explore examples of ranked voting, we will summarize the voters’ preference rankings using a table in which the +top row shows the number of ballots that ranked the options in the same order. Let’s practice interpreting the +information in this type of table. +EXAMPLE 11.5 +Interpreting the Sample Preference Summary +Refer to the table below containing voters’ preference rankings to answer the following questions. +Number of Ballots +100 +200 +150 +75 +Option A +1 +4 +3 +4 +Option B +2 +3 +4 +2 +Option C +4 +2 +1 +1 +Option D +3 +1 +2 +3 +1. +How many voters ranked the options in the following order: Option A in fourth place, Option B in second place, +Option C in first place, and Option D in third place? +2. +How many ballots in total were collected? +3. +How many voters indicated that Option C was their first choice? +Solution +1. +The column farthest to the right displays this ordering. The top entry in this column is 75; so, there were 75 voters +who ranked the options in this way. +2. +The sum of the top row gives the total number of ballots collected: +. So, there were 525 +ballots collected. +3. +In the Option C row, there are two entries of 1 which indicate a first choice for that option. These occur in the last +two columns. The sum of the top entries in these columns is +. So, 225 voters indicated Option C as +their first choice. +YOUR TURN 11.5 +A kindergarten class votes on their favorite colors using a ranked ballot. Use the results in the following table to +answer the questions. +11.1 • Voting Methods +1137 + +Number of Ballots +4 +6 +4 +7 +Red +2 +6 +2 +5 +Blue +1 +2 +1 +4 +Green +6 +5 +5 +2 +Yellow +5 +4 +6 +3 +Purple +4 +3 +4 +1 +Pink +3 +1 +3 +6 +1. How many students voted in total? +2. How many students voted for yellow as their favorite color? +3. How many students in the class indicated that blue was their favorite color while green was their least favorite +color? +Now that we’ve covered how to read a summary of preference rankings, let’s practice using the ranked-choice method to +determine the winner of an election. Recall that ranked-choice voting is still the Hare Method where the candidate with +the very least number of votes is eliminated each round until a majority is attained. The difference here is that the voters +have completed a ranked ballot, so they don't have to visit the polls multiple times. Here are the steps for ranked-choice +voting. +Steps to Determine Winner by Ranked-Choice Voting +To determine the winner when ranked-choice voting occurs, we take these three steps: +Step 1: Determine the number of votes needed to achieve a majority. This is the least whole number greater than 50 +percent of the total votes. +Step 2: Count the number of first place votes for each candidate. If a candidate has a majority, that candidate wins the +election and we are done! Otherwise, eliminate the candidate(s) with the fewest votes and complete Step 3. +Step 3: Reallocate the votes to the remaining candidates, and repeat Step 2. +Be methodical to avoid arithmetic errors. Make sure that each time you count the number of first place votes they +sum to the number of ballots. +VIDEO +How Does Ranked-Choice Voting Work? (https://openstax.org/r/ranked-choice_voting1) +EXAMPLE 11.6 +Most Popular Color in Kindergarten +Let’s review the kindergarten class color preferences again, and this time determine which color would be selected based +on these results using the ranked-choice method. +1138 +11 • Voting and Apportionment +Access for free at openstax.org + +Number of Ballots +4 +6 +4 +7 +Red +2 +6 +2 +5 +Blue +1 +2 +1 +4 +Green +6 +5 +5 +2 +Yellow +5 +4 +6 +3 +Purple +4 +3 +4 +1 +Pink +3 +1 +3 +6 +Solution +Step 1: Determine whether any candidate received a majority. There were 21 ballots. Fifty percent of 21 is +. A majority is 11. +Step 2: Count the number of first place votes for each candidate. If a candidate has a majority, that candidate wins the +election. Otherwise, eliminate the candidate(s) with the fewest votes. +• +Red: 0 +• +Blue: +• +Green: 0 +• +Yellow: 0 +• +Purple: 7 +• +Pink: 6 +Notice that +, which is the total number of ballots. Confirming this helps to catch any arithmetic or +counting errors. No candidate has a majority with 11 or more votes. We must eliminate red, green, and yellow which had +the fewest votes with 0 each. The remaining votes that must be counted for Round 2 are given in the table below. +Number of Ballots +4 +6 +4 +7 +Blue +1 +2 +1 +4 +Purple +4 +3 +4 +1 +Pink +3 +1 +3 +6 +Step 3: Reallocate the votes to the remaining candidates. We can do this by numbering the choices as 1, 2, and 3 in such +a way that the order of preference is retained as seen in the table below: +Number of Ballots +4 +6 +4 +7 +Blue +1 +2 +1 +2 +Purple +3 +3 +3 +1 +Pink +2 +1 +2 +3 +Step 4: Repeat the process from Step 2. Count the number of first place votes for each candidate. If a candidate has a +majority, that candidate wins the election. Otherwise, eliminate the candidate(s) with the fewest votes. +11.1 • Voting Methods +1139 + +• +Blue: +• +Purple: 7 +• +Pink: 6 +Confirm that +. Great! No candidate has 11 or more votes. We must eliminate pink which had the fewest +votes with 6. The remaining votes that must be counted for Round 2 are shown in the table below. +Number of Ballots +4 +6 +4 +7 +Blue +1 +2 +1 +2 +Purple +3 +3 +3 +1 +Step 5: Repeat the process from Step 3. Reallocate the votes to the remaining candidates. We can do this by numbering +the choices as 1 and 2; +Number of Ballots +4 +6 +4 +7 +Blue +1 +1 +1 +2 +Purple +2 +2 +2 +1 +Step 6: Repeat the process from Step 2 one last time. Count the number of first place votes for each candidate. If a +candidate has a majority, that candidate wins the election. Otherwise, eliminate the candidate(s) with the fewest votes. +• +Blue: +• +Purple: 7 +Blue has a majority and wins the election! +VIDEO +Determine Winner of Election by Ranked-Choice Method (aka Instant Runoff) (https://openstax.org/r/ranked- +choice_voting2) +YOUR TURN 11.6 +1. Suppose that 58 Star Wars fans were asked to vote for their favorite Star Wars character. They were given a +ranked ballot, and the results are shown in the following table. Use ranked-choice voting to determine the +winner. +Number of Ballots +7 +6 +10 +8 +4 +5 +6 +7 +2 +3 +Han Solo +3 +5 +1 +3 +2 +4 +2 +2 +1 +4 +Princess Leia +2 +1 +3 +6 +4 +3 +3 +1 +3 +2 +Luke Skywalker +6 +6 +6 +1 +5 +5 +5 +4 +6 +1 +Chewbacca +4 +4 +5 +4 +6 +1 +6 +5 +4 +5 +1140 +11 • Voting and Apportionment +Access for free at openstax.org + +Number of Ballots +7 +6 +10 +8 +4 +5 +6 +7 +2 +3 +Yoda +1 +2 +4 +2 +3 +2 +1 +3 +2 +3 +R2-D2 +5 +3 +2 +5 +1 +6 +4 +6 +5 +6 +Borda Count Voting +Ranked-choice voting is one type of ranked voting that simulates multiple runoffs based on ranked ballots. Another type +of ranked voting is the Borda count method, which uses ranked ballots that award candidates points corresponding to +the number of candidates ranked lower on each ballot. +To understand how this works, let’s review the favorite colors of our kindergarten class from the table below. Let’s focus +on the votes represented by the first column of the preference summary. +Number of Ballots +4 +6 +4 +7 +Red +2 +6 +2 +5 +Blue +1 +2 +1 +4 +Green +6 +5 +5 +2 +Yellow +5 +4 +6 +3 +Purple +4 +3 +4 +1 +Pink +3 +1 +3 +6 +Each student had six options. This first column tells us that four students ranked blue as their first choice, red as their +second choice, pink as their third choice, purple as their fourth choice, yellow as their fifth choice, and green as their +sixth choice. Blue was ranked higher than +other colors. For each of the four students who completed their +ballot in this way, blue would receive five points. Since there were four ballots with this ordering, blue would receive +points from the first column. To determine the total points for each candidate, we have to find the sum of the +points they received in each column. +To determine the winner of a contest using the Borda count method, we must compare total number of points earned by +each candidate. The candidate with the most points is the winner. Each row of the preference summary corresponds to a +single candidate. To find the number of points received by a particular candidate in the preference summary, or their +Borda score, we will need to focus on the row in which that candidate appears. +Before we practice determining the winner of a Borda count election, let’s examine how to find the Borda score for a +single candidate. +EXAMPLE 11.7 +Most Popular Color in Kindergarten Revisited +Let’s review the ballots from the kindergarten class again, as shown in the table below. This time, let’s determine the +Borda score received by the color purple. +11.1 • Voting Methods +1141 + +Number of Ballots +4 +6 +4 +7 +Red +2 +6 +2 +5 +Blue +1 +2 +1 +4 +Green +6 +5 +5 +2 +Yellow +5 +4 +6 +3 +Purple +4 +3 +4 +1 +Pink +3 +1 +3 +6 +Solution +Step 1: Find the number of points received by the candidate in each column. +Column 1: +Column 2: +Column 3: +Column 4: +Step 2: Find the sum of the points received in each column. This is the total number of points received by this candidate: +This process can also be combined into one step as shown here. +Purple received 69 points in this election. +When calculating a Borda score in one step, be careful to use the correct order of operations. Perform the +subtraction inside each pair of parentheses first, then perform each multiplication, and then perform each addition. +YOUR TURN 11.7 +1. Consider the color preferences of the kindergarten class once more. Using the Borda count method, determine +the total number of points the color blue received. +Now let’s determine the winner of an election by comparing the Borda scores for each of the candidates. +EXAMPLE 11.8 +Determine the Winner by Two Ranked Voting Methods +Use the table below, which displays a sample preference summary, to answer the questions that follow. +1142 +11 • Voting and Apportionment +Access for free at openstax.org + +Number of Ballots +95 +90 +110 +115 +Option A +4 +4 +1 +1 +Option B +2 +2 +2 +2 +Option C +3 +1 +3 +4 +Option D +1 +3 +4 +3 +1. +Use the ranked-choice voting method to determine the winner of the election. +2. +Use the Borda count method to determine the winner of the election. +Solution +1. +Step 1: Determine the number of votes needed to achieve a majority. The number of ballots is +. Fifty percent of 410 is +. So, 206 votes or more is a majority. +Step 2: Count the number of first place votes for each candidate. +◦ +Option A has +◦ +Option B has 0 +◦ +Option C has 90 +◦ +Option D has 95 +Since Option A has a majority, Option A is the winner by the ranked-choice method. +2. +The Borda scores would be: +◦ +Option A: +◦ +Option B: +◦ +Option C: +◦ +Option D: +Since Option B has a Borda score of 820 points, Option B is the winner by the Borda count method. +YOUR TURN 11.8 +Answer the following questions using the table below, which summarizes Imaginarian voter preferences. +Number of Ballots +12 +19 +27 +29 +31 +21 +Candidate A +1 +1 +2 +2 +3 +3 +Candidate B +2 +3 +1 +3 +1 +2 +Candidate C +3 +2 +3 +1 +2 +1 +1. Use the ranked-choice voting method to determine the winner of the election. +2. Use the Borda count method to determine the winner of the election. +3. Compare the results of the two methods. Did the same candidate win? What observations can you make +about the results? +The Borda count method may seem too complicated to even consider using for Imaginaria, but each voting method has +its own pros and cons. The Borda count method, for example, favors compromise candidates over divisive candidates. A +compromise candidate is not the first choice of most of the voters, but is more acceptable to the population as a whole +11.1 • Voting Methods +1143 + +than the other candidates. A divisive candidate is simultaneously the first choice of a large portion of the voters and the +last choice of another large portion of the voters. +In Example 11.8, Candidate A was ranked first by 225 voters, but was ranked last by 185 voters. No voters ranked +Candidate A as second or third. It appears that, although Candidate A had the majority of first place votes, there was a +significant minority who strongly disliked them. Candidate A was a divisive candidate. Candidate B, on the other hand, +was the second choice of every voter, making Candidate B a good compromise. The Borda count method chose +Candidate B, a compromise candidate, that was more acceptable to the population as a whole. This scenario is cited by +both opponents and proponents of the Borda count method. +VIDEO +Determine Winner of Election by Borda Count Method (https://openstax.org/r/Borda_count_method) +Pairwise Comparison and Condorcet Voting +We have discussed two kinds of ranked voting methods so far: ranked-choice and Borda count. A third type of ranked +voting is the pairwise comparison method, in which the candidates receive a point for each candidate they would beat +in a one-on-one election and half a point for each candidate they would tie. If one candidate earns more points than the +others, then that candidate wins. This method is one of several Condorcet voting methods, which are methods in which +candidates are ranked and then compared pairwise to each other, a candidate having to beat all others in order to win. +These methods vary in the way candidates are scored, and there is not always a clear winner. A candidate who wins each +possible pairing is known as a Condorcet candidate. These terms are named after the Marquis de Condorcet, a French +philosopher and mathematician who preferred the pairwise comparison method to the Hare method and made public +arguments in its favor. +PEOPLE IN MATHEMATICS +Marquis de Condorcet +Condorcet voting methods are named for the Marquis de Condorcet, a French philosopher and mathematician known +for, among other accomplishments, writing “Sur l'admission des femmes au droit de Cité” (“On the Admission of +Women to the Rights of Citizenship”), in 1789, the first published essay on the political rights of women. +For more details visit this Web site. +If you include a Condorcet voting method in the constitution of Imaginaria, the election supervisors may want to use a +pairwise comparison matrix like the one in Figure 11.3. It’s a tool used to list the number of wins associated with each +pairing of two candidates. Each candidate will receive a point for each win and a half a point for each tie. Each pairing is +listed twice, once for the number of wins of a candidate over a particular challenger and once for the number of wins of +the challenger over that candidate. +Figure 11.3 Pairwise Comparison Matrix for Three Candidates +Steps to Determine a Winner by Pairwise Comparison Method Using a Matrix +To determine the winner when the pairwise comparison method is used, we take these three steps: +Step 1: On the matrix, indicate a losing matchup by crossing out a box, +, and tie match ups by drawing a slash +1144 +11 • Voting and Apportionment +Access for free at openstax.org + +through the box, +. +Step 2: Award each candidate 1 point for a win, half a point for a tie, and 0 points for a loss. +Step 3: Identify the winner, which is the candidate with the most points. +VIDEO +Determine Winner of Election by Using the Pairwise Comparison Method (https://openstax.org/r/ +pairwise_comparison_method) +Before you decide on the pairwise comparison method for Imaginaria, review what’s involved in constructing a pairwise +comparison matrix from a summary of ranked ballots. Then we can use the matrix to determine the winner of the +election. Does the winner using the Borda method still win? +EXAMPLE 11.9 +Construct and Use a Pairwise Comparison Matrix +Consider the summary of ranked ballots shown in the table below. Determine the winner of an election using the +pairwise comparison method. +Number of Ballots +95 +90 +110 +115 +Option A +4 +4 +1 +1 +Option B +2 +2 +2 +2 +Option C +3 +1 +3 +4 +Option D +1 +3 +4 +3 +1. +Construct a pairwise comparison matrix for the sample summary of ranked ballots in the table above. +2. +Use the pairwise comparison method to determine a winner. +3. +Recall that in Example 11.8, Candidate A won by the ranked-ballot method, and Candidate B won by the Hare +method. Did the same candidate win using the pairwise comparison method? +4. +Is the winner a Condorcet candidate? +Solution +There are four candidates on the ballots. We will need a row and a column for each candidate in addition to the +headings, so we will draw a five by five matrix. +Figure 11.4 Pairwise Comparison Matrix for Four Candidates +1. +Step 1: Refer to Figure 11.4 to determine the values that belong in each cell. +11.1 • Voting Methods +1145 + +◦ +A over B: A is preferred to B in columns 3 and 4. So, A scores +points. +◦ +A over C: A is preferred to C in columns 3 and 4. So, A scores 225 points again. +◦ +A over D: Similarly, A scores 225 points. +◦ +B over A: B is preferred to A in columns 1 and 2. So, B scores +points. +Step 2: Continuing in this way, we complete the pairwise comparison matrix, as shown in Figure 11.5. +Figure 11.5 Pairwise Comparison Matrix for Four Candidates with Vote Counts +2. +Step 1: Losing pairings are crossed off with an +. In the event of a tie, we will draw a slash, +. +Step 2: Determine the number of points for each candidate by analyzing their row of wins. Each win is 1 point, each +loss, +, is 0 points, and each tie, +, is half a point. Construct an additional column for each candidate’s points. +Figure 11.6 Pairwise Comparison Matrix for Four Candidates with Pairwise Winners and Points Column Added +Step 3: The winner by the pairwise comparison method is Option A with 3 points. +3. +Option A was not the winner by the Hare method. +4. +The winner, Option A, is a Condorcet candidate because Option A won each pairwise comparison. +1146 +11 • Voting and Apportionment +Access for free at openstax.org + +Notice that the pairwise vote totals are not used to determine the points. Vote totals are only used to determine a +win or a loss. Avoid the common error of adding the values in each row to get the points. +YOUR TURN 11.9 +According to Variety magazine, there were 8,469 eligible Oscar voters in 2020. To make our matrix easier to work +with, we've rounded this number up to 8,700. We'd never do this in a real voting situation. Suppose that the voter +preferences for the ballot for Best Director had been as shown in the table below. +Number of Ballots +2,400 +2,100 +1,900 +1,200 +1,100 +Martin Scorsese, The Irishman +3 +1 +4 +5 +2 +Todd Phillips, Joker +1 +4 +5 +2 +4 +Sam Mendes, 1917 +5 +5 +3 +4 +5 +Quentin Tarantino, Once Upon a Time in Hollywood +4 +2 +1 +3 +3 +Bong Joon-ho, Parasite +2 +3 +2 +1 +1 +Refer to the table to answer each question. +1. Construct a pairwise comparison matrix for the Best Director ballots. +2. Who is the “Best Director” according to the pairwise comparison method? +3. Is the winner a Condorcet candidate? +Three Key Questions +Before you decide if you want to use the pairwise comparison method for Imaginarian elections, let’s consider three +questions that might affect your decision. +I. +Is there always a winner? +II. +If there is a winner, is the winner always a Condorcet candidate? +III. +If there is a Condorcet candidate, does that candidate always win? +Let’s think about why these questions might be important to you if you chose the pairwise comparison method. First, if +no candidate meets the criteria to win an election, you will need a backup plan such as a runoff election. Second, if the +winner is not a Condorcet candidate, then there is at least one candidate who beat the winner in a pairwise matchup and +the supporters of that candidate might question the validity of the election. Finally, if there is a Condorcet candidate who +beat every other candidate in a pairwise matchup, it is reasonable to conclude that it would be unfair for anyone else to +win. The rest of the examples in this section should illustrate these key concepts. +EXAMPLE 11.10 +Rock, Paper, Scissors by Pairwise Comparison +Suppose that three people are playing the game Rock, Paper, Scissors. On the count of three, each person shows a hand +signal for rock, paper, or scissors. Each hand signal beats another hand signal. The group keeps having a tie because +Person A always picks rock, Person B always picks paper which beats rock, and Person C always picks scissors which +beats paper, and is beaten by rock! This leads to a disagreement about which choice is best. They decide to use the +pairwise comparison method determine the winner. Their preference rankings are given in the following table. +11.1 • Voting Methods +1147 + +Voters +A +B +C +Rock (R) +1 +3 +2 +Paper (P) +2 +1 +3 +Scissors (S) +3 +2 +1 +Solution +Construct the comparison matrix: +Figure 11.7 Rock, Paper, Scissors Pairwise Comparison Matrix +There is a tie! There is no winner. +Example 11.10 illustrates the answer to the first key question. The pairwise comparison method does not always result in +a winner. For example, much like the game of Rock, Paper, Scissors, it is possible for a cyclic pattern to emerge in which +each candidate beats the next until the last candidate who beats the first. +Figure 11.8 Rock, Paper, Scissors Cyclic Outcome +YOUR TURN 11.10 +1. A pairwise comparison matrix is given. Determine the winner by the pairwise comparison method. If there is not +a winner, explain why. If there is a winner, tell whether the winner is a Condorcet candidate. +1148 +11 • Voting and Apportionment +Access for free at openstax.org + +A Pairwise Comparison Matrix +Now, you have the answer to the second key question. The pairwise comparison matrix in YOUR TURN 11.10 is an +example of a scenario where a winner is not a Condorcet candidate. +The answer to the third question is not as clear. If there is a Condorcet candidate, does that candidate win? So far, we +have not come across a contradictory example where the Condorcet candidate didn't win, but we cannot know with +certainty that it is not possible by looking at examples. Instead, we will need to use some reasoning. Let’s review some +particular cases of elections with a certain number of candidates, and then we will try to generalize the scenario to an +election with +candidates. +EXAMPLE 11.11 +Does the Condorcet Candidate Win? +1. +Suppose there is an election with five candidates—A, B, C, D, and E—and that Candidate C is a Condorcet candidate. +How many points did Candidate C win? +2. +What is the greatest number of points that any one of the other candidates could win? +3. +Is it possible for Candidate C to lose or tie? +Solution +1. +In any pairwise election with five candidates, each candidate must compete against four other candidates. It follows +that the most points a single candidate can win is four points, which would occur if the candidate won every +matchup. As a Condorcet candidate, Candidate C won all the pairwise matchups against Candidates A, B, D, and E, +earning four points. +2. +The rest of the candidates lost to Candidate C. The most points a particular candidate could win if they won +matchups with each of the other three candidates is three points. +3. +Since Candidate C has four points and the rest of the candidates have three points or less, Candidate C is the winner. +Therefore, it is not possible for Candidate C to tie or lose. +YOUR TURN 11.11 +Suppose there is an election with 26 candidates, A through Z, and that Candidate C is a Condorcet candidate. +1. How many points did Candidate C win? +2. What is the greatest number of points that any one of the other candidates could win? +3. Is it possible for Candidate C to lose or tie? +11.1 • Voting Methods +1149 + +Let’s consider a general case where there are +candidates. One of the candidates is a Condorcet candidate. Since the +Condorcet candidate wins all matchups, the Condorcet candidate wins +points. Since each of the other candidates +lost to the Condorcet candidate, the most a single candidate could win is +. Since the Condorcet candidate won +points and each other candidate won +points or fewer, the Condorcet candidate is the winner. You have your answer +to the third key question! If there is a Condorcet candidate, that candidate is always the winner. +Approval Voting +The last type of voting system you will consider for your budding democracy is an approval voting system. In this +system, each voter may approve any number of candidates without rank or preference for one over another (among the +approved candidates), and the candidate approved by the most voters wins. This voting system has aspects in common +with plurality voting and Condorcet voting methods, but it has characteristics that distinguish it from both. An approval +voting ballot lists the candidates and provides the option to approve or not approve each candidate. +The term “approval voting” was not used until the 1970s Brams, Steven J.; Fishburn, Peter C. (2007), Approval Voting, +Springer-Verlag, p. xv, ISBN 978-0-387-49895-9, although its use has been documented as early as the 13th century +(Brams, Steven J. (April 1, 2006). The Normative Turn in Public Choice (PDF) (Speech). Presidential Address to Public +Choice Society. New Orleans, Louisiana.) Approval voting has the appeal of being simpler than ranked voting methods. It +also allows an individual voter to support more than one candidate equally. This has appeal for those who do not want a +split vote among a few mainstream candidates to lead to the election of a fringe candidate. It also has appeal for those +who want an underdog to have a chance of success because voters will not worry about wasting their vote on a +candidate who is not believed likely to win. +EXAMPLE 11.12 +Rock, Paper, Scissors, Lizard, Spock +Suppose that Person A/B/C were just about to give up on their game of Rock, Paper, Scissors when they were joined by +Person D who reminded them that their updated version, Rock, Paper, Scissors, Lizard, Spock was a far superior game +with the added rules that Lizard eats Paper, Paper disproves Spock, Spock vaporizes Rock, Rock crushes Lizard, Lizard +poisons Spock, Spock smashes Scissors, and Scissors decapitates Lizard. +Figure 11.9 Rock, Paper, Scissors, Lizard, Spock Dominance +Person D encourages their friends to hold a new election. This time, for the sake of simplicity, the group decides to use +approval voting to determine the best move in the game. The summary of approval ballots for Rock, Paper, Scissors, +Lizard, Spock is given in the table below. +Voters +A +B +C +D +Rock +Yes +No +No +No +Paper +No +Yes +No +No +Scissors +No +No +Yes +No +1150 +11 • Voting and Apportionment +Access for free at openstax.org + +Voters +A +B +C +D +Lizard +No +No +No +Yes +Spock +Yes +Yes +Yes +Yes +Solution +Count the number of approval votes for each candidate by counting the number of “Yes” votes in each row of the table. +• +Rock: 1 +• +Paper: 1 +• +Scissors: 1 +• +Lizard: 1 +• +Spock: 4 +Spock is the winning candidate, approved by four voters! +YOUR TURN 11.12 +1. The Chionilis family is trying to decide on a restaurant again, but now they don’t want to deal with multiple +runoffs or even ranking. They will use the approval voting method shown in the following table. Each family +member will approve their top two choices. Rainbow China won when multiple runoffs were used. Find the +winner for tonight’s dinner. +Options +A +R +S +T +G +E +M +D +Rainbow China +Yes +No +No +Yes +No +No +Yes +Yes +Dough Boys Pizza +Yes +Yes +Yes +Yes +No +Yes +No +Yes +Taco City +No +No +Yes +No +Yes +Yes +No +No +Caribbean Flavor +No +Yes +No +No +Yes +No +Yes +No +EXAMPLE 11.13 +The Chionilis Family Is Hungry Again! +The eight members of the Chionilis family—Annette, Rene, Seema, Titus, Galen, Elena, Max and Demitri—have another +decision to make. Approval voting worked out nicely the last time. They are going to use it again, but this time, Annette, +Rene, Seema, and Galen are feeling a little indecisive. They can't narrow their choice down to two. They will approve their +three top choices, but the other family members will only approve two. These choices are reflected in the following table. +Determine the restaurant that will be chosen. +Options +A +R +S +T +G +E +M +D +Rainbow China +Yes +Yes +Yes +Yes +Yes +No +Yes +Yes +Dough Boys Pizza +Yes +Yes +Yes +Yes +No +Yes +No +Yes +11.1 • Voting Methods +1151 + +Options +A +R +S +T +G +E +M +D +Taco City +Yes +No +Yes +No +Yes +Yes +No +No +Caribbean Flavor +No +Yes +No +No +Yes +No +Yes +No +Solution +Count the number of approval votes for each restaurant by counting the number of “Yes” votes in each row. +• +Rainbow China: 7 +• +Dough Boys Pizza: 6 +• +Taco City: 4 +• +Caribbean Flavor: 3 +This time, Rainbow China won! +YOUR TURN 11.13 +1. What would be the outcome of the election if every member of the Chionilis family approved their top three +choices from the table below? +Options +A +R +S +T +G +Rainbow China +1 +3 +3 +1 +3 +Dough Boys Pizza +2 +2 +1 +2 +1 +Taco City +3 +4 +2 +4 +2 +Caribbean Flavor +4 +1 +4 +3 +4 +Compare and Contrast Voting Methods to Identify Flaws +Wow! We have covered a lot of options for the voting methods. Now, you need to decide which one is best for +Imaginaria. Imaginarians might consider characteristics of certain voting systems desirable and others undesirable. In +some cases, voters may consider these undesirable traits to be flaws in a voting system that are significant enough to +motivate them to reject that system. If you are feeling a bit overwhelmed by this decision, maybe it would help to read +about the experiences of others who have faced similar questions. +Consider the 2000 U.S. presidential election in which Green Party candidate Ralph Nader and Reform Party candidate Pat +Buchanan were on the ballet running against the mainstream candidates, Democrat Al Gore and Republican George W. +Bush. The voting results for Florida are given in Table 11.3. +Candidate +Party +Votes +Percentage +(G) George W. Bush +Republican +2,912,790 +48.85% +(A) Al Gore +Democrat +2,912,253 +48.84% +Table 11.3 Florida Results in the 2000 U.S. Presidential Election +(source: https://www.fec.gov/resources/cms-content/documents/ +FederalElections2000_PresidentialGeneralElectionResultsbyState.pdf) +1152 +11 • Voting and Apportionment +Access for free at openstax.org + +Candidate +Party +Votes +Percentage +(R) Ralph Nader +Green +97,488 +1.63% +(P) Pat Buchanan +Reform +17,484 +0.29% +(H) Harry Brown +Libertarian +16,415 +0.28% +(O) 7 Other Candidates +Other +6,680 +0.11% +Total +5,963,110 +Table 11.3 Florida Results in the 2000 U.S. Presidential Election +(source: https://www.fec.gov/resources/cms-content/documents/ +FederalElections2000_PresidentialGeneralElectionResultsbyState.pdf) +In more than one state, Buchanan was able to split the Republican vote enough to allow Gore to win that state. Nader +split the Democrat vote in Florida and New Hampshire by enough votes to prevent Gore from winning those states. Had +Gore won either state, he would have had enough electoral votes to win the election. Instead, Bush won. This is an +example of a flaw in the plurality system of voting: the spoiler. +A spoiler is a less popular candidate who takes votes from a more popular candidate with similar positions, swinging the +race to another candidate with vastly different views that they would not support. This encourages voters not to vote for +the candidate that they perceive to be the best, but instead for the candidate they can live with who they perceive to +have a better chance of winning. Some voters may prefer a method such as approval voting, which does not have this +trait in common with plurality voting. +EXAMPLE 11.14 +The Spoiler Controversy +Because the vote counts for George W. Bush and Al Gore differed by only 537 votes, many Democrats blamed Ralph +Nader and the Green Party for their loss. Let’s consider how the election results might have differed if the approval +voting method had been used. +Use Table 11.3 and the following assumptions to extrapolate the results of an approval method election: +• +100 percent of Pat Buchanan supporters would approve George W. Bush. +• +100 percent of Ralph Nader supporters would approve Al Gore. +• +72 percent of Libertarians would approve George W. Bush. +• +28 percent of Libertarians would approve Al Gore (as was roughly the known percentage at the time according to +the Cato Institute). +• +50 percent of the supporters of other candidates would approve George Bush while 50 percent would approve Al +Gore. +Solution +Step 1: Create a summary of approval ballots based on the given assumptions. For the Libertarian candidate, 72 percent +of 16,415 of the votes is +and 28 percent is +. For the other candidates, 50 +percent of the votes is +. +Number of Votes +2,912,790 +(G) +2,912,253 +(A) +97,488 +(R) +17,484 +(B) +11,819 +(72% H) +4,596 +(28% H) +3,340 +(50% O) +3,340 +(50% O) +(G) George W. Bush +Yes +No +No +Yes +Yes +No +No +Yes +(A) Al Gore +No +Yes +Yes +No +No +Yes +Yes +No +11.1 • Voting Methods +1153 + +Number of Votes +2,912,790 +(G) +2,912,253 +(A) +97,488 +(R) +17,484 +(B) +11,819 +(72% H) +4,596 +(28% H) +3,340 +(50% O) +3,340 +(50% O) +(R) Ralph Nader +No +No +Yes +No +No +No +No +No +(P) Pat Buchanan +No +No +No +Yes +No +No +No +No +(H) Harry Brown +No +No +No +No +Yes +Yes +No +No +(O) 7 Other Candidates +No +No +No +No +No +No +Yes +Yes +Step 2: Count the number of approval votes for each candidate. +• +George W. Bush: +• +Al Gore: +• +Ralph Nader: +• +Pat Buchanan: +• +Harry Brown: +• +Other Candidates: +In this scenario, Al Gore is the winner. +YOUR TURN 11.14 +1. Extrapolate the results of an approval method election using Table 11.3, the assumptions from Example 11.14, +and the additional assumptions that the supporters of Al Gore would all approve Ralph Nader and that half of +the supporters of George Bush would approve Pat Buchanan while half would approve Harry Brown. +The results in Example 11.14 and Your Turn 11.14 highlight one of the characteristics of approval voting. Ralph Nader +moved up from a distant third place finish to a close second place finish when Al Gore’s supporters approved him on +their ballots. In this way, fringe candidates have a better chance of winning, which some voters consider a flaw but +others consider a benefit. +Another aspect of approval voting systems that is a concern to many voters is that candidates in approval elections +might encourage their loyal supporters to approve them and only them to avoid giving support to any other candidate. +If this occurred, the election in effect becomes a traditional plurality election. This is a flaw that cannot occur in an +instant runoff system since all candidates are ranked. +EXAMPLE 11.15 +Three Habitable Planets +In the future, humans have explored distant solar systems and found three habitable planets which could be colonized. +Since it will take all available resources to colonize one planet, humans must agree on the planet. Planet A has the most +comfortable climate and most plentiful resources, but it is the farthest from Earth making travel to the planet a +challenge. Planet B is half the distance but will require more resources to make comfortable. Planet C is the least suitable +of the three and terraforming will be required, but it is close enough to make travel between Earth and Planet C possible +on a more regular basis. The table below provides the voter preferences for the colonization of each planet. +1154 +11 • Voting and Apportionment +Access for free at openstax.org + +Percentage of Voters +45% +15% +40% +Planet A +1 +3 +3 +Planet B +2 +1 +2 +Planet C +3 +2 +1 +If the entire population were able to vote, determine the winning planet using each of the methods listed below. +1. +Plurality +2. +Ranked-choice method +3. +Borda count +Solution +1. +The plurality method only considers the top choice of each voter. By this system, Planet A has 45 percent of the vote, +Planet B has 15 percent of the vote, and Planet C has 40 percent of the vote. Planet A wins. +2. +Using either instant runoff or a two-round system, Planet B with only 15 percent of the vote will be eliminated in the +first round. In Round 2, the 15 percent that voted for Planet B would vote for their second choice, Planet C. This +leaves Planet A with 45 percent and Planet C with 55 percent. Planet C has a majority and wins the election. +3. +To find the Borda score for each candidate, imagine there are exactly 100 voters. Then the summary of ranked +ballots looks like: +Out of 100 Voters +45 +15 +40 +Planet A +1 +3 +3 +Planet B +2 +1 +2 +Planet C +3 +2 +1 +The Borda score for each candidate is as follows: +Planet A: +Planet B: +Planet C: +Planet B wins. +YOUR TURN 11.15 +The juniors at a high school in Central Florida are voting for a theme park to visit for an end of the year field trip. The +options are Disney’s four theme parks: Animal Kingdom, Magic Kingdom, EPCOT Center, or Hollywood Studios. +Percentage of Voters +35% +25% +30% +Animal Kingdom +1 +3 +4 +Magic Kingdom +2 +1 +3 +11.1 • Voting Methods +1155 + +Percentage of Voters +35% +25% +30% +Epcot Center +3 +4 +1 +Hollywood Studios +4 +2 +2 +Determine the outcome of the vote by each of the given voting methods. +1. Plurality +2. Ranked-choice method +3. Borda count +The election in Example 11.15 involves a scenario in which there are two extreme candidates, Planet A and Planet B, and +a moderate candidate, Planet C. The supporters of the extreme candidates prefer the moderate candidate to the other +extremist ones. This makes Planet C a compromise candidate. In this case, both the plurality method and ranked-choice +voting resulted in the election of one of the extreme candidates, but the Borda count method elected the compromise +candidate in this scenario. Depending on a person’s perspective, this may be perceived as a flaw in either ranked-choice +and plurality systems, or the Borda count method. +In Fairness in Voting Methods, we will analyze the fairness of each voting system in greater detail using objective +measures of fairness. +TECH CHECK +Voting Calculators +It is possible to create Excel spreadsheets that complete the calculations necessary to determine the winner of an +election by various voting methods. In some cases, this work has already been done and posted online. As you +practice applying the various voting methods that could be used in Imaginaria, quick Internet search will lead to sites +such as Ms. Hearn Math (https://openstax.org/r/calculators) with free specialty calculators. +These sites can be a great way to check your results! +Check Your Understanding +1. Name three voting methods that use a ranked ballot. +2. Determine whether the following statement is true or false: The same ranked ballots may result in a different +winner depending on which voting method is used. +3. Determine whether the following statement is true or false. A majority candidate is always a Condorcet candidate. +4. The _______________ method is a system of voting using ranked ballots in which each candidate is awarded points +corresponding to the number of candidates ranked lower on each ballot. +5. The ________________ method is a system of voting using ranked ballots (or multiple elections) in which each +candidate receives a point for each candidate they would beat in a one-on-one election and half a point for each +candidate they would tie. +6. The __________________ method is a runoff voting system in which only the candidate(s) with the very least votes are +eliminated. +7. Explain the differences between two-round voting and ranked-choice voting. +SECTION 11.1 EXERCISES +For the following exercises, identify the winning candidate based on the described voter profile, if possible. If it is not +possible, state so. Explain your reasoning. +1156 +11 • Voting and Apportionment +Access for free at openstax.org + +1. In a plurality election, the candidates have the following vote counts: A 125, B 132, C 149, D 112. +2. In the first round of a ranked-choice election with three candidates—A, B, and C. Candidate A received 55 first +place rankings; Candidate B received 25; and Candidate C received 30. +3. The pairwise matchup points for each candidate were: A 1, B +, D +. +4. In a Borda count election, the candidates have the following Borda scores: A 15, B 11, C 12, D 16. +5. There is a pairwise comparison election with candidates A, B, and C. Candidate A had the most first choice +rankings, Candidate B has the highest Borda score, and Candidate C is a Condorcet candidate. +6. In the first round of a ranked-choice election with three candidates—A, B, and C—Candidate A received 20 first +place rankings, Candidate B received 25, and Candidate C received 30. +For the following exercises, use the table. +O'Malley +De La Fuente +Clinton +Sanders +Other +110,227 +67,331 +17,174,432 +13,245,671 +322,276 +Popular Vote in the 2016 U.S. Democratic Presidential Primary +(source: Federal Election Commission, Federal Elections 2016 Report) +7. Calculate the number of votes required to have a majority of the popular vote in the 2016 U.S. Democratic +Presidential Primary. +8. Which candidate had a plurality? Did this candidate have a majority? +For the following exercises, use the given table. +Candidate +Votes +Bush +281,189 +Trump +13,783,037 +Cruz +7,455,780 +Rubio +3,354,067 +Carson +822,242 +Kasich +4,198,498 +Other +337,714 +Popular Vote in the 2016 +U.S. Republican Presidential +Primary (source: Federal +Election Commission, +Federal Elections 2016 +Report) +9. Calculate the number of votes required to have a majority of the popular vote in the 2016 U.S. Republican +Presidential Primary. +10. Which candidate had a plurality? Did this candidate have a majority? +For the following exercises, use Table 11.4 and Table 11.5. +11. Suppose the Republican Primary in 2016 was a two-round system. Would there be a second round? Why or why +not? If so, which candidates would advance to the second round? +12. Suppose the Democratic Primary in 2016 was a two-round system. Would there be a second round? Why or why +not? If so, which candidates would advance to the second round? +11.1 • Voting Methods +1157 + +13. Suppose the Democratic Primary in 2016 used the Hare method. Would there be a second round? Why or why +not? +14. Suppose the Republican Primary in 2016 used the Hare method. Would there be a second round? Why or why +not? +For the following exercises, use the following table and the Hare method. +Options +A +B +C +D +E +Candidate 1 +1 +3 +3 +1 +3 +Candidate 2 +2 +1 +1 +2 +4 +Candidate 3 +3 +4 +2 +4 +1 +Candidate 4 +4 +2 +4 +3 +2 +15. How many votes are needed to win by the Hare method? +16. How many votes does each candidate receive in Round 1? +17. Which candidates advance to Round 2? +18. How many votes does each remaining candidate receive in Round 2? +19. Will there be a third round? Why or why not? +20. Which candidate wins the election? +For the following exercises, use the sample summary of ranked ballots in the given table. +Number of Ballots +10 +20 +15 +5 +Option A +1 +4 +3 +4 +Option B +2 +3 +4 +2 +Option C +4 +2 +1 +3 +Option D +3 +1 +2 +1 +Sample Summary of Ranked Ballots +21. How many votes were recorded, and how many are required to have a majority? +22. How many voters indicated that Option A was their first choice? +23. How many voters indicated that Option B was their first choice? +24. How many voters indicated that Option A was their last choice? +25. How many voters indicated that Option B was their last choice? +26. Use ranked-choice voting to determine the two candidates in the final round and the number of votes they each +receive in that round. +27. Is there a winning candidate? If so, which candidate? Justify your answer. +Suppose that 55 Star Wars fans were asked to vote for their favorite new Star Wars character. They were given a ranked +ballot, and the results are shown in the table. Use this table and ranked-choice voting for the following exercises. +1158 +11 • Voting and Apportionment +Access for free at openstax.org + +Number Of Ballots +7 +6 +10 +8 +4 +5 +6 +7 +2 +Finn +3 +5 +1 +3 +2 +4 +2 +2 +1 +Rey +2 +1 +3 +2 +4 +3 +3 +1 +3 +Poe +1 +6 +4 +1 +5 +5 +5 +4 +2 +BB8 +4 +2 +2 +4 +6 +1 +6 +5 +4 +Rose +6 +4 +5 +6 +3 +2 +1 +3 +6 +Kylo +5 +3 +6 +5 +1 +6 +4 +6 +5 +Favorite New Heroes in Star Wars Sequels Ballot Preferences +28. How many votes does each candidate get on the first round of voting? +29. How many votes are required to get a majority? +30. Which candidates remain in the final round, and how many votes do they have? +31. Who is the winner of the election? +Refer to Table 11.6 for the following exercises. +32. Find the Borda score for each candidate. +33. Compare your results from question 32 to those from question 26. Compare the winner and the second-place +candidate using the Borda count method to those using the ranked-choice method. Are they the same? +Refer to Table 11.7 for the following exercises. +34. Find the Borda score for each candidate. +35. Compare your results from question 34 to those from question 30. Compare the winner and the second-place +candidate using the Borda count method to those using the ranked-choice method. Are they the same? +For the following exercises, use the table below. +Number of Ballots +100 +80 +110 +105 +55 +Option A +1 +1 +4 +4 +2 +Option B +2 +2 +2 +3 +1 +Option C +4 +4 +1 +1 +4 +Option D +3 +3 +3 +2 +3 +36. Do any candidates appear to be divisive candidates? Justify your answer. +37. Do any candidates appear to be compromise candidates? Justify your answer. +38. How many votes are required for a majority? +39. Which candidate is eliminated first by the ranked-choice method? +40. Which candidate is eliminated second by the ranked-choice method? +41. Which candidate is the winner by the ranked-choice method? +42. What are the Borda scores for each candidate? +43. Which candidate is the winner by the Borda count method? +44. Which method resulted in a win for the compromise candidate: ranked-choice voting or the Borda count +method or both? +Use the pairwise comparison matrix in the given figure for the following exercises. +11.1 • Voting Methods +1159 + +Pairwise Comparison Matrix for Candidates Q, R, S and T +45. Analyze the pairwise comparison matrix. Display the pairings in a table and indicate the winner of each +matchup by marking an +through the losing matchups and a single slash +through the ties. +46. Calculate the points received by each candidate in the pairwise comparison matrix. +47. Determine whether there is a winner of the pairwise comparison election represented by the matrix. If there is +a winner, determine whether the winner is a Condorcet candidate. +Use the pairwise comparison matrix in the given figure for the following exercises. +Pairwise Comparison Matrix for Candidates U, V, W, X, and Y +48. Analyze the pairwise comparison matrix. Display the pairings in a table and indicate the winner of each +matchup. +49. Calculate the points received by each candidate in the pairwise comparison matrix. +50. Determine whether there is a winner of the pairwise comparison election represented by the matrix. If there is +a winner, determine whether the winner is a Condorcet candidate and explain your reasoning. +51. In J.K. Rowling’s Harry Potter series, Albus Dumbledore was the headmaster of Hogwarts for many years. +Imagine that an election is to be held to find his successor. Severus Snape, the head of Slytherin House, will be +running against the heads of Gryffindor and Ravenclaw, Minerva McGonagall and Filius Flitwick. Use the +preference rankings for each candidate in the following table to construct a pairwise comparison matrix. +1160 +11 • Voting and Apportionment +Access for free at openstax.org + +Percentage of Vote +25% +40% +35% +(S) Snape +1 +3 +3 +(M) McGonagall +3 +1 +2 +(F) Flitwick +2 +2 +1 +52. Analyze the pairwise comparison matrix you constructed for question 51. Display the pairings in a table and +indicate the winner of each matchup. +53. Use the pairwise comparison matrix from questions 51 and 52 to find the number of points earned by each +candidate. Who is the winner by the pairwise comparison method? +54. Is the winner of the Hogwarts headmaster election a Condorcet candidate? Explain how you know. +55. The women of The Big Bang Theory decide to hold their own approval voting election to determine the best +option in Rock, Paper, Scissors, Lizard, Spock. Use the summary of their approval ballots in the table below to +determine the number of votes for each candidate. Determine the winner, or state that there is none. +Voters +Penny +Bernadette +Amy +Rock +Yes +No +No +Paper +Yes +Yes +No +Scissors +Yes +Yes +Yes +Lizard +No +No +No +Spock +Yes +No +Yes +For the following exercises, use the table below. +Percentage of Vote +40% +35% +25% +Candidate A +1 +3 +2 +Candidate B +2 +1 +3 +Candidate C +3 +2 +1 +56. Which candidate is the winner by the ranked-choice method? +57. Suppose that they used the approval method and each voter approved their top two choices. Which candidate +is the winner by the approval method? +58. Which candidate is the winner by the Borda count method? +11.1 • Voting Methods +1161 + +11.2 Fairness in Voting Methods +Figure 11.10 Citizens strive to ensure their voting system is fair. (credit: “Governor Votes Early” by Maryland GovPics/ +Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Compare and contrast fairness of voting using majority criterion. +2. +Compare and contrast fairness of voting using head-to-head criterion. +3. +Compare and contrast fairness of voting using monotonicity criterion. +4. +Compare and contrast fairness of voting using irrelevant alternatives criterion. +5. +Apply Arrow’s Impossibility Theorem when evaluating voting fairness. +Now that we’ve covered a variety of voting methods and discussed their differences and similarities, you might be +leaning toward one method over another. You will need to convince the other founders of Imaginaria that your +preference will be the best for the country. Before your collaborators approve the inclusion of a voting method in the +constitution, they will want to know that the voting method is a fair method. In this section, we will formally define the +characteristics of a fair system. We will analyze each voting previously discussed to determine which characteristics of +fairness they have, and which they do not. In order to guarantee one ideal, we must often sacrifice others. +The Majority Criterion +One of the most fundamental concepts in voting is the idea that most voters should be in favor of a candidate for a +candidate to win, and that a candidate should not win without majority support. This concept is known as the majority +criterion. +With respect to the four main ranked voting methods we have discussed—plurality, ranked-choice, pairwise comparison, +and the Borda count method—we will explore two important questions: +1. +Which of these voting systems satisfy the majority criterion and which do not? +2. +Is it always “fair” for a voting system to satisfy the majority criterion? +Keep in mind that this criterion only applies when one of the candidates has a majority. So, the examples we will analyze +will be based on scenarios in which a single candidate has more than 50 percent of the vote. +1162 +11 • Voting and Apportionment +Access for free at openstax.org + +EXAMPLE 11.16 +Roommates Choose Fast Food +It’s final exams week and seven college students are hungry. They must get food, but from which drive thru? Their +preferences are listed in the table below. The majority have listed McDonald’s as their top choice. Let’s calculate what the +results of the election will be using various voting methods. +Voters +A +B +C +D +E +F +G +(M) McDonald’s +1 +1 +1 +1 +5 +5 +5 +(B) Burger King +2 +2 +2 +2 +2 +2 +2 +(T) Taco Bell +4 +4 +3 +4 +3 +3 +1 +(O) Pollo Tropical +3 +5 +4 +3 +4 +1 +3 +(I) Pizza Hut +5 +3 +5 +5 +1 +4 +4 +1. +Which restaurant is the winner using the plurality voting method? +2. +Which restaurant is the winner using the ranked-choice voting method? +3. +Does the majority criterion apply? If so, for which of voting method(s), if any, did the majority criterion fail? +Solution +1. +For plurality voting, we only need to count the first-place votes for each candidate. In this case, McDonald’s has four +first place votes, which is a majority and wins the election automatically. +2. +For ranked-choice voting, McDonald’s also wins because it has a majority at the end of Round 1. +3. +The majority criterion does apply because one candidate had a majority of the first-place votes. The majority +criterion did not fail by either method, because in each case the majority candidate won. +YOUR TURN 11.16 +Use the information in the table to answer the following questions. +Voters +L +M +N +O +P +Option A +2 +2 +3 +2 +1 +Option B +1 +4 +1 +4 +2 +Option C +3 +1 +4 +1 +3 +Option D +4 +1 +2 +3 +4 +1. Which option is the winner when using the plurality voting method? +2. Which option is the winner when using the ranked-choice voting method? +3. Does the majority criterion apply? If so, for which voting method(s), if any, did the majority criterion fail? +From Example 11.16, it appears that the plurality and ranked-choice voting methods satisfy the majority criterion. In +general, the majority candidate always wins in a plurality election because the candidate that has more than half of the +votes has more votes than any other candidate. The same is true for ranked-choice voting; and there will never be a +need for a second round when there is a majority candidate. Let’s examine how some of the other voting methods stand +11.2 • Fairness in Voting Methods +1163 + +up to the majority criterion. +EXAMPLE 11.17 +Roommates Choose Fast Food +Those seven college students are hungry again! Their preferences haven’t changed, as shown below. Let’s calculate if the +results change when we use different voting methods. +VOTERS +A +B +C +D +E +F +G +(M) McDonald’s +1 +1 +1 +1 +5 +5 +5 +(B) Burger King +2 +2 +2 +2 +2 +2 +2 +(T) Taco Bell +4 +4 +3 +4 +3 +3 +1 +(O) Pollo Tropical +3 +5 +4 +3 +4 +1 +3 +(I) Pizza Hut +5 +3 +5 +5 +1 +4 +4 +1. +Which restaurant is the winner using the pairwise comparison voting method? +2. +Which restaurant is the winner using the Borda count voting method? +3. +Does the majority criterion apply? If so, for which voting method(s), if any, did the majority criterion fail? +Solution +1. +For pairwise comparison, notice that McDonald’s is a Condorcet candidate because it wins every pairwise +comparison. So, McDonald’s is the winner. +2. +For the Borda count, we must calculate the Borda score for each candidate: McDonald’s is 16, Burger King is 21, Taco +Bell is 13, Pollo Tropical is 12, Pizza Hut is 8. The winner is Burger King! +3. +Yes, the majority criterion applies because McDonald’s has the majority of first place votes. The majority criterion +only fails using the Borda method. +YOUR TURN 11.17 +Use the information in the following table to find the winner using each of the voting methods in parts 1 and 2, then +answer the question in part 3. +Voters +L +M +N +O +P +Option A +2 +2 +3 +2 +2 +Option B +1 +4 +1 +4 +1 +Option C +3 +1 +4 +1 +3 +Option D +4 +3 +2 +3 +4 +1. Pairwise comparison +2. Borda count +3. Does the majority criterion apply? If so, for which voting method(s) did the majority criterion fail? +Example 11.17 demonstrates a concept that we also saw in Borda Count Voting—the Borda method frequently favors the +1164 +11 • Voting and Apportionment +Access for free at openstax.org + +compromise candidate over the divisive candidate. This can happen even when the divisive candidate has a majority, as +it did in this example. Although a majority of the voters were in favor of McDonald’s, a significant minority was strongly +opposed to McDonald’s, ranking it last. Since the Borda score includes all rankings, this strong opposition has an impact +on the outcome of the election. +Pairwise comparison will always satisfy the majority criterion because the candidate with the majority of first-place votes +wins each pairwise matchup. While it is possible for the majority candidate to win by the Borda count method, it is not +guaranteed. So, the Borda method fails the majority criterion. A summary of each voting method as it relates to the +majority criterion is found in the following table. +Voting Method +Majority Criterion +Plurality +Satisfies +Ranked-choice +Satisfies +Pairwise comparison +Satisfies +Borda count +Violates +If you prefer the Borda method, you might argue that its failure to satisfy the majority criterion is actually one of its +strengths. As we saw in Example 11.16 and Example 11.17, the majority have the power to vote for their own benefit at +the expense of the minority. While four students were very enthusiastic about McDonald’s, three students were strongly +opposed to McDonald’s. It is reasonable to say that the better option would be Burger King, the compromise candidate, +which everyone ranked highly and no one strongly opposed. The Borda method is designed to favor a candidate that is +acceptable to the population as a whole. In this way, the Borda method avoids a downfall of strict majority rule known as +the tyranny of the majority, which occurs when a minority of a population is treated unfairly because their situation is +different from the situation of the majority. +The people of Imaginaria should know that the power of the majority to vote their will has serious implications for other +groups. For example, according to the UCLA School of Law Williams Institute, the LGBTQ+ community in the United +States makes up approximately 4.5 percent of the population. When elections occur that include issues that affect the +LGBTQ+ community, members of the LGBTQ+ community depend on the 95.5 percent of the population who do not +identify as LGBTQ+ to consider their perspectives when voting on issues such as same-sex marriage, the use of public +restrooms by transgender people, and adoption by same-sex couples. +PEOPLE IN MATHEMATICS +Robert Dahl +In his book, Democracy and Its Critics, Robert Dahl wrote, “If a majority is not entitled to do so, then it is thereby +deprived of its rights; but if a majority is entitled to do so, then it can deprive the minority of its rights.” Dahl was a +renowned political theorist, but he is also considered to be a mathematician since his work utilizes ideas from an area +of mathematics known as Game Theory (Mathematics Genealogy Project, NDSU Department of Mathematics with the +American Mathematical Society). +WHO KNEW? +Three Branches of Government +Concerns about the consequences of majority rule are not new. In 1788, John Adams warned of the consequences of +majority rule and he argued for three branches of government as a way to temper them. In the early 1800s, a young +French aristocrat named Alexis de Tocqueville toured the United States and wrote Democracy in America, which +focused on the impact of democracy on political and civil societies. He observed, even then, the dominance of the +white majority over the Indigenous people and enslaved people, which was perpetuated by majority rule. +11.2 • Fairness in Voting Methods +1165 + +VIDEO +Separation of Powers and Checks and Balances (https://openstax.org/r/separation_of_powers) +Head-to-Head Criterion +Another fairness criterion you must consider as you select a voting method for Imaginaria is the Condorcet criterion, +also known as the head-to-head criterion. An election method satisfies the Condorcet criterion provided that the +Condorcet candidate wins the election whenever a Condorcet candidate exists. A Condorcet method is any voting +method that satisfies the Condorcet criterion. +Recall from Three Key Questions that not every election has a Condorcet candidate; the Condorcet criterion will not apply +to every election. Also recall that a Condorcet candidate cannot lose an election by pairwise comparison. So, the pairwise +comparison voting method is said to satisfy the Condorcet criterion. +EXAMPLE 11.18 +Spending Tax Refund +A survey asked a random sample of 100 people in the United States to rank their priorities for spending their tax refund. +The options were (V) go on vacation, (S) put into savings, (D) pay off debt, or (T) other. The pairwise comparison matrix +for the results is in Figure 11.11. Determine whether the Condorcet criterion applies. +Figure 11.11 Pairwise Comparison Matrix for Tax Refund Spending +Solution +The Condorcet criterion only applies when there is a Condorcet candidate. “Pay off debt” (D) is a Condorcet candidate +because D wins every matchup. Yes, the Condorcet criterion applies to this election. +YOUR TURN 11.18 +1. Determine whether the Condorcet criterion applies based on the summary of ranked ballots given in the table +below. +Votes +3 +2 +Option A +1 +3 +Option B +2 +1 +Option C +3 +2 +1166 +11 • Voting and Apportionment +Access for free at openstax.org + +EXAMPLE 11.19 +Spending Tax Refund +Let’s return to the survey about tax refund spending from Example 11.18. We know that the Condorcet criterion applies +because Option D, “Pay off debt,” is a Condorcet candidate, which wins every pairwise match up. +Use the information in the ballot summary from the table below to find the winner and determine whether the +Condorcet criterion is satisfied in this election when each of the following voting methods are used. +Votes +33 +32 +31 +4 +On a vacation (V) +1 +3 +3 +2 +Put into savings (S) +3 +1 +2 +1 +Pay off debt (D) +2 +2 +1 +4 +Other (T) +4 +4 +4 +3 +1. +Plurality +2. +Ranked-choice voting +3. +Borda count +Solution +1. +V wins 33 first place votes; S, 36; D, 31; and T, 0. So candidate S, “Put into savings,” has a plurality and wins. Since the +Condorcet candidate D didn’t win, the Condorcet criterion is violated. +2. +Use the steps outlined in Ranked-Choice Voting for determining the winner of an election by ranked-choice voting, +the application of the Hare method in which instant runoffs are used. +Step 1: The number of votes needed to achieve a majority is 51. +Step 2: As illustrated in part 1, no candidate has a majority of first-place votes; so the candidate with the fewest +votes, T, must be eliminated. +Step 3: Reallocate votes to the remaining candidates for the second round: +Votes +33 +32 +31 +4 +On a vacation (V) +1 +3 +3 +2 +Put into savings (S) +3 +1 +2 +1 +Pay off debt (D) +2 +2 +1 +3 +Step 4: Repeat the process from Step 2. Count the first-place votes for each candidate: V has 33 votes, S has 36 +votes, D has 31. votes. Eliminate candidate D, “Pay off debt,” for the third round. +Step 5: Repeat the process from Step 3. Reallocate votes to the remaining candidates for the third round: +Votes +33 +32 +31 +4 +On a vacation (V) +1 +2 +2 +2 +Put into savings (S) +2 +1 +1 +1 +Step 6: Repeat the process from Step 2 one last time. Count the first-place votes for each candidate: V has 33, S has +11.2 • Fairness in Voting Methods +1167 + +67. Candidate S, “Put into savings,” has a majority and wins. Since the Condorcet candidate D candidate D, “Pay off +debt,” didn’t win, the Condorcet criterion is violated. +3. +Calculate the Borda score for each candidate. +V: +S: +D: +T: +Candidate D, “Pay off debt,” has the highest Borda score and wins. Since D was the Condorcet candidate, this election +satisfies the Condorcet criterion. +YOUR TURN 11.19 +Use the summary of ranked ballots below to find the winner and determine whether the Condorcet criterion is +satisfied when each of the following voting methods are used. Recall that Option A is a Condorcet candidate. +Votes +3 +2 +Option A +1 +3 +Option B +2 +1 +Option C +3 +2 +1. Plurality +2. Ranked-choice voting +3. Borda count +As we have seen, the plurality method, ranked-choice voting, and the Borda count method each fail the Condorcet +criterion in some circumstances. Of the four main ranked voting methods we have discussed, only the pairwise +comparison method satisfies the Condorcet criterion every time. A summary of each voting method as it relates to the +Condorcet criterion is found in the following table. +Voting Method +Condorcet Criterion +Plurality +Violates +Ranked-choice +Violates +Pairwise comparison +Satisfies +Borda count +Violates +Monotonicity Criterion +The citizens of Imaginaria might be surprised to learn that it is possible for a voter to cause a candidate to lose by +ranking that candidate higher on their ballot. Is that fair? Most voters would say, “Absolutely not!!” This is an example of +a violation of the fairness criterion called the monotonicity criterion, which is satisfied when no candidate is harmed by +up-ranking nor helped by down-ranking, provided all other votes remain the same. +Consider a scenario in which voters are permitted a first round that is not binding, and then they may change their vote +before the second round. Such a first round can be called a “straw poll.” Now, let’s suppose that a particular candidate +1168 +11 • Voting and Apportionment +Access for free at openstax.org + +won the straw poll. After that, several voters are convinced to increase their support, or up-rank, that winning candidate +and no voters decrease that support. It is reasonable to expect that the winner of the first round will also win the second. +Similarly, if some of the voters decide to decrease their support, or down-rank, a losing candidate, it is reasonable to +expect that candidate will still lose in the second round. +You might be wondering why it’s called the monotonicity criterion. In mathematics, the term monotonicity refers to the +quality of always increasing or always decreasing. For example, a person’s age is monotonic because it always increases, +whereas a person’s weight is not monotonic because it can increase or decrease. If the only changes to the votes for a +particular candidate after a straw poll are in one direction, this change is considered monotonic. +If you are going to make an informed decision about which voting method to use in Imaginaria, you need to know which +of the four main ranked voting methods we have discussed—plurality, ranked-choice, pairwise comparison, and the +Borda count method—satisfy the monotonicity criterion. +EXAMPLE 11.20 +Favorite Dog Breed by Plurality +The local animal shelter is having a vote-by-donation charity event. For a $10 donation, an individual can complete a +ranked ballot indicating their favorite large dog breed: standard poodle, golden retriever, Labrador retriever, or bulldog. +Use the summary of ballots below to answer each question. +Votes +42 +53 +61 +24 +(S) Standard Poodle +1 +3 +2 +1 +(G) Golden Retriever +3 +1 +4 +4 +(L) Labrador Retriever +4 +2 +1 +2 +(B) Bulldog +2 +4 +3 +3 +1. +Determine the winner of the election by plurality. +2. +Suppose that the 53 voters in the second column increased their ranking of the winner by 1. Determine the winner +by plurality with the new rankings. +3. +Does this election violate the monotonicity criterion? +4. +Do you think the result of part 3 is also true for plurality voting and the monotonicity criterion in general? Why or +why not? +Solution +1. +The number of votes for each candidate are: S 66, G 53, L 61, and B 0. The winner is the standard poodle. +2. +If the 53 voters in the second column rank S as 2 and L as 3, then the number of votes for each candidate are: S with +66, G with 53, L with 61, and B with 0. The winner is still the standard poodle. +3. +This election does not violate the monotonicity criterion because the winner was not hurt by up-ranking. +4. +In general, increasing the ranking for a winner of a plurality election will either leave them with the same or more +first place votes while leaving the other candidates with the same or fewer first place votes. So a plurality election +will never violate the monotonicity criterion. +YOUR TURN 11.20 +Use the Favorite Large Dog Breed Ballot Summary to answer each question. +1. Determine the winner of the election by Borda count. +2. Suppose that the 61 voters in the third column increased their ranking of the winner by 1. Determine the +winner by Borda count with the new rankings. +3. Does this election violate the monotonicity criterion? +4. Do you think the result of part 3 is true for Borda count and the monotonicity criterion in general? Why or why +11.2 • Fairness in Voting Methods +1169 + +not? +EXAMPLE 11.21 +Favorite Dog Breed by Pairwise Comparison +Earlier, we discovered that the summary of ranked ballots shown in the table below results in the pairwise comparison +matrix in Figure 11.12. Use this information to answer the questions. +Number of Ballots +95 +90 +110 +115 +Option A +4 +4 +1 +1 +Option B +2 +2 +2 +2 +Option C +3 +1 +3 +4 +Option D +1 +3 +4 +3 +Figure 11.12 Analyzed Pairwise Comparison Matrix for Sample Summary of Ranked Ballots +1. +Determine the winner of the election by the pairwise comparison method. +2. +Suppose that the 95 voters in the first column increased their ranking of the winner by 1. Determine the winner by +the pairwise comparison method with the new rankings. +3. +Does this election violate the monotonicity criterion? +4. +Do you think the result of part 3 is true for the pairwise comparison method and the monotonicity criterion in +general? Why or why not? +Solution +1. +By the pairwise comparison method, Option A wins with three points. +2. +If the 95 voters in the first column of Figure 11.12 increased their ranking of the winner by 1, then C would fall into +fourth place and A would move up to third place on those ballots. This would only affect the matchup between A +and C, and the result would be that A would gain 95 votes while C would lose 95 votes. This means A would have 320 +votes and C would have 90. Since A already was ahead of C, this just puts A further ahead and causes no change to +the election results. +3. +Since the winner A is not hurt by an up-rank, and the loser C is not helped by a down-rank, this election is fair by the +1170 +11 • Voting and Apportionment +Access for free at openstax.org + +monotonicity criterion. +4. +Yes, the monotonicity criterion would be satisfied by the pairwise comparison method, because an up-rank of the +winner can never decrease the number of pairwise wins. Similarly, a down-rank can never increase the number of +pairwise wins. +YOUR TURN 11.21 +Use the Favorite Large Dog Breed Ballot Summary to answer each question. +1. Determine the winner of the election by the ranked-choice method. +2. Suppose that the 24 voters in the last column of the table increased their ranking of the winner by 1. +Determine the winner by the ranked-choice method with the new rankings. +3. Does this election violate the monotonicity criterion? +4. Do you think the result in question 3 is true for the ranked-choice method and the monotonicity criterion in +general? Why or why not? +The last few examples illustrate that the plurality method, pairwise comparison voting, and the Borda count method +each satisfy the monotonicity criterion. Of the four main ranked voting methods we have discussed, only the ranked- +choice method violates the monotonicity criterion. A summary of each voting method as it relates to the Condorcet +criterion is found in the table below. +Voting Method +Monotonicity Criterion +Plurality +Satisfies +Ranked-choice +Violates +Pairwise comparison +Satisfies +Borda count +Satisfies +Irrelevant Alternatives Criterion +We have covered a lot about voting fairness, but there is one more fairness criterion that you and the other Imaginarians +should know. Consider this well-known anecdote that is sometimes attributed to the American philosopher Sidney +Morgenbesser: +A man is told by his waiter that the dessert options this evening are blueberry pie or apple pie. The man orders the apple +pie. The waiter returns and tells him that there is also a third option, cherry pie. The man says, “In that case, I would like +the blueberry pie.” (Gaming the Vote: Why Elections Aren’t Fair (and What We Can Do About It), William Pound stone, p. +50, ISBN 0-8090-4893-0) +This story illustrates the concept of the Irrelevant Alternatives Criterion, also known as the Independence of Irrelevant +Alternatives Criterion (IIA), which means that the introduction or removal of a third candidate should not change or +reverse the rankings of the original two candidates relative to one another. In particular, if a losing candidate is removed +from the race or if a new candidate is added, the winner of the race should not change. +EXAMPLE 11.22 +Apple, Blueberry, or Cherry? +Suppose that 30 students in a class are going to vote on whether to have apple, blueberry, or cherry pie. Use the +summary of ranked ballots in below to answer each question. +11.2 • Fairness in Voting Methods +1171 + +Number of Ballots +14 +12 +4 +(A) Apple Pie +1 +3 +3 +(B) Blueberry Pie +2 +1 +2 +(C) Cherry Pie +3 +2 +1 +1. +Determine the winner of the election by plurality. +2. +Which candidate would win a plurality election if cherry pie were removed from the ballot? +3. +Does this election violate the IIA? +Solution +1. +The number of first place votes for each candidate is: A with 14, B with 12, and C with 4. Apple pie has the most first- +place votes and wins the election. +2. +If cherry pie is removed from the ballot, then the four voters in the third column now rank blueberry pie as their first +choice. So the four votes for C now belong to B. This means that blueberry pie has 16 votes compared to the 14 +votes for apple pie. Blueberry pie now wins the plurality election. +3. +Yes, the election violates the IIA because the removal of a losing candidate from the ballot changed the winner of +the election. +YOUR TURN 11.22 +The local animal shelter is having another vote-by-donation charity event! This time, for a $10 donation, an +individual can complete a ranked ballot indicating their favorite small dog breed, miniature, from the ones provided: +miniature poodle, Yorkshire terrier, or Chihuahua. Use the summary of ballots below to answer each question. +Votes +53 +42 +24 +11 +(P) Miniature Poodle +1 +3 +3 +2 +(Y) Yorkshire Terrier +2 +2 +1 +1 +(C) Chihuahua +3 +1 +2 +3 +1. Determine the winner of the election by the ranked-choice method. +2. Determine the winner of the election if poodles are removed from the ballot. +3. Does this election violate IIA? +EXAMPLE 11.23 +Best Fourth Wall Breaking Stare on The Office +The NBC sitcom The Office ran for nine years and has been one of the most popular streamed television shows of all +time. One of the trademarks of the show was that characters would often break the fourth wall to communicate with the +audience just by staring directly into the camera. In fact, there is a website dedicated to "The Office stares" +(https://openstax.org/r/theofficestaremachine) where you can watch over 700 of these stares! Suppose that 36 fans were +asked which character had the best "The Office stare." Use the ballot summary in below to answer each question. +1172 +11 • Voting and Apportionment +Access for free at openstax.org + +Number of Ballots +9 +11 +7 +6 +3 +(J) Jim Halpert (John Krasinski) +1 +2 +4 +2 +4 +(P) Pam Beesly-Halpert (Jenna Fischer) +4 +1 +2 +4 +3 +(D) Dwight Schrute (Rainn Wilson) +2 +3 +3 +1 +2 +(M) Michael Scott (Steve Carell) +3 +4 +1 +3 +1 +1. +Determine the winner of the election by the pairwise comparison method. +2. +Determine the winner of the election by the pairwise comparison method if Michael Scott is removed from the +ballot. +3. +Does this election violate IIA? +Solution +1. +Construct and analyze a pairwise comparison matrix: +Figure 11.13 Pairwise Comparison Matrix for Jim, Pam, Dwight, and Michael +Jim Halpert wins with 2 points. +2. +If Michael Scott is removed, the summary of ranked ballots becomes: +Number of Ballots +9 +11 +7 +6 +3 +(J) Jim Halpert (John Krasinski) +1 +2 +3 +2 +3 +(P) Pam Beesly-Halpert (Jenna Fischer) +3 +1 +1 +3 +2 +(D) Dwight Schrute (Rainn Wilson) +2 +3 +2 +1 +1 +Construct and analyze a pairwise comparison matrix: +11.2 • Fairness in Voting Methods +1173 + +Figure 11.14 Pairwise Comparison Matrix for Jim, Pam, and Dwight +Pam wins with +points. +3. +Yes, this violates the IIA, because the winning candidate was hurt by the elimination of a losing candidate. +YOUR TURN 11.23 +Use the initial ballot summary from Example 11.23 to answer the questions. +1. Determine the winner of the election by the Borda count method. +2. Determine the winner of the election by the Borda count method if Michael Scott is removed from the ballot. +3. Does this election violate IIA? +We have seen that all four of the main voting systems we are working with fail the Irrelevant Alternatives Criterion (IIA). +A summary of each voting method as it relates to the IIA criterion is found in the table below. +Voting Method +Irrelevant Alternatives Criterion +Plurality +Violates +Ranked-choice +Violates +Pairwise comparison +Violates +Borda count +Violates +WHO KNEW? +Electronic Voting: Does Your Vote Count? +In order for an election to be fair, voting must be accessible to everyone and every vote must be counted. When +hundreds of thousands to millions of votes must be collected and counted in a short period of time, deciding it can be +challenging to be on counting procedures that are accurate and secure. Electronic voting machines or even Internet +voting can speed up the process, but how reliable are these methods? This has been a subject for debate for years. +In a press release on August 3, 2007, California Secretary of State Debra Bowen explained the results of an extensive +review of electronic voting systems in her state. She said that transparency and auditability were key. She went on to +say, “I think voters and counties are the victims of a federal certification process that hasn’t done an adequate job of +ensuring that the systems made available to them are secure, accurate, reliable and accessible. Congress enacted the +Help America Vote Act, which pushed many counties into buying electronic systems that—as we’ve seen for some +time and we saw again in the independent UC review—were not properly reviewed or tested to ensure that they +1174 +11 • Voting and Apportionment +Access for free at openstax.org + +protected the integrity of the vote.” Secretary Bowden subsequently ordered that voting machines must have tighter +security to be used in California. (DB07:042, Secretary of State Debra Bowen Moves to Strengthen Voter Confidence in +Election Security Following Top-to-Bottom Review of Voting Systems, https://sos.ca.gov/elections.) In some instances, +the use of electronic voting in parliamentary elections has been discontinued completely for security reasons. For +example, according to the National Democratic Institute, the Netherlands returned to all paper ballots and hand +counting in 2006. Will you use voting machines or Internet voting in Imaginaria? +So far, every one of the voting methods we have analyzed has failed one or more of the fairness criteria in one +election or another. +Voting +Method +Majority +Criterion +Condorcet Criterion (Head- +to-Head Criterion) +Monotonicity +Criterion +(Independence of) Irrelevant +Alternatives Criterion +Plurality +Satisfies +Violates +Satisfies +Violates +Ranked- +choice +Satisfies +Violates +Satisfies +Violates +Pairwise +comparison +Satisfies +Satisfies +Violates +Violates +Borda count +Violates +Violates +Satisfies +Violates +You might be wondering if there is a voting system you could recommend for Imaginaria that satisfies all the fairness +criteria. If there is one, it remains to be discovered and it is not a voting system that is based solely on preference +rankings. In 1972, Harvard Professor of Economics Kenneth J. Arrow received the Nobel Prize in Economics for proving +Arrow’s Impossibility Theorem, which states that any voting system, either existing or yet to be created, in which the +only information available is the preference rankings of the candidates, will fail to satisfy at least one of the following +fairness criteria: the majority criterion, the Condorcet criterion, the monotonicity criterion, and the independence of +irrelevant alternatives criterion. This theorem only applies to a specific category of voting systems—those for which the +preference ranking is the only information collected. There are other types of voting systems to which the Impossibility +Theorem does not apply. For example, there is a class of voting systems called Cardinal voting systems that allow for +rating the candidates in some way. +“Rating” is different from “ranking” because a voter can give different candidates the same rating. Consider the five-star +rating systems used by various industries, or the thumbs up/thumbs down rating system used on YouTube. Could there +be a Cardinal voting system that does not violate any of the fairness criteria we have discussed? It’s possible, but more +research must be done in order to prove it. +PEOPLE IN MATHEMATICS +Kenneth J. Arrow +In 1972, Kenneth J. Arrow, a Harvard Professor of Economics, received the Nobel Prize in Economics jointly with Sir +John Hicks, another world renowned economist, for their contributions to economic theory. In particular, Professor +Arrow proved mathematically that no ranked voting system meets all four of the fairness criteria discussed in this +section. The statement of this fact is known as Arrow’s Impossibility Theorem. Visit this site for more details on +Professor Arrow (https://openstax.org/r/prizes/economic-sciences/1972/arrow/facts/) +Check Your Understanding +8. Which fairness criterion is violated by all four of the main ranked voting methods presented in this chapter? +9. Which of the four main ranked voting methods presented in this chapter satisfies the Condorcet criterion? +11.2 • Fairness in Voting Methods +1175 + +10. Which of the four main ranked voting methods presented in this section violates the majority criterion? +11. Which of the four main ranked voting methods presented in this section violates the monotonicity criterion? +12. According to Arrow’s Impossibility Theorem, which of the four main ranked voting methods presented in this +chapter violate at least one of the fairness criteria? +13. Determine whether the following statement is true or false and explain your reasoning: Any ranked election that +violates the majority criterion also violates the Condorcet criterion. +14. Determine whether the following statement is true or false and explain your reasoning: Any ranked election that +violates the Condorcet criterion also violates the majority criterion. +15. Does Arrow’s Impossibility Theorem apply to approval voting? Why or why not? +Determine whether each statement is true or false. Explain your reasoning. +16. Arrow’s Impossibility Theorem guarantees that ranked voting systems always lead to unfair elections. +17. Approval voting is in the class of voting systems called Cardinal Voting systems. +SECTION 11.2 EXERCISES +For the following exercises, identify which fairness criteria, if any, are violated by characteristics of the described voter +profile. Explain your reasoning. +1. In a plurality election, the candidates have the following vote counts: A 125, B 132, C 149, D 112. The pairwise +matchup points for each candidate would have been: A 1, B 3, C 1, D 1. +2. In a Borda count election, the candidates have the following Borda scores: A 1245, B 1360, C 787. Candidate A +received 55 percent of the first-place rankings. +3. In a pairwise comparison election, the four candidates initially received the following points for winning +matchups: A 2, B +, +, C 1, D +. When candidate C dropped out of the election, the remaining candidates +received: A 1, B +, D +. +4. In a Borda count election, the candidates have the following Borda scores: A 15, B 11, C 12, D 16. The pairwise +matchup points for the same voter profiles would have been: A 2, B 0, C 1, D 3. +5. In a Borda count election, the candidates have the following Borda scores: A 15, B 11, C 12, D 16. When +Candidate E was added to the ballot, the Borda scores became: A 25, B 21, C 15, D 24, E 18. +6. In a pairwise comparison election, Candidate C was a Condorcet candidate in a straw poll. When the actual +election took place, several voters up-ranked Candidate C on their ballots, but no other changes were made to +the voter preferences, and Candidate B won the election. +For the following exercises, use the table below. +Votes +49 +51 +Candidate A +3 +1 +Candidate B +1 +2 +Candidate C +2 +3 +7. In a pairwise comparison election, Candidate A was in first place, Candidate B was in second place, and +Candidate C was in third place. When the actual election tool place, the only changes were that several voters +down-ranked Candidate B on their ballots, but the outcome remained the same. +8. Determine the Borda score for each candidate and the winner of the election using the Borda count method. +9. Is there a majority candidate? If so, which candidate? +10. Does this Borda method election violate the majority criterion? Justify your answer. +11. Is there a Condorcet candidate? If so, which candidate? +12. Does this Borda method election violate the Condorcet criterion? Justify your answer. +13. If Candidate C is removed from the ballot, which candidate wins by the Borda count method? +14. Does this Borda count method election violate IIA? Justify your answer. +Use the table below for the following exercises. +1176 +11 • Voting and Apportionment +Access for free at openstax.org + +Votes +7 +9 +12 +15 +5 +Candidate A +4 +4 +1 +1 +4 +Candidate B +1 +1 +2 +2 +3 +Candidate C +3 +2 +3 +4 +1 +Candidate D +2 +3 +4 +3 +2 +15. Determine Borda score for each candidate and the winner of the election using the Borda count method. +16. Is there a majority candidate? If so, which candidate? +17. Does this Borda method election violate the majority criterion? Justify your answer. +18. Is there a Condorcet candidate? If so, which candidate? +19. Does the Borda method election violate the Condorcet criterion? Justify your answer. +20. Can an election that fails the majority criterion satisfy the Condorcet criterion? Why or why not? +For the following exercises, use the table below. +Number of Ballots +10 +7 +5 +5 +4 +Candidate A +1 +3 +3 +3 +4 +Candidate B +3 +2 +1 +4 +1 +Candidate C +2 +4 +2 +1 +2 +Candidate D +4 +1 +4 +2 +3 +21. Determine the Borda score for each candidate and the winner of the election using the Borda count method. +22. Is there a majority candidate? +23. Does the election violate the majority criterion? Justify your answer. +24. Determine the winner by pairwise comparison. +25. Is there a Condorcet candidate? +26. Does the Borda election violate the Condorcet criterion? Justify your answer. +27. Determine the winner by the ranked-choice method. +28. Does the ranked-choice election violate the majority criterion? Justify your answer. +29. Does the ranked-choice election violate the Condorcet criterion? Justify your answer. +30. Can an election that fails the Condorcet criterion satisfy the majority criterion? Why or why not? +Use the table below for the following exercises. +Number of Ballots +49 +48 +3 +Candidate A +1 +3 +3 +Candidate B +2 +1 +2 +Candidate C +3 +2 +1 +31. Determine the winner of the election using the plurality method. +32. Determine the winner by pairwise comparison. +33. Is there a Condorcet candidate? +34. Does this plurality election violate the Condorcet criterion? Justify your answer. +35. If Candidate C is removed from the ballot, which candidate wins by plurality? +11.2 • Fairness in Voting Methods +1177 + +36. Does this plurality violate the IIA? Explain your reasoning. +Use the sample ballot summary below for the following exercises. +Number of Ballots +16 +20 +24 +8 +Candidate A +1 +3 +2 +1 +Candidate B +3 +1 +4 +4 +Candidate C +4 +2 +1 +2 +Candidate D +2 +4 +3 +3 +37. Determine the winner of the election using the ranked-choice method. +38. Determine the winner by pairwise comparison. +39. Is there a Condorcet candidate? +40. Does this ranked-choice election violate the Condorcet criterion? Justify your answer. +41. If the four voters in the last column rank Candidate C ahead of A, which candidate wins by the ranked-choice +method? +42. Does this ranked-choice election violate the monotonicity criterion? Explain your reasoning. +Use the sample ballot summary below for the following exercises. +Number of Ballots +15 +12 +9 +3 +Candidate A +1 +3 +3 +2 +Candidate B +2 +2 +1 +1 +Candidate C +3 +1 +2 +3 +43. Determine the winner of the election using the ranked-choice method. +44. How could it be demonstrated that this ranked-choice election violates IIA? +45. Determine the winner of the election by the Borda method. +46. Does this Borda method election violate the IIA? Why or why not? +47. Does this Borda method election violate the monotonicity criterion? Why or why not? +Use the pairwise comparison matrix in the given figure for the following exercises. +48. Which candidate wins the pairwise election? +49. Determine the winner by pairwise comparison if N were removed from the ballot. +50. Determine the winner by pairwise comparison if M were removed from the ballot. +1178 +11 • Voting and Apportionment +Access for free at openstax.org + +51. Does this pairwise election satisfy the IIA? +11.3 Standard Divisors, Standard Quotas, and the Apportionment +Problem +Figure 11.15 Every person at a party gets their fair slice of the cake. (credit: “apple spice cake” by Mark Bonica/Flickr, CC +BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Analyze the apportionment problem and applications to representation. +2. +Evaluate applications of standard divisors. +3. +Evaluate applications of standard quotas. +The Apportionment Problem +In the new democracy of Imaginaria, there are four states: Fictionville, Pretendstead, Illusionham, and Mythbury. Each +state will have representatives in the Imaginarian Legislature. You might now have an agreement on which voting +method your citizens will use to elect representatives. However, before that process can even begin, you must decided on +how many representatives each state will receive. This decision will present its own challenges. +When sharing your birthday cake, it’s only fair that everyone gets the same portion size, right? You were portioning the +cake by dividing it up equally and giving everyone a slice. A great thing about cake is that you can slice it any way you +want, but how do you apportion, or divide and distribute, items that can't be sliced? Suppose that you have a box of 16 +Ring Pops™, gem-shaped lollipops on a plastic ring. You are going to share the box with four other kids. Dividing the 16 +Ring Pops™ among the group of five leads to a problem; after each person in the group gets three Ring Pops™, there is +still one left! Who gets the last one? The apportionment problem is how to fairly divide, or apportion, available +resources that must be distributed to the recipients in whole, not fractional, parts. +The apportionment problem applies to many aspects of life, including the representatives in the Imaginarian legislature. +The table below provides a short list of examples of resources that must be apportioned in whole parts, and the +recipients of those resources. +Resource +Recipients +Covid-19 Vaccines +Nations around the world +Airport Terminals +Airlines +Faculty Positions at a University +Departments +Public Schools +Communities +11.3 • Standard Divisors, Standard Quotas, and the Apportionment Problem +1179 + +Resource +Recipients +U.S. House of Representatives Seats +States +Parliamentary Seats +Political Parties +Fair division of a resource is not necessarily equal division of the resource like when distributing cake slices. When +distributing airport terminals amongst airlines, there are many factors to consider such as the size of the airline, the +number and types of aircraft they have, and the demand for the service. In most cases, fairness is defined as being +proportional; two quantities are proportional if they have the same relative size. In the case of the Covid-19 vaccine, the +expectation would be that countries with larger populations get more doses of the vaccine. In the Imaginarian +legislature, the expectation may be that the states with larger populations will receive the larger number of +representatives. This concept is referred to as a part-to-part ratio. +Suppose that a supermarket has a special on pies, two for $5. The first customer purchases four pies for $10, and the +second customer purchases eight pies for $20. The dollar to pie ratio for the first customer is +and the dollar to pie ratio for the second customer is +. So, +the dollar to pie ratio is constant. Although the customers do not spend the same amount of money, the amount each +spent was proportional to the number of pies purchased. +Now suppose that the supermarket changed the special to $5 for the first pie, and $2 for each additional pie. In that +case, four pies would cost +, while 8 pies would cost +. The dollar to pie ratios would be +and +, respectively. This special does not result in a +constant part to part ratio. The dollars spent are not proportional to the number of pies purchased. +VIDEO +What Is a Ratio? (https://openstax.org/r/ratios_proportions) +What Are the Different Types of Ratios? (https://openstax.org/r/ratio_types) +EXAMPLE 11.24 +Ratio of Faculty to Students at a College +The following table provides a comparison of the number of faculty members in each department at a particular college +to the student head count in that department and the number of class sections in that department in the Spring +semester. Use this information to answer the questions. +Department +Mathematics +English +History +Science +(S) Student Head Count +4800 +2376 +1536 +2880 +(C) Class Sections +120 +108 +48 +96 +(T) Total Faculty +30 +27 +12 +24 +(F) Full-Time Faculty +10 +9 +4 +8 +(P) Part-Time Faculty +20 +18 +8 +16 +1. +Determine the ratios for each department: S to C, C to T, S to T, F to P +2. +What are the units of the ratios that you found? +3. +Which of these pairs, if any, has a constant part to part ratio? State the ratio. +4. +Does it appear that the total number of faculty positions were allocated to each department based on student head +count, the number of class sections, or neither? Justify your answer. +1180 +11 • Voting and Apportionment +Access for free at openstax.org + +Solution +1. +Divide the first quantity by the second, for each department, as shown in the table below. +2. +Answers are provided in last column of the table. +Department +eMathematics +English +History +Science +Units of Ratios Found +S to C +Students per class section +C to T +Class sections per faculty member +S to T +Students per faculty member +F to P +Full-time faculty member per part- +time faculty member +3. +The ratio of class sections to faculty members is a constant ratio of four. The ratio of full-time faculty to part-time +faculty is a constant ratio of +. +4. +It appears that the faculty positions were allocated based on the number of class sections because there is a +constant ratio of four class sections per faculty member. +YOUR TURN 11.24 +The SAT is to be administered at a high school. In preparation, pencils have been distributed to each of the +classrooms based on the room capacity. Use the information in the following table to answer each question. +Room Number +B +C +D +E +Room Capacity (number of student desks) +24 +18 +32 +22 +Number of Pencils +36 +27 +48 +33 +1. Find the part to part ratio of desks to pencils for each room. Represent it as both a reduced fraction and a +decimal rounded to the nearest hundredth as needed. +2. Find the part to part ratio of pencils to desks for each room. Represent it as both a reduced fraction and a +decimal rounded to the nearest hundredth as needed. +3. Have the pencils been distributed proportionally? If so, what is the constant ratio of pencils to desks? Give the +units. +There are some useful relationships between quantities that are proportional to each other. When there is a constant +ratio between two quantities, the one quantity can be found by multiplying the other by that ratio. Remember the +supermarket special on pies, 2 pies for $5? The ratio of dollars to pies is +and the ratio of +pies to dollars is +. These two values are reciprocals of each other, +and +. +This means that multiplying by one has the same effect as dividing by the other. This also means that knowing either +constant ratio allows us to calculate the price given the number of pies. To find the cost of 20 pies, multiply by the ratio +of dollars to pies or divide by the ratio of pies to dollars. +• +• +These patterns are true in general. +11.3 • Standard Divisors, Standard Quotas, and the Apportionment Problem +1181 + +FORMULA +Let +be a particular item and +another such that there is a constant ratio of +to +• +and +• +• +EXAMPLE 11.25 +Ratio of Faculty to Students at a College +Refer to the information given in Example 11.24. +1. +If there are 32 class sections each semester in the Fine Art department, and the same ratio is used to determine the +number of faculty members, how many faculty members would you expect to see in the Fine Art department? +2. +If the Health Sciences department has 6 full-time faculty members, how many part-time faculty members are in the +department? +Solution +1. +Multiply the number of class sections by the ratio of faculty members per class section to find the number of faculty. +Since there are 4 faculty members per class, the number of faculty members in 32 classes should be +faculty members. +2. +Multiply the number of full-time faculty by the ratio of part-time to full-time to find the number of part-time. Since +ratio of full-time faculty to part-time faculty at the college is +or 1 full-time per 2 part time, the ratio of part-time to +full-time is +part-time to 1 full-time; so the number of part-time faculty in a department with 6 full-time faculty +should be +part-time faculty. +YOUR TURN 11.25 +Refer again to the table providing information on classroom capacities and pencil distribution. +Room Number +B +C +D +E +Room Capacity (number of student desks) +24 +18 +32 +22 +Number of Pencils +36 +27 +48 +33 +1. Determine the number of pencils that would be allocated to a classroom F with 28 desks by multiplying the +number of desks by the ratio of pencils to desks. +2. Determine the number of pencils that would be allocated to a classroom F with 28 desks by dividing the +number of desks by the ratio of desks to pencils. +3. Assuming pencils continued to be distributed in the same manner, if there were another classroom, G, that +received 51 pencils, how many desks would we expect to find in the room? +The apportionment application that will be important to the founders of Imaginaria occurs in representative +democracies in which elected persons represent a group. The United Kingdom, France, and India each have a +parliament, and the United States has a Congress, just as Imaginaria will have a legislature! The citizens of a country +must decide what portion of the representatives each group, such as a state or province or even a political party, will +have. A larger portion of representatives means greater influence over policy. +1182 +11 • Voting and Apportionment +Access for free at openstax.org + +EXAMPLE 11.26 +Ratio of U.S. Representatives to State Population +Table 11.4 contains a list of the five U.S. states with the greatest number of representatives in the U.S. House of +Representatives, along with the population of that state in 2021. Use the information in the table to answer the +questions. +State +Representative Seats +State Population +(CA) California +53 +39,613,000 +(TX) Texas +36 +29,730,300 +(NY) New York +27 +19,300,000 +(FL) Florida +27 +21,944,600 +(PA) Pennsylvania +18 +12,804,100 +Table 11.4 The First through Fifth Ranked States by Number of +Representatives (sources: https://www.census.gov/popclock/ [state +population], https://www.britannica.com/topic/United-States- +House-of-Representatives-Seats-by-State-1787120 [representative +seats]) +1. +What is the ratio of State Population to Representative Seats for each state to the nearest hundred thousand? +2. +What is the ratio of Representative Seats to State Population for each state rounded to seven decimal places? +3. +What is the ratio of Representative Seats to State Population for each state rounded to six decimal places? +4. +Does there appear to be a constant ratio? Justify your answer. +Solution +1. +CA 700,000; TX 800,000; NY 700,000; FL 800,000; PA 700,000 +2. +CA 0.0000013; TX 0.0000012; NY 0.0000014; FL 0.0000012; PA 0.0000014 +3. +CA 0.000001; TX 0.000001; NY 0.000001; FL 0.000001; PA 0.000001 +4. +The ratio of State Population to Representative Seats seems to be either 700,000 or 800,000. There does appear to +be a constant ratio of about 0.000001 of Representative Seats to State Population if we round off to the sixth +decimal place. +YOUR TURN 11.26 +Table 11.9 contains a list of the five U.S. states ranked sixth through tenth in the number of representatives in the +U.S. House of Representatives, along with the population of that state in 2021. Use the information in the table to +answer the questions. +State +Representative Seats +State Population +(IL) Illinois +18 +12,804,100 +(OH) Ohio +16 +11,714,600 +Table 11.5 The Sixth through Tenth Ranked U.S. States by Number of +Representatives (sources: https://www.census.gov/popclock/ [state +population], https://www.britannica.com/topic/United-States-House- +of-Representatives-Seats-by-State-1787120 [representative seats]) +11.3 • Standard Divisors, Standard Quotas, and the Apportionment Problem +1183 + +State +Representative Seats +State Population +(MI) Michigan +14 +9,992,430 +(GA) Georgia +14 +10,830,000 +(NC) North Carolina +13 +10,701,000 +Table 11.5 The Sixth through Tenth Ranked U.S. States by Number of +Representatives (sources: https://www.census.gov/popclock/ [state +population], https://www.britannica.com/topic/United-States-House- +of-Representatives-Seats-by-State-1787120 [representative seats]) +1. What is the ratio of State Population to Representative Seats for each state to the nearest hundred thousand? +2. What is the ratio of Representative Seats to State Population for each state rounded to seven decimal places? +3. What is the ratio of Representative Seats to State Population for each state rounded to six decimal places? +4. Does there appear to be a constant ratio? Are the results similar to the top five states? +VIDEO +Math Antics – Rounding (https://openstax.org/r/rounding) +You might be wondering why the ratio doesn't appear to be quite the same depending on the rounding of the values. +We will see that the key to this variation lies in the fractions. Just like the five children sharing 16 Ring Pops™, there are +going to be leftovers and there are many methods for deciding what to do with those leftovers. +The Standard Divisor +There are two houses of congress in the United States: the Senate and the House of Representatives. Each state has two +senators, but the number of representatives depends on the population of the state. The number of representative seats +in the U.S. House of Representatives is currently set by law to be 435. In order to distribute the seats fairly to each state, +the ratio of the population of the U.S. to the number of representative seats must be calculated. The ratio of the total +population to the house size is called the standard divisor, and it is the number of members of the total population +represented by one seat. +Although apportionment applies to many other scenarios, such as the pencil distribution during the SAT, the terminology +of apportionment is based on the House of Representatives scenario. Thus, several government-related terms take on a +more general meaning. The states are the recipients of the apportioned resource, the seats are the units of the +resource being apportioned, the house size is the total number of seats to be apportioned, the state population is the +measurement of the state's size, and the total population is the sum of the state populations. +FORMULA +EXAMPLE 11.27 +The Standard Divisor of the U.S. House of Representatives 2021 +As of this writing, the Census.gov website U.S. Population clock showed a population of 332,693,997. There are 435 seats +in the U.S. House of Representatives. Find the standard divisor rounded to the nearest tenth. +Solution +Dividing +people by 435 seats, there are 758,960.6 people per representative. +1184 +11 • Voting and Apportionment +Access for free at openstax.org + +YOUR TURN 11.27 +1. By the end of the first U.S. Congress in 1791, there were 13 states, 65 representative seats, and approximately +3,929,214 citizens. Find the standard divisor rounded to the nearest tenth. +Whether the standard divisor is less than, equal to, or greater than 1 depends on the ratio of the population to the +number of seats. +• +The standard divisor will be equal to 1 if the total population is equal to the number of seats. This would mean that +each member of the population is allocated their own personal seat. +• +The standard divisor will be a number between 0 and 1 when the total population is less than the number of seats. +This means that each member of the population is allocated more than one seat. +• +The standard divisor will be a number greater than 1when the total population is greater than the number of seats. +This means that a certain number of members of the population will share 1 seat. +If the population is five children and the house consists of five pieces of candy, the standard divisor is +meaning each child gets one candy. If the population is five children and ten pieces of +candy, the standard divisor is +meaning that each child gets more than one candy. If the +population is five children and four pieces of candy, the standard divisor is +meaning +that each child gets less than one candy. +If the seats in the Imaginarian legislature are distributed to the states based on population, then the house size will be +less than the population and we should expect the standard divisor to be a number greater than 1. +EXAMPLE 11.28 +School Resource Officers in Brevard County, Florida +The public schools in a certain county have been allotted 349 school resource officers to be distributed among 327 public +schools attended by approximately 271,500 students. +1. +Identify the states, seats, house size, state population, and total population in this apportionment scenario. +2. +Describe the ratio the standard divisor represents in this scenario and calculate the standard divisor to the nearest +tenth. +Solution +1. +The states are the schools in that county. The seats are the school resource officers. The house size is the number of +school resource officers, which is 349. The state population is the number of students in a particular school, which +was not given. The total population consists of the sum of the school populations, which is 271,500. +2. +The standard divisor is the ratio of the total population to the house size, which is the number of students served by +each resource officer. Divide +students per officer. +YOUR TURN 11.28 +The Hernandez family and the Higgins family went trick-or-treating together for Halloween last year. They returned +with 313 pieces of candy, which they will now apportion to the families. The Hernandez family has three children and +the Higgins family has four children. +1. Identify the states, seats, house size, state population, and total population in this apportionment scenario. +2. Describe the ratio the standard divisor represents in this scenario and calculate the standard divisor to four +decimal places. +The Standard Quota +Once the standard divisor for the Imaginarian legislature is calculated, the next task is to determine the number of seats +that each state should receive, which is referred to as the state’s standard quota. Unless all the states have the same +population, each state will receive a different number of seats because the quantities will be proportionate to the state +populations. To determine those amounts, we will use an idea we learned earlier. Recall that, when the number of units +11.3 • Standard Divisors, Standard Quotas, and the Apportionment Problem +1185 + +of item +is proportionate to the number of units of item +, we have: +In this case, we are trying to calculate the number of seats a state should be apportioned, the state’s standard quota. So +So +would refer to seats allocated to a particular state, while +would refer to the state population. This means that +the ratio of +to +is the ratio of the total population to house size, which is the standard divisor. So in apportionment +terms, we have the following formula. +FORMULA +EXAMPLE 11.29 +The Standard Quota of the U.S. House of Representatives 2021 +Example 27 outlined that the Census.gov website U.S. Population clock showed a population of 330,147,881, there are +435 seats in the U.S. House of Representatives, and the standard divisor was 758,960.6 people per representative. The +state of California has a population of approximately 39,613,000. Use these values to determine the standard quota for +California to two decimal places. +Solution +representatives +YOUR TURN 11.29 +1. By the end of the first U.S. Congress in 1791, there were 13 states, 65 representative seats, and approximately +3,929,214 citizens. In that year, the state of Delaware had a population of approximately 59,000. people. Use this +information and the standard divisor you found in Your Turn 11.27 to find Delaware's standard quota rounded to +two decimal places. +EXAMPLE 11.30 +Apportionment of Laptops in a Science Department +The science department of a high school has received a grant for 34 laptops. They plan to apportion them among their +six classrooms based on each classroom’s student capacity. Use the values in the table below to find the standard quota +for each classroom. +Room +Students +A +30 +B +25 +C +28 +D +32 +E +24 +F +27 +1186 +11 • Voting and Apportionment +Access for free at openstax.org + +Solution +Step 1: Identify the state population, total population, and the house size. The states are the classrooms, and the state +populations are listed in the table. The total population is the sum of the state populations, which is 166. The house size +is the number of seats, or laptops, to be allocated, which is 34. +Step 2: Calculate the standard divisor by dividing the total population by the house size. +students per laptop. +Step 3: Calculate the standard quota by dividing the state population by the standard divisors , as shown in the table +below. +Room +Room Capacity +Room’s Standard Quota +A +30 +laptops +B +25 +laptops +C +28 +laptops +D +32 +laptops +E +24 +laptops +F +27 +laptops +Step 4: Find the sum of the standard quotas. +. This is only slightly off +from the number of laptops—34—which can be caused by rounding off in previous steps. This is a good indication that +the calculations were correct. If you find that the value of the sum of the standard quotas is significantly different from +the house size (number of seats), it is possible that the standard divisor was calculated using too few decimal places. +Calculate the standard divisor and standard quotas again but round off to a greater number of decimal places. +YOUR TURN 11.30 +1. This year the Hernandez family and the Higgins family were joined by the Ho family for Halloween trick-or- +treating. The Hernandez family has three children, the Higgins family has four children, and the Ho family has +two children. This time, they collected 527 pieces of candy, which they are going to apportion based on the +number of children. Find the standard quota for each family. Round all values to four decimal places. If +traditional rounding methods are applied to determine the actual number whole number values of pieces of +candy received by each family, do the values sum to 527? +Check Your Understanding +In the following questions, assume there is a constant ratio between units of A and units of B. Two students are having +a discussion. Determine who is correct: Student 1, Student 2, both, or neither. +18. Student 1 says that the number of units of A is the product of the number of units of B times the ratio of A to B. +Student 2 says that the number of units of A is the quotient of the number of units of B divided by the ratio of A +to B. +19. Student 1 says that the number of units of A is the quotient of the number of units of B divided by the ratio of B +to A. +Student 2 says that the number of units of B is the quotient of the number of units of A divided by the ratio of A +to B. +20. Suppose there are 110 of item A and 55 of item B. +11.3 • Standard Divisors, Standard Quotas, and the Apportionment Problem +1187 + +Student 1 says the ratio of A to B is +Student 2 says the ratio of B to A is +. +21. Suppose the constant ratio of A to B is 0.75. Student 1 says the ratio of A to B is +Student 2 says the ratio of B +to A is +. +22. Student 1 says that +. Student 2 says that +. +23. Student 1 says that +. +Student 2 says that +. +24. Student 1 says that +. +Student 2 says that +SECTION 11.3 EXERCISES +For the following exercises, identify the states, the seats, and the state population (the basis for the apportionment) in +the given scenarios. +1. A parent has 25 pieces of candy to split among their four children. They will earn the candy based on how many +minutes of chores they children did this week. +2. The board of trustees of a college has recently approved the installation of 70 new emergency blue lights in five +parking lots. The number of lights in each lot will be proportionate to the size of the parking lot, which is to be +measured in acres. +3. The reading coach at an elementary school has 52 prizes to distribute to students as a reward for time spent +reading. +4. “Top officials from Operation Warp Speed, the [U.S.] government’s program to fast-track the development and +delivery of COVID-19 vaccines, announced they’ve allocated 6.4 million doses of COVID-19 vaccines to states +based on their total populations.” (The Coronavirus Crisis, by Pien Huang, Shots Health News From NPR, +npr.org, November 24, 2020) +5. Refer to question 4, except suppose that the COVID-19 vaccine allocations were based on the most vulnerable +population, residents aged 65 and over. +For the following exercises, use the given information to find the standard divisor to the nearest hundredth. Include the +units. +6. The total population is 2,235 automobiles, and the number of seats is 14 warehouses. +7. The total population is 135 hospitals, and the number of seats is 200 respirators. +For the following exercises, use the given information to find the standard quota. Include the units. +8. The state population is eight residents in a unit, and the standard divisor is 1.75 residents per parking space. +9. The state population is 52 ICU patients each week, and the number of seats is 6.5 patients per respirator. +10. The total population is 145 basketball players, the number of seats is 62 trophies, and the state population is 14 +basketball players on Team Tigers. +11. The total population is 12 giraffes, the number of seats is nine water troughs, and the state population is three +giraffes in Enclosure C. +For the following exercises, use the table below, which shows student head count, class section, and total faculty in +each of four college departments. +Department +(M) Math +(E) English +(H) History +(S) Science +(O) College Overall +(T) Student Head Count +4800 +2376 +1536 +2880 +87118 +(C) Class Sections +120 +108 +48 +96 +3712 +(F) Faculty Members +30 +27 +12 +24 +928 +12. Determine the F to T ratios for each department rounded to four decimal places as needed. What are the units? +13. Determine the C to F ratios for each department rounded to four decimal places as needed. What are the units? +14. What is the F to T ratio for the college overall? Include units. How does it compare to the F to T ratios for +1188 +11 • Voting and Apportionment +Access for free at openstax.org + +individual departments? +15. What is the overall C to F ratio? Include units. How does it compare to the C to F ratios for individual +departments? +16. Does there appear to be a constant F to T ratio? If so, what is the ratio? If not, what implications does this have +about the different departments? +17. Does there appear to be a constant C to F ratio? If so, what is the ratio? If not, what implications does this have +about the different departments? +18. If the departments are the states, the students are the population, and the faculty members are the seats, use +the College Overall column to determine the standard divisor for the apportionment of the faculty rounded to +two decimal places as needed. Include the units. +19. If the departments are the states, the classes are the population, and the faculty members are the seats, use +the Overall College column to determine the standard divisor rounded to two decimal places as needed. +Include the units. +20. Use the standard divisor from question 18 to find the standard quota for each department rounded to two +decimal places as needed. What are the units? +21. Use the standard divisor from question 19 to find the standard quota for each department rounded to two +decimal places as needed. +For the following exercises, use this information: Wakanda, the domain of the Black Panther, King T’Challa has six +fortress cities. In Wakandan, the word “birnin” means “fortress city.” King T’Challa has found 111 Vibranium artifacts +that must be distributed among the fortress cities of Wakanda. He has decided to apportion the artifacts based on the +number of residents of each birnin. The table below displays the populations of major Wakandan cities. +Fortress +Cities +Birnin +Djata (D) +Birnin +T’Chaka (T) +Birnin +Zana (Z) +Birnin +S’Yan (S) +Birnin +Bashenga (B) +Birnin +Azzaria (A) +Total +Population +Residents +26,000 +57,000 +27,000 +18,000 +64,000 +45,000 +237,000 +22. Identify the states, the seats, and the state population (the basis for the apportionment) in this scenario. +23. Find the standard divisor for the apportionment of the Vibranium artifacts. Round to the nearest tenth as +needed. Include the units. +24. Find each birnin’s standard quota for the apportionment of the Vibranium artifacts. Round to the nearest +hundredth as needed. What are the units? +25. Find the sum of the standard quotas. Is it reasonably close to the number of artifacts available for distribution? +For the following exercises, suppose that 6.4 million doses of COVID-19 vaccine are to be distributed among U.S. states. +The vaccines will either be distributed based on the total state population or based on the number of people over 65 +years old, as shown in the table below. +State +State Population +State Population Age 65+ +Percentage of State Population 65+ +(CA) California +39,613,000 +5,669,000 +14.3% +(TX) Texas +29,730,300 +3,602,000 +12.6% +(NY) New York +19,300,000 +3,214,000 +16.4% +(FL) Florida +21,944,600 +4,358,000 +20.5% +(PA) Pennsylvania +12,804,100 +2,336,000 +18.2% +(US) United States +330,151,000 +52,345,000 +15.8% +26. Find the standard divisor for the apportionment of the vaccine doses by population using the estimate for the +total U.S. population. Round to the nearest tenth as needed. Include the units. +27. Find each state's standard quota for the apportionment of the vaccine doses. Round to the nearest tenth as +11.3 • Standard Divisors, Standard Quotas, and the Apportionment Problem +1189 + +needed. What are the units? +28. Find the standard divisor for the apportionment of the vaccine doses by population age 65 and older using the +estimate for the total U.S. population of people aged 65 and older. Round to the nearest tenth as needed. +Include the units. +29. Find each state's standard quota for the apportionment of the vaccine doses by total state population. Round to +the nearest tenth as needed. What are the units? +30. Compare the standard quota for each state based on the entire state population to the standard quota for each +state based on the portion of the population age 65 and older. Which states would receive more doses of +vaccine if the apportionment were based on the population of people age 65 and older? +31. Approximately 15.8 percent of U.S. residents are age 65 and older. +a. +Which of the five states listed have a percentage of residents age 65 and older greater than 15.8 percent? +b. +Which of the five states listed have a percentage of residents age 65 and older less than 15.8 percent? +c. +Explain the correlation. +For the following exercises, use this information: Children from five families—the Chorro family, the Eswaran family, the +Javernick family, the Lahde family, and the Stolly family—joined a town-wide Easter egg hunt. When they returned with +their baskets, they had 827 eggs! They decided to share their eggs among the families based on the number of children +in each family, as shown in the table below. +Family +Number of Children +(C) Chorro +3 +(E) Eswaran +2 +(J) Javernick +4 +(L) Lahde +1 +(S) Stolly +5 +32. Identify the states, the seats, and the state population (the basis for the apportionment) in this scenario. +33. Find the standard divisor for the apportionment of the Easter eggs. Round to five decimal places as needed. +Include the units. +34. Find each family’s standard quota for the apportionment of the Easter eggs. Round to the nearest hundredth as +needed. What are the units? +35. Find the sum of the standard quotas from exercise 34. Is the sum reasonably close to the number of Easter +eggs available for distribution? +1190 +11 • Voting and Apportionment +Access for free at openstax.org + +11.4 Apportionment Methods +Figure 11.16 Schoolchildren depend on apportionment of resources like laptops among schools and classroom. (credit: +“Richmond Public Schools” by Virginia Department of Education/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Describe and interpret the apportionment problem. +2. +Apply Hamilton’s Method. +3. +Describe and interpret the quota rule. +4. +Apply Jefferson’s Method. +5. +Apply Adams’s Method. +6. +Apply Webster’s Method. +7. +Compare and contrast apportionment methods. +8. +Identify and contrast flaws in various apportionment methods. +A Closer Look at the Apportionment Problem +In Standard Divisors, Standard Quotas, and the Apportionment Problem we calculated the standard divisor and the +standard quotas in various apportionment scenarios. The results of those calculations routinely led to fractions and +decimals of units. However, the seats in the House of Representatives, laptops in a classroom, or a variety of other +resources, are indivisible, meaning they cannot be divided up into fractional parts. This leaves a decision to be made. For +example, if the standard quota for the number of laptops to be distributed to a classroom is 12.44 units, how do we deal +with the fractional part of 0.44? It is unclear if the classroom should receive 12 units, 13 units, or some other value. Let’s +try traditional rounding to the nearest whole number value. +EXAMPLE 11.31 +Installing Emergency Lights +The board of trustees of a college has recently approved the installation of 70 new emergency blue lights in three +parking lots. The number of lights in each lot will be proportionate to the size of the parking lot, which isto be measured +in acres. The total number of acres is 34; so the standard divisor is +. The standard quota for each lot is listed +in the table below. Use this information to answer each question. +11.4 • Apportionment Methods +1191 + +Lot +Acres +Lot’s Standard Quota +A +15 +emergency blue lights +B +9 +emergency blue lights +C +10 +emergency blue lights +1. +Use traditional rounding to determine the number of lights assigned to each lot. +2. +Find the sum of the values from part 1. +3. +Does the sum found in part 2 equal the number of lights available? +Solution +1. +If traditional rounding is used, there will be 31, 19, and 21 lights distributed to each lot, respectively. +2. +The total of these values is 71. +3. +No, the total from part 2 is one more than the number of lights available. In other words, one of the parking lots +must get 1 fewer light than apportioned. +YOUR TURN 11.31 +The science department of a high school has received a grant for 34 laptops. They plan to apportion them among +their six classrooms based on each classroom’s student capacity. Use the standard quotas in the table below to +answer each question. +Room +Room Capacity +Room’s Standard Quota +A +30 +laptops +B +25 +laptops +C +28 +laptops +D +32 +laptops +E +24 +laptops +F +27 +laptops +1. Use traditional rounding to determine the number of laptops assigned to each classroom. +2. Find the sum of the values from part 1. +3. Does the sum found in part 2 equal the number of laptops available? +Example 11.31 demonstrates that we cannot successfully apportion indivisible resources by rounding off each standard +quota using traditional rounding. This leaves us with a problem. What is a fair way to distribute the fractional parts of the +standard quotas? We will refer to this as the apportionment problem. Several methods for making this decision will be +discussed. +EXAMPLE 11.32 +A-10C Thunderbolt II Aircraft +In 2015, the U.S. Air Force had a fleet of approximately 281 A-10C Thunderbolt II aircraft. Suppose that the Air Force +1192 +11 • Voting and Apportionment +Access for free at openstax.org + +administration wanted to distribute 27 aircrafts across six bases based on the number of qualified pilots stationed at +those bases. Use the information in the table below to answer each question. +Base +Pilots +(A) Alpha +13 +(B) Bravo +12 +(C) Charlie +5 +(D) Delta +16 +(E) Echo +7 +(F) Foxtrot +9 +1. +Identify the states, the seats, and the state population (the basis for the apportionment) in this scenario. +2. +Find the standard divisor for the apportionment of the aircraft. Round to four decimal places as needed. Include the +units. +3. +Find each Air Force base’s standard quota for the apportionment of the aircraft. Round to the nearest hundredth as +needed. What are the units? +4. +How does this example demonstrate the apportionment problem? Will traditional rounding solve the problem? +Solution +1. +The states are the bases, the seats are the aircraft, and the state populations are the pilots at a given base. +2. +pilots per aircraft. +3. +A +, B +, C +, D +, E +, F +. The units are +aircraft. +4. +This example demonstrates the apportionment problem because it is not possible to send a fractional number of +aircraft to an Air Force base. On the other hand, if we use traditional rounding methods to get whole numbers, the +results are +aircraft will be apportioned, which is one less than the number of aircraft that +were supposed to beapportioned. +YOUR TURN 11.32 +1. The reading coach at an elementary school has 13 gift cards to distribute to their three students as a reward for +time spent reading. When they calculated the standard quota for each student based on the number of minutes +they student had read, the results were: 4.49 gift cards, 4.03 gift cards, and 4.48 gift cards. How does this +demonstrate the apportionment problem? +Hamilton's Method of Apportionment +One of the problems encountered when standard quotas are transformed into whole numbers using traditional +rounding is that it is possible for the sum of the values to be greater than the number of seats available. A reasonable +way to avoid this is to always round down, even when the first decimal place is five or greater. For example, a standard +quota of 12.33 and a standard quota of 12.99 would both round down to 12. This is called the lower quota. +EXAMPLE 11.33 +Lower Quota for Apportionment of Aircraft +The Air Force administration wants to distribute 27 aircrafts across six bases based on the number of qualified pilots +stationed at those bases. The standard quotas for each base are listed in the table below. Use this information to answer +11.4 • Apportionment Methods +1193 + +the questions. +Base +Standard Quota +(A) Alpha +aircraft +(B) Bravo +aircraft +(C) Charlie +aircraft +(D) Delta +aircraft +(E) Echo +aircraft +(F) Foxtrot +aircraft +1. +Give the lower quota for each Air Force base. +2. +Find the sum of the lower quotas. By how much does this sum fall short of the actual number of aircraft? +Solution +1. +Round down. The lower quota for each Air Force base is 5, 5, 2, 6, 3, 3, respectively. +2. +The sum is 24. This is 3 fewer than the actual number of aircraft. +YOUR TURN 11.33 +The apportionment of 70 new emergency blue lights in three parking lots is based on acreage. The standard quota +for each lot is listed in the table below. Use this information to answer each question. +Lot +Acres +Lot’s Standard Quota +A +15 +emergency blue lights +B +9 +emergency blue lights +C +10 +emergency blue lights +1. Give the lower quota for each parking lot. +2. Find the sum of the lower quotas. +3. By how much does this sum fall short of the actual number of emergency lights? +If the standard quotas are all rounded down, their sum will always be less than or equal to the house size. Then, it would +only remain to find a fair way to distribute any remaining seats. Alexander Hamilton, who was a general in the American +Revolution, author of the Federalist Papers, and the first secretary of the treasury, took this approach to apportionment. +Steps for Hamilton’s Method of Apportionment +There are five steps we follow when applying Hamilton’s Method of apportionment: +1. +Find the standard divisor. +2. +Find each state’s standard quota. +3. +Give each state the state’s lower quota (with each state receiving at least 1 seat). +4. +Give each remaining seat one at a time to the states with the largest fractional parts of their standard quotas until +1194 +11 • Voting and Apportionment +Access for free at openstax.org + +no seats remain. +5. +Check the solution by confirming that the sum of the modified quotas equals the house size. +VIDEO +Hamilton Method of Apportionment (https://openstax.org/r/Hamiltons_method) +EXAMPLE 11.34 +Hawaiian School Districts +Suppose that the Hawaii State Department of Education has a budget for 616 schools and is doing a research study to +determine the equitable number of schools to have in each of the five counties based on the residents under 19 years +old, This data is provided in the table below. Using the Hamilton method, calculate how many schools would be funded +in each state. +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Residents under age 19 +46,310 +224,230 +20 +16,560 +38,450 +325,570 +Solution +Step 1: Calculate the standard divisor. Divide the total population, 325,570, by the house size, 616 seats. The standard +divisor is 528.52. +Step 2: Find each state’s standard quota: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Standard +Quota +616 +Step 3: Find each state’s lower quota and their sum: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Lower Quota +87 +424 +1 +31 +72 +615 +Step 4: Compare the sum of the states’ lower quotas, 615, to the house size, 616. One seat remains to be apportioned +and must be given to the state with the largest fractional part: Maui with 0.75. So, the final Hamilton quotas are as +follows: Hawaii 87, Honolulu 424, Kalawao 1, Kauai 31, and Maui 73. +Step 5: Find the total to confirm the sum of the quotas equals the house size, 616. Then +. +The apportionment is complete. +11.4 • Apportionment Methods +1195 + +YOUR TURN 11.34 +1. In the country of Imaginaria, there will be four states: Fictionville, Pretendstead, Illusionham, and Mythbury. +Suppose there will be 35 seats in the legislature of Imaginaria. Use Hamilton’s method of apportionment to +determine the number of seats in each state based on the populations in the following table. +Fictionville +Pretendstead +Illusionham +Mythbury +Total +Population +71,000 +117,000 +211,000 +1,194,000 +1,593,000 +TECH CHECK +Apportionment Calculators +Check out websites such as Ms. Hearn Math (https://openstax.org/r/hamilton-calculator) for a free Hamilton +apportionment calculator. +This can be a useful tool to confirm your results! +The Quota Rule +A characteristic of an apportionment that is considered favorable is when the final quota values all either result from +rounding down or rounding up from the standard quotas. The value that results from rounding down is called the lower +quota, and the value that results from rounding up is called the upper quota. +As we explore more methods of apportionment, we will consider whether they satisfy the quota rule. If a scenario exists +in which a particular apportionment allocates a value greater than the upper quota or less than the lower quota, then +that apportionment violates the quota rule and the apportionment method that was used violates the quota rule. +EXAMPLE 11.35 +Which Apportionment Method Satisfies the Quota Rule? +Several apportionment methods have been used to allocate 125 seats to ten states and the results are shown in the +table below. Determine which apportionments do not satisfy the quota rule and justify your answer. +State A +State B +State C +State D +State E +State F +State G +Standard Quota +41.26 +16.00 +5.77 +2.64 +7.82 +10.47 +0.21 +Lower Quota +41 +16 +5 +2 +7 +10 +0 +Upper Quota +42 +17 +6 +3 +8 +11 +1 +Method X +43 +16 +5 +2 +7 +10 +1 +Method Y +41 +16 +6 +2 +8 +10 +1 +Method Z +42 +16 +7 +3 +7 +9 +1 +Solution +Look for states such that the number of seats allocated differs from the lower or upper quota. Method X violates the +quota rule because State A receives 43 seats instead of 41 or 42. Method Z violates the quota rule because State C +1196 +11 • Voting and Apportionment +Access for free at openstax.org + +receives 7 seats instead of 5 or 6 and State F receives 9 instead of 10 or 11. +YOUR TURN 11.35 +1. Apportionment Method V has been used to allocate 125 seats to ten states as shown in the table below. +Determine whether the apportionment satisfies the quota rule and justify your answer. +State A +State B +State C +State D +State E +State F +State G +Standard Quota41.26 +10.70 +16.00 +13.11 +17.00 +5.77 +2.64 +Lower Quota +41 +10 +16 +13 +17 +5 +2 +Upper Quota +42 +11 +17 +14 +18 +6 +3 +Method V +42 +11 +17 +13 +18 +4 +2 +It is possible for an apportionment method to satisfy the quota rule in some scenarios but violate it in others. However, +because the Hamilton method always begins with the lower quota and either adds one to it or keeps it the same, the +final Hamilton quota will always consist of values that are either lower quota values or upper quota values. When an +apportionment method has this characteristic, it is said to satisfy the quota rule. So, we can say: +The Hamilton method of apportionment satisfies the quota rule. +Although the Hamilton method of apportionment satisfies the quota rule, it can result in some unexpected outcomes, +which has caused it to pass in and out of favor of the U.S. government over the years. There are several apportionment +methods that have been popular alternatives, such as Jefferson’s method of apportionment that the founders of +Imaginaria should consider. +Jefferson’s Method of Apportionment +Another approach to dealing with the fractional parts of the standard quotas is to modify the standard divisor so that the +total of the resulting modified lower quotas is the necessary number of seats. This is the approach used by Jefferson. +In Jefferson’s method, the change to the standard divisor is made so that the total of the modified lower quotas equals +the house size. The change in the standard divisor to get the modified divisor is relatively small. There is not a formula +for this. The modified divisor is found by “guess and check.” It is important to remember that increasing the divisor +decreases the quotas, but decreasing the divisor increases the quotas. So, if you need a larger quota, try reducing the +divisor, and if you need a smaller quota, try increasing the divisor. +EXAMPLE 11.36 +Modifying a Standard Divisor +Suppose the population of a state is 50 and the standard divisor is 12.5. +1. +Find the state’s standard quota. +2. +Increase the standard divisor by 2 units and use the modified divisor to determine the modified quota for the state. +3. +Decrease the modified divisor from part 2 by 1.5 units and use the new modified divisor to determine the modified +quota for the state. +4. +Choose any value of divisor between the value of the modified divisor from part 2 and the value of the modified +divisor from part 3 and use it to determine the modified quota for the state. +5. +Which modified quota was the largest, the modified quota from part 2, from part 3, or from part 4? Explain why. +Solution +1. +The state’s standard quota is +. +2. +The modified divisor is 14.5. The modified quota is +. +3. +The modified divisor is 13. The modified quota is +. +4. +One value between 13 and 14.5 is 13.5. With a modified divisor of 13.5, the modified quota is +. +5. +The modified quota from part 3 was the largest because the divisor was the smallest of the three. Dividing the same +11.4 • Apportionment Methods +1197 + +number by a smaller value gives a larger result. +YOUR TURN 11.36 +Suppose the population of a state is 12 and the standard divisor is 0.225. +1. Find the state’s standard quota. +2. Decrease the standard divisor by 0.200 units and use the modified divisor to determine the modified quota +for the state. +3. Increase the modified divisor from part 2 by 0.100 units and use the new modified divisor to determine the +modified quota for the state. +4. Choose any value of divisor between the value of the modified divisor from part 2 and the value of the +modified divisor from part 3 and use it to determine the modified quota for the state. +5. Which modified quota was the smallest, the modified quota from part 2, from part 3, or from part 4? Explain +why. +When you use Jefferson’s method, you might have to adjust the divisor several times find modified lower quotas that +sum to the house size. First, guess what the divisor should be based on the sum of the lower quotas and then increase +or decrease it from there based on whether the sum needs to be smaller or larger respectively. If the result still does not +produce lower quotas that sum to the house size, adjust again. Keep a record of the values that didn't work to help you +narrow your search. +Steps for Jefferson’s Method of Apportionment +We take four steps to apply Jefferson’s Method of apportionment: +Step 1: Find the standard divisor. +Step 2: Find each state’s quota. This will be the standard quota the first time Step 2 is completed and the standard +divisor is used, but Step 2 may be repeated as needed using a modified divisor and resulting in modified quotas. +Step 3: Find the states’ lower quotas (with each state receiving at least one seat), and their sum. +Step 4: If the sum from Step 3 equals the number of seats, the apportionment is complete. If the sum of the lower +quotas is less than the number of seats, reduce the standard divisor. If the sum of the lower quotas is greater than the +number of seats, increase the standard divisor. Return to Step 2 using the modified divisor. +EXAMPLE 11.37 +Hawaiian State Representative Districts +Suppose that the Hawaii State Department of Education has a budget for 616 schools and is doing a research study to +determine the equitable number of schools to have in each of five counties based on the residents under the age of 19. +With the data in the table below, apply Jefferson’s method to apportion the schools to the counties. +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Residents under Age 19 +46,310 +224,230 +20 +16,560 +38,450 +325,570 +Solution +Step 1: The process for finding the standard divisor, standard quotas, and lower quotas is the same in the Hamilton and +Jefferson methods of apportionment. We walked through the Hamilton Method in Example 11.34, and following these +steps resulted in lower quotas as shown in the table below. +1198 +11 • Voting and Apportionment +Access for free at openstax.org + +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Standard +Quota +616 +Lower Quota +87 +424 +1 +31 +72 +615 +Step 2: Compare the sum of the states’ lower quotas, 615, to the house size, 616. Since 615 is less than 616, use a +modified divisor that is less than the standard divisor of 528.52. Try 526.00. +Step 3: Find each state’s modified quota, lower quota, and the sum of the lower quotas based on the modified divisor of +526: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Modified +Quota +616 +Lower Quota +88 +426 +1 +31 +72 +618 +Step 4: The new sum of the lower quotas is 2 units greater than 616. We have overshot the goal. So, increase the divisor +to a value between 526.00 and 528.52. Try 527.00. +Step 5: Repeat the process of finding the quotas. Find each state’s modified quota, lower quota, and the sum of the +lower quotas based on the modified divisor of 526.00: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Modified +Quota +616 +Lower Quota +87 +425 +1 +31 +72 +616 +Step 6: The new sum of the lower quotas equals the house size. The apportionment is complete. +The apportionment is: Hawaii County 87, Honolulu County 425, Kalawao County 1, Kauai 31, and Maui 72 schools. +When using Jefferson’s method, the modified divisors you use may be different from what another person chooses, but +final apportionment values will be the same. +11.4 • Apportionment Methods +1199 + +YOUR TURN 11.37 +1. Let’s return to the Imaginarian states of Fictionville, Pretendstead, Illusionham, and Mythbury. Suppose that +there are going to be 35 seats in the legislature. This time use Jefferson’s method of apportionment to determine +the number of seats in each state based on the populations in the table below. How many seats would each state +receive? +Fictionville +Pretendstead +Illusionham +Mythbury +Total +Population +71,000 +117,000 +211,000 +1,194,000 +1,593,000 +Notice that, in this apportionment, Mythbury received more than the upper quota. Since this apportionment of +representatives to Imaginarian states by Jefferson’s method does not satisfy the quota rule, we say that: +Jefferson’s method violates the quota rule. +We have discussed two apportionment methods: one that satisfies the quota rule and one that does not. Before you +decide which method to use in Imaginaria, there are a couple more options to consider. +VIDEO +Jefferson Apportionment Method (https://openstax.org/r/Jeffersons_method) +TECH CHECK +Apportionment Calculators +It is possible to create Excel spreadsheets that complete the calculations necessary to complete a Jefferson +Apportionment. In some cases, this work has already been done and posted online. Check out websites such Ms. +Hearn Math (https://openstax.org/r/jefferson-calculator) for a free Jefferson apportionment calculator. +This can be a useful tool to confirm your results! +Adams’s Method of Apportionment +Adams’s method of apportionment is another method of apportionment that is based on a modified divisor. However, +instead of basing the changes on the sum of the lower quotas, as Jefferson did, Adams used the upper quotas. +To apply Adams’s Method of apportionment, there are four steps we follow: +1. +Find the standard divisor. +2. +Find each state’s quota. This will be the standard quota the first time Step 2 is completed, and the standard divisor is +used, but Step 2 may be repeated as needed using a modified divisor and resultingin modified quotas. +3. +Find the states’ upper quotas and their sum. +4. +If the sum from Step 3 equals the number of seats, the apportionment is complete. If the sum of the upper quotas +is less than the number of seats, reduce the standard divisor. If the sum of the upper quotas is greater than the +number of seats, increase the standard divisor. Return to Step 2 using the modified divisor. +1200 +11 • Voting and Apportionment +Access for free at openstax.org + +EXAMPLE 11.38 +Hawaiian School Districts +As in earlier examples, suppose that the Hawaii State Department of Education has a budget for 616 schools and is +doing a research study to determine the equitable number of schools to have in each of the five counties based on the +residents under the age of 19. Use the data in the following table and the Adams method to apportion the schools to the +counties. +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Residents under Age 19 +46,310 +224,230 +20 +16,560 +38,450 +325,570 +Solution +Step 1: The steps of finding the standard divisor and each state’s quota are the same in the Jefferson and Adams +methods. As in Example 11.37, the standard divisor is 528.52. +Step 2: Find each state’s upper quota and their sum: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Standard +Quota +616 +Upper Quota +88 +425 +1 +32 +73 +619 +Step 3: Compare the sum of the states’ upper quotas, 619, to the house size, 616. Since 619 is greater than 616, we need +to reduce the size of the quotas. Use a modified divisor that is greater than the standard divisor of 528.52. Try 534.00. +Step 4: Find each state’s modified quota, upper quota, and the sum of the upper quotas based on the modified divisor of +534: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Modified +Quota +616 +Upper Quota +88 +420 +1 +32 +72 +613 +Step 5: The new sum of the upper quotas is 3 units less than 616. Larger quotas are needed. So, decrease the divisor to a +value between 534.00 and 528.52. Try 532.00. +Step 6: Find each state’s modified quota, upper quota, and the sum of the upper quotas based on the modified divisor of +532.00: +County +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Modified +Quota +616 +Upper Quota +88 +422 +1 +32 +73 +616 +Step 7: The new sum of the upper quotas equals the house size. The apportionment is complete. +The apportionment is Hawaii County 88, Honolulu County 422, Kalawao County 1, Kauai 32, and Maui 73 schools. +11.4 • Apportionment Methods +1201 + +When using Adams’s method, just as with Jefferson’s method, the modified divisors you use may be different from what +another person chooses, but final apportionment values will be the same. +YOUR TURN 11.38 +1. There are four states in Imaginaria: Fictionville, Pretendstead, Illusionham, and Mythbury. Assume there will be +35 seats in the legislature of Imaginaria. Use Adams's method of apportionment to determine the number of +seats in each state based on the populations in the table below. How many seats would each state receive? +Fictionville +Pretendstead +Illusionham +Mythbury +Total +Population +71,000 +117,000 +211,000 +1,194,000 +1,593,000 +In this apportionment, Mythbury received less than the state’s lower quota. So, this apportionment is an example of a +scenario in which the Adams’s method violates the quota rule. +Adams’s method of apportionment violates the quota rule. +So far, only Hamilton’s method satisfies the quota rule, but there is one more apportionment method you should +consider for Imaginaria. +VIDEO +Adams Method Apportionment Calculator (https://openstax.org/r/Adams_method) +TECH CHECK +Apportionment Calculators +Check out websites such as Ms. Hearn Math (https://openstax.org/r/adams-calculator) for a free Adams Method +apportionment calculator. +This can be a useful tool to confirm your results! +Webster’s Method of Apportionment +Webster’s method of apportionment is another method of apportionment that is based on a modified divisor. However, +instead of basing the changes on the sum of the lower quotas, as Jefferson did or the sum of the upper quotas as Adams +did, Webster used traditional rounding. +To apply Webster’s method of apportionment, there are four steps we take: +1. +Find the standard divisor. +2. +Find each state’s quota. This will be the standard quota the first time Step 2 is completed, and the standard divisor is +used, but Step 2 may be repeated as needed using a modified divisor and resulting in modified quotas. +3. +Round each state’s quota to the nearest whole number and find the sum of these values. +4. +If the sum of the rounded quotas equals the number of seats, the apportionment is complete. If the sum of the +rounded quotas is less than the number of seats, reduce the divisor. If the sum of the rounded quotas is greater +than the number of seats, increase the divisor. Return to Step 2 using the modified divisor. +When using Webster’s method, just as with Jefferson’s method, the modified divisors you use may be different from what +another person chooses, but final apportionment values will be the same. +1202 +11 • Voting and Apportionment +Access for free at openstax.org + +EXAMPLE 11.39 +Hawaiian School Districts +Use the data in the table below to apportion 616 schools to Hawaiian counties. This time, use Webster’s method. +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Residents under Age 19 +46,310 +224,230 +20 +16,560 +38,450 +325,570 +Solution +To apply Webster’s method of apportionment, there are four steps we take: +Step 1: The processes of finding the standard divisor and standard quota are the same in the Jefferson, Adams, and +Webster’s methods. As in the previous examples, the standard divisor is 528.52. +Step 2: Find each state’s rounded quota and their sum: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Standard +Quota +616 +Rounded +Quota +88 +424 +1 +31 +73 +617 +Step 3: Compare the sum of the states’ rounded quotas, 617, to the house size, 616. Since 617 is greater than 616, we +need to reduce the size of the quotas. Use a modified divisor that is greater than the standard divisor of 528.52. Try +534.00. +Step 4: Find each state’s modified quota, rounded quota, and the sum of the rounded quotas based on the modified +divisor of 534: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Modified +Quota +616 +Upper Quota +87 +420 +1 +31 +72 +612 +Step 5: The new sum of the rounded quotas is 4 units less than 616. Larger quotas are needed. So, decrease the divisor +to a value between 534.00 and 528.52. Try 530.00. +Step 6: Find each state’s modified quota, rounded quota, and the sum of the rounded quotas based on the modified +divisor of 530.00: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Modified +Quota +616 +Upper Quota +87 +423 +1 +31 +73 +615 +Step 7: The new sum of the rounded quotas is 1 unit less than 616. Larger quotas are needed. So, decrease the divisor to +a value between 528.52 and 530.00. Try 529.50. +11.4 • Apportionment Methods +1203 + +Step 8: Find each state’s modified quota, rounded quota, and the sum of the rounded quotas based on the modified +divisor of 529.50: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Modified +Quota +616 +Upper Quota +87 +423 +1 +31 +73 +615 +Step 9: The new sum is still only 1 unit less than 616. Larger quotas are needed, but not much larger. So, decrease the +divisor to a value between 528.52 and 529.50. Try 529.30. +Step 10: Find each state’s modified quota, rounded quota, and the sum of the rounded quotas based on the modified +divisor of 529.30: +Hawaii +Honolulu +Kalawao +Kauai +Maui +Total +Modified +Quota +616 +Upper Quota +87 +424 +1 +31 +73 +616 +Step 11: The new sum of the rounded quotas equals the house size. The apportionment is complete. +The apportionment is Hawaii County 87, Honolulu County 424, Kalawao County 1, Kauai 31, and Maui 73 schools. +YOUR TURN 11.39 +1. If you use Webster’s method to apportion 35 legislative seats to the 4 states of Imaginaria, Fictionville, +Pretendstead, Illusionham, and Mythbury, with the populations given in the table below, what is the resulting +apportionment? +Fictionville +Pretendstead +Illusionham +Mythbury +Total +Population +71,000 +117,000 +211,000 +1,194,000 +1,593,000 +So far, we know that the Hamilton method satisfies the quota rule, while the Jefferson and Adams methods do not. The +apportionments in the Example and Your Turn above are both scenarios in which the Webster method satisfies the quota +rule. Does it always? We have a little more work to do to find out. However, one thing is clear. Not all apportionment +methods have the same results. Before you make such an important decision for Imaginaria, it’s important to think +about the differences in the apportionments that result from these four methods. How will the differences affect the +citizens of Imaginaria? +TECH CHECK +Apportionment Calculators +Check out websites such as Ms. Hearn Math (https://openstax.org/r/webster-calculator) for a free Webster Method +apportionment calculator. +1204 +11 • Voting and Apportionment +Access for free at openstax.org + +This can be a useful tool to confirm your results! +Comparing Apportionment Methods +Recall that the four apportionment methods discussed in this chapter differ in two main ways: +• +Whether or not a modified divisor is used +• +The type of rounding of the quotas that is used +How might these differences affect Imaginarians? In the next two examples, we will compare the results when different +apportionment methods are applied to the same scenario. +EXAMPLE 11.40 +Hawaiian School Districts with Different Apportionment Methods +Let’s use the results from Example 11.34, Example 11.37, Example 11.38, and Example 11.39 to compare the four +apportionment methods we have discussed. The following table summarizes the results of the results of the Hamilton, +Jefferson, Adams and Webster methods when applied to the apportionment of 616 schools to Hawaiian counties. +Hawaii +Honolulu +Kalawao +Kauai +Maui +Under 19 years old +46,310 +224,230 +20 +16,560 +38,450 +Hamilton +87 +424 +1 +31 +73 +Jefferson +87 +425 +1 +31 +72 +Adams +88 +422 +1 +32 +73 +Webster +87 +424 +1 +31 +73 +1. +Do any of the apportionment methods result in the same apportionment? If so, which ones? +2. +Which apportionment method would the citizens of the largest county likely favor most and least? Justify your +answer. +3. +As a group, which apportionment method would the citizens of the other four counties likely favor most and least? +Justify your answer. +Solution +1. +Yes, the Hamilton and Webster methods result in the same apportionment. +2. +The largest county is Honolulu. The citizens would likely favor the Jefferson method of apportionment most since +they received the most seats by that method. They would likely favor the Adams method of apportionment least +because they received the least number of seats by that method. +3. +As a group, the other four counties received 192 seats by either the Hamilton or Webster method, 194 seats by the +Adams method, and 191 seats by the Jefferson method. They would likely favor the Adams method the most and +favor the Jefferson methods the least. +YOUR TURN 11.40 +In Your Turn 11.34, 11.37, 11.38, and 11.39, you apportioned 35 legislative seats among the four states of Imaginaria +using the Hamilton, Jefferson, Adams, and Webster methods of apportionment. To understand how the differences +in the apportionments might affect Imaginarians, answer these questions. +1. Which apportionment method would the citizens of the largest state likely favor most and least? Justify your +answer. +2. As a group, which apportionment method would the citizens of the other three states likely favor most and +11.4 • Apportionment Methods +1205 + +least? Justify your answer. +The Adams method favored the smaller states and the Jefferson method favored the larger states in the previous +example, but is this the case in general? +Since the Jefferson method begins with the lower quotas, any adjustment to the quotas will be an increase. As you have +seen, this is accomplished by using a modified divisor that is smaller than the standard divisor. The next example +compares the impact of a decreasing divisor on the modified quotas of large states to the impact of the same size +decrease on small states. +EXAMPLE 11.41 +Effect of Decreasing Divisors on Modified Quotas +The following table displays the effect of reducing the size of the divisor. Observe the effect this has on the modified +quotas of smaller states versus larger states and use the table answer each question. +Modified Quotas +State +Population +Divisor: 10,500 +Divisor: 10,000 +Divisor: 9,500 +A +10,000 +0.95 +1 +1.05 +B +100,000 +9.52 +10 +10.53 +C +1,000,000 +95.24 +100 +105.26 +1. +When the divisor decreases from 10,500 to 10,000, how many representatives are gained by each state based on the +lower quota? +2. +When the divisor decreases from 10,000 to 9,500, how many representatives are gained by each state based on the +lower quota? +3. +Which state gains the most representatives each time the divisor is decreased? +Solution +1. +Since a state must have at least one seat, State A begins with 1 seat and still has one seat. State B begins with 9 +seats and increases to 10 seats. State C begins with 95 seats and increases to 100 seats. So, State A gains 0, B gains +1, and C gains 5 seats. +2. +State A begins with 1 and still has 1. State B begins with 10 and still has 10. State C begins with 100 and increases to +105. So, State A gains 0, State B gains 0, and State C gains 5. +3. +State C, the largest state, gains the most representatives each time the divisor is decreased. +This example demonstrates that the Jefferson method is biased toward states with larger populations because the +modified divisor is smaller than the standard divisor. On the other hand, the Adams’s method, which begins with the +upper quotas, must increase the standard divisor in order to reduce the quotas. Once again, the effect on the number of +seats is greater for the larger states, but this time they are decreased. This means that the Adams’s method favors states +with smaller populations. +YOUR TURN 11.41 +The following table displays the effect of increasing the size of the divisor. Observe the effect this has on the +modified quotas of smaller states versus larger states and use the table to answer each question. +1206 +11 • Voting and Apportionment +Access for free at openstax.org + +Modified Quotas +State +Population +Divisor: 11,500 +Divisor: 12,000 +Divisor: 12,500 +A +10,000 +0.87 +0.83 +0.8 +B +100,000 +8.7 +8.33 +8 +C +1,000,000 +86.96 +83.33 +80 +1. When the divisor increases from 11,500 to 12,000, how many representatives are lost by each state based on +the upper quota? +2. When the divisor increases from 12,000 to 12,500, how many representatives are lost by each state based on +the upper quota? +3. Which state loses the most representatives each time the divisor is increased? +Flaws in Apportionment Methods +As we have seen, different apportionment methods can have the same results in some scenarios but different results in +others. Citizens of states which receive fewer seats with a particular apportionment method will view the apportionment +method as flawed and argue in favor of a different method. This inevitably creates debates regarding the use of one +method over another. Methods that favor larger states are likely to be challenged by smaller states, methods that favor +smaller states are likely to be challenged by larger states, and methods that violate the quota rule are likely to be +challenged by states of any size depending on the circumstances. +Suppose that the State of Hawaii House of Representatives had 51 representatives, each with their own district. Imagine +that redistricting were underway, and the representative districts were to be apportioned to each of five counties based +on population. The following table shows the apportionment that would result from the use of the Jefferson, Adams, and +Webster methods of apportionment. +Hawaii +Honolulu +Kalawao +Kauai +Maui +Population +201,500 +974,600 +100 +72,300 +167,400 +Lower Quota +7 +35 +0 +2 +6 +Upper Quota +8 +36 +1 +3 +7 +Jefferson +7 +35 +1 +2 +6 +Adams +7 +34 +1 +3 +6 +Webster +7 +34 +1 +3 +6 +From the table, you can see that Hawaii, Kalawao, and Maui receive the same number of seats regardless of the method +used. However, citizens of Honolulu would likely reject the Adams and Webster methods arguing that they violate the +quota rule. Similarly, citizens of Kauai would probably reject the Jefferson method based on the argument that it unfairly +favors the larger states. This scenario demonstrates that the Adams and Webster methods violate the quota rule, but the +Jefferson method also violates the quota rule at times. The Hamilton method is the only method that satisfies the +quota rule in all scenarios. It also consistently favors neither larger nor smaller states. Unfortunately, it can have some +strange and results in certain circumstances, which you will see in the next section. +11.4 • Apportionment Methods +1207 + +WHO KNEW? +Gerrymandering: A Subtle Way to Impact Apportionment +In addition to your choice of voting method and your choice of apportionment method, there is another important +decision to make which could potentially have a huge impact on the fairness of elections in Imaginaria—the creation +of electoral districts. In example above, we imagined that there were 51 state legislators in Hawaii, each representing +their own district. But how did the legislators decide on the boundaries for these districts? Typically, boundaries are +drawn so that each district has approximately the same number of residents, but the percentage of residents in each +district with a particular political affiliation can swing the power from one group to another. When the districts are +drawn to impact the power of a political party, ethnic or racial group, or other group, this is called gerrymandering. +For example, districts can be drawn so that a particular group is spread thinly across districts, increasing the +likelihood that they will not have strong representation. +Figure 11.17 This cartoon map conveys the idea that the drawing of the map may impact election outcomes. (credit: +“The Gerry Mander”/Wikimedia Commons, Public Domain) +"The Gerry-mander" first appeared in this cartoon-map in the Boston Gazette, March 26, 1812, and was soon +reproduced in several other Massachusetts newspapers in response to election district changes initiated by governor +Eldridge Gerry. Note that while the practice is named after him, Gerry was not the first to employ it. +PEOPLE IN MATHEMATICS +Jonathan Mattingly +Jonathan Mattingly is a mathematician who was featured in a Nature article titled “The Mathematicians Who Want to +Save Democracy” (https://openstax.org/r/mattingly). Mattingly is a mathematician at Duke University in North +Carolina and he runs election simulations based on alternate versions of electoral districts in order to analyze the +effects of gerrymandering. He has even been asked to testify as an expert witness in court. Mattingly and other +mathematicians who are working on the problem will potentially have an impact on the redistricting that will occur as +a result of the 2020 census. (Carrie Arnold, “The Mathematicians Who Want to Save Democracy,” Nature 546, 200–202, +2017.) +1208 +11 • Voting and Apportionment +Access for free at openstax.org + +Check Your Understanding +25. Which of the four apportionment methods discussed in this section does not use a modified divisor? +26. Which of the four apportionment methods discussed in this section satisfies the quota rule? +27. Which of the four apportionment methods discussed in this section is biased toward states with larger +populations? +28. Which of the four apportionment methods discussed in this section is biased toward states with smaller +populations? +29. Which of the four apportionment methods discussed in this section begin the apportionment with a state’s upper +quota and adjust down? +30. Which of the four apportionment methods discussed in this section begin the apportionment with a state’s lower +quota and adjust up? +31. Which of the four apportionment methods discussed in this section use traditional rounding? +32. Does the change from a standard divisor to a modified divisor tend to change the number of seats for larger or +smaller states more? +SECTION 11.4 EXERCISES +For the following exercises, use the standard quotas given in the table below. +State A +State B +State C +State D +State E +State F +Total Seats +Scenario X +17.63 +26.62 +10.81 +16.01 +13.69 +15.24 +100 +Scenario Y +12.37 +7.59 +71.71 +6.75 +5.76 +20.81 +125 +Scenario Z +3.53 +31.56 +2.95 +5.12 +9.84 +NA +53 +1. Round the standard quota for each state in Scenario X using traditional rounding. Find the sum of the modified +quotas. What is the difference between the sum and the house size? +2. Round the standard quota for each state in Scenario Y using traditional rounding. Find the sum of the modified +quotas. What is the difference between the sum and the house size? +3. Round the standard quota for each state in Scenario Z using traditional rounding. Find the sum of the modified +quotas. What is the difference between the sum and the house size? +4. Find the lower quota for each state in Scenario Y. If each state is allocated its lower quota, how many seats +remain to be allocated? +5. Find the lower quota for each state in Scenario X. If each state is allocated its lower quota, how many seats +remain to be allocated? +6. Find the lower quota for each state in Scenario Z. If each state is allocated its lower quota, how many seats +remain to be allocated? +7. Find the upper quota for each state in Scenario X and determine how much the sum of the upper quotas +exceeds the house size. +8. Find the upper quota for each state in Scenario Y and determine how much the sum of the upper quotas +exceeds the house size. +9. Find the upper quota for each state in Scenario Z and determine how much the sum of the upper quotas +exceeds the house size. +10. Determine the Hamilton apportionment for Scenario Y. +11. Determine the Hamilton apportionment for Scenario X. +12. Determine the Hamilton apportionment for Scenario Z. +For the following exercises, use the information in the table below, which gives standard and final quotas for Methods +X, Y, and Z. +11.4 • Apportionment Methods +1209 + +State A +State B +State C +State D +State E +Standard Quota +1.67 +3.33 +5.00 +6.67 +8.33 +Method X +2 +2 +5 +7 +9 +Method Y +1 +3 +5 +7 +9 +Method Z +1 +3 +5 +6 +10 +13. Does the apportionment resulting from method X satisfy the quota rule? Why or why not? +14. Does the apportionment resulting from method Z satisfy the quota rule? Why or why not? +15. Does the apportionment resulting from method Y satisfy the quota rule? Why or why not? +In the movie Black Panther, the hero lives in the fictional country of Wakanda. Imagine that 111 Vibranium artifacts +must be distributed among the fortress cities, or birnin, of Wakanda based on the population of each birnin. Use the +population and standard quota information in the table below for the following exercises. +Birnin +Djata (D) +Birnin +T'Chaka (T) +Birnin +Zana (Z) +Birnin +S'Yan (S) +Birnin +Bashenga (B) +Birnin +Azzaria (A) +Total +Residents +26,000 +57,000 +27,000 +18,000 +64,000 +45,000 +237,000 +Standard +Quota +12.18 +26.70 +12.65 +8.43 +29.98 +21.08 +111 +16. Modify the standard quota for each state using traditional rounding. Find the sum of the modified quotas. What +is the difference between the sum and the house size? +17. Find the standard lower quota for each state. If each state is allocated its lower quota, how many seats remain +to be allocated? +18. Find the standard upper quota for each state and determine how much the sum of the upper quotas exceeds +the house size. +19. Use the Hamilton method to apportion the artifacts. +20. Find the modified lower quota for each state using a modified divisor of 2,000. Is the sum of the modified +quotas too high, too low, or equal to the house size? +21. Find the modified lower quota for each state using a modified divisor of 2,100. Is the sum of the modified +quotas too high, too low, or equal to the house size? +22. Use the Jefferson method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If +so, indicate the value of the modified divisor. +23. Does the Jefferson method result in an apportionment that satisfies or violates the quota rule in this scenario? +24. Find the modified upper quota for each state using a modified divisor of 2,250. Is the sum of the modified +quotas too high, too low, or equal to the house size? +25. Find the modified upper quota for each state using a modified divisor of 2,150. Is the sum of the modified +quotas too high, too low, or equal to the house size? +26. Use the Adams method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so, +indicate the value of the modified divisor. +27. Does the Adams method result in an apportionment that satisfies or violates the quota rule in this scenario? +28. Which method of apportionment, Jefferson or Adams, is a resident of Birnin T'Chaka likely to prefer? Justify your +answer. +29. Use the Webster method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If +so,indicate the value of the modified divisor. +30. Does the Webster method result in an apportionment that satisfies or violates the quota rule in this scenario? +31. Which of the four methods of apportionment from this section (Hamilton, Jefferson, Adams, or Webster) are the +residents of Birnin S'Yan likely to prefer? Justify your answer. +Children from five families—the Chorro family, the Eswaran family, the Javernick family, the Lahde family, and the Stolly +1210 +11 • Voting and Apportionment +Access for free at openstax.org + +family—joined a town Easter egg hunt. When they returned with their baskets, they had 827 eggs! They decided to +share their eggs amongst the families based on the number of children in each family. Use the population and +standard quota information in the table below for the following exercises. +(C) Chorro +(E) Eswaran +(J) Javernick +(L) Lahde +(S) Stolly +Total +Children +3 +2 +4 +1 +5 +15 +Standard Quota +155.04 +103.36 +206.72 +103.36 +258.40 +827 +32. Modify the standard quota for each state using traditional rounding. Find the sum of the modified quotas. What +is the difference between the sum and the house size? +33. Find the standard lower quota for each state. If each state is allocated its lower quota, how many seats remain +to be allocated? +34. Find the standard upper quota for each state, and determine how much the sum of the upper quotas exceeds +the house size. +35. Use the Hamilton method to apportion the Easter eggs. +36. Find the modified lower quota for each state using a modified divisor of 0.01800. Is the sum of the modified +quotas too high, too low, or equal to the house size? +37. Find the modified lower quota for each state using a modified divisor of 0.01810. Is the sum of the modified +quotas too high, too low, or equal to the house size? +38. Use the Jefferson method to apportion the Easter eggs. Determine whether it is necessary to modify the divisor. +If so, indicate the value of the modified divisor. +39. Does the Jefferson method result in an apportionment that satisfies or violates the quota rule in this scenario? +40. Find the modified upper quota for each state using a modified divisor of 0.0182. Is the sum of the modified +quotas too high, too low, or equal to the house size? +41. Find the modified upper quota for each state using a modified divisor of 0.01816. Is the sum of the modified +quotas too high, too low, or equal to the house size? +42. Use the Adams method to apportion the Easter eggs. Determine whether it is necessary to modify the divisor. If +so, indicate the value of the modified divisor. +43. Does the Adams method result in an apportionment that satisfies or violates the quota rule in this scenario? +44. Use the Webster method to apportion the Easter eggs. Determine whether it is necessary to modify the divisor. +If so, indicate the value of the modified divisor. +45. Does the Webster method result in an apportionment that satisfies or violates the quota rule in this scenario? +For the following exercises, use this information: Suppose that the State of Delaware received 2,000 packs of COVID-19 +vaccines, with ten doses per pack. These (unopened) packs must be distributed to the three counties based on total +population. Use the population information in the table below to determine how many packs of vaccine will be +distributed to each county based on the given apportionment method. +(N) New Castle +(K) Kent +(S) Sussex +Residents +558,753 +180,786 +234,225 +46. Hamilton’s Method +47. Jefferson’s Method +48. Adams's Method +49. Webster’s Method +50. Notice that the apportionments found in questions 46, 47, 48, and 49 all satisfy the quota rule. Does this +contradict the statement, “The Jefferson, Adams, and Webster methods of apportionment all violate the quota +rule”? Why or why not? +11.4 • Apportionment Methods +1211 + +11.5 Fairness in Apportionment Methods +Figure 11.18 In this seating chart for the House of Representatives, each color indicates representatives from a +particular state. +Learning Objectives +After completing this section, you should be able to: +1. +Describe and illustrate the Alabama paradox. +2. +Describe and illustrate the population paradox. +3. +Describe and illustrate the new-states paradox. +4. +Identify ways to promote fairness in apportionment methods. +Apportionment Paradoxes +The citizens of Imaginaria will want the apportionment method to be as fair as possible. There are certain characteristics +that they would reasonably expect from a fair apportionment. +• +If the house size is increased, the state quotas should all increase or remain the same but never decrease. +• +If one state’s population is growing more rapidly than another state’s population, the faster growing state should +not lose a seat while a slower growing state maintains or gains a seat. +• +If there is a fixed number of seats, adding a new state should not cause an existing state to gain seats while others +lose them. +However, apportionment methods are known to contradict these expectations. Before you decide on the right +apportionment for Imaginarians, let’s explore the apportionment paradox, a situation that occurs when an +apportionment method produces results that seem to contradict reasonable expectations of fairness. +There is a lot that the founders of Imaginaria can learn from U.S. history. The constitution of the United States requires +that the seats in the House of Representatives be apportioned according to the results of the census that occurs every +decade, but the number of seats and the apportionment method is not stipulated. Over the years, several different +apportionment methods and house sizes have been used and scrutinized for fairness. This scrutiny has led to the +discovery of several of these apportionment paradoxes. +The Alabama Paradox +At the time of the 1880 U.S. Census, the Hamilton method of apportionment had replaced the Jefferson method. The +number of seats in the House of Representatives was not fixed. To achieve the fairest apportionment possible, the house +sizes were chosen so that the Hamilton and Webster methods would result in the same apportionment. The chief clerk of +the Census Bureau calculated the apportionments for house sizes between 275 and 350. There was a surprising result +that became known as the Alabama paradox, which is said to occur when an increase in house size reduces a state’s +quota. Alabama would receive eight seats with a house size of 299, but only receive seven seats if the house size +increased to 300. (Michael J. Caulfield (Gannon University), "Apportioning Representatives in the United States Congress - +Paradoxes of Apportionment," Convergence (November 2010), DOI:10.4169/loci003163) +1212 +11 • Voting and Apportionment +Access for free at openstax.org + +EXAMPLE 11.42 +The 1880 Alabama Quota +The 1880 census recorded the population of Alabama as 1,513,401 and that of the U.S. as 62,979,766. +1. +Calculate the standard divisor and standard quota for the State of Alabama based on a house size of 299. +2. +Calculate the standard divisor and standard quota for the State of Alabama based on a house size of 300. +3. +Did the standard quota increase or decrease when the house size increased? +4. +Consider the Hamilton method of apportionment. Explain how Alabama’s final quota could be smaller with a larger +standard quota. +Solution +1. +The standard divisor is +citizens per seat. The standard quota for +Alabama is +seats. +2. +The standard divisor is +citizens per seat. The standard quota for +Alabama is +seats. +3. +The standard quota increased. +4. +In each case, the state would receive the lower quota of 7 and then be awarded one more seat if the fractional part +of the standard quota were high enough relative to the fractional parts of the other states’ standard quotas. When +the house size was 299, Alabama received one of the remaining seats after the lower quotas were distributed. When +the house size was 300, Alabama did not receive one of the remaining seats after the lower quotas were distributed. +It must have been the case that either the fractional part 0.2090 ranked lower amongst the other fractional parts of +the state quotas than the fractional part 0.1850 did, or there were fewer remaining seats, or both. +After the 1900 census, the Census Bureau again calculated the apportionment based on various house sizes. It was +determined that Colorado would receive three seats with a house size of 356, but only two seats with a house size of 357. +YOUR TURN 11.42 +The 1900 census recorded the population of Colorado as 539,700 and that of the U.S. as 76,212,168. +1. Calculate the standard divisor and standard quota for the State of Colorado based on a house size of 356. +2. Calculate the standard divisor and standard quota for the State of Colorado based on a house size of 357. +3. Did the standard quota increase or decrease when the house size increased? +4. Consider the Hamilton method of apportionment. Explain how Colorado’s final quota could be smaller with a +larger standard quota. +EXAMPLE 11.43 +Hamilton’s Method and the Alabama Paradox +Suppose that States A and B each have a population of 6, while State C has a population of 2. +1. +Use the Hamilton method to apportion 10 seats. +2. +Use the Hamilton method to apportion 11 seats. +3. +Does this example demonstrate the Alabama paradox? If so, how? +Solution +1. +Step 1: The total population is 14. The standard divisor is +individuals per seat. +Step 2: The states’ standard quotas are as follows: A +, B +, and C +Step 3: The states’ lower quotas are as follows: A 4, B 4, and C 1. +Step 4: The sum of the lower quotas is 9, which means there is one seat remaining to be apportioned. State C has +the highest fractional part and receives the additional seat. +Step 5: The final apportionment is as follows: A 4, B 4, and C 2, which sums to 10. +2. +Step 1: The total population is 14. The standard divisor is +individuals per seat. +11.5 • Fairness in Apportionment Methods +1213 + +Step 2: The states’ standard quotas are: A +, B +, and C +. +Step 3: The states’ lower quotas are: A 4, B 4, and C 1. +Step 4: The sum of the lower quotas is 9, which means there are two seats remaining to be apportioned. A and B +have the highest fractional parts and receive the additional seats. +Step 5: The final apportionment is: A 5, B 5, and C 1. +3. +Yes, this demonstrates the Alabama paradox because State C receives two seats if the house size is 10, but only one +seat if the house size is 11. +YOUR TURN 11.43 +Suppose that the founders of Imaginaria decide to have a parliament that apportions seats to four political parties +based on the portion of the vote each party has earned. Also, suppose that Party A has 56.7 thousand votes, Party B +has 38.5 thousand votes, Party C has 4.2 thousand votes, and Party D has 0.6 thousand votes. +1. Use the Hamilton method to apportion 323 seats. +2. Use the Hamilton method to apportion 324 seats. +3. Does this example demonstrate the Alabama paradox? If so, how? +The Population Paradox +It is important for the founders of Imaginaria to keep in mind that the populations of states change as time passes. +Some populations grow and some shrink. Some populations increase by a large amount while others increase by a small +amount. These changes may necessitate a reapportionment of seats, or the recalculation of state quotas due to a +change in population. It would be reasonable for Imaginarians to expect that the state with a population that has grown +more than others will gain a seat before the other states. Once again, this is not always the case with the Hamilton +method of apportionment. The population growth rate of a state is the ratio of the change in the population to the +original size of the population, often expressed as a percentage. This value is positive if the population is increasing and +negative if the population is decreasing. The population paradox occurs when a state with an increasing population +loses a seat while a state with a decreasing population either retains or gains seats. More generally, the population +paradox occurs when a state with a higher population growth rate loses seats while a state with a lower population +growth rate retains or gains seats. +Notice that the population paradox definitions has two parts. If either part applies, then the population paradox has +occurred. The first part of the definition only applies when one state has a decreasing population. The second part of the +definition applies in all situations, whether there is a state with a decreasing population or not. It will be easier to identify +situations that involve a decreasing population. The other situations requires the calculation of a growth rate. The reason +that we don't have to calculate a growth rate when one state has a decreasing population and the other has an +increasing population is that increasing population has a positive growth rate which is always greater than the negative +growth rate of a decreasing population. +A state must lose a seat in order for the population paradox to apply. It is not enough for a state with a lower +growth rate to gain a seat while a state with a higher growth rate retains the same number of seats. +EXAMPLE 11.44 +Apportionment of Respirators to Hospitals +Suppose that 18 respirators are to be apportioned to three hospitals based on their capacities. The Hamilton method is +used to allocate the respirators in 2020, then to reallocate based on new capacities in 2021. The results are shown in the +table below. How does this demonstrate the population paradox? +1214 +11 • Voting and Apportionment +Access for free at openstax.org + +Hospital +Capacity in +2020 +Respirators +in 2020 +Change in +Capacity +Growth Rate = +Capacity in +2021 +Respirators +in 2021 +A +825 +9 +57 +882 +9 +B +613 +7 +13 +626 +6 +C +239 +2 +3 +242 +3 +Solution +Hospital B lost a respirator while hospital C gained one, even though hospital B had a higher growth rate than hospital C. +YOUR TURN 11.44 +1. Suppose that 18 respirators are to be apportioned to three hospitals based on their capacities. The respirators +are allocated based on the Hamilton method in 2020, then reallocated based on new capacities in 2021. The +results are shown in the table below. How does this demonstrate the population paradox? +Hospital +Capacity +in 2020 +Respirators +in 2020 +Change in +Capacity +Growth Rate +Capacity +in 2021 +Respirators +in 2021 +A +237 +5 +6 +243 +6 +B +889 +21 +69 +958 +21 +C +674 +16 +18 +692 +15 +FORMULA +The growth rate of a population can be calculated by subtracting the previous population size from the current +population size, and then dividing the difference by the previous population size. +Make sure to calculate the subtraction before the division. If you are entering the values in a calculator, it is helpful +to put parentheses around the subtracted terms. +EXAMPLE 11.45 +The Congress of Costaguana +The country of Costaguana has three states: Azuera with a population of 894,000; Punta Mala with a population of +696,000; and Esmeralda with a population of 215,000. There are 38 seats in the Congress of Costaguana. The +apportionment of the seats is determined by Hamilton’s method to be: 19 for Azuera, 15 for Punta Mala, and 4 for +Esmerelda. A census reveals that the population has grown and the seats must be reapportioned. If Azuera now has +953,000 residents, Punta Mala now has 706,000 residents, and Esmerelda now has 218,000 residents, how many seats +11.5 • Fairness in Apportionment Methods +1215 + +will each state receive upon reapportionment? How is this an example of the population paradox? +Solution +The Hamilton reapportionment is: 19 for Azuera, 14 for Punta Mala, and 5 for Esmerelda. This is an example of the +population paradox because Punta Mala lost a seat to Esmerelda, even though Punta Mala’s population grew by 1.44 +percent while Esmerelda’s only grew by 1.40 percent. +YOUR TURN 11.45 +1. The country of Elbonia has three states: Mudston with a population of 866,000; WallaWalla with a population of +626,000; and Dilberta with a population of 256,000. There are 38 seats in the Congress of Elbonia. The +apportionment of the seats is determined by Hamilton’s method to be: 19 for Mudston, 14 for WallaWalla, and 5 +for Dilberta. A census reveals that the population has grown and the seats must be reapportioned. If Mudston +now has 921,000 residents, WallaWalla now has 640,000 residents, and Dilberta now has 260,000 residents, how +many seats will each state receive upon reapportionment? How is this an example of the population paradox? +The New-States Paradox +As a founder of Imaginaria, you might also consider the possibility that Imaginaria could annex nearby lands and +increase the number of states. This occurred several times in the United States such as when Oklahoma became a state +in 1907. The House size was increased from 386 to 391 to accommodate Oklahoma’s quota of five seats. When the seats +were reapportioned using Hamilton’s method, New York lost a seat to Maine despite the fact that their populations had +not changed. This is an example of the new-state paradox, which occurs when the addition of a new state is +accompanied by an increase in seats to maintain the standard ratio of population to seats, but one of the existing states +loses a seat in the resulting reapportionment. +EXAMPLE 11.46 +New State of Oscuridad +The country of San Lorenzo has grown from two states to three. The house size of the congress has been increased by +eight and the seats have been reapportioned to accommodate the new state of Oscuridad. The constitution mandates +the use of the Hamilton method of apportionment. Use this information and the following table to answer the questions. +State +Population (in hundreds) +Original Apportionment +Reapportionment +Clara +7,100 +39 +40 +Velasco +9,080 +51 +50 +Oscuridad +1,500 +Not Applicable +8 +1. +What was the original house size? +2. +What is the new house size? +3. +How is this reapportionment an example of the new-states paradox? +Solution +1. +2. +3. +The original state of Velasco lost a seat to the original state of Clara when the new state of Oscuridad was added. +YOUR TURN 11.46 +The country of Narnia has grown from two states to three. The house size of the congress has been increased by five +1216 +11 • Voting and Apportionment +Access for free at openstax.org + +and the seats have been reapportioned to accommodate the new state of Chippingford. The constitution mandates +the use of the Hamilton method of apportionment. Use this information and the following table to answer the +questions. +State +Population (in hundreds) +Original Apportionment +Reapportionment +Beruna +7600 +39 +40 +Beaversdam +9720 +51 +50 +Chippingford +1000 +Not Applicable +5 +1. What was the original house size? +2. What is the new house size? +3. How is this reapportionment an example of the new-states paradox? +EXAMPLE 11.47 +The Growing Country of Gulliversia +The country of Gulliversia has two states: Lilliput with a population of 700,000 and Brobdingnag with a population of +937,000. The constitution of Gulliversia requires that the 90 congressional seats be apportioned by Hamilton’s method. +Lilliput has received 38 seats while Brobdingnag has received 52 seats. Recently, the island of Houyhnhnmsland with a +population of 170,000 has joined the union, becoming a state of Gulliversia. When Houyhnhnmsland is included, nine +additional seats must be apportioned to maintain the same ratio of seats to citizens. Use Hamilton’s method to +reapportion the 99 seats to the three states. How is the resulting apportionment an example of the new-states paradox? +Solution +The reapportionment gives 39 seats to Lilliput, 51 seats to Brobdingnag, and 9 seats to Houyhnhnmsland. This is an +example of the new-states paradox because the original state of Brobdingnag lost a seat to the original state of Lilliput +when the new state was added to the union. +YOUR TURN 11.47 +1. The country of Neverland has two states: Neverwood with a population of 760,000 and Mermaids Lagoon with a +population of 943,000. The constitution of Neverland requires that the 84 congressional seats be apportioned by +Hamilton’s method. Neverwood has received 37 seats while Mermaids Lagoon has received 47 seats. Recently, +the island of Marooners Rock with a population of 190,000 has joined the union, becoming a state of Neverland. +When Marooners Rock is included, nine additional seats must be apportioned to maintain the same ratio of seats +to citizens. Use Hamilton’s method to reapportion the 93 seats to the three states. How is the resulting +apportionment an example of the new-states paradox? +When a new state is added, it is necessary to determine the amount that the house size must be increased to retain the +original ratio of population to seats, in other words to keep the original standard divisor. To calculate the new house size, +divide the new population by the original standard divisor, and round to the nearest whole number. +FORMULA +11.5 • Fairness in Apportionment Methods +1217 + +EXAMPLE 11.48 +Oklahoma Joins the Union +Oklahoma was admitted as the 46th state on November 16, 1907. Before Oklahoma joined the union, the U.S. population +was approximately 75,030,000 and the House of Representatives had 386 seats. The new state had a population of +approximately 970,000. Use this information to estimate the original standard divisor to the nearest hundred, the new +population, the new house size, and the number of seats Oklahoma should receive. +Solution +Step 1: +Step 2: New Population +Step 3: +Step 4: There are +new seats to be apportioned to Oklahoma. +YOUR TURN 11.48 +1. New Mexico was admitted as the 47th state on January 6, 1912. Before New Mexico joined the union, the U.S. +population was approximately 76,000,000 and the House of Representatives had 391 seats. The new state had a +population of approximately 300,000. Use this information to estimate the original standard divisor to the +nearest hundred, the new population, the new house size, and the number of seats New Mexico should receive. +The Search for the Perfect Apportionment Method +The ideal apportionment method would simultaneously satisfy the following four fairness criteria. +• +Satisfy the quota rule +• +Avoid the Alabama paradox +• +Avoid the population paradox +• +Avoid the new-states paradox +We have seen that the Hamilton method allows the Alabama paradox, the population paradox, and the new-states +paradox in some apportionment scenarios. Let’s explore the results of the other methods of apportionment we have +discussed in some of the same scenarios. +EXAMPLE 11.49 +Orange Grove and the New-States Paradox +The incorporated town of Orange Grove consists of two subdivisions: The Oaks with 1,254 residents and The Villages +with 10,746 residents. A council with 100 members supervises the municipality’s operations. The council votes to annex +an unincorporated subdivision called The Lakes with a population of 630. They plan to increase the size of the council to +maintain the ratio of seats to residents such that the new council will have 100 seats plus the number of seats given to +The Lakes. Use each of the following apportionment methods and indicated number of additional seats to find the +original and new apportionment and determine whether the new-state paradox occurs. +1. +Jefferson’s method with five additional seats. +2. +Adams’s method with six additional seats. +3. +Webster’s method with five additional seats +Solution +1. +Using a modified divisor of 119, the original apportionment would have been: The Oaks 10 and The Villages 90. +Using a modified divisor of 119, the new apportionment would be: The Oaks 10, The Villages 90, and The Lakes 5. +The new-state paradox does not occur. +2. +Using a modified divisor of 121, the original apportionment would have been: The Oaks 11 and The Villages 89. +Using a modified divisor of 121, the new apportionment would be: The Oaks 11, The Villages 89, and The Lakes 6. +The new-state paradox does not occur. +1218 +11 • Voting and Apportionment +Access for free at openstax.org + +3. +Using the standard divisor of 120, the original apportionment would have been: The Oaks 10 and The Villages 90. +Using a modified divisor of 119.5, the new apportionment would be: The Oaks 10, The Villages 90, and The Lakes 5. +The new-state paradox does not occur. +YOUR TURN 11.49 +1. Suppose that in 2016, States A, B, and C had populations of 13 million, 12 million, and 112 million, respectively. In +2020, State A has grown by 1 million residents, State B has lost 1 million residents, and State C has gained 2 +million residents. Compare the apportionments in 2016 to 2020 using each method given below. Which of the +four methods violate(s) the population paradox in this scenario? +We have seen in our examples that neither the population paradox nor the new-states paradox occurred when using the +Jefferson, Adams, and Webster methods. It turns out that, although all three of these divisor methods violate the quota +rule, none of them ever causes the population paradox, new-states paradox, or even the Alabama paradox. On the other +hand, the Hamilton method satisfies the quota rule, but will cause the population paradox, the new-states paradox, and +the Alabama paradox in some scenarios. +In 1983, mathematicians Michel Balinski and Peyton Young proved that no method of apportionment can simultaneously +satisfy all four fairness criteria. +There are other apportionment methods that satisfy different subsets of these fairness criteria. For example, the +mathematicians, Balinski and Young who proved the Balinski-Young Impossibility Theorem created a method that both +satisfies the quota rule and is free of the Alabama paradox. (Balinski, Michel L.; Young, H. Peyton (November 1974). “A +New Method for Congressional Apportionment.” Proceedings of the National Academy of Sciences. 71 (11): 4602–4606.) +However, no method may always follow the quota rule and simultaneously be free of the population paradox. (Balinski, +Michel L.; Young, H. Peyton (September 1980). "The Theory of Apportionment" (PDF). Working Papers. International +Institute for Applied Systems Analysis. WP-80-131.) +So, as you and your fellow founders of Imaginaria make the important decision about the right apportionment method +for Imaginaria, do not look for a perfect apportionment method. Instead, look for an apportionment method that best +meets the needs and concerns of Imaginarians. +Check Your Understanding +In the following questions, a scenario is given. Determine whether the scenario is an example of a quota rule violation, +the Alabama paradox, the population paradox, the new-states paradox, or none of these. +33. A city purchased five new firetrucks and apportioned them among the existing fire stations. Although your +neighborhood fire station has the same proportion of the city’s firetrucks as before the new ones were +purchased, it now has one fewer. +34. The school resources officers in a county were reapportioned based on the most recent census. The number of +students at Chapel Run Elementary went up while the number of students at Panther Trail Elementary went +down. However, Chapel Run now has one fewer resources officer while Panther Trail has one more than it did +previously. +35. The standard quota for States A, B, and C are 1.19, 2.73, and 5.71 respectively. State A received 1 seat, State B +received 3 seats, and State C received 4 seats. +36. A corporation that owns several hospitals purchased an additional hospital, causing the doctor to patient ratio +to decrease. +37. When the city of Cocoa annexed an adjacent unincorporated community, the number of seats on the city +council was increased to maintain the standard ratio of citizens to seats, but one existing community of Cocoa +still lost a seat on the city council to another existing community of Cocoa when the new community was +added. +In the following questions, a type of scenario is described. Indicate which paradox could arise in a scenario of this kind. +38. A reapportionment occurs because the populations of the states change, and the house size remains the same. +39. A reapportionment occurs because the house size increases, and the populations of the states remain the +same. +40. A reapportionment occurs because an additional state is added to the union, the populations of the original +states remain the same, and the house size is increased to correspond to the population of the new state. +11.5 • Fairness in Apportionment Methods +1219 + +SECTION 11.5 EXERCISES +In the following exercises, determine whether the scenario violates the Alabama paradox. Justify your answer. +1. A company with an office in each of four cities must distribute 145 new Chromebooks to the four offices. It is +determined that Office A will receive 42, Office B will receive 17, Office C will receive 35, and Office D will receive +51. At the last minute, it is discovered that there are 146 Chromebooks. When they are reapportioned, Office A +receives 42, Office B receives 16, Office C receives 36, and Office D receives 52. +2. A county with three towns has 30 garbage trucks to apportion. Attenborough receives 6 trucks, Breckenridge +receives 8 trucks, and Cabbotsville receives 16 trucks, in proportion to their populations. When one additional +truck is purchased, the reallocation results in 5 trucks for Attenborough, 9 for Breckenridge, and 8 for +Cabbotsville though there has been no change in populations. +3. A group of 60 dentists work for a company that runs five offices. The dentists have been apportioned to the +offices by the number of patients. Office A receives 14 dentists, Office B receives 11, Office C receives 11, Office +D receives 12, and Office E receives 12. When a new dentist joins the group, the new apportionment gives the +following: A 14, B 11, C 11, D 12, and E 13. +In the following exercises, determine whether the scenario violates the population paradox. Justify your answer. +4. A company with locations in three cities plans to give 200 achievement awards that shall be apportioned to the +three cities by population. City A with 9,150 employees is allotted 61 awards, City B with 6,040 employees is +allotted 40 awards, and City C with 14,810 employees is allotted 99 awards to distribute. Then it is discovered +that the number of employees is out of date. The awards are reallocated based on the new populations: City A +with 9,180; City B with 6,040; and City C with 14,930. It turns out that the apportionment remains the same. +5. A soccer club must apportion soccer balls to the teams among four age brackets based on the number of teams +in each bracket. Bracket U8 receives 32 balls, U12 receives 46 balls, U15 receives 29 balls, and U18 receives 25 +balls. Mid-season, the balls are reapportioned. U8 has decreased by 10 percent, U12 has increased by 20 +percent, U15 remains the same, and U18 has increased by 10 percent. The reapportionment gives 33 balls to +U8, 46 balls to U12, 29 balls to U15, and 14 balls to U18. +6. READ (Reading Education Assistance Dogs) trains and certifies therapy animals to serve in classrooms and help +children develop a love of reading in a low-stress environment. Suppose that 20 therapy teams are apportioned +to three schools based on their populations. When the population of School A decreases by 10 percent, the +population of School B increases by 10 percent, and the population of School C remains constant, the +apportionment remains the same. +In the following exercises, determine whether the scenario violates the new-states paradox. Justify your answer. +7. A charity organization has 851 volunteers in Country A and 3449 volunteers in Country B. There are 43 lead +organizers that must be apportioned to the two locations. Country A receives 9 while Country B receives 34. +When the operations are expanded to Country C with 725 volunteers, 7 new lead organizers are added to the +team. When the lead organizers are reapportioned, Country A receives 9, Country B receives 34, and Country C +receives 7 lead organizers. +8. A country has two states. There are 22 seats in the legislature. State A has 4 seats, and State B has 18 seats. +When State C joins the union, the number of representatives is increased by five. Under the new +apportionment, State A receives 3 seats, State B receives 19 seats, and State C receives 5 seats. +9. In the Garunga Solar System, there are four inhabited planets. Three of the planets are members of the United +Association of Garungan Planets (UAGP). The UAGP has 67 seats. Planet Angluertka has 40 seats, planet +Bangluertka has 27 seats, and planet Clangluertka has 13 seats. When planet Danggluertka decides to join the +UAGP, 14 seats are added to accommodate them proportionately. The new allocation of seats gives 40 seats to +Angluertka, 26 seats to Bangluertka, 14 seats to Clangluertka, and 14 seats to Danggluertka. +In the following exercises, use the Hamilton method of apportionment to answer the questions. +10. When the number of seats changed from 147 to 148, the standard quotas changed from A 44.24, B 17.35, C +37.12, and D 48.29 to A 45.54, B 17.47, C 37.37, and D 48.62. +a. +How did the increase in seats impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +11. When the number of seats changed from 126 to 127, the standard quotas changed from A 9.57, B 29.49, C +33.89, D 28.43, and E 24.61 to A 9.65, B 29.72, C 34.16, D 38.66, and E 24.81. +a. +How did the increase in seats impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +12. When the number of seats changed from 25 to 26, the standard quotas changed from A 2.21, B 5.25, C 11.27, +1220 +11 • Voting and Apportionment +Access for free at openstax.org + +and D 6.27 to A 2.30, B 5.46, C 11.72, and D 6.52. +a. +How did the increase in seats impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +13. When the number of seats changed from 25 to 26, the standard quotas changed from A 2.43, B 5.42, and C 8.15 +to A 2.46, B 5.47, and C 8.07. +a. +How did the increase in seats impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +14. The house size is 18. When the population of State A increases by 11.76 percent, State B increases by 16.22 +percent, and State C increases by 12.18 percent, the standard quotas change from A 2.46, B 6.00, and C 9.53 to +A 2.41, B 6.25, and C 9.34. +a. +How did the change in populations impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +15. The house size is 100. When the population of State A increases by 20 percent, State B increases by 10 percent, +State C increases by 30 percent, and the populations of States D, E, and F remain the same. The standard +quotas change from A 12.50, B 25.00, C 9.38, D 18.75, E 12.50 and F 21.88 to A 13.91, B 25.51, C 11.30, D 17.39, E +11.59, and F 20.29. +a. +How did the change in populations impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +16. The house size is 17. When the population of State A increases by 12.73 percent, State B increases by 15.63 +percent, and State C increases by 12.90 percent, the standard quotas change from A 2.53, B 5.90, and C 8.57 to +A 2.51, B 5.99, and C 8.50. +a. +How did the change in populations impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +17. The house size is 38 seats. When the population of State A increases by 6.60 percent, State B increases by 1.44 +percent, and State C increases by 1.40 percent, the standard quotas change from A 18.82, B 14.65, and C 4.53 to +A 19.59, B 14.29, and C 4.41. +a. +How did the change in populations impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +18. The house size is 24 seats. When the population of State A increases by 28 percent, State B increases by 26 +percent, and State C increases by 15 percent, the standard quotas change from A 3.38, B 6.32, and C 14.30 to A +3.63, B 6.67, and C 13.71. +a. +How did the change in populations impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +19. The house size was 60. There were three states with standard quotas of A 4.18, B 15.38, and C 40.44. A fourth +state was annexed, and the house size was increased to 65. The new standard quotas are A 4.17, B 15.33, C +40.32, and D 5.18. +a. +How did the additional state impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +20. The house size was 50. There were three states with standard quotas of A 9.41, B 24.42, and C 16.17. A fourth +state was annexed, and the house size was increased to 66. The new standard quotas are A 9.36, B 24.30, C +16.09, and D 16.25. +a. +How did the additional state impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +21. The house size was 27. There were three states with standard quotas of A 6.39, B 11.40, and C 9.21. A fourth +state was annexed, and the house size was increased to 35. The new standard quotas are A 6.38, B 11.37, C +9.19, and D 8.06. +a. +How did the additional state impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +22. The house size was 100. There were three states with standard quotas of A 26.09, B 30.43, and C 43.48. A fourth +state was annexed, and the house size was increased to 122. The new standard quotas are A 26.14, B 30.50, C +43.57, and D 21.78. +a. +How did the additional state impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +11.5 • Fairness in Apportionment Methods +1221 + +In the following exercises, use the information in the table below. +State +A +B +C +D +E +F +G +H +I +J +K +L +P +Q +R +Population +624 +1,219 +979 +3,462 +7,470 +4,264 +5,300 +263 +809 +931 +781 +676 +150 +250 +350 +Original +House Size +38 +204 +126 +50 +Updated +House Size +39 +205 +127 +51 +23. Consider States A, B, and C. +a. +Determine the apportionment for States A, B, and C with the original house size using the Hamilton +method. +b. +Determine the apportionment for States A, B, and C with the updated house size using the Hamilton +method. +c. +Does the change in the house size and use of the Hamilton method cause the Alabama paradox? Explain +your reasoning. +24. Consider States D, E, F, and G. +a. +Determine the apportionment for States D, E, F, and G with the original house size using the Hamilton +method. +b. +Determine the apportionment for States D, E, F, and G with the updated house size using the Hamilton +method. +c. +Does the change in the house size and use of the Hamilton method cause the Alabama paradox? Explain +your reasoning. +25. Consider States H, I, J, K, and L. +a. +Determine the apportionment for States H, I, J, K, and L with the original house size using the Hamilton +method. +b. +Determine the apportionment for States H, I, J, K, and L with the updated house size using the Hamilton +method. +c. +Does the change in the house size and use of the Hamilton method cause the Alabama paradox? Explain +your reasoning. +26. Consider States P, Q, and R. +a. +Determine the apportionment for States P, Q, and R with the original house size using the Hamilton +method. +b. +Determine the apportionment for States P, Q, and R with the updated house size using the Hamilton +method. +c. +Does the change in the house size and use of the Hamilton method cause the Alabama paradox? Explain +your reasoning. +In the following exercises, use the information in the table below. +State +Original Population +Updated Population +Population Growth Rate +House Size +A +889 +958 +7.76% +B +674 +692 +2.67% +42 +C +237 +243 +2.53% +D +12,032 +14,124 +E +10,789 +9,726 +135 +1222 +11 • Voting and Apportionment +Access for free at openstax.org + +State +Original Population +Updated Population +Population Growth Rate +House Size +F +995 +2,304 +G +901 +1,156 +28.3% +H +1,683 +2,125 +26.3% +24 +I +3,808 +4,369 +14.7% +X +56 +63 +Y +125 +141 +16 +Z +182 +213 +P +6,534 +6,534 +Q +7,832 +7,810 +40 +R +13,959 +13,992 +S +20,515 +21,164 +27. Calculate the population growth rates +, , and +for States D, E, and F. Give answer as a percentage rounded to +one decimal place. +28. Calculate the population growth rates +, , , and +for States P, Q, R, and S. Give answer as a percentage +rounded to one decimal place. +29. Calculate the population growth rates +, +, and +for States X, Y, and Z. Give answer as a percentage rounded to +one decimal place. +30. Consider States A, B, and C. +a. +Determine the Hamilton apportionment for States A, B, and C with the original population. +b. +Determine the Hamilton apportionment for States A, B, and C with the updated population. +c. +Does the increase in population of States A, B, and C from the original population to the updated +population and the use of the Hamilton method cause the population paradox? Explain your reasoning. +31. Consider States D, E, and F. +a. +Determine the Hamilton apportionment for States D, E, and F with the original population. +b. +Determine the Hamilton apportionment for States D, E, and F with the updated population. +c. +Does the increase in population and the use of the Hamilton method cause the population paradox? +Explain your reasoning. +32. Consider States G, H, and I. +a. +Determine the Hamilton apportionment for States G, H, and I with the original population. +b. +Determine the Hamilton apportionment for States G, H, and I with the updated population. +c. +Does the increase in population and the use of the Hamilton method cause the population paradox? +Explain your reasoning. +33. Consider States X, Y, and Z. +a. +Determine the Hamilton apportionment for States X, Y, and Z with the original population. +b. +Determine the Hamilton apportionment for States X, Y, and Z with the updated population. +c. +Does the increase in population and the use of the Hamilton method cause the population paradox? +Explain your reasoning. +34. Consider States P, Q, R, and S. +a. +Determine the Hamilton apportionment for States P, Q, R, and S with the original population. +b. +Determine the Hamilton apportionment for States P, Q, R, and S with the updated population. +11.5 • Fairness in Apportionment Methods +1223 + +c. +Does the increase in population and the use of the Hamilton method cause the population paradox? +Explain your reasoning. +In the following exercises, use the information in the table below. +State +Population +Original House Size +New House Size +A +627 +B +1,287 +25 +C +973 +32 +D +815 +E +520 +F +1,510 +50 +G +1,060 +H +950 +P +1,222 +100 +Q +473 +R +225 +K +1,688 +L +7,912 +48 +M +1,448 +T +150 +U +250 +V +350 +50 +W +450 +35. Consider States A, B, and C. +a. +Calculate the standard divisor based on the original house size. +b. +Use the Hamilton method to apportion the seats. +36. Consider States E, F, and G. +a. +Calculate the standard divisor based on the original house size. +b. +Use the Hamilton method to apportion the seats. +37. Consider States P and Q. +a. +Calculate the standard divisor based on the original house size. +b. +Use the Hamilton method to apportion the seats. +38. Consider States K and L. +1224 +11 • Voting and Apportionment +Access for free at openstax.org + +a. +Calculate the standard divisor based on the original house size. +b. +Use the Hamilton method to apportion the seats. +39. Suppose that States A, B, and C annex State D and increase the house size proportionately. +a. +Calculate the standard divisor based on the new house size. +b. +Use the Hamilton method to reapportion the seats. +c. +Does the new-states paradox occur? +40. Suppose that States E, F, and G annex State H and increase the house size proportionately. +a. +Determine the new house size, +, that is necessary. +b. +Calculate the standard divisor based on the new house size. +c. +Use the Hamilton method to reapportion the seats. +d. +Does the new-states paradox occur? +41. Suppose that States P and Q annex State R and increase the house size proportionately. +a. +Determine the new house size, , that is necessary. +b. +Calculate the standard divisor based on the new house size. +c. +Use the Hamilton method to reapportion the seats. +d. +Does the new-states paradox occur? +42. Suppose that States K and L annex State M and increase the house size proportionately. +a. +Determine the new house size, +, that is necessary. +b. +Calculate the standard divisor based on the new house size. +c. +Use the Hamilton method to reapportion the seats. +d. +Does the new-states paradox occur? +43. Suppose that States T, U, and V annex State W and increase the house size proportionately. +a. +Calculate the standard divisor based on the original house size and only States T, U, and V. +b. +Use the Hamilton method to apportion the seats to T, U, and V. +c. +Determine the new house size when State W is annexed. +d. +Calculate the standard divisor based on the new house size. +e. +Use the Hamilton method to reapportion the seats. +f. +Does the new-states paradox occur? (Refer to part b.) +44. Suppose 24 seats are apportioned to States A, B, and C with populations of 16, 15, and 125 respectively. Then +the populations of States A, B, and C change to 17, 15, and 126 respectively. +a. +Demonstrate that the population paradox occurs when the Hamilton method is used. +b. +Determine whether the population paradox occurs when the Webster method is used. Justify your answer. +45. Suppose that 10 seats are apportioned to States A, B, and C with populations 6, 6, and 2 respectively. Then the +number of seats is increased to 11. Demonstrate that the Alabama paradox occurs when the Hamilton method +is used. +11.5 • Fairness in Apportionment Methods +1225 + +Chapter Summary +Key Terms +11.1 Voting Methods +• +majority +• +plurality +• +runoff election +• +runoff voting system +• +two-round system +• +Hare Method +• +preference ranking +• +ranked ballot +• +ranked-choice voting (RCV) +• +instant runoff voting (IRV) +• +Borda count method +• +Borda score +• +Compromise candidate +• +divisive candidate +• +pairwise comparison method +• +Condorcet voting methods +• +Condorcet candidate +• +approval voting system +• +approval voting ballot +• +spoiler +11.2 Fairness in Voting Methods +• +majority criterion +• +tyranny of the majority +• +Condorcet criterion +• +Condorcet method +• +monotonicity criterion +• +up-rank +• +down-rank +• +independence of irrelevant alternatives criterion (IIA) +• +Arrow’s Impossibility Theorem +• +cardinal voting system +11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem +• +apportion +• +apportionment problem +• +proportional +• +part-to-part ratio +• +representative democracies +• +standard divisor +• +states +• +seats +• +house size +• +state population +• +total population +• +standard quota +11.4 Apportionment Methods +• +apportionment problem +• +lower quota +• +upper quota +1226 +11 • Chapter Summary +Access for free at openstax.org + +11.5 Fairness in Apportionment Methods +• +apportionment paradox +• +Alabama paradox +• +reapportionment +• +population growth rate +• +population paradox +• +new-state paradox +Key Concepts +11.1 Voting Methods +• +In plurality voting, the candidate with the most votes wins. +• +When a voting method does not result in a winner, runoff voting can be used to do so. +• +Ranked-choice voting, also known as instant runoff voting, is one type of ranked voting system. +• +The Borda count method is a type of ranked voting system in which each candidate is given a Borda score based on +the number of candidates ranked lower than them on each ballot. +• +When pairwise comparison is used, the winner will be the Condorcet candidate if one exists. +• +Approval voting allows voters to give equally weighted votes to multiple candidates. +• +When a voter finds a characteristic of a particular voting method unappealing, they may consider that characteristic +a flaw in the voting method and look for an alternative method that does not have that characteristic. +11.2 Fairness in Voting Methods +• +There are several common measures of voting fairness, including the majority criterion, the head-to head criterion, +the monotonicity criterion, and the irrelevant alternatives criterion. +• +According to Arrow’s Impossibility Theorem, each voting method in which the only information is the order of +preference of the voters will violate one of the fairness criteria. +11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem +• +The apportionment problem is how to fairly divide and distribute available resources to recipients in whole, not +fractional, parts. +• +To distribute the seats in the U.S. House of Representatives fairly to each state, calculations are based on state +population, total population, and house size, or the total number of seats to be apportioned. +• +The standard divisor is the ratio of the total population to the house size, and the standard quota is the number of +seats that each state should receive. +11.4 Apportionment Methods +• +Hamilton’s method of apportionment uses the standard divisor and standard lower quotas, and it distributes any +remaining seats based on the size of the fractional parts of the standard lower quota. Hamilton’s method satisfies +the quota rule and favors neither larger nor smaller states. +• +Jefferson’s method of apportionment uses a modified divisor that is adjusted so that the modified lower quotas sum +to the house size. Jefferson’s method violates the quota rule and favors larger states. +• +Adams’s method of apportionment uses a modified divisor that is adjusted so that the modified upper quotas, sum +to the house size. Adams’s method violates the quota rule and favors smaller states. +• +Webster’s method of apportionment uses a modified divisor that is adjusted so that the modified state quotas, +rounded using traditional rounding, sum to the house size. Webster’s method violates the quota rule but favors +neither larger nor smaller states. +11.5 Fairness in Apportionment Methods +• +Several surprising outcomes can occur when apportioning seats that voters may find unfair: Alabama paradox, +population paradox, and new-state paradox. +• +Apportionment methods are susceptible to apportionment paradoxes and may violate the quota rule. +• +The Balinsky-Young Impossibility Theorem indicates that no apportionment can satisfy all fairness criteria. +Videos +11.1 Voting Methods +• +How Does Ranked-Choice Voting Work? (https://openstax.org/r/ranked-choice_voting1) +• +Determine Winner of Election by Ranked-Choice Method (aka Instant Runoff) (https://openstax.org/r/ranked- +11 • Chapter Summary +1227 + +choice_voting2) +• +Determine Winner of Election by Borda Count Method (https://openstax.org/r/Borda_count_method) +• +Determine Winner of Election by Pairwise Comparison Method (https://openstax.org/r/w1NNK7Dn3E8) +11.2 Fairness in Voting Methods +• +Separation of Powers and Checks and Balances (https://openstax.org/r/separation_of_powers) +11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem +• +What Is a Ratio? (https://openstax.org/r/ratios_proportions) +• +What Are the Different Types of Ratios? (https://openstax.org/r/ratio_types) +• +Math Antics – Rounding (https://openstax.org/r/rounding) +11.4 Apportionment Methods +• +Hamilton Method of Apportionment (https://openstax.org/r/Hamiltons_method) +• +Jefferson Apportionment Method (https://openstax.org/r/Jeffersons_method) +• +Adams Method Apportionment Calculator (https://openstax.org/r/Adams_method) +Formula Review +11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem +Let +be a particular item and +another such that there is a constant ratio of +to +. +• +and +• +• +11.5 Fairness in Apportionment Methods +Projects +The First Census: Challenges of Collecting Population Data +As you and your fellow founders write the Imaginarian constitution, you must also design systems to collect accurate +information about the population of Imaginaria. Why is this important and what are the challenges you will face? Let’s +find out! Research and answer each question. (Questions adapted from “Authorizing the First Census–The Significance of +Population Data,” Census.gov, United States Census Bureau) +1. +The First Congress laid out a plan for collecting this data in Chapter II, An Act Providing for the Enumeration of the +Inhabitants of the United States, which was approved March 1, 1790. What group did Congress select to carry out +the first enumeration? Why did they choose this group? What might be the advantages and disadvantages to this +approach? +2. +Choose at least two challenges faced by the U.S. government during the first enumeration and explain how the +information gathered might have helped address them. +3. +President James Madison was a U.S. Representative who participated in the census debate in the first congress. +What were the main points of his remarks and how were they relevant to the overall debate about the first +enumeration? +4. +What issues do you think the U.S. Census Bureau encounters today as it continues to collect and process data about +the U.S. population that might be significant to you and the other founders of Imaginaria? +The Party System: How Many Political Parties Are Enough? +The electoral system of Imaginaria will likely involve multiple political parties. The way these parties interact with the +system may be determined by the founders of the new democracy. Let’s explore the ways in which political parties +interact with the governments of various democracies by researching and answering each of the following questions. +1228 +11 • Chapter Summary +Access for free at openstax.org + +1. +What concerns did the founders of the United States have about political parties? Have any of their concerns +become a reality? Were political parties addressed in the U.S. Constitution? How did they become such a large part +of politics in the U.S.? As a founder of Imaginaria, would you address political parties in your constitution? +2. +What is frontloading? Does our current system of frontloading impact fair representation? Why do two small, racially +homogeneous states hold their primaries first? Do you think this impacts the final results? +3. +How does the interaction political parties and the electoral system in the United States differ from that of other +countries? Give at least three specific examples. Of these three, which would you be most likely to use as a model +for Imaginaria? +Technology and Voting: How Has the Digital Age Impacted Elections? +Unlike most other countries, Imaginaria will be founded in the digital age. Let’s explore the impact this might have on +how you choose to set up your electoral system. Research and answer each question. +1. +Participation in an electoral system if very important. In what ways does the Internet positively affect participation? +In what ways does it negatively affect participation. What roll, if any, should government of Imaginaria play in +tempering or promoting the Impact of the Internet? +2. +Find at least three examples of governments that utilize Internet voting around the world. What concerns have +slowed the spread of this technology? What are the advantages and disadvantages of Internet voting? Would you be +in favor of Internet voting for Imaginaria? Why or why not? +11 • Chapter Summary +1229 + +Chapter Review +Voting Methods +1. In a plurality election, the candidates have the following vote counts: A 125, B 132, C 149, and D 112. Which +candidate has the plurality and wins the election? +For the following exercises, use the table below. +Options +A +B +C +D +E +Candidate 1 +1 +3 +3 +1 +3 +Candidate 2 +2 +1 +1 +2 +4 +Candidate 3 +3 +4 +2 +4 +1 +Candidate 4 +4 +2 +4 +3 +2 +2. Which candidate has a plurality? +3. Does the plurality candidate have a majority? +4. Determine the winner of the election by the Hare method based on the sample preference summary in the +table. +For the following exercises, use the table below. +Number of Ballots +10 +20 +15 +5 +Option A +1 +4 +3 +4 +Option B +2 +3 +4 +2 +Option C +4 +2 +1 +3 +Option D +3 +1 +2 +1 +5. Use ranked-choice voting to determine the two options in the final round and the number of votes they each +receive in that round. +6. Is there a winning option? If so, which option? Justify your answer. +For the following exercises, use the table below. +Number of Ballots +100 +80 +110 +105 +55 +Candidate A +1 +1 +4 +4 +2 +Candidate B +2 +2 +2 +3 +1 +Candidate C +4 +4 +1 +1 +4 +Candidate D +3 +3 +3 +2 +3 +7. What are the Borda scores for each candidate? +8. Which candidate is the winner by the Borda count method? +Use Pairwise Comparison Matrix for Candidates U, V, W, X, and Y to answer Questions 9 and 10. +9. Calculate the points received by each candidate in pairwise comparison matrix. +10. Determine the winner of the pairwise comparison election represented by matrix. If there is a winner, +1230 +11 • Chapter Summary +Access for free at openstax.org + +determine whether the winner is a Condorcet candidate and explain your reasoning. If there is no winner, +indicate this. +11. The ladies of The Big Bang Theory decide to hold their own approval voting election to determine the best option +in Rock, Paper, Scissors, Lizard, Spock. Use the summary of their approval ballots in the table below to determine +the number of votes for each candidate. Determine the winner, or state that there is none. +VOTERS +Penny +Bernadette +Amy +Rock +Yes +No +No +Paper +Yes +Yes +No +Scissors +Yes +Yes +Yes +Lizard +No +No +No +Spock +Yes +No +Yes +For the following exercises, use the table below. +Percentage of Vote +40% +35% +25% +Candidate A +1 +3 +2 +Candidate B +2 +1 +3 +Candidate C +3 +2 +1 +12. Which candidate is the winner by the ranked-choice method? +13. Suppose that you used the approval method and each voter approved their top two choices. Which candidate is +the winner by the approval method? +14. Which candidate is the winner by the Borda count method? +Fairness in Voting Methods +15. In a Borda count election, the candidates have the following Borda scores: A 1245, B 1360, and C 787. Candidate A +received 55% of the first place rankings. Identify which fairness criteria, if any, are violated by characteristics of the +described voter profile in this Borda count election. Explain your reasoning. +For the following exercises, use the table below. +Number of Ballots +8 +10 +12 +4 +Option A +1 +3 +2 +1 +Option B +3 +1 +4 +4 +Option C +4 +2 +1 +2 +Option D +2 +4 +3 +3 +16. Determine Borda score for each candidate, and the winner of the election using the Borda count method. +17. Is there a majority candidate? If so, which candidate? +18. Does the Borda method election violate the majority criterion? Justify your answer. +19. In a Borda count election, the candidates have the following Borda scores: A 15, B 11, C 12, and D 16. The +pairwise match up points for the same voter profiles would have been A 2, B 0, C 1, and D 3. Identify which +11 • Chapter Summary +1231 + +fairness criteria, if any, are violated by characteristics of the described voter profile in this Borda election. +Explain your reasoning. +20. Determine the winner of the election using the ranked-choice method. +21. If the four voters in the last column rank C ahead of A, which candidate wins by the ranked-choice method? +22. Does this ranked-choice election violate the monotonicity criterion? Explain your reasoning. +For the following exercises, use the table below. +Number of Ballots +15 +12 +9 +3 +Option A +1 +3 +3 +2 +Option B +2 +2 +1 +1 +Option C +3 +1 +2 +3 +23. Determine the winner of the election by the Borda method. +24. Does this Borda method election violate the IIA? Why or why not? +25. Which of the ranked voting methods in this chapter, if any, meets the majority criterion, the head-to-head +criterion, the monotonicity criterion, and the irrelevant alternatives criterion? +Standard Divisors, Standard Quotas, and the Apportionment Problem +26. Identify the states, the seats, and the state population (the basis for the apportionment) in the given scenario: The +reading coach at an elementary school has 52 prizes to distribute to their students as a reward for time spent +reading. +27. Use the given information to find the standard divisor to the nearest hundredth. Include the units. The total +population is 2,235 automobiles, and the number of seats is 14 warehouses. +28. Use the given information to find the standard quota. Include the units. The state population is eight residents in a +unit, and the standard divisor is 1.75 residents per parking space. +Apportionment Methods +29. Which of the four apportionment methods discussed in this section does not use a modified divisor? +30. Determine the Hamilton apportionment for Scenario X in the table below. +State A +State B +State C +State D +State E +State F +Total Seats +Scenario X +17.63 +26.62 +10.81 +16.01 +13.69 +15.24 +100 +31. Does the apportionment resulting from Method X in the table below satisfy the quota rule? Why or why not? +State A +State B +State C +State D +State E +Standard Quota +1.67 +3.33 +5.00 +6.67 +8.33 +Apportionment Method X +2 +2 +5 +7 +9 +For the following exercises, use the table below and the following information: In Wakanda, the domain of the Black +Panther, King T’Challa, has six fortress cities. In Wakandan, the word “birnin” means “fortress city.” King T’Challa has +found 111 Vibranium artifacts that must be distributed among the fortress cities of Wakanda. He has decided to +apportion the artifacts based on the number of residents of each birnin. +1232 +11 • Chapter Summary +Access for free at openstax.org + +Fortress +Cities +Birnin +Djata (D) +Birnin +T'Chaka (T) +Birnin +Zana (Z) +Birnin +S'Yan (S) +Birnin +Bashenga (B) +Birnin +Azzaria (A) +Total +Residents +26,000 +57,000 +27,000 +18,000 +64,000 +45,000 +237,000 +Standard +Quota +12.18 +26.70 +12.65 +8.43 +29.98 +21.08 +111 +32. Does the Jefferson method result in an apportionment that satisfies or violates the quota rule in this scenario? +33. Find the modified upper quota for each state using a modified divisor of 2,250. Is the sum of the modified +quotas too high, too low, or equal to the house size? +34. Use the Adams method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so, +indicate the value of the modified divisor. +35. Does the Adams method result in an apportionment that satisfies or violates the quota rule in this scenario? +36. Use the Webster method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If +so, indicate the value of the modified divisor. +37. Does the Webster method result in an apportionment that satisfies or violates the quota rule in this scenario? +38. Which of the four methods of apportionment from this section are the residents of Birnin S'Yan likely to prefer? +Justify your answer. +39. Does the change from a standard divisor to a modified divisor tend to change the number of seats for larger or +smaller states more? +40. Which of the four apportionment methods—Jefferson, Adams, Hamilton, or Webster—satisfies the quota rule? +Fairness in Apportionment Methods +41. A city purchased five new firetrucks and apportioned them among the existing fire stations. Although your +neighborhood fire station has the same proportion of the city’s firetrucks as before the new ones were purchased, +it now has one fewer. Is this scenario an example of a quota rule violation, the Alabama paradox, the population +paradox, the new-states paradox, or none of these? +42. When the number of seats changed from 25 to 26, the standard quotas changed from A 2.21, B 5.25, C 11.27, and +D 6.27 to A 2.30, B 5.46, C 11.72, and D 6.52. +a. +How did the increase in seats impact the apportionment? +b. +Is this apportionment an example of a paradox? Justify your answer. +43. The school resources officers in a county were reapportioned based on the most recent census. The number of +students at Chapel Run Elementary went up while the number of students at Panther Trail Elementary went down, +but Chapel Run now has 1 fewer resources officers while Panther Trail has one more than it did previously. Is this +scenario an example of a quota rule violation, the Alabama paradox, the population paradox, the new-states +paradox, or none of these? +For the following exercises, the house size is 24 seats. When the population of A increases by 28 percent, B increases by +26 percent, and C increases by 15 percent, the standard quotas change from A 3.38, B 6.32, and C 14.30 to A 3.63, B +6.67, and C 13.71. +44. How did the change in populations impact the apportionment? +45. Is this apportionment an example of a paradox? Justify your answer. +46. When the city of Cocoa annexed an adjacent unincorporated community, the number of seats on the city council +was increased to maintain the standard ratio of citizens to seats, but one existing community of Cocoa still lost a +seat on the city council to another existing community of Cocoa when the new community was added. Is this +scenario an example of a quota rule violation, the Alabama paradox, the population paradox, the new-states +paradox, or none of these? +For the following exercises, the house size was 27. There were three states with standard quotas of A 6.39, B 11.40, and +C 9.21. A fourth state was annexed, and the house size was increased to 35. The new standard quotas are A 6.38, B +11.37, C 9.19, and D 8.06. +47. How did the additional state impact the apportionment? +48. Is this apportionment an example of a paradox? Justify your answer. +For the following exercises, suppose 11 seats are apportioned to States A, B, and C with populations of 50, 129, and 181 +11 • Chapter Summary +1233 + +people, respectively. Then the populations of States A, B, and C change to 57, 151, and 208, respectively. +49. Demonstrate that the population paradox occurs when the Hamilton method is used. +50. Demonstrate that the population paradox does not occur when the Jefferson method is used. Justify your +answer. +51. Demonstrate that the population paradox does not occur when the Adams method is used. Justify your answer. +52. Demonstrate that the population paradox does not occur when the Webster method is used. Justify your +answer. +Chapter Test +For the following exercises, use the table below. +Number of Ballots +12 +17 +15 +13 +Option A +1 +2 +4 +3 +Option B +3 +1 +3 +2 +Option C +4 +3 +1 +4 +Option D +2 +4 +2 +1 +1. Determine the winner of the election by plurality. +2. Determine the Borda scores for each candidate to determine the winner by Borda count method. +3. Create and analyze a pairwise comparison matrix based on the preference summary to determine the winner of +the election by pairwise comparison. +4. From the table below use ranked-choice voting to determine the winner of the election. +Number of Ballots +28 +5 +30 +5 +16 +16 +Option L +3 +2 +1 +1 +2 +3 +Option R +1 +1 +3 +2 +3 +2 +Option E +2 +3 +2 +3 +1 +1 +For the following exercises, identify which fairness criteria, if any, are violated by characteristics of the described voter +profile. Explain your reasoning +5. In a Borda count election, the candidates have the following Borda scores: A 1345, B 1260, C 685. Candidate B +received 51% of the first-place rankings. +6. In a plurality election, the candidates have the following percentages of first place votes: A 25, B 21, C 30, D 24. +The pairwise matchup points for the same voter profiles would have been A 3, B 0, C 2, D 2. +For the following exercises, use the table below. +Number of Ballots +13 +14 +11 +12 +Option A +2 +1 +3 +3 +Option B +3 +2 +4 +1 +Option C +4 +4 +1 +2 +Option D +1 +3 +3 +4 +7. Determine the winner by ranked-choice voting if two of the voters in the second column up-rank the original +1234 +11 • Chapter Summary +Access for free at openstax.org + +winner. Refer to Question 4. Which fairness criterion, if any, is violated? +8. Determine the winner by ranked-choice voting if candidate R is removed from the election. Refer to Question 4. +Which fairness criterion, if any, is violated? +For the following exercises, use this information: The incorporated town of Orange Grove consists of two subdivisions: +The Oaks with 1,254 residents, and The Villages with 10,746 residents. A council with 100 members supervises the +municipality's operations with representation proportionate to the number of residents. +9. Identify the states, the seats, and the state population (the basis for apportionment) in the given scenario. +10. Determine the standard divisor for the apportionment +11. Determine each state's standard quota rounded to two decimal places. +For the following exercises, use this information: Air Force administration wanted to distribute 27 aircraft across 6 +bases based on the number of qualified pilots stationed at those bases. The standard quota is 2.2963. The standard +quotas for each base are listed in the table below. +Air Force Base +(A) Alpha +(B) Bravo +(C) Charlie +(D) Delta +(E) Echo +(F) Foxtrot +Pilots +13 +12 +5 +16 +7 +9 +Standard Quota +5.66 +5.23 +2.18 +6.97 +3.05 +3.92 +12. Determine the states' lower quotas and the states' upper quotas. +13. Use Adams's method to apportion the aircraft. +14. Use Jefferson's method to apportion the aircraft. +15. The apportionment of 616 schools to 5 Hawaiian counties by various methods is displayed in the table below. +County +Hawaii +Honolulu +Kalawao +Kauai +Maui +Lower Quota +87 +424 +0 +31 +72 +Upper Quota +88 +425 +1 +32 +73 +Jefferson +87 +425 +1 +31 +72 +Adams +88 +422 +1 +32 +73 +Webster +87 +424 +1 +31 +73 +Apportionment by which methods, if any, fail to satisfy the quota rule? Explain your reasoning. +For the following exercises, use this information: The incorporated town of Orange Grove consists of two subdivisions: +The Oaks with 1,254 residents, and The Villages with 10,746 residents. A council with 100 members supervises the +municipality's operations. The Hamilton method was used to apportion the council seats. The Oaks has 10 seats on the +council, while The Villages has 90 seats. The council votes to annex an unincorporated subdivision called The Lakes with +a population of 630. They plan to increase the size of the council to maintain the ratio of seats to residents such that +the new council will have 100 seats plus the number of seats given to The Lakes. +16. What is the standard divisor from the original apportionment? +17. What is the new house size? +18. Use the Hamilton method to reapportion the seats. +19. Is the reapportionment an example of the new-states paradox? If so, how? +For the following exercises, use this information: determine whether the reapportionment violates the Alabama +paradox, the population paradox, or neither. Justify your answer. +20. States A, B, C, and D received 21, 25, 26, and 28 seats respectively. When the population remains the same, but +house size is increased, the reapportionment is A 20, B 26, C 27, and D 29. +21. States A, B, C, and D received 21, 25, 26, and 28 seats respectively. When the house size remains the same, the +population of state A increased, the population of state B decreased, and the populations of states C and D +11 • Chapter Summary +1235 + +remained the same, the reapportionment is A 20, B 26, C 26, and D 28. +1236 +11 • Chapter Summary +Access for free at openstax.org + +Figure 12.1 Networks connect cities around the globe. (credit: "Globalization" by faith.e.murphy/Flickr, Public Domain) +Chapter Outline +12.1 Graph Basics +12.2 Graph Structures +12.3 Comparing Graphs +12.4 Navigating Graphs +12.5 Euler Circuits +12.6 Euler Trails +12.7 Hamilton Cycles +12.8 Hamilton Paths +12.9 Traveling Salesperson Problem +12.10 Trees +Introduction +In this chapter, you will learn the fundamental skills needed to work with graphs used in an area of mathematics known +as graph theory. You can think of these graphs as a kind of map. Maps have served many purposes over the course of +history. You probably use GPS maps to navigate to various destinations. A scientist from ancient Greece named Ptolemy +wanted an accurate map of the world to make more accurate astrological predications. In recent years, neurobiologists +have mapped the cerebral cortex to better understand the human brain. Social network analysts map online interactions +to assist advertisers in reaching target audiences. Like other maps, the graphs you will study in this chapter can serve +many purposes, but they do not have a lot of the details you might expect in a map such as size, shape, and distance +between objects. All of that is stripped away so that we can focus on one element of maps, the connections between +objects. +12 +GRAPH THEORY +12 • Introduction +1237 + +12.1 Graph Basics +Figure 12.2 Cell phone networks connect individuals. (credit: "Business people using their phones" by Rawpixel +Ltd./Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify parts of a graph. +2. +Model applications of graph basics. +When you hear the word, graph, what comes to mind? You might think of the +-coordinate system you learned about +earlier in this course, or you might think of the line graphs and bar charts that are used to display data in news reports. +The graphs we discuss in this chapter are probably very different from what you think of as a graph. They look like a +bunch of dots connected by short line segments. The dots represent a group of objects and the line segments represent +the connections, or relationships, between them. The objects might be bus stops, computers, Snapchat accounts, family +members, or any other objects that have direct connections to each other. The connections can be physical or virtual, +formal or casual, scientific or social. Regardless of the kind of connections, the purpose of the graph is to help us +visualize the structure of the entire network to better understand the interactions of the objects within it. +Parts of a Graph +In a graph, the objects are represented with dots and their connections are represented with lines like those in Figure +12.3. Figure 12.3 displays a simple graph labeled G and a multigraph labeled H. The dots are called vertices; an +individual dot is a vertex, which is one object of a set of objects, some of which may be connected. We often label +vertices with letters. For example, Graph G has vertices a, b, c, and d, and Multigraph H has vertices, e, f, g, and h. Each +line segment or connection joining two vertices is referred to as an edge. H is considered a multigraph because it has a +double edge between f and h, and a double edge between h and g. Another reason H is called a multigraph is that it has +a loop connecting vertex e to itself; a loop is an edge that joins a vertex to itself. Loops and double edges are not allowed +in a simple graph. +To sum up, a simple graph is a collection of vertices and any edges that may connect them, such that every edge +connects two vertices with no loops and no two vertices are joined by more than one edge. A multigraph is a graph in +which there may be loops or pairs of vertices that are joined by more than one edge. In this chapter, most of our work +will be with simple graphs, which we will call graphs for convenience. +1238 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.3 A Graph and a Multigraph +It is not necessary for the edges in a graph to be straight. In fact, you can draw an edge any way you want. In graph +theory, the focus is on which vertices are connected, not how the connections are drawn (see Figure 12.4). In a graph, +each edge can be named by the two letters of the associated vertices. The four edges in Graph X in Figure 12.4 are ab, +ac, ad, and ae. The order of the letters is not important when you name the edge of a graph. For example, ab refers to +the same edge as ba. +A graph may have vertices that are not joined to other vertices by edges, such as vertex f in Graph X in Figure 12.4, +but any edge must have a vertex at each end. +Figure 12.4 Different Representations of the Same Graph +EXAMPLE 12.1 +Identifying Edges and Vertices +Name all the vertices and edges of graph F in Figure 12.5. +Figure 12.5 Graph F +Solution +The vertices are v, w, x, y, and z. The edges are vw, vx, wx, wz, xy, and xz. +When listing the vertices and edges in a graph, work in alphabetical order to avoid accidentally listing the same item +12.1 • Graph Basics +1239 + +twice. When you are finished, count the number of vertices or edges you listed and compare that to the number of +vertices or edges on the graph to ensure you didn’t miss any. +YOUR TURN 12.1 +1. Name all the vertices and edges of Graph A. +Graph A +Since the purpose of a graph is to represent the connections between objects, it is very important to know if two vertices +share a common edge. The two vertices at either end of a given edge are referred to as neighboring, or adjacent. For +example, in Figure 12.5, vertices x and w are adjacent, but vertices y and w are not. +EXAMPLE 12.2 +Identifying Vertices That Are Not Adjacent +Name all the pairs of vertices of graph F in Figure 12.5 that are not adjacent. +Solution +The pairs of vertices that are not adjacent in graph F are v and y, v and z, w and y, and y and z. +YOUR TURN 12.2 +1. Name all the pairs of vertices of graph A in Figure 12.6 that are not adjacent. +PEOPLE IN MATHEMATICS +Sergey Brin and Laurence Page +The “Google boys,” Sergey Brin and Laurence Page, transformed the World Wide Web in 1998 when they used the +mathematics of graph theory to create an algorithm called Page Rank, which is known as the Google Search Engine +today. The two computer scientists identified webpages as vertices and hyperlinks on those pages as edges because +hyperlinks connect one website to the next. The number of edges influences the ranking of a website on the Google +Search Engine because the websites with more links to “credible sources” are ranked higher. ("Page Rank: The Graph +Theory-based Backbone of Google," (https://openstax.org/r/The_Graph_Theory) September 20, 2011, Cornell +University, Networks Blog. +1240 +12 • Graph Theory +Access for free at openstax.org + +Analyzing Geographical Maps with Graphs +Figure 12.6 Commercial airlines' route systems create a global network. +When graphs are used to model and analyze real-world applications, the number of edges that meet at a particular +vertex is important. For example, a graph may represent the direct flight connections for a particular airport as in Figure +12.7. Representing the connections with a graph rather than a map shifts the focus away from the relative positions and +toward which airports are connected. In Figure 12.7, the vertices are the airports, and the edges are the direct flight +paths. The number of flight connections between a particular airport and other South Florida airports is the number of +edges meeting at a particular vertex. For example, Key West has direct flights to three of the five airports on the graph. +In graph theory terms, we would say that vertex FYW has degree 3. The degree of a vertex is the number of edges that +connect to that vertex. +Figure 12.7 Direct Flights between South Florida Airports +EXAMPLE 12.3 +Determining the Degree of a Vertex +Determine the degree of each vertex of Graph J in Figure 12.7. If graph J represents direct flights between a set of +airports, do any of the airports have direct flights to two or more of the other cities on the graph? +12.1 • Graph Basics +1241 + +Figure 12.8 Graph J +Solution +For each vertex, count the number of edges that meet at that vertex. This value is the degree of the vertex. In Figure +12.9, the dashed edges indicate the edges that meet at the marked vertex. +Figure 12.9 Degrees of Vertices of Graph J +Vertex a has degree 3, vertex b has degree 1, vertices c and d each have degree 2, and vertex e has degree 0. Airports a, +c, and d have direct flights to two or more of the other airports. +YOUR TURN 12.3 +1. Name a vertex of Graph A in Figure 12.6 with degree 4. +Graphs are also used to analyze regional boundaries. The states of Utah, Colorado, Arizona, and New Mexico all meet at +a single point known as the “Four Corners,” which is shown in the map in Figure 12.10. +1242 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.10 Map of the Four Corners +In Figure 12.11, each vertex represents one of these states, and each edge represents a shared border. States like Utah +and New Mexico that meet at only a single point are not considered to have a shared border. By representing this map as +a graph, where the connections are shared borders, we shift our perspective from physical attributes such as shape, size +and distance, toward the existence of the relationship of having a shared boundary. +Figure 12.11 Graph of the Shared Boundaries of Four Corners States +EXAMPLE 12.4 +Graphing the Midwestern States +A map of the Midwest is given in Figure 12.12. Create a graph of the region in which each vertex represents a state and +each edge represents a shared border. +12.1 • Graph Basics +1243 + +Figure 12.12 Map of Midwestern States +Solution +Step 1: For each state, draw and label a vertex as in Figure 12.13. +Figure 12.13 Vertex Assigned to Each Midwestern State +Step 2: Draw edges between any two states that share a common land border as in Figure 12.14. +1244 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.14 Edge Assigned to Each Pair of Midwestern States with Common Border +The graph is given in Figure 12.15. +Figure 12.15 Final Graph Representing Common Boundaries between Midwestern States +YOUR TURN 12.4 +1. The figure shows a map of the Island of Oahu in the State of Hawaii divided into regions. Draw a graph in which +each vertex represents one of the regions and each edge represents a shared land border. +12.1 • Graph Basics +1245 + +Map of Oahu +VIDEO +Graph Theory: Create a Graph to Represent Common Boundaries on a Map (https://openstax.org/r/Graph_Theory) +Graphs of Social Interactions +Geographical maps are just one of many real-world scenarios which graphs can depict. Any scenario in which objects are +connected to each other can be represented with a graph, and the connections don’t have to be physical. Just think +about all the connections you have to people around the world through social media! Who is in your network of Twitter +followers? Whose Snapchat network are you connected to? +EXAMPLE 12.5 +Graphing Chloe’s Roblox Friends +Roblox is an online gaming platform. Chloe is interested to know how many people in her network of Roblox friends are +also friends with each other so she polls them. Explain how a graph or multigraph might be drawn to model this +scenario by identifying the objects that could be represented by vertices and the connections that could be represented +by edges. Indicate whether a graph or a multigraph would be a better model. +Solution +The objects that are represented with vertices are Roblox friends. A Roblox friendship between two friends will be +represented as an edge between a pair of vertices. There will be no double edges because it is not possible for two +friends to be linked twice in Roblox; they are either friends or they are not. Also, a player cannot be a friend to themself, +so there is no need for a loop. Since there are no double edges or loops, this is best represented as a graph. +YOUR TURN 12.5 +1. In a particular poker tournament, five groups of five players will play at a table until one player wins, then the +five winning players will play each other at a table in a final round. Explain how a graph or multigraph might be +1246 +12 • Graph Theory +Access for free at openstax.org + +drawn to model this scenario by identifying the objects that could be represented by vertices and the +connections that could be represented by edges. Indicate whether a graph or a multigraph would be a better +model. +WHO KNEW? +Using Graph Theory to Reduce Internet Fraud +Could graphs be used to reduce Internet fraud? At least one researcher thinks so. Graph theory is used every day to +analyze our behavior, particularly on social network sites. Alex Buetel, a computer scientist from Carnegie Mellon +University in Pittsburgh, Pennsylvania, published a research paper in 2016 that discussed the possibilities of +distinguishing the normal interactions from those that might be fraudulent using graph theory. Buetel wrote, “To +more effectively model and detect abnormal behavior, we model how fraudsters work, catching previously undetected +fraud on Facebook, Twitter, and Tencent Weibo and improving classification accuracy by up to 68%.” In the same +paper, the researcher discusses how similar techniques can be used to model many other applications and even, +“predict why you like a particular movie.” (Alex Beutel, "User Behavior Modeling with Large-Scale Graph Analysis," +http://reports-archive.adm.cs.cmu.edu/anon/2016/CMU-CS-16-105.pdf, May 2016, CMU-CS-16-105, Computer Science +Department, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA) +Check Your Understanding +1. A simple graph has no loops. +a. +True +b. +False +2. It is not possible to have a vertex of degree 0. +a. +True +b. +False +3. A graph with three vertices has at most three edges. +a. +True +b. +False +4. A multigraph with three vertices has at most three edges. +a. +True +b. +False +5. If vertex a is adjacent to vertex b, and vertex b is adjacent to vertex c, then vertex a must be adjacent to vertex c. +a. +True +b. +False +SECTION 12.1 EXERCISES +For the following exercises, use the figure showing Diagrams A, B, C, and D. +1. Identify any multigraphs. +2. Identify any graphs. +3. Identify any multigraph that has a loop. +4. Identify any multigraph that has a double edge. +12.1 • Graph Basics +1247 + +For the following exercises, use the figure showing Graphs S, T, U, V, and W. +5. Determine the number of vertices in Graph T. +6. Determine the number of vertices in Graph U. +7. Identify the graph with the most vertices. +8. Identify the graphs with four vertices. +9. Identify the graph with the most edges. +10. Identify the graph with the fewest edges. +11. Name the vertices in Graph S. +12. Name the vertices in Graph V. +13. Determine the number of edges in Graph U. +14. Determine the number of edges in Graph T. +15. Name the edges in Graph V. +16. Name the edges in Graph W. +17. Identify any pairs of vertices in Graph T that are not adjacent. +18. Identify any vertices in Graph V that are not adjacent. +19. Find the degree of vertex a in graph U. +20. Find the degree of vertex a in graph S. +21. Which two graphs have a vertex of degree 4? +For the following exercises, use the figure showing Graphs A, B, C, and D. +22. Identify the graph with 1 vertex of degree 6 and 6 vertices of degree 1. +23. Identify the graph with 1 vertex of degree 6 and 6 vertices of degree 2. +24. Identify the graph with exactly 1 vertex of degree 0. +25. Identify the graph with exactly 2 vertices of degree 1. +26. Name all the pairs of adjacent vertices in Graph B. +27. Name all the pairs of adjacent vertices in Graph A. +28. Identify the graph in which the sum of the degrees of the vertices is 14. +For each of the following exercises, a scenario is given. Explain how a graph or multigraph might be drawn to model +the scenario by identifying the objects that could be represented by vertices and the connections that could be +represented by edges. Indicate whether a graph or a multigraph would be a better model. +29. The Centers for Disease Control and Prevention (CDC) is tracking the spread of a virus. The CDC is attempting to +determine where the virus began by identifying where each known carrier contracted the virus. +30. The faculty members in a college mathematics department have a “telephone tree” that assigns each faculty +member two other faculty members to call with information in the event of an emergency. The chairperson of +the department contacts two faculty members. Each of these faculty members contacts two faculty members, +and so on, until all faculty members have been notified. +31. There is a theory that any two people on earth are no more than six social connections apart, which is known as +“six degrees of separation” or the “six handshake rule.” It means that there is a chain of a “friend of a friend” +that connects any two people through at most six steps. +32. The U.S. Postal Service has a network of post offices that move mail around the United States. Mail trucks from +one post office follow routes deliver mail to other locations. These mail trucks also pick up mail to bring back to +their home post office. +In the following exercises , draw a graph to represent the given scenario. +1248 +12 • Graph Theory +Access for free at openstax.org + +33. Draw a graph to model the scenario described in Your Turn 12.5. +34. In the game of chess, a king can move one space in any direction, vertically, horizontally, or diagonally, as +indicated in the figure. Draw a graph to represent the possible movements of a king on a three-by-three section +of a chess board such that each edge represents one move, and each vertex represents a position where the +king might be at the beginning or end of a turn. Assume that the king can have unlimited turns. +35. Chloe is interested to know how many in her network of Roblox friends are also friends with each other, so she +polls them. The following is a list of each of Chloe’s friends and their common friends. Create a graph of the +Roblox friends in Chloe’s network, including Chloe. +• +Aria is friends with no one else in Chloe’s network. +• +Benicio is friends with Dakshayani, Eun-ah, and Fukashi. +• +Dakshayani is friends with Benicio. +• +Eun-ah is friends with Benicio and Fukashi. +• +Fukashi is friends with Benicio and Eun-ah. +12.2 Graph Structures +Figure 12.16 Neuroimaging shows brain activity. (credit: "MRI Scan" by NIH Image Gallery/Flickr, Public Domain) +Learning Objectives +After completing this section, you should be able to: +1. +Describe and interpret relationships in graphs. +2. +Model relationships with graphs. +Graph theory is used in neuroscience to study how different parts of the brain connect. Neurobiologists use functional +magnetic resonance imaging (fMRI) to measure levels of blood in different parts of the brain, called nodes. When nodes +are active at the same time, it suggests there is a functional connection between them so they form a network. This +network can be represented as a graph where the vertices are the nodes and the functional connections are the edges +between them. (Mikey Taylor, "Graph Theory & Machine Learning in Neuroscience," Medium.com, June 24, 2020. +12.2 • Graph Structures +1249 + +Importance of the Degrees of Vertices +One reason scientists study these networks is to determine how successful the communication within a network +continues to be when it experiences failures in nodes and connections. Graphs can be used to study the resilience of +these networks. (Mikey Taylor, "Graph Theory & Machine Learning in Neuroscience," Medium.com, June 24, 2020) +EXAMPLE 12.6 +Using Graphs to Understand Relationships +The graphs in Figure 12.17 represent neural networks, where the vertices are the nodes, and the edges represent +functional connections between them. Which graph do you think would represent a network with the highest resistance +to failure? Which graph would probably be the most vulnerable to failure? How might this relate to the degrees of the +vertices? +Figure 12.17 Models of Neural Networks +Solution +Graph B represents a network that would have the highest resistance to failure because there are more edges +connecting the nodes indicating there are more connections between nodes than on either of the other graphs. This +would make the network more resistant to failure because there are more options to work around a faulty edge or node. +Graph A would probably be the most susceptible to failure because the network depends on a few particular edges and +vertices for connections. There are more vertices of higher degree in Graph B than in Graph A because there are more +edges connecting the nodes. +YOUR TURN 12.6 +Sociologists use graphs to study connections between people and to identify characteristics that make communities +more resilient, more likely to stay connected over time. In the graphs shown, the edges represent social connections +and the vertices are individuals. +Models of Communities +1. Which graph do you think represents the most resilient community? What is the sum of the degrees of all the +vertices in this graph? +2. Which graph do you think represents the community that is least likely to stay connected over time? What is +the sum of the degrees of all the vertices in this graph? +3. Is the sum of the degrees higher or lower in a more resilient community? +1250 +12 • Graph Theory +Access for free at openstax.org + +Relating the Number of Edges to the Degrees of Vertices +In the applications of graph theory to neuroscience and sociology in Example 12.6 and YOUR TURN 12.6, there was a +correlation between the degrees of vertices and the resilience of a network. Researchers in many fields have also +observed a direct relationship between the number of edges in a graph and the degrees of the vertices. To begin to +understand this relationship, consider a graph with five vertices and zero edges as in Figure 12.18. +Figure 12.18 Graph with Five Vertices and zero Edges +Instead of being marked with a name, each vertex in Figure 12.18 is marked with its degree. In this case, all of the +degrees are 0 so the sum of the degrees is also zero. Suppose that we add an edge between any two existing vertices +and indicate the degrees of the vertices. This gives us a graph with five vertices and one edge like the graph in Figure +12.19. +Figure 12.19 Graph with Five Vertices and One Edge +Note that the degrees of two vertices increased, each by 1. So, the sum of the degrees is now 2. Suppose that we +continue in this way, adding one edge at a time and making note of the number of edges and the sum of the degrees of +the vertices as in Figure 12.20. +Figure 12.20 Comparing Number of Edges to Sum of Degrees +Figure 12.20 demonstrates a characteristic that is true of all graphs of any shape or size. When the number of edges is +increased by one, the sum of the degrees increases by two. This happens because each edge has two ends and each end +increases the degree of one vertex by one unit. As a result, the sum of the degrees of the vertices on any graph is always +twice the number of edges. This relationship is known as the Sum of Degrees Theorem. +FORMULA +For the Sum of Degrees Theorem, +or +EXAMPLE 12.7 +Finding the Sum of Degrees +Suppose that a graph has five edges. +1. +Find the sum of the degrees of the vertices. +2. +Draw two different graphs that demonstrate this conclusion. +12.2 • Graph Structures +1251 + +Solution +1. +The sum of the degrees is twice the number of edges: 2 × 5=10. The sum of the degrees is 10. +2. +See Figure 12.21. +Figure 12.21 +YOUR TURN 12.7 +Suppose that the sum of the degrees of a graph is six. +1. Find the number of edges. +2. Draw two graphs that demonstrate your conclusion. +Completeness +Suppose that there were five strangers in a room, A, B, C, D, and E, and each one would be introduced to each of the +others. How many introductions are necessary? One way to begin to answer this question is to draw a graph with each +vertex representing an individual in the room and each edge representing an introduction as in Figure 12.22. +Figure 12.22 Model of Introductions between Five Strangers +Let’s approach the problem by thinking about how many new people Person A would meet, then Person B, and so on, +making sure not to repeat any introductions. The first graph in Figure 12.22 shows Person A meeting Persons B, C, D, +and E, for a total of 4 introductions. The next graph shows that Person B still has to meet Persons C, D, and E, for a total +of 3 more introductions. The next graph shows that Person C still has to meet Persons D and E, which is 2 more +introductions. The next graph shows that Person D only remains to meet Person E, which is 1 more introduction. The +final graph has +edges representing 10 introductions. The last graph is an example of a complete +graph because each pair of vertices is joined by an edge. Another way of saying this is that the graph is complete +because each vertex is adjacent to every other vertex.Figure 12.23 shows complete graphs with three, four, five, and six +vertices. +Figure 12.23 Complete Graphs with Up to Six Vertices +Suppose we want to know the number of introductions necessary in a room with six people. This would be represented +by a complete graph with six vertices, and the total number of introductions would be +, the +number of edges in the graph. In fact, you can always find the number of introductions in a room with +people by +adding all the whole numbers from 1 to +. +1252 +12 • Graph Theory +Access for free at openstax.org + +FORMULA +The number of edges in a complete graph with +vertices is the sum of the whole numbers from 1 to +, +. +Suppose that we want to determine how many introductions are necessary in a room with 500 strangers. In other words, +suppose that we want to determine the number of edges in a complete graph with 500 vertices. Adding up all the +numbers from 1 to 499 could take a long time! In the next example, we use the Sum of Degrees Theorem to make the +problem more manageable. +EXAMPLE 12.8 +Using the Sum of Degrees Theorem +Use the Sum of Degrees Theorem to determine the number of introductions required in a room with +1. +6 strangers. +2. +10 strangers. +3. +strangers. +Solution +1. +Since there are 6 strangers, there are 6 vertices. Since each individual must meet 5 other individuals, there are 5 +edges meeting at each vertex which means each vertex has degree 5. Since there are 6 vertices of degree 5, the sum +of degrees is +. By the Sum of Degrees Theorem, the number of edges is half the sum of the degrees, which +is +. So, the total number of introductions is 15. +2. +Since there are 10 strangers, there are 10 vertices. Since each individual must meet 9 other individuals, there are 9 +edges meeting at each vertex which means each vertex has degree 9. Since there are 10 vertices of degree 9, the +sum of degrees is +. By the Sum of Degrees Theorem, the number of edges is half the sum of the degrees, +which is +. So, the total number of introductions is 45. +3. +Since there are +strangers, there are +vertices. Since each individual must meet +other individuals, there are +edges meeting at each vertex which means each vertex has degree +. Since there are +vertices of degree +, the sum of degrees is +. By the Sum of Degrees Theorem, the number of edges is half the sum of the +degrees, which is +. So, the total number of introductions is +. +YOUR TURN 12.8 +1. Determine the number of introductions necessary in a room with 500 strangers using the Sum of Degrees +Theorem. +Now we have a shorter way to calculate the number of introductions in a room with +strangers, and the number of +edges on a complete graph with +vertices. Let’s update our formula. +FORMULA +The number of edges in a complete graph with +vertices is +. +Subgraphs +Sometimes a graph is a part of a larger graph. For example, the graph of South Florida Airports from Figure 12.7 is part +of a larger graph that includes Orlando International Airport in Central Florida, which is shown in Figure 12.24 +12.2 • Graph Structures +1253 + +Figure 12.24 Orlando and South Florida Airports +The graph in Figure 12.24 includes an additional vertex, MCO, and additional edges shown with dashed lines. The graph +of direct flights between South Florida airports from Figure 12.7 is called a subgraph of the graph that also includes +direct flights between Orlando and the same South Florida airports in Figure 12.24. In general terms, if Graph B consists +entirely of a set of edges and vertices from a larger Graph A, then B is called a subgraph of A. +EXAMPLE 12.9 +Identifying a Subgraph +In Figure 12.25, Graph G is given, along with four diagrams. Determine whether each diagram is or is not a subgraph of +Graph G and explain why. +Figure 12.25 Graph G and Potential Subgraphs +Solution +• +Diagram J is not a subgraph of Graph G because edge ec is not in Graph G. +• +Diagram K is a subgraph of Graph G because all of its vertices and edges were part of Graph G. +• +Diagram L is not a graph at all, because there is a line extending from vertex a that does not connect a to another +vertex. So, Diagram L cannot be a subgraph. +• +Diagram M is not a subgraph of Graph G because it contains a vertex f, which is not part of G. +YOUR TURN 12.9 +1. Draw a subgraph of Graph F from Figure 12.3 that has exactly four vertices and five edges. +1254 +12 • Graph Theory +Access for free at openstax.org + +Identifying and Naming Cycles +Figure 12.26 The water cycle begins and ends with water. (credit: "Diagram of the water cycle" by NASA, Public Domain) +When you think of a cycle in everyday life, you probably think of something that begins and ends the same way. For +example, the water cycle (Figure 12.26) begins with water in a lake or ocean, which evaporates into water vapor, +condenses into clouds, and then returns to earth as rain or some other form of precipitation that settles into lakes or +oceans. A cycle in graph theory is similar in that it begins and ends in the same way: It is a series of connected edges +that begin and end at the same vertex but otherwise never repeat any vertices. +In a cycle, there are always the same number of vertices as edges, and all vertices must be of degree 2. Cycles are often +referred to by the number of vertices. For example, a cycle with 5 vertices can be called a 5-cycle. Cycles can also be +named after polygons based on the number of edges. For example a 5-cycle is also called a pentagon. Table 12.1 lists +these names for cycles of size 3 through size 10. +Cycle Category +Number of Edges +Example +triangle +3 +quadrilateral +4 +pentagon +5 +hexagon +6 +heptagon +7 +octagon +8 +Table 12.1 Categories of Cycles +12.2 • Graph Structures +1255 + +Cycle Category +Number of Edges +Example +nonagon +9 +decagon +10 +Table 12.1 Categories of Cycles +There are many more polygon names, including a megagon that has a million edges and a googolgon that has +edges, but usually we just say +-gon when the number +is past 10. For example, a cycle with 11 edges could be called an +11-gon. +Notice that the 10-cycle, or decagon, appears to cross over itself in Table 12.1. Remember, graphs can be drawn +differently but represent the same connections. In Figure 12.27, the same decagon is transformed into a graph that does +not appear to overlap itself. We have done this without changing any of the connections so both diagrams represent the +same relationships, and both diagrams are considered decagons. +Figure 12.27 Transformation of a Decagon +WHO KNEW? +Googol to Google +Did the name "googolgon" ring a bell for you? If so, there is a good reason for it. The creators of Google.com +renamed their search engine Google, a misspelled reference to the number "googol" alluding to the enormous +number of calculations their algorithm can tackle. A googol is a 1 followed by 100 zeroes, or +. (William L. Hosch, +"Google, American Company," https://www.britannica.com/topic/Google-Inc, Encyclopedia Britannica online) +Cyclic Subgraphs and Cliques +When cycles appear as subgraphs within a larger graph, they are called cyclic subgraphs. Cyclic subgraphs are named +by listing their vertices sequentially. The vertex where you begin is not important. Graph K in Figure 12.28 has two +triangle cycles (g, h, j) and (h, i, j), and one quadrilateral cycle (g, h, i, j). +Figure 12.28 Cycles in Graph K +The same cycle can be named in many ways, but we must keep in mind that listing two vertices consecutively +indicates an edge exists between them. For example, (g, h, i, j) can also be named (h, i, j, g) or (j, i, h, g). However, +you cannot name it (g, i, h, j,) which doesn't reflect the correct order of the vertices. We can see that the order (g, i, +h, j) is incorrect because the cycle has no edges gi or hj. +1256 +12 • Graph Theory +Access for free at openstax.org + +EXAMPLE 12.10 +Identifying Cyclic Subgraphs +Identify the types of cyclic subgraphs in Graph H in Figure 12.29 and name them. +Figure 12.29 Graph H +Solution +In order to avoid missing a cycle, begin by analyzing the vertex a, then proceed in alphabetical order. Vertex a is part of +two cycles, the quadrilateral cycle (a, b, f, g,) and the hexagonal cycle (a, b, c, d, e, f). Next look at vertex b. It is part of the +hexagonal cycle (b, c, d, e, f, g). After checking each of the remaining vertices, we see there are no other cycles. +In order to avoid naming the same cycle twice, consider naming cycles beginning with the letter closest to the +beginning of the alphabet and then progressing clockwise. +YOUR TURN 12.10 +1. Which types of cycles are in Graph G in Figure 12.32? Use the vertices of the graph to give a name for one of each +type that you find. +We have seen that sociologists use graphs to study the structures of social networks. In sociology, there is a principle +known as Triadic Closure. It says that if two individuals in a social network have a friend in common, then it is more likely +those two individuals will become friends too. Sociologists refer to this as an open triad becoming a closed triad. This +concept can be visualized as graphs in Figure 12.30. (Chakraborty, Dutta, Mondal, and Nath, "Application of Graph Theory +in Social Media, International Journal of Computer Sciences and Engineering, 6(10):722-729) In the open triad in Figure +12.30, person a and person b each has a friendship with person c. In the closed triad, person a and person a have also +developed a friendship. Notice that the graph representing the closed triad is a three-cycle, or a triangle, in graph theory +terms. +Figure 12.30 Graphs of Open Triad and Closed Triad +Another topic of interest to sociologists, as well as computer scientists and scientists in many fields, is the concept of a +clique. In a social group, a clique is a subgroup who are all friends. A triad is an example of a clique with three people, +but there can be cliques of any size. In graph theory, a clique is a complete subgraph. +EXAMPLE 12.11 +Identifying Triads and Cliques +The graphs in YOUR TURN 12.6 represent social communities. The vertices are individuals and the edges are social +connections. Use the graphs to answer the questions. +1. +Which graph has the most triads? Name the triangles. +2. +Which graph has the fewest triads? Name the triangles. +3. +How might an increase in the number of triads in a graph affect the resilience of a community? Explain your +reasoning. +12.2 • Graph Structures +1257 + +4. +Identify a clique with more than three vertices in Graph E by naming its vertices. +Solution +1. +Graph E has the most triads. The triangles are (a, b, c), (a, e, g), (c, f, i), (e, g, h), (e, h, i), and (f, h, i). +2. +Graph D has the fewest triads. There are no triangles in the graph. +3. +More triads means vertices of greater degree, which indicates greater resilience. Specifically, if an edge is removed +from a closed triad, there the two individuals who are no longer adjacent are still connected by way of the third +member of the triad. +4. +The vertices a, e, c, and g form a clique with four vertices. +YOUR TURN 12.11 +1. From Exercise 35 in Section 12.1 Exercises, Chloe is interested to know how many in her network of Roblox +friends are also friends with each other, so she polls them. The following is a list of each of Chloe’s friends and +their common friends. Find a pentagon cycle in the graph of the Roblox friends in Chloe’s network that includes +Chloe. +• +Aria is friends with no one else in Chloe’s network. +• +Benicio is friends with Dakshayani, Eun-ah, and Fukashi. +• +Dakshayani is friends with Benicio. +• +Eun-ah is friends with Benicio and Fukashi. +• +Fukashi is friends with Benicio and Eun-ah. +Three-Cycles in Complete Graphs +Just as complete graphs have a predictable number of edges, complete graphs have a predictable number of cyclic +subgraphs. Let’s look at the three-cycles within complete graphs with up to six vertices, which are shown in Figure 12.31. +Figure 12.31 Complete Graphs with Up to Six Vertices +Let's list the names of all the triangles in each graph. Since every pair of vertices is adjacent, any three vertices on a +complete graph form a triangle. There is only one triangle in the complete graph with three vertices, (a, b, c). For the +rest of the graphs, it is important that we take an organized approach. Start with the vertex that is first alphabetically, +listing any triangles that include that vertex also in alphabetical order. Then, proceed to the next vertex in the alphabet, +and list any triangles that include that vertex, except those that are already listed. Keep going in this way as shown in +Table 12.2. +Complete +Graph With: +3 +Vertices +4 Vertices +5 Vertices +6 Vertices +All a triangles +(a, b, c) +(a, b, c), (a, +b, d), +(a, c, d) +(a, b, c), (a, b, d), (a, b, e) +(a, c, d), (a, c, e) +(a, d, e) +(a, b, c), (a, b, d), (a, b, e), (a, b, f) +(a, c, d), (a, c, e), (a, c, f) +(a, d, e), (a, d, f) +(a, e, f) +Other b +triangles +None +(b, c, d) +(b, c, d), (b, c, e), +(b, d, e) +(b, c, d), (b, c, e), (b, c, f) +(b, d, e), (b, d, f) +(b, e, f) +Table 12.2 Listing Triangles in Complete Graphs +1258 +12 • Graph Theory +Access for free at openstax.org + +Complete +Graph With: +3 +Vertices +4 Vertices +5 Vertices +6 Vertices +Other c +triangles +None +None +(c, d, e) +(c, d, e), (c, d, f) +(c, e, f) +Other d +triangles +None +None +None +(d, e, f) +Total +1 +Table 12.2 Listing Triangles in Complete Graphs +Look at the last row in Table 12.2. Do you see a pattern emerge for counting triangles in a complete graph? Without +drawing a complete graph with 7 vertices, we can predict that it will have +triangles inside it. This pattern also appears in a +famous diagram known to Western mathematicians as "Pascal’s Triangle." Figure 12.32 displays the first 11 rows of +Pascal’s Triangle. Row 0 of Pascal’s Triangle only has the number 1 in it. The first and last entries in each of the other +rows are also 1s. Otherwise, all the entries are formed by adding two entries from the previous row. For example, in row +6, entry 1 is 6, which was found by adding 1 and the 5 from the previous row, and entry 2 is 15, which was found by +adding the 5 and the 10 from the previous row, as shown in Table 12.2. +Figure 12.32 Pascal’s Triangle +IMPORTANT! When you count the rows and entries of Pascal’s Triangle begin with row 0 and entry 0, not row 1 and +entry 1. +VIDEO +The Mathematical Secrets of Pascal's Triangle by Wajdi Mohamed Ratemi (https://openstax.org/r/ +Wajdi_Mohamed_Ratemi) +If we begin to list the third entries in each row of Pascal's triangle from the top down, we see 1, 4, 10, 20, 35, and so on. +12.2 • Graph Structures +1259 + +Notice that these values are exactly the number of triangles in a complete graph of sizes 3, 4, 5, 6, and 7, respectively. +Specifically, entry 3 in row 7 is 35, the number of triangles in a complete graph of size 7. Let's practice using Pascal’s +triangle to find the number of triangles in a complete graph of a particular size. +STEPS TO FIND THE NUMBER OF TRIANGLES IN A COMPLETE GRAPH OF SIZE n +Step 1 Calculate the entries in row n of Pascal's Triangle using sums of pairs of entries in previous rows as needed. +Step 2 The number of triangles is entry 3 in row n. Remember to count the entries 0, 1, 2, and 3, from left to right. +EXAMPLE 12.12 +Using Pascal’s Triangle to Find the Number of Triangles in a Complete Graph +Use Pascal’s Triangle in Figure 12.32 to find the number of triangles in a complete graph with 11 vertices. +Solution +Step 1: Identify the row number and calculate its entries. Since the complete graph has 11 vertices, we will need to look +in row 11 of Pascal’s Triangle. Use row 10 in Figure 12.32 to find the entries in row 11 as shown in Figure 12.33. +Figure 12.33 Rows 10 and 11 of Pascal’s Triangle +Step 2: Find entry number 3. The number of triangles is in entry 3. Remember to count the entries beginning with entry +0 as shown in Figure 12.34. +Figure 12.34 Identifying an Entry in Pascal’s Triangle +Entry 3 in row 11 is 165. So, there are 165 triangles in a complete graph with 11 vertices. +YOUR TURN 12.12 +1. A sociologist is studying a community of 13 individuals in which every person has a social connection to every +other person. How many closed triads are there in the community? +TECH CHECK +Since the entries in Pascal’s Triangle are useful in many applications, many resources such as online and traditional +calculator functions have been developed to assist in calculating the values. The website dCode.xyz has a tool +(https://openstax.org/r/has_a_tool) that automatically calculates entries in Pascal’s Triangle using these steps: +Step 1: Make sure to select Index 0. +Step 2: Select your preference, display the triangle until a particular row (line), display only one row (line), or calculate +the value at coordinates (which means give the value of a single entry). +Step 3: Enter row number (line number) you are interested, or enter the row number and entry number in +parentheses: (row number, entry number). +Step 4: Select Calculate. +Step 5: Retrieve the answer from the results box. +1260 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.35 Pascal's Triangle Tool on dCode.xyz +Check Your Understanding +6. A complete graph with four vertices must have exactly four edges. +a. +True +b. +False +7. In a cycle, every vertex has degree two. +a. +True +b. +False +8. The number of vertices in a graph is twice the number of edges. +a. +True +b. +False +9. The cycle (a, b, c, d) can also be called (c, b, a, d). +a. +True +b. +False +10. The cycle (a, b, c, d) can also be called (a, b, d, c). +a. +True +b. +False +11. The only cliques that are cycles are cliques of three vertices. +a. +True +b. +False +12. A student found that the sum of the degrees of the vertices in a graph was 13. Why is that impossible? +13. A student finds the number of edges in a complete graph by taking half the number of vertices, then subtracting +12.2 • Graph Structures +1261 + +one from the number of vertices, then multiplying these two values together. Will the student get the correct +result? +SECTION 12.2 EXERCISES +Use the figure to answer the following exercises. +1. List any graphs that are subgraphs of Graph U. +2. List any graphs that are subgraphs of Graph V. +3. Explain why Graph T is not a subgraph of Graph S. +4. List any graphs that have a quadrilateral cyclic subgraph and name the quadrilaterals. +Use the Sum of Degrees Theorem as needed to answer the following exercises. +5. How many edges are in a graph if the sum of the degrees of its vertices is 18. +6. Draw two possible graphs to demonstrate that the sum of the degrees of the vertices in a graph with 7 edges is +14. Label the degrees of the vertices. +7. What is the sum of the degrees of the vertices of a graph that has 122 edges? +8. A graph has 3 vertices of degree 2, 4 vertices of degree 1, and 2 vertices of degree 3. How many edges are in +the graph? +9. There are 6 vertices and 11 edges in a graph, 2 of degree 5, 1 of degree 4, and 2 of degree 3. What is the degree +of the remaining vertex? +10. A complete graph has 5 vertices. What is the sum of the degrees of the vertices? +11. A complete graph has 120 edges. How many vertices does it have? +Use the figure to answer the following exercises. Identify the graph or graphs with the given characteristic. +12. The sum of the degrees of the vertices is 16. +13. The graph is complete. +14. The graph has no cyclic subgraphs +15. The graph contains at least one octagon. +16. The graph contains exactly two 3-cycles. +Use the figure to answer the following exercises. Identify the graph or graphs with the given characteristic. +17. The graph is a subgraph of Graph 6. +18. The sum of the degrees of the vertices is 16. +19. The graph is complete. +20. The graph has no pentagons. +21. The graph contains at least one octagon. +22. The graph contains exactly two cyclic subgraphs. +1262 +12 • Graph Theory +Access for free at openstax.org + +23. Use Table 12.1 to name a heptagon in Graph 8. +Use the figure to answer the following exercises. Identify the graph or graphs with the given characteristic. +24. The sum of the degrees of the vertices is 16. +25. The graph is complete. +26. The graph contains exactly one quadrilateral. +27. The graph contains at least one octagon. +28. The graph contains exactly one cyclic subgraph. +Use Table 12.1 to answer the following exercises about the below figure. +29. List every quadrilateral in Graph 12. +30. List every quadrilateral in Graph 10. +31. Determine the number of quadrilaterals in Graph 9. +Use the figure to answer the following exercises. +32. Name five triangles in the graph. +33. Identify a clique with more than three vertices in the graph by listing its vertices. +For the following exercises, draw a graph with the given characteristics. +34. 4 vertices, 6 edges, a subgraph that is a 4-cycle. +35. 11 vertices, the only cyclic subgraphs are triangles. +36. Complete graph, no quadrilateral subgraph +37. 4 vertices, 4 edges, no cyclic subgraphs +38. Complete graph, sum of the degrees of the vertices is 20. +39. 7 vertices, largest clique has 5 vertices. +40. In chess, a knight can move in any direction, but it must move two spaces then turn and move one more space. +The eight possible moves a knight can make from a space in the center of a five-by-five grid are shown in the first +figure. Draw a graph that represents all the legal moves of a knight on a three-by-three grid starting from the +lower left corner as shown in the second figure where the vertices will represent the spaces occupied by the knight +and the edges will indicate a move from one space to the next. +12.2 • Graph Structures +1263 + +41. What kind of cycle is the resulting graph you drew for Exercise 40? +42. Use Pascal’s triangle to find number of triangles in a complete graph with 14 vertices. +43. Do you think that a graph representing network of friends on Facebook is likely to be complete or not? Explain +your reasoning. +44. Would the graph of a family tree, in which edges represent parent-child lineage, contain any cycles? Why or why +not? +45. We have seen that the number of triangles in a complete graph with 7 vertices can be calculated using the pattern +. This pattern gets very long for complete +graphs with more vertices, but we have seen sums from 1 to a number before, and we had a shortcut! Recall from +Example 12.8 that +. This makes finding the sum from 1 up to +shorter. We +can write each of the sums we need in the form +. +To find the sum from 1 to 5, since +, use +To find the sum from 1 to 4, since +, use +To find the sum from 1 to 3, since +, use +To find the sum from 1 to 2, since +, use +To find the sum from 1 to 1, (Sounds silly, but it helps to see the pattern!) since +, use +Use these to rewrite the calculation for the number of triangles in a graph with 7 vertices. +Similarly, for the number of triangles in a complete graph with 8 vertices, instead of +1264 +12 • Graph Theory +Access for free at openstax.org + +use the shorter formula +. +Use this pattern to find the number of triangles in a complete graph with 10 vertices. In which row and entry of +Pascal’s triangle can you also find this result? +46. Use the fact that the sum from 1 to +is +to write a formula for the number of triangles in a complete +graph with n vertices. +12.3 Comparing Graphs +Figure 12.36 A flat map represents the surface of Earth in two dimensions. (credit: modification of work "World Map" by +Scratchinghead/Wikimedia Commons, Public Domain) +Figure 12.37 A globe represents the surface of Earth in three dimensions. (credit: "Globe" by Wendy Cope/Flickr, CC BY +2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify the characteristics used to compare graphs. +2. +Determine when two graphs represent the same relationships +3. +Explore real-world examples of graph isomorphisms +4. +Find the complement of a graph +Maps of the same region may not always look the same. For example, a map of Earth on a flat surface looks distorted at +the poles. When the same regions are mapped on a spherical globe, the countries that are closer to the polls appear +12.3 • Comparing Graphs +1265 + +smaller without the distortion. Despite these differences, the two maps still communicate the same relationships +between nations such as shared boundaries, and relative position on the earth. In essence, they are the same map. This +means that for every point on one map, there is a corresponding point on the other map in the same relative location. In +this section, we will determine exactly what characteristics need to be the shared by two graphs in order for us to +consider them “the same.” +When Are Two Graphs Really the Same Graph? +In arithmetic, when two numbers have the same value, we say they are equal, like ½ = 0.5. Although ½ and 0.5 look +different, they have the same value, because they are assigned the same position on the real number line. When do we +say that two graphs are equal? +Figure 12.38 shows Graphs A and F, which are identical except for the labels. Graphs are visual representations of +connections. As long as two graphs indicate the same pattern of connections, like Graph A and Graph F, they are +considered to be equal, or in graph theory terms, isomorphic. +Figure 12.38 Identical Graphs with Different Labels +Two graphs are isomorphic if either one of these conditions holds: +1. +One graph can be transformed into the other without breaking existing connections or adding new ones. +2. +There is a correspondence between their vertices in such a way that any adjacent pair in one graph corresponds to +an adjacent pair in the other graph. +It is important to note that if either one of the isomorphic conditions holds, then both of them do. When we need to +decide if two graphs are isomorphic, we will need to make sure that one of them holds. For example, Figure 12.39 shows +how Graph T can be bent and flipped to look like Graph Z, which means that Graphs T and Z satisfy condition 1 and are +isomorphic. +Figure 12.39 Graph T Transformed Into Graph Z +Also, notice that the vertices that were adjacent in the first graph are still adjacent in the transformed graph as shown in +Figure 12.40. For example, vertex 3 is still adjacent to vertex 4, which means they are still neighboring vertices joined by +a single edge. +Figure 12.40 Adjacent Vertices Are Still Adjacent +When Are Two Graphs Really Different? +Verifying that two graphs are isomorphic can be a challenging process, especially for larger graphs. It makes sense to +check for any obvious ways in which the graphs might differ so that we don’t spend time trying to verify that graphs are +1266 +12 • Graph Theory +Access for free at openstax.org + +isomorphic when they are not. If two graphs have any of the differences shown in Table 12.3, then they cannot be +isomorphic. +Unequal number of vertices +Unequal number of edges +Unequal number of vertices of a particular degree +Different cyclic subgraphs +Table 12.3 Characteristics of Graphs That Are Not Isomorphic +Recognizing Isomorphic Graphs +Isomorphic graphs that represent the same pattern of connections can look very different despite having the same +underlying structure. The edges can be stretched and twisted. The graph can be rotated or flipped. For example, in +Figure 12.41, each of the diagrams represents the same pattern of connections. +Figure 12.41 Four Representations of the Same Graph +Looking at Figure 12.41, how can we know that these graphs are isomorphic? We will start by checking for any obvious +differences. Each of the graphs in Figure 12.41 has four vertices and five edges; so, there are no differences there. Next, +we will focus on the degrees of the vertices, which have been labeled in Figure 12.42. +Figure 12.42 Graphs with Vertices of the Same Degrees +As shown in Figure 12.42, each graph has two vertices of degree 2 and two vertices of degree 3; so, there are no +differences there. Now, let’s check for cyclic subgraphs. These are highlighted in Figure 12.43. +12.3 • Comparing Graphs +1267 + +Figure 12.43 Graphs with the Same Cyclic Subgraphs +As shown in Figure 12.43, each graph has two triangles and one quadrilateral; so, no differences there either. It is +beginning to look likely that these graphs are isomorphic, but we will have to look further to be sure. +To know with certainty that these graphs are isomorphic, we need to confirm one of the two conditions from the +definition of isomorphic graphs. With smaller graphs, you may be able to visualize how to stretch and twist one graph to +get the other to see if condition 1 holds. Imagine the edges are stretchy and picture how to pull and twist one graph to +form the other. If you can do this without breaking or adding any connections, then the graphs are really the same. +Figure 12.44 demonstrates how to change graph A4 to get A3, graph A3 to get A2, and graph A2 to get A1. +Figure 12.44 Transforming Graphs +Now that we have used visual analysis to see that condition 1 holds for graphs A1, A2, A3, and A4 in Figure 12.43, we +know that they are isomorphic. In Figure 12.44, one of the edges of graph A4 crossed another edge of the graph. By +transforming it into graph A3, we have “untangled” it. Graphs that can be untangled are called planar graphs. The +complete graph with five vertices is an example of a nonplanar graph-that means that, no matter how hard you try, you +can’t untangle it. But, when you try to figure out if two graphs are the same, it can be helpful to untangle them as much +as possible to make the similarities and differences more obvious. +EXAMPLE 12.13 +Identifying Isomorphic Graphs +Which of the three graphs in Figure 12.45 are isomorphic, if any? Justify your answer. +Figure 12.45 Three Similar Graphs +Solution +Step 1: Check for differences in number of vertices, number of edges, degrees of vertices, and types of cycles to see if an +isomorphism is possible. +1268 +12 • Graph Theory +Access for free at openstax.org + +• +Vertices: They all have the same number of vertices, 4. +• +Edges: They all have the same number of edges, 4. +• +Degrees: Graph B1 and Graph B2 each have a vertex of degree 3, while Graph B3 does not. So, Graph B3 is not +isomorphic to either of the other two graphs, but Graph B1 and Graph B2 could possibly be isomorphic. +• +Cycles: Focus on any cycles in Graph B1 and Graph B2. Each graph has a triangle as shown in Figure 12.46. +Figure 12.46 Cycles of Graph B1 and Graph B2 +Step 2: If no differences were found and an isomorphism is possible, verify one of the conditions in the definition of +isomorphic. +Since we were able to determine that Graph B1 and Graph B2 have no obvious differences, and they are relatively small +graphs, we will attempt to transform one graph into the other, which would verify condition 1. Graph B1 can be +transformed into Graph B2 without breaking or adding connections as shown in Figure 12.47. Begin by untangling graph +B1. Then rotate or flip as needed to see that the graphs match. +Figure 12.47 Transformation of Graph B1 +So, Graph B1 and Graph B2 have the same structure and are isomorphic. +YOUR TURN 12.13 +1. Name four differences between Graph C1 and Graph C2 that show they cannot be isomorphic. +Two Distinct Graphs +Have you ever noticed that many popular board games may look different but are really the same game? A good +example is the many variations of the board game Monopoly®, which was submitted to the U.S. Patent Office in 1935. +Although the rules have been revised a bit, a very similar game board is still in use today. There have been many +versions of Monopoly over the years. Many have been stylized to reflect a popular theme, such as a show or sports team, +while retaining the same game board structure. If we were to represent these different versions of the game using a +graph, we would find that the graphs are isomorphic. . Let's analyze some game boards using graph theory to determine +if they have the same structure despite having different appearances. +12.3 • Comparing Graphs +1269 + +Figure 12.48 Two Game Boards +EXAMPLE 12.14 +Deciding If Graphs are Isomorphic +A teacher uses games to teach her students about colors and numbers as shown in Figure 12.48. +In the Colors Game, shown in Figure A, each player begins in the space marked START and proceeds in a +counterclockwise direction. On each turn, the player spins a spinner marked 1 and 2 and moves forward the number of +spaces shown on the spinner. If the player lands on a space marked with any color other than white, the player must +move forward or back to the other space of the same color. The first player to land in or pass the space marked END +wins. +In the Clock Game, shown in Figure B, each player begins in the space marked START and proceeds in a clockwise +direction. On each turn, the player spins a spinner marked 1 and 2 and moves forward the number of spaces shown on +the spinner. In the same turn, the player must read the number in the space and move forward the number of spaces +indicated by a positive value or backward the number of spaces indicated by a negative value. Then the turn ends. The +first player to land in or pass the space marked END wins. +Draw a graph or multigraph to represent each game in which the vertices are the spaces and the edges represent the +ability of a player to move between the spaces either by a spin or as dictated by a marked color or number. (We will +ignore the direction of motion for simplicity.) Transform one of the graphs to show that it is isomorphic to the other and +explain what this tells you about the games. +Solution +The graph representing the Clock Game can be transformed as shown in Figure 12.49 so that we can see that the graphs +are isomorphic. There is a correspondence between their vertices in such a way that any adjacent pair in one graph +corresponds to an adjacent pair in the other graph. +1270 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.49 Graphs for Clock Game and Colors Game +This tells us that the games are essentially the same game with the same moves even though the games appear to be +different. +YOUR TURN 12.14 +1. Let's compare the games in Figures A and B. In the game Diamonds, shown in Figure A, each player begins in the +space marked START and proceeds in a counterclockwise direction. On each turn, the player rolls a six-sided die +and moves forward the number of spaces shown on the die. If the player lands on a space marked with a +diamond, the player jumps to the next space marked with a diamond and stops there. The first player to land in +or pass the space marked END wins. In the Dots game, shown in Figure B, each player begins in the space +marked 1 and proceeds in numerical order through the numbered spaces. On each turn, the player rolls a six- +sided die and moves forward the number of spaces shown on the die. When a player lands on a space marked +with a green dot, they immediately move to the space indicated by the arrow. Construct a graph to represent +each gameboard such that each vertex represented a space on the game board and each edge represented the +ability of a player to move directly from one space to the next. For example, in the Dots game, the vertex +representing space 3 would be adjacent to 2, 4, and 6. Identify at least three differences between a graph based +on Diamonds and a graph based on Dots to show the graphs are not isomorphic. +12.3 • Comparing Graphs +1271 + +Two Game Boards +PEOPLE IN MATHEMATICS +Elizabeth Magie +Game designer, engineer, comedian, and political activist, Elizabeth Magie designed a game called “Landlord’s Game” +to educate fellow citizens about the dangers of monopolies and the benefits of wealth redistribution through a land +tax. Magie, whose father had campaigned with Abraham Lincoln, was a proponent of a land tax, an idea popularized +by Henry George’s 1879 book, Progress and Poverty. She designed the game to be played by two sets of rules for +comparison. In one version, the goal was to dominate opponents by creating monopolies, leaving one wealthy player +standing in the end. In the other version, all the players were rewarded when a monopoly was created through a +simulated land tax. She patented this game for the first time in 1904. Does the game board in Figure 12.50 look +familiar? +1272 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.50 Landlord’s Game Patent (credit: "Landlord's Game Patent" Wikimedia Commons, Public Domain) +That’s right! A modified version of the game was obtained from friends by a man named Charles Darrow who +renamed it Monopoly and sold to Parker Brothers. Parker Brothers later paid Magie $500 for the right to her patent +on it. The property tax version was left behind, and the modern game of Monopoly was born. (Mary Pilon, Monopoly’s +Lost Female Inventor, September 1, 2018, National Women’s History Museum, “Monopoly’s Lost Female Inventor,” +(https://openstax.org/r/womens_history) +Identifying and Naming Isomorphisms +When two graphs are isomorphic, meaning they have the same structure, there is a correspondence between their +vertices, which can be named by listing corresponding pairs of vertices. This list of corresponding pairs of vertices in +such a way that any adjacent pair in one graph corresponds to an adjacent pair in the other graph is called an +isomorphism. Consider the isomorphic graphs in Figure 12.38. In Figure 12.51, we could replace the labels Graph F with +the labels from Graph A and have an identical graph, as in Figure 12.52. +Figure 12.51 Identical Graphs with Different Labels +12.3 • Comparing Graphs +1273 + +Figure 12.52 Corresponding Vertices +So, we can identify an isomorphism between Graph A and Graph F by listing the corresponding pairs of vertices: b-g, c-h, +d-i, and e-j. Notice that b is adjacent to c and g is adjacent to h. This must be the case since b corresponds to g and c +corresponds to h. The same is true for other pairs of adjacent vertices. +An isomorphism between graphs is not necessarily unique. There can be more than one isomorphism between two +graphs. We can see how to form a different isomorphism between Graph A and Graph F from Figure 12.33 by rotating +Graph F clockwise and comparing the rotated version of F to Graph A as in YOUR TURN 12.13. Now, we can see that a +second isomorphism exists, which has the correspondence: b-j, d-h, c-i, and e-g as shown in Figure 12.53. +Figure 12.53 Transforming Graph F into Graph A +When you name isomorphisms, one way to check that your answer is reasonable is to make sure that the degrees of +corresponding vertices are equal. +VIDEO +Determine If Two Graphs Are Isomorphic and Identify the Isomorphism (https://openstax.org/r/Determine_If_Two) +EXAMPLE 12.15 +Identifying Isomorphisms +In Example 12.13, we showed that the Graphs B1 and B2 in Figure 12.45 are isomorphic. In Figure 12.54, labels have been +assigned to the vertices of Graphs B1 and B2. Identify an isomorphism between them by listing corresponding pairs of +vertices. +Figure 12.54 Two Isomorphic Graphs +Solution +Figure 12.47 showed how to transform Graph B1 to get Graph B2. In Figure 12.55, we will do the same, but this time we +will include the labels. +1274 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.55 Transform Graph B1 into Graph B2 +From YOUR TURN 12.13, we can see the corresponding vertices: a-q, d-p, c-r, and b-s, which is an isomorphism of the two +graphs. +YOUR TURN 12.15 +1. Graphs E and T are isomorphic. Name two isomorphisms. +Graph E and Graph T +EXAMPLE 12.16 +Recognizing an Isomorphism +Determine whether Graphs G and S in Figure 12.56 are isomorphic. If not, explain how they are different. If so, name the +isomorphism. +Figure 12.56 Graph G and Graph S +Solution +Step 1: Check for any differences. +• +Vertices: Graphs G and S each have six vertices. +• +Edges: Graphs G and S each have six edges. +• +Degrees: Each graph also has two vertices of degree 1, two vertices of degree 2, and two vertices of degree 3. +• +Cycles: From Figure 12.57, we can see that Graph G contains a quadrilateral cycle (b, f, d, c) but Graph S has no +quadrilaterals. Also, Graph S contains a triangle cycle (m, r, o) but Graph G has no triangles. This means that the +graphs are not isomorphic. +Figure 12.57 Comparing Subgraphs +12.3 • Comparing Graphs +1275 + +Step 2: This step is not necessary because we now know Graphs G and S are not isomorphic. +YOUR TURN 12.16 +1. Three students have been asked to determine if Graphs E and T are isomorphic and justify their answers. +Graph E and Graph T +Javier believes that Graphs E and T are not isomorphic because Graph E contains a triangle cycle and Graph T +does not. He supports his conclusion with the graphs shown. +Javier’s Highlighted Cycle +Maubi also believes that Graphs E and T are not isomorphic, but says that it is because Graph T has a +quadrilateral cycle (p, q, r, s) while Graph E does not. Caden believes that Graphs E and T are isomorphic. She +said one isomorphism is a-p, b-q, c-s, and d-r. Determine who is correct, who is incorrect, and explain how you +know. +Complementary Graphs +Suppose that you are a camp counselor at Camp Woebegone and you are holding a camp Olympics with four events. The +campers have signed up for the events. You drew a graph in Figure 12.58 to help you visualize which events have +campers in common. +Figure 12.58 Graph of Events with Campers in Common +Graph E in Figure 12.58 shows that some of the same campers will be in events a and b, as well as b and d, c and d, and a +and c. What do you think the graph would look like that represented the events that do not have campers in common? It +would have the same vertices, but any pair of adjacent edges in Graph E, would not be adjacent in the new graph, and +vice versa. This is called a complementary graph, as shown in Figure 12.59. Two graphs are complementary if they have +the same set of vertices, but any vertices that are adjacent in one, are not adjacent in the other. In this case, we can say +that one graph is the complement of the other. +1276 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.59 Graphs of Camp Olympics +One way to find the complement of a graph is to draw the complete graph with the same number of vertices and remove +all the edges that were in the original graph. Let’s say you wanted to find the complement of Graph E from Figure 12.59, +and you didn’t already know it was Graph F. You could start with the complete graph with four vertices and remove the +edges that are in Graph E as shown in Figure 12.60. +Figure 12.60 Use Complete Graph to Find Complement +EXAMPLE 12.17 +Finding a Complement +A particular high school has end-of-course exams in (E3) English 3, (E4) English 4, (M) Advanced Math, (C) Calculus, (W) +World History, (U) U.S. History, (B) Biology, and (P) Physics. No English 3 students are taking English 4, World History, or +Biology; no English 4 students are also in Calculus, Advanced Math, U.S. History, or Physics; no Physics students are also +taking Advanced Math; No World History students are also taking U.S. History; and no Advanced Math students are also +taking Calculus. +1. +Create a graph in which the vertices represent the exams, and an edge between a pair of vertices indicates that +there are no students taking both exams. +2. +Find the complement of the graph in part 1. +3. +Explain what the graph in part 2 represents. +Solution +1. +In Figure 12.61, we have drawn a vertex for each exam and edges between any vertices that have no students in +common. +Figure 12.61 Graph of Exams with No Students in Common +2. +One way to get the complement of the graph in Figure 12.61 is to draw a complete graph with the same number of +vertices and remove the edges they have in common as shown in Figure 12.62. +12.3 • Comparing Graphs +1277 + +Figure 12.62 Remove Unwanted Edges from Complete Graph +The final graph of the complement is in Figure 12.63. +Figure 12.63 Graph of Exams with Students in Common +3. +In the graph in Figure 12.63, the vertices are still the exams, and a pair of adjacent vertices represents a pair of +exams that have students in common. +YOUR TURN 12.17 +1. Find the complement of the Graph H. +When two graphs are really the same graph, they have the same missing edges. So, when two graphs have a lot of +edges, it may actually be easier to determine if they are isomorphic by looking at which edges are missing rather than +which edges are included. In other words, we can determine if two graphs are isomorphic by checking if their +complements are isomorphic. +EXAMPLE 12.18 +Using a Complement to Find an Isomorphism +Use Figure 12.64 to answer each question. +Figure 12.64 Graph K +1. +Find the complement of Graph K. +2. +Identify an isomorphism between the complement of Graph K from part 1, and the complement of Graph H in YOUR +1278 +12 • Graph Theory +Access for free at openstax.org + +TURN 12.17. +3. +Confirm that the correspondence between the vertices you found in part 2 also gives an isomorphism between +Graph H from YOUR TURN 12.17, and Graph K from Figure 12.64. +Solution +1. +The complement of Graph K can be found by removing the edges of Graph K from a complete graph with the same +vertices as shown in Figure 12.65. +Figure 12.65 Find Complement of Graph K +2. +An isomorphism between the complement of Graph K and the complement of Graph H is A-M, C-L, E-N, B-O, and D- +P, which is confirmed by transforming the complement of Graph K in Figure 12.66. +Figure 12.66 Isomorphism between Complements of K and H +3. +Figure 12.67 shows how Graph K can be transformed into Graph H to confirm that the correspondence is A-M, C-L, +E-N, B-O, and D-P also gives an isomorphism between Graph K and Graph H. +Figure 12.67 Isomorphism between K and H +This means we now have three conditions that guarantee two graphs are isomorphic. +First Way: One graph can be transformed into the other without breaking existing connections or adding new ones. +Second Way: There is a correspondence between their vertices in such a way that any adjacent pair in one graph +corresponds to an adjacent pair in the other graph. +Third Way: Their complements are isomorphic. +If any one of these statements is true, then they are all true. If any one of these statements is false, then they are all +false. +YOUR TURN 12.18 +1. Suppose that a Graph M is a complete graph and Graph N is the complement of M. What are the degrees of the +vertices of Graph N? How do you know? +WORK IT OUT +Here is an activity you can do with a few of your classmates that will build your graph comparison skills. +12.3 • Comparing Graphs +1279 + +Step 1: Draw a planar graph with the following characteristics: exactly five vertices, one vertex of degree four, at least +two vertices of degree three, and exactly eight edges. Give names to the vertices. Make sure you do not use the same +letters or numbers to label your vertices as your classmates do. +Step 2: Analyze your graph. What is the degree of each vertex? Does your graph have any cyclic subgraphs? If so, list +them and indicate their sizes. +Step 3: Draw and analyze the complement of your graph. How many edges and vertices does it have? What is the +degree of each vertex? Does the complementary graph have any cyclic subgraphs? If so, list them and indicate their +sizes. +Step 4: Compare your graphs to each of your classmates’ graphs. Does your graph have the same number of edges +and vertices as the graph of your classmate? Does your graph have the same size cyclic subgraphs as the graph of +your classmate? How does the complement of your graph compare to the complement of the graph of your +classmate? Determine if your graph is isomorphic to your classmates’ graph. If so, give a correspondence that +demonstrates the isomorphism. If not, explain how you know. +Check Your Understanding +For the following exercises, determine whether each statement is always true, sometimes true, or never true. +14. Two graphs are isomorphic, and the graphs have the same structure. +15. A graph with four vertices is isomorphic to a graph with five vertices. +16. The sums of the degrees of the vertices of two graphs are equal, but the two graphs are not isomorphic. +17. Two graphs are isomorphic, but the graphs have a different number of edges. +18. One graph can be transformed to look like a second graph without removing or adding any connections, and +the two graphs are isomorphic. +19. Two graphs are isomorphic, and there is more than one isomorphism between the two graphs. +20. Two graphs have the same number of vertices, but there is no isomorphism between them. +21. There is a correspondence between the vertices of Graph A and the vertices of Graph B such that the adjacent +vertices in Graph A always correspond to vertices of Graph B, but the two graphs are not isomorphic +22. Two graphs have the same number of edges, the same number of vertices, vertices of the same degree, and +have all the same subgraphs, but they are not isomorphic +23. Two graphs are isomorphic, and the sum of the degrees of the vertices of one equals the sum of the degrees of +the other graph. +24. If two graphs are isomorphic, then their complements are isomorphic. +SECTION 12.3 EXERCISES +Use the figure to answer the following exercises. A pair of graphs is given. Identify three differences between them that +demonstrate the graphs are not isomorphic. +1. G and H +2. G and I +3. G and J +4. G and K +5. G and L +6. H and I +7. H and J +8. H and K +9. H and L +10. I and J +11. I and K +1280 +12 • Graph Theory +Access for free at openstax.org + +12. I and L +13. J and K +14. K and L +Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a +subgraph of the other graph, or the two graphs are isomorphic. +15. P and Q +16. P and R +17. P and U +18. Q and R +19. Q and S +20. Q and T +21. Q and V +22. S and V +23. T and U +24. U and V +Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Either give a reason that the +graphs are not isomorphic, or show how one of the graphs can be transformed to look like the other. +25. A and B +26. A and C +27. A and D +28. B and C +29. B and D +30. C and D +Use the figure to answer the following exercises. Determine if one graph is a subgraph of the other graph, or the two +graphs are isomorphic. If they are isomorphic, name an isomorphism. If one is a subgraph of the other, indicate a +correspondence between the vertices that demonstrates the relationship. +31. W and Z +32. X and Y +33. X and Z +Use the figure to answer the following exercises. +12.3 • Comparing Graphs +1281 + +34. Find the complement of Graph W. +35. Find the complement of Graph Y. +36. Find the complement of Graph X. +37. Find an isomorphism between the complement of W and the complement of Y if one exists. If not, explain how +you know. +38. Are W and Y isomorphic? Explain how you know. +39. Find an isomorphism between the complement of W and the Complement of X if one Exists. If not, explain how +you know. +40. Are W and X isomorphic? Explain how you know. +41. Find the complement of the graph in the given figure representing direct flights between south Florida airports. +Explain what the graph represents. +1282 +12 • Graph Theory +Access for free at openstax.org + +12.4 Navigating Graphs +Figure 12.68 Visitors navigate a garden maze. (credit: “Longleat Maze” by Niki Odolphie/Wikimedia, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Describe and identify walks, trails, paths, and circuits. +2. +Solve application problems using walks, trails, paths, and circuits. +3. +Identify the chromatic number of a graph. +4. +Describe the Four-Color Problem. +5. +Solve applications using graph colorings. +Now that we know the basic parts of graphs and we can distinguish one graph from another, it is time to really put our +graphs to work for us. Many applications of graph theory involve navigating through a graph very much like you would +navigate through a maze. Imagine that you are at the entrance to a maze. Your goal is to get from one point to another +as efficiently as possible. Maybe there are treasures hidden along the way that make straying from the shortest path +worthwhile, or maybe you just need to get to the end fast. Either way, you definitely want to avoid any wrong turns that +would cause unnecessary backtracking. Luckily, graph theory is here to help! +Walks +Suppose Figure 12.69 is a maze you want to solve. You want to get from the start to the end. +Figure 12.69 Your Maze +You can approach this task any way you want. The only rule is that you can’t climb over the wall. To put this in the context +of graph theory, let’s imagine that at every intersection and every turn, there is a vertex. The edges that join the vertices +must stay within the walls. The graph within the maze would look like Figure 12.70. +12.4 • Navigating Graphs +1283 + +Figure 12.70 The Graph in Your Maze +One approach to solving a maze is to just start walking. It is not the most efficient approach. You might cross through +the same intersection twice. You might backtrack a bit. It’s okay. We are just out for a walk. It might look something like +the black sequence of vertices and edges in Figure 12.71. +Figure 12.71 The Walk Through the Graph in Your Maze +This type of sequence of adjacent vertices and edges is actually called a walk (or directed walk) in graph theory too! +A walk can be identified by naming the sequence of its vertices (or by naming the sequence of its edges if those are +labeled). Let’s take the graph out of the context of the maze and give each vertex a name and each edge of the walk a +direction as in Figure 12.72. +Figure 12.72 The Graph without Your Maze +The name of this walk from p to r is p → q → o → n → i → j → c → d → c → j → k → s → r. When a particular edge on our +graph was traveled in both directions, it had arrows in both directions and the letters of vertices that were visited more +than once had to be repeated in the name of the walk. +Not every list of vertices or list of edges makes a path. The order must take into account the way in which the edges +and vertices are connected. The list must be a sequence, which means they are in order. No edges or vertices can be +skipped and you cannot go off the graph. +1284 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.73 A Walk or Not a Walk? +The highlighted edges Graph Y in Figure 12.73 represent a walk between f and b. The highlighted edges in Graph X do +not represent a walk between f and b, because there is a turn at a point that is not a vertex. This is like climbing over a +wall when you are walking through a maze. Another way of saying this is that b → d → f is not a walk, because there is no +edge between b and d. +A point on a graph where two edges cross is not a vertex. +EXAMPLE 12.19 +Naming a Walk Through A House +Figure 12.74 shows the floor plan of a house. Use the floor plan to answer each question. +Figure 12.74 Floor Plan of a House +1. +Draw a graph to represent the floor plan in which each vertex represents a different room (or hallway) and edges +represent doorways between rooms. +2. +Name a walk through the house that begins in the living room, ends in the garage and visits each room (or hallway) +at least once. +Solution +1. +Step 1: We will need a vertex for each room and it is convenient to label them according to the names of the rooms. +Visualize the scenario in your head as shown in Figure 12.75. You don’t have to write this step on your paper. +12.4 • Navigating Graphs +1285 + +Figure 12.75 Assigning Vertices to Rooms +Step 2: Draw a graph to represent the scenario. Start with the vertices. Then connect those vertices that share a +doorway in the floorplan as shown in Figure 12.76. +Figure 12.76 Graph of the Floor plan +2. +Step 1: Draw a path that begins at vertex L, representing the living room, and ending at vertex G, representing the +garage making sure to visit every room at least once. There are many ways this can be done. You may want to +number the edges to keep track of their order. One example is shown in Figure 12.77. +Figure 12.77 Draw the Path from L to G +Step 2: Name the path that you followed by listing the vertices in the order you visited them. +L → R → L → K → L → H → B → H → M → H → G +YOUR TURN 12.19 +1. Recall from the Graph Basics section that a graph in which there are multiple edges between the same pair of +vertices is called a multigraph. The figure shows a map of the four bridges that link Staten Island to Brooklyn and +New Jersey and a multigraph representing the map. In the multigraph, the edges are labeled instead of the +vertices. The edges represent the bridges, G (Goethals Bridge), B (Bayonne Bridge), C (Outerbridge Crossing), +and V (Verrazzano-Narrows Bridge). The vertices represent New Jersey, Staten Island, and Brooklyn. +1286 +12 • Graph Theory +Access for free at openstax.org + +Map and Multigraph of Staten Island Bridges +A path can be named by a sequences of edges instead of a sequence of vertices. Determine which of the +following sequence of edges is a walk. +a. +B → V → C → G → C +b. +V → C → B → G → C +c. +C → V → G → B → B +d. +G → V → B → V → C +Paths and Trails +A walk is the most basic way of navigating a graph because it has no restrictions except staying on the graph. When +there are restrictions on which vertices or edges we can visit, we will call the walk by a different name. For example, if we +want to find a walk that avoids travelling the same edge twice, we will say we want to find a trail (or directed trail). If we +want to find a walk that avoids visiting the same vertex twice, we will say, we want to find a path (or directed path). +Walks, trails, and paths are all related. +1. +All paths are trails, but trails that visit the same vertex twice are not paths. +2. +All trails are walks, but walks in which an edge is visited twice would not be trails. +We can visualize the relationship as in Figure 12.78. +Figure 12.78 Walks, Trails, and Paths +Let’s practice identifying walks, trails, and paths using the graphs in Figure 12.79. +12.4 • Navigating Graphs +1287 + +Figure 12.79 Graphs A and K +EXAMPLE 12.20 +Identifying Walks, Paths, and Trails +Consider each sequence of vertices from Graph A in Figure 12.79. Determine if it is only a walk, both a walk and a path, +both a walk and a trail, all three, or none of these. +1. +b → c → d → e → f +2. +c → b → d → b → e +3. +c → f → e → d → b → c +4. +b → e → f → c → b → d +Solution +1. +First, check to see if the sequence of vertices is a walk by making sure that the vertices are consecutive. As you can +see in Figure 12.80, there is no edge between vertex c and vertex d. +Figure 12.80 +This means that the sequence is not a walk. If it is not a walk, then it can’t be a path and it cannot be a trail, so, it is +none of these. +2. +First, check to see if the sequence is a walk. As you can see in Figure 12.81, the vertices are consecutive. +Figure 12.81 +This means that the sequence is a walk. Since the vertex b is visited twice, this walk is not a path. Since edge +is +traveled twice, this walk is not a trail. So, the sequence is only a walk. +3. +First, check to see if the sequence is a walk. We can see in Figure 12.82 that the vertices are consecutive, which +means it is a walk. +Figure 12.82 +Next, check to see if any vertex is visited twice. Remember, we do not consider beginning and ending at the same +1288 +12 • Graph Theory +Access for free at openstax.org + +vertex to be visiting a vertex twice. So, no vertex was visited twice. This means we have a walk that is also a path. +Next check to see if any edge was visited twice; none were. So, the sequence is a walk, a path, and a trail. +4. +First, check to see if the sequence is a walk. We can see in Figure 12.83 that the vertices are consecutive, which +means it is a walk. +Figure 12.83 +Next check to see if any vertex is visited twice. Since vertex b is visited twice, this is not a path. Finally, check to see if +any edges are traveled twice. Since no edges are traveled twice, this is a trail. So, the sequence of vertices is a walk +and a trail. +YOUR TURN 12.20 +Consider each sequence of vertices from Graph K in Figure 12.106. Determine if it is only a walk, both a walk and a +path, both a walk and a trail, all three, or none of these. +1. n → q → o → p +2. p → n → q → o → n → m +3. m → n → o → p → q +VIDEO +Walks, Trails, and Paths in Graph Theory (https://openstax.org/r/walks_trails_paths) +Circuits +In many applications of graph theory, such as creating efficient delivery routes, beginning and ending at the same +location is a requirement. When a walk, path, or trail end at the same location or vertex they began, we call it closed. +Otherwise, we call it open (does not begin and end at the same location or vertex). Some examples of closed walks, +closed trails, and closed paths are given in in the following table. +12.4 • Navigating Graphs +1289 + +DESCRIPTION +EXAMPLE +CHARACTERISTICS +A closed walk is a walk that begins and ends at the same vertex. +d → f → b → c → f → d +Alternating +sequence of +vertices and +edges +Begins and ends +at the same vertex +A closed trail is a trail that begins and ends at the same vertex. It +is commonly called a circuit. +d → f → b → c → f → e → d +No repeated +edges +Begins and ends +at the same vertex +A closed path is a path that begins and ends at the same vertex. +It is also referred to as a directed cycle because it travels +through a cyclic subgraph. +d → f → b → c → d +No repeated +edges or vertices +Begins and ends +at the same vertex +Table 12.4 Closed Walks, Trails, and Paths +Since walks, trails, and paths are all related, closed walks, circuits, and directed cycles are related too. +1. +All circuits are closed walks, but closed walks that visit the same edge twice are not circuits. +2. +All directed cycles are circuits, but circuits in which a vertex is visited twice are not directed cycles. +We can visualize the relationship as in Figure 12.84. +1290 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.84 Closed Walks, Circuits, and Directed Cycles +The same circuit can be named using any of its vertices as a starting point. For example, the circuit d → f → b → c → d can +also be referred to in the following ways. +a → b → c → d → a is the same as +Let’s practice working with closed walks, circuits (closed trails), and directed cycles (closed paths). In the graph in Figure +12.85, the vertices are major central and south Florida airports. The edges are direct flights between them. +Figure 12.85 Major Central and South Florida Airports +EXAMPLE 12.21 +Determining a Closed Walk, Circuit, or Directed Cycle +Suppose that you need to travel by air from Miami (MIA) to Orlando (MCO) and you were restricted to flights represented +on the graph. For the trip to Orlando, you decide to purchase tickets with a layover in Key West (EYW) as shown in Figure +12.86, but you still have to decide on the return trip. Determine if your roundtrip itinerary is a closed walk, a circuit, and/ +or a directed cycle, based on the return trip described in each part. +12.4 • Navigating Graphs +1291 + +Figure 12.86 MIA to EYW to MCO +1. +You returned to Miami (MIA) by reversing your route. +2. +Your direct flight back left Orlando (MCO) but was diverted to Fort Lauderdale (FLL)! From there you flew to Tampa +(TPA) before returning to Miami (MIA). +Solution +1. +The whole trip was MIA → EYW → MCO → EYW → MIA. This is a closed walk, because it is a walk that begins and ends +at the same vertex. It is not a circuit, because it repeats edges. If it is not a circuit, then it cannot be a directed cycle. +2. +The whole trip was MIA → EYW → MCO → FLL → TPA → MIA. This is a closed walk, because it is a walk that begins and +ends at the same vertex. It is a circuit because no edges were repeated. It is also a directed cycle because no +vertices were repeated either. So, it is all three! +YOUR TURN 12.21 +Suppose that you want to travel from Palm Beach (PBI) to any other city listed in Figure 12.113 and then return to +Palm Beach. +1. Why is it impossible to find an itinerary that is a circuit? +2. How is this related to the degree of the vertex PBI? +VIDEO +Closed Walks, Closed Trails (Circuits), and Closed Paths (Directed Cycles) in Graph Theory (https://openstax.org/r/ +closed_walks) +Graph Colorings +In this section so far, we have looked at how to navigate graphs by proceeding from one vertex to another in a sequence +that does not skip any vertices, but in some applications we may want to skip vertices. Remember the camp Olympics at +Camp Woebegone in Comparing Graphs? You were planning a camp Olympics with four events. The campers signed up +for the events. You drew a graph to help you visualize which events have campers in common. The vertices of Graph E in +Figure 12.87 represent the events and adjacent vertices indicate that there are campers who are participating in both. +Figure 12.87 Graphs of Camp Olympics +In this case, we do NOT want events represented by two adjacent vertices to occur in the same timeslot, because that +would prevent the campers who wanted to participate in both from doing so. We can use the graph in Figure 12.87 to +count the timeslots we need so there are no conflicts. Let’s assign each timeslot a different color. We could categorize +events that happen at 1 pm as Red; 2 pm, Purple; 3 pm, Blue; and 4 pm, Green. Then assign different colors to any pair +of adjacent vertices to ensure that the events they represent do not end up in the same timeslot. Figure 12.88 shows +several of the ways to do this while obeying the rule that no pair of adjacent vertices can be the same color. +1292 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.88 Graph Colorings +In Figure 12.88, the graphs with vertices colored so that no adjacent vertices are the same color are called graph +colorings. Notice that Graph 3 has the fewest colors, which means it shows us how to have the fewest number of +timeslots. The events marked in red, a and d, can be held at the same time because they are not adjacent and do not +have conflicts. Also, the events marked in purple, b and c, can be held at the same time. We would not need green or +blue timeslots at all! +A graph that uses +colors is called an +-coloring. The smallest number of colors needed to color a particular graph is +called its chromatic number. +Graph colorings can be used in many applications like the scheduling scenario at camp Woebegone. Let's look at how +they work in more detail. Figure 12.89 shows two different colorings of a particular graph. Coloring A is called a four- +coloring, because it uses four colors, red (R), green (G), blue (B), and purple (P). Coloring B is called a three-coloring +because it uses three. The colors allow us to visually subdivide the graphs into groups. The only rule is that adjacent +vertices are different colors so that they are in different groups. +Figure 12.89 Two Colorings of the Same Graph +It turns out that a three-coloring is the best we can do with the graph in Figure 12.89. No matter how many different +patterns you try with only two colors, you will never find one in which the adjacent vertices are always different colors. In +other words, the graph has a chromatic number of three. For large graphs, computer assistance is usually required to +find the chromatic numbers. There is no formula for finding the chromatic number of a graph, but there are some facts +that are helpful in Table 12.5. +12.4 • Navigating Graphs +1293 + +Fact +Example +Recall that planar graphs are +untangled; that is, they can be +drawn on a flat surface so that no +two edges are crossing. If a graph +is planar, it can be colored with +four colors or possibly fewer. +Each vertex in a complete graph is +adjacent to every other vertex +forcing us to color them all +different colors.The chromatic +number of a complete graph is +the same as the number of +vertices. +If a graph has a clique, a complete +subgraph, then each vertex in the +clique must have a different color +and the vertices outside the clique +may or may not have even more +colors. The chromatic number of +a graph is at least the number of +vertices in its biggest clique. +Table 12.5 Facts about Graph Colorings +Creating Colorings to Solve Problems +Let’s see how these facts can help us color the graph in Figure 12.90. +• +Since the graph is planar, the chromatic number is no more than four. +• +The graph is not complete, but it has complete subgraphs of three vertices. In other words, it has triangles like the +one shown with blue vertices in Figure 12.90. This means that the chromatic number is at least three. +Figure 12.90 A Graph with Triangles +We know we can color this graph in three or four colors. It is usually best to start by coloring the vertex of highest degree +as shown in Figure 12.91. In this case, we used red (R). The color is not important. +1294 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.91 Color Highest Degree Vertex First +We want to color as many of the vertices the same color as possible; so, we look at all the vertices that are not adjacent +to the red vertex and begin to color them, red starting with the one among them of highest degree. Since the only +vertices that are not adjacent are both degree 2, choose either one and color it red as shown in Figure 12.92. +Figure 12.92 Color More Red from Highest to Lowest Degree +Now, there is only one vertex left that is not adjacent to a red; so, color it red. Of the remaining vertices, the highest +degree is four; so, color one of the vertices of degree four in a different color. These two steps are shown in Figure 12.93. +Figure 12.93 Complete the Reds and Start Another Color +Repeat the same procedure. There are three remaining vertices that are not adjacent to a blue. Color as many blue as +possible, with priority going to vertices of higher degree as shown in Figure 12.94. +Figure 12.94 Repeat Procedure for Color Blue +All the remaining vertices are adjacent to blue. So, it is time to repeat the procedure with another color as shown in +Figure 12.95. +12.4 • Navigating Graphs +1295 + +Figure 12.95 Repeat Procedure for Color Purple +All the vertices are now colored with a three-coloring so we know the chromatic number is at most three, but we knew +the chromatic number was at least three because the graph has a triangle. So, we are now certain it is exactly three. +VIDEO +Coloring Graphs Part 1: Coloring and Identifying Chromatic Number (https://openstax.org/r/ +Coloring_and_Identifying) +Suppose the vertices of the graph in Figure 12.96 represented the nine events in the Camp Woebegone Olympics, and +edges join any events that have campers in common, but nonadjacent vertices do not. +Figure 12.96 Coloring for Nine Event Camp Woebegone Olympics +Any vertices of the same color are nonadjacent; so, they have no conflicts. Since this graph is a three-coloring, all nine +events could be scheduled in just three timeslots! +EXAMPLE 12.22 +Understanding Chromatic Numbers +In Example 12.17, we discussed a high school, which holds end-of-course exams in (E3) English 3, (E4) English 4, (M) +Advanced Math, (C) Calculus, (W) World History, (U) U.S. History, (B) Biology, and (P) Physics. We were given a list of +courses that had no students in common. We used that information to find the graph in Figure 12.63, which shows +edges between exams with students in common. Use the graph we found in Example 12.17 to answer each question. +Figure 12.97 Graph of Exams with Students in Common +1. +The graph contains a clique of size 4 formed by the vertices P, E3, C, and U. What does this tell you about the +chromatic number? +2. +The graph is not planar, meaning that you cannot untangle it. What does this tell you about the chromatic number? +3. +Create a coloring by coloring vertex of highest degree first, coloring as many other vertices as possible each color +from highest to lowest degree, then repeating this process for the remaining vertices. +1296 +12 • Graph Theory +Access for free at openstax.org + +4. +Do you know what the minimum number of timeslots is? If so, what is it and how do you know? If not, what are the +possibilities? +Solution +1. +We would need four different colors just for the clique with four vertices; so, the chromatic number is at least four. +2. +It is possible for the chromatic number to be greater than four. +3. +The process is shown in Table 12.6. +This is the original graph. Vertex B has +highest degree. +Color vertex B. Vertex E3 is the only +remaining vertex that is NOT adjacent +to B. +Color vertex E3 the same color. +Vertices P, U, W, and C are the +remaining vertices with highest +degree. Pick one to color. +Color vertex P a new color. Vertices E4 +and M are NOT adjacent to P, and M +has higher degree. +Color vertex M the same color. Vertex +E4 is the only remaining vertex NOT +adjacent to P or M. +Color vertex E4 the same color. +Vertices U, W, and C are the +remaining vertices with highest +degree. Pick one to color. +Color U a new color. Vertex W is the +only remaining vertex NOT adjacent to +U. +Color vertex W the same color. Vertex +C is the only remaining vertex that has +NOT been colored. +Color vertex C a new color. The +coloring is final. We used four +colors. +Table 12.6 Coloring the Graph +The last graph in Table 12.6 is the final coloring. +4. +Yes, the minimum number of times slots is the chromatic number. We knew the chromatic number had to be at least +12.4 • Navigating Graphs +1297 + +four because there was a clique with four vertices. Now we have found a four-coloring of the graph which tells us +that the chromatic number is at most four. So, we know four must be the chromatic number. +The procedure used in Example 22 to color the graph is not guaranteed to result in a graph that has the minimum +number of colors possible, but it is usually results in a coloring that is close to the chromatic number. +YOUR TURN 12.22 +The given figure shows the same graph colored in several ways. The largest clique of the graph has three vertices. +Use this information to answer each question. +Possible Colorings of the Same Graph +1. Is the graph planar? What does your answer tell you about the chromatic number? +2. What is the lowest possible chromatic number of this graph based on the size of any cliques? +3. What are the possible chromatic numbers? +4. Which graph or graphs meet the definition of a valid graph coloring? +5. Draw an +-coloring for the graph where +is the answer to Exercise 1. +The Four Color Problem +Figure 12.98 Using only four colors, no two adjacent regions have the same color. +The idea of coloring graphs to solve problems was inspired by one of the most famous problems in mathematics, the +“four color problem.” The idea was that, no matter how complicated a map might be, only four colors were needed to +color the map so that no two regions that shared a boundary would be the same color. For many years, everyone +suspected this to be true, because no one could create a map that needed more than four colors, but they couldn’t prove +it was true in general. Finally, graphs were used to solve the problem! +VIDEO +The Four Color Map Theorem – Numberphile (https://openstax.org/r/The_Four) +We saw how maps can be represented as graphs in Graph Basics. Figure 12.99 from Example 12.4 shows a map of the +midwestern region of the United States. +1298 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.99 Map of Midwestern States +Figure 12.100 shows how this map can be associated with a graph in which each vertex represents a state and each edge +indicates the states that share a common boundary. Figure 12.101 shows the final graph. +Figure 12.100 Edge Assigned to Each Pair of Midwestern States with Common Border +Figure 12.101 Final Graph Representing Common Boundaries between Midwestern States +Notice that the graph representing the common boundaries between midwestern states is planar, meaning that it can be +drawn on a flat surface without edges crossing. As we have seen, any planar graph has a chromatic number of four or +less. This very well-known fact is called the Four-Color Theorem, or Four-Color Map Theorem. +12.4 • Navigating Graphs +1299 + +VIDEO +Coloring Graphs Part 2: Coloring Maps and the Four Color Problem (https://openstax.org/r/Coloring_Maps) +PEOPLE IN MATHEMATICS +Four-Color Theorem +In the 19th century, Francis Guthrie was coloring a map of British counties and noticed that he could color so that no +two adjacent counties were the same color using only four colors. This seemed to be the case for other maps as well, +but he could not figure out why. He shared this with his brother, Frederick, who subsequently took the problem to his +professor, August De Morgan, one of the most famous mathematicians of all time. De Morgan never was able to solve +the problem, but he shared it with his colleagues. The problem continued to intrigue and perplex mathematicians for +over a hundred years, inspiring discoveries in several areas of mathematics. It was finally proven by Kenneth Appel +and Wolfgang Haken from the University of Illinois in 1976 using a combination of computer-aided calculations and +graph theory. This was the first time in history that a mathematical theorem was proven using a computer! (Jesus +Najera, “The Four-Color Theorem,” Cantor’s Paradise, a Medium publication, October 22, 2019) +EXAMPLE 12.23 +Using Four Colors or Fewer to Color a Graph +Find a coloring of the graph in Figure 12.99, which uses four colors or fewer. Use the resulting coloring as a guide to +recolor the map in Figure 12.101. How many colors did you use? Does this support the conclusion of the Four-Color +Theorem? If so, how? +Solution +The steps to color the graph are shown in Table 12.7. +Step 1: Graph with degrees of vertices +labeled. +Vertex IA has highest degree. +Step 2: Color vertex IA any color. +Vertices ND, KS, MI, OH, and IN are +NOT adjacent to IA. +MI and IN have highest degree, 3. +Step 3: Color either MI or IN the +same color as IA. +Vertex ND and KS are the only +remaining vertices not adjacent to +a red vertex. +They both have degree 2. +Table 12.7 Steps to Color Graph of Midwestern States +1300 +12 • Graph Theory +Access for free at openstax.org + +Step 4: Since KS and ND are not +adjacent to each other, we can save a +step and color both red. The highest +degreeof remaining vertices is four. +Step 5: Choose one of the vertices of +degree 4, SD, to color with a new color, +blue. The vertices WI, IL, MO, IN,and +OH are NOT adjacent to blue. +WI, IL, and MO have the highest +degree. +Step 6: Choose one of WI, IL, or +MO to color. +We color WI blue. +MO, IN and OH are NOT adjacent +to blue. +MO has the highest degree of +these. +Step 7: Color MO blue. All remaining +vertices are adjacent to blue. +Choose a new color. Four vertices +remain, MN, NE, IL, and OH. MN, NE, +and IL have the highest degree. +Step 8: Since MN, IL, and NE are not +adjacent, save steps and color all three +the new color, purple. +Vertex OH is the only remaining vertex +that has NOT been colored. +Step 9: Since vertex OH is not +adjacent to purple, color it purple. +This is the final graph. +We used three colors. +Table 12.7 Steps to Color Graph of Midwestern States +The final graph in Table 12.7 shows how we would color the map. In Figure 12.102 we have colored the map to +correspond to the colors on the graph. +12.4 • Navigating Graphs +1301 + +Figure 12.102 Midwestern States in Three Colors +We used three colors to color the graph. This supports the Four-Color Theorem, because the graph is planar and its +chromatic number is less than four. +YOUR TURN 12.23 +1. The first figure shows a map of the Island of Oahu in the State of Hawaii divided into regions, and the second +figure shows a graph of the map. What is the chromatic number? Find a graph coloring that reflects the +chromatic number of the graph. +Map of Oahu +1302 +12 • Graph Theory +Access for free at openstax.org + +Graph Representing Common Boundaries between Regions of Oahu +WHO KNEW? +Coloring A Möbius Strip +Have you ever heard of a möbius strip? It is a flat object with only one side. This might sound theoretical, but you can +make one for yourself. Take a narrow strip of paper, make a half twist in it, and tape the ends together. To prove to +yourself that it has only one side, pick a point anywhere on the paper and start drawing a line. You can draw the line +continuously without lifting your pen from the paper and eventually, you will get back to the point at which you +started. Since you never had to flip over the paper, you have just proved it has one side! +Now, you might be wondering what this has to do with Graph Theory. It turns out that a map drawn on a möbius strip +cannot necessarily be drawn using four colors. This doesn’t contradict the Four-Color Theorem, which only applies to +graphs that are planar. A möbius strip does not live on a “plane,” or flat surface, which means the maps are not always +planar like those drawn on a flat piece of paper. In the case of a möbius strip, it is the Six-Color Map Theorem! Figure +12.103 is an example of a möbius strip with a map that requires five colors. +Figure 12.103 A Five-Coloring on a Möbius Strip +VIDEO +Neil deGrasse Tyson Explains the Möbius Strip (https://openstax.org/r/Neil_de_Grasse) +Check Your Understanding +For the following exercises, determine whether each statement is always true or sometimes true. +25. A trail is a path. +26. A trail is a walk. +27. A walk is a path. +28. A circuit is a trail. +29. A directed cycle is a path. +30. A circuit is a directed cycle. +31. A directed cycle is a circuit. +32. If a graph has an +-coloring, then its chromatic number is +. +33. If the chromatic number of a graph is +, then it has an +-coloring. +34. If a graph is planar, then it has a chromatic number of at most four. +12.4 • Navigating Graphs +1303 + +For the following exercise, fill in the blanks to make the statement true. +35. A walk that ____________ is a trail. +36. A trail that ____________ is a circuit. +37. A circuit that ___________ is a directed cycle. +38. A closed walk that ____________ is a circuit. +39. A complete graph with +vertices has a chromatic number of_____. +40. A graph with a clique with +-vertices has a chromatic number of _________ +. +SECTION 12.4 EXERCISES +For the following exercises, identify each sequence of vertices from the figure as a walk, trail, and/or path. Select all +that apply. +1. A → B → F → G → K → J → F → B +2. G → K → O → N → J → K → L +3. F → J → K → G → B → A +4. I → J → K → L → K → J → N +5. M → N → O → K → L → H +6. A → F → K → P +7. N → J → F → B → C → G → F → E +8. E → F → J → I → E +For the following exercises, identify each sequence of vertices from Figure 12.134 as a closed walk, circuit (closed trail), +and/or directed cycle (closed path). Select all that apply. +9. A → B → F → G → K → J → F → B → A +10. G → K → O → N → J → K → L +11. F → J → N → O → K → J → I → E → F +12. I → J → K → G → F → E → I +13. M → N → O → K → J → I → M +14. N → J → F → B → C → G → F → J → N +15. A → B → G → F → E → A +16. E → F → G → K → J → F → B → A → E +For the following exercises, use the graphs shown. Identify the graph or graphs with the given characteristics. +17. The colors do not follow the definition of a graph coloring. +18. The graph is a two-coloring. +19. The graph is a three-coloring. +20. The graph is a four-coloring. +21. The graph is planar. +22. The graph has a chromatic number of two. +23. Fewer colors could be used to color the graph. +24. The chromatic number is more than four. +25. The Four-Color Theorem applies to the graph. +1304 +12 • Graph Theory +Access for free at openstax.org + +For the following exercises, complete the sequence of vertices from the graph to create the indicated kind of closed +walk. +26. Closed path: a → b → □ → □ → □ → a +27. Closed path that visits edge ed twice: e → d → □ → □ → □ → e +28. Directed cycle: e → a → □ → □ → □ → e +29. Directed cycle: b → d → □ → □ → □ → b +30. Circuit that visits vertex d twice: b → c → d → □ → □ → □ → b +31. Circuit that visits vertex f twice: g → h → f → □ → □ → □ → g +For the following exercises, use the graphs shown. +32. Find the chromatic number +for graph 1 and give an +-coloring. +33. Find the chromatic number +for graph 2 and give an +-coloring. +34. Find the chromatic number +for graph 3 and give an +-coloring. +35. Find the chromatic number +for graph 4 and give an +-coloring. +For the following exercises, indicate the smallest and largest possible chromatic number of the graph described. +36. The graph has 15 vertices and contains a clique with 9 vertices. +37. A planar graph with 100 vertices and a clique with 3 vertices. +38. A planar graph with more than 2 vertices and no cliques. +39. A complete graph with 2,123 vertices. +40. In chess, a knight can move in any direction, but it must move two spaces then turn and move one more space. +The 8 possible moves a knight can make from a space in the center of a five-by-five grid are shown in the first +figure. A knight’s tour is a sequence of moves by a knight on a chessboard (of any size) such that the knight visits +every square exactly once. If the knight’s tour brings the knight back to its starting position on the board, it is +called a closed knight’s tour. Otherwise, it is called an open knight’s tour. Determine if the closed knight’s tour in +the second figure is most accurately described as a closed walk, a circuit, or a directed cycle. Explain your +reasoning. +12.4 • Navigating Graphs +1305 + +41. An open knight’s tour on a five-by-five board is shown in the figure. Is it most accurately described as a walk, a trail, +or a path? +42. A knight’s tour is not possible on a four-by-four board like the one shown in the figure. Find an open tour of the +board in which the knight is permitted to travel through a given space more than once to achieve the goal of +visiting every space. Is your tour most accurately described as a walk, trail, or path? Explain why. +The neighborhood of Pines West has three cul-de-sacs that meet at an intersection as shown in the figure. A postal +delivery person starts at the intersection and visits each house in a cul-de-sac once, returns to the intersection, visits +each house in the next cul-de-sac, and so on, returning to the intersection when finished. +43. Describe how the route can be represented as a graph. If there is no backtracking, in other words, the person +never reverses direction, is the route followed by the postal delivery person best described as a walk, trail, path, +closed walk, circuit, or directed cycle? Explain your reasoning. +1306 +12 • Graph Theory +Access for free at openstax.org + +44. Draw the graph for the route described in the neighborhood. Find its chromatic number. Find a map coloring +that shows the chromatic number. +45. When radio towers are within range of each other, they must use different frequencies. In the graph, the vertices +represent the towsers, and the edges indicate the towers are in range of each other. Use graph colorings to +determine the number of frequencies needed to avoid two towers in the same range being assigned the same +frequency. Give an example of a coloring to support your conclusion. +46. A Sudoku puzzle consists of a nine-by-nine grid which is subdivided into nine three-by-three smaller grids. Given +an incomplete grid, the goal is to complete it in such a way that each row contains the numbers 1 through 9, each +column contains the numbers 1 through 9, and each three by three grid contains the numbers 1 through 9 as +shown in the figure. If a graph were drawn with a vertex to represent each entry in the grid, what should the edges +represent so that a nine-coloring of the graph would be a solution to the puzzle? +12.4 • Navigating Graphs +1307 + +At a wedding reception, the bride and groom would like their family and friends to get to know each other better. They +have decided that if two people know each other, they will not seat them at the same table. The following is a list of +guests and who they know. Guests A, B, C, and D all know each other; guests E, F, G, and H all know each other; guests +I, J, and K all know each other; guests P, Q and R all know each other, guests L, M, and O all know each other. +Additionally, I knows D and I knows G, J knows M, K knows P, N knows L, and N knows O. +47. Draw a graph to show the relationships. +48. Determine the minimum number of tables needed to seat the guests so that they will sit with people they don’t +know. +49. Give a coloring to support your conclusion. +The map of the states of Imaginaria is given. Use the figure to answer the following exercises. +50. Draw a graph to represent the map. +51. Determine the chromatic number of the graph you found. +52. Give a graph coloring that supports your answer in the previous question. +53. Color the provided map to correspond with the coloring you found in the previous question. +1308 +12 • Graph Theory +Access for free at openstax.org + +12.5 Euler Circuits +Figure 12.104 Delivery trucks move goods from place to place. (credit: “Mack Midliner” by Jason Lawrence/Flickr, CC BY +2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Determine if a graph is connected. +2. +State the Chinese postman problem. +3. +Describe and identify Euler Circuits. +4. +Apply the Euler Circuits Theorem. +5. +Evaluate Euler Circuits in real-world applications. +The delivery of goods is a huge part of our daily lives. From the factory to the distribution center, to the local vendor, or +to your front door, nearly every product that you buy has been shipped multiple times to get to you. If the cost and time +of that delivery is too great, you will not be able to afford the product. Delivery personnel have to leave from one +location, deliver the goods to various places, and then return to their original location and do all of this in an organized +way without losing money. How do delivery services find the most efficient delivery route? The answer lies in graph +theory. +Connectedness +Before we can talk about finding the best delivery route, we have to make sure there is a route at all. For example, +suppose that you were tasked with visiting every airport on the graph in Figure 12.105 by plane. Could you accomplish +that task, only taking direct flight paths between airports listed on this graph? In other words, are all the airports +connected by paths? Or are some of the airports disconnected from the others? +12.5 • Euler Circuits +1309 + +Figure 12.105 Major Central and South Florida Airports +In Figure 12.105, we can see TPA is adjacent to PBI, FLL, MIA, and EYW. Also, there is a path between TPA and MCO +through FLL. This indicates there is a path between each pair of vertices. So, it is possible to travel to each of these +airports only taking direct flight paths and visiting no other airports. In other words, the graph is connected because +there is a path joining every pair of vertices on the graph. Notice that if one vertex is connected to each of the others in a +graph, then all the vertices are connected to each other. So, one way to determine if a graph is connected is to focus on a +single vertex and determine if there is a path between that vertex and each of the others. If so, the graph is connected. If +not, the graph is disconnected, which means at least one pair of vertices in a graph is not joined by a path. Let’s take a +closer look at graph X in Figure 12.106. Focus on vertex a. There is a path between vertices a and b, but there is no path +between vertex a and vertex c. So, Graph X is disconnected. +Figure 12.106 Connected vs. Disconnected +When you are working with a planar graph, you can also determine if a graph is connected by untangling it. If you draw +it so that so that none of the edges are overlapping, like we did with Graph X in Figure 12.106, it is easier to see that the +graph is disconnected. +Figure 12.107 Untangling Graph X +Versions 2 and 3 of Graph X in Figure 12.107 each have the same number of vertices, number of edges, degrees of the +vertices, and pairs of adjacent vertices as version 1. In other words, each version retains the same structure as the +original graph. Since versions 2 and 3 of Graph X, do not have overlapping edges, it is easier to identify pairs of vertices +that do not have paths between them, and it is more obvious that Graph X is disconnected. In fact, there are two +completely separate, disconnected subgraphs, one with the vertices in {a, b, e}, and the other with the vertices {c, d, f} +These sets of vertices together with all of their edges are called components of Graph X. A component of a graph is a +subgraph in which there is a path between each pair of vertices in the subgraph, but no edges between any of the +vertices in the subgraph and a vertex that is not in the subgraph. +Now let’s focus on Graph Y from Figure 12.106. As in Graph X, there is a path between vertices a and b, as well as +between vertices a and e, but Graph Y is different from Graph X because of vertex g. Not only is there a path between +vertices a and g, but vertex g bridges the gap between a and c with the path a → b → g → c. Similarly, there is a path +1310 +12 • Graph Theory +Access for free at openstax.org + +between vertices a and d and vertices a and f: a → b → g → d, a → b → g → d → f. Since there is a path between vertex a +and every other vertex, Graph Y is connected. You can also see this a bit more clearly by untangling Graph Y as in Figure +12.108. Even when Y is drawn so that the edges do not overlap, the graph cannot be drawn as two separate, +unconnected pieces. In other words, Graph Y has only one component with the vertices {a, b, c, d, e, f}. +We can give an alternate definition of connected and disconnected using the idea of components. A graph is connected +if it has only one component. A graph is disconnected if it has more than one component. These alternate definitions are +equivalent to the previous definitions. This means that you can confirm a graph is connected or disconnected either by +checking to see if there is a path between each vertex and each other vertex, or by identifying the number of +components. A graph is connected if it has only one component. +Figure 12.108 Untangling Graph Y +EXAMPLE 12.24 +Determining If a Graph Is Connected or Disconnected +Use Figure 12.109 to answer each question. +Figure 12.109 Graph E +1. +Find a path between vertex a and every other vertex on the graph, if possible. +2. +Identify all the components of Graph E. +3. +Determine whether the graph is connected or disconnected and explain how you know. +Solution +1. +The paths are a → d → b, a → d → c, and a → d. +2. +There is only one component in Graph E. It has the vertices {a, b, c, d}. +3. +The graph is connected, because there is a path between vertex a and every other vertex. We can also so that Graph +E is connected because it has only one component. +YOUR TURN 12.24 +1. Determine whether each graph is connected or disconnected and identify the set of vertices that make up each +of its components. +12.5 • Euler Circuits +1311 + +VIDEO +Connected and Disconnected Graphs in Graph Theory (https://openstax.org/r/connected_disconnected_graphs) +EXAMPLE 12.25 +Applying Connectedness +The U.S. Interstate Highway System extends throughout the 48 contiguous states. It also has routes in the states of +Hawaii and Alaska, and the commonwealth of Puerto Rico. Consider a graph representing the U.S. Interstate Highway +System, in which there is a vertex for each of the 50 states and Puerto Rico, and an edge is drawn between any two +vertices representing states that are connected by a highway in that system. Would this graph be connected or +disconnected? Explain your reasoning. +Solution +Hawaii, Alaska, and Puerto Rico are geographically separate from the 48 contiguous states, and each from each other. +This means that there are vertices on the graph with no path joining them to the other vertices. So the graph is +disconnected. +YOUR TURN 12.25 +1. “Six degrees of separation” is the idea that any two people on Earth are connected by at most six social +connections. Assume that this is true. Consider a graph in which each vertex is a person on Earth, and each edge +is a social connection. Would this graph be connected or disconnected? Explain your reasoning. +Origin of Euler Circuits +The city of Konigsberg, modern day Kaliningrad, Russia, has waterways that divide up the city. In the 1700s, the city had +seven bridges over the various waterways. The map of those bridges is shown in Figure 12.110. The question as to +whether it was possible to find a route that crossed each bridge exactly once and return to the starting point was known +as the Konigsberg Bridge Problem. In 1735, one of the most influential mathematicians of all time, Leonard Euler, solved +the problem using an area of mathematics that he created himself, graph theory! +Figure 12.110 Map of the Bridges of Konigsberg in 1700s (credit: “Konigsberg Bridge” by Merian Erben/Wikimedia +Commons, Public Domain) +Euler drew a multigraph in which each vertex represented a land mass, and each edge represented a bridge connecting +them, as shown in Figure 12.111. Remember from Navigating Graphs that a circuit is a trail, so it never repeats an edge, +1312 +12 • Graph Theory +Access for free at openstax.org + +and it is closed, so it begins and ends at the same vertex. Euler pointed out that the Konigsberg Bridge Problem was the +same as asking this graph theory question: Is it possible to find a circuit that crosses every edge? Since then, circuits (or +closed trails) that visit every edge in a graph exactly once have come to be known as Euler circuits in honor of Leonard +Euler. +VIDEO +Recognizing Euler Trails and Euler Circuits (https://openstax.org/r/Euler_trails_circuits) +Euler was able to prove that, in order to have an Euler circuit, the degrees of all the vertices of a graph have to be even. +He also proved that any graph with that characteristic must have an Euler circuit. So, saying that a connected graph is +Eulerian is the same as saying it has vertices with all even degrees, known as the Eulerian circuit theorem. +Figure 12.111 Graph of Konigsberg Bridges +To understand why the Euler circuit theorem is true, think about a vertex of degree 3 on any graph, as shown in Figure +12.112. +Figure 12.112 A Vertex of Degree 3 +First imagine the vertex of degree 3 shown in Figure 12.112 is not the starting vertex. At some point, each edge must be +traveled. The first time one of the three edges is traveled, the direction will be toward the vertex, and the second time it +will be away from the vertex. Then, at some point, the third edge must be traveled coming in toward the vertex again. +This is a problem, because the only way to get back to the starting vertex is to then visit one of the three edges a second +time. So, this vertex cannot be part of an Euler circuit. +Next imagine the vertex of degree 3 shown in Figure 12.112 is the starting vertex. The first time one of the edges is +traveled, the direction is away from the vertex. At some point, the second edge will be traveled coming in toward the +vertex, and the third edge will be the way back out, but the starting vertex is also the ending vertex in a circuit. The only +way to return to the vertex is now to travel one of the edges a second time. So, again, this vertex cannot be part of an +Euler circuit. +For the same reason that a vertex of degree 3 can never be part of an Euler circuit, a vertex of any odd degree cannot +either. We can use this fact and the graph in Figure 12.113 to solve the Konigsberg Bridge Problem. Since the degrees of +the vertices of the graph in Figure 12.112 are not even, the graph is not Eulerian and it cannot have an Euler circuit. This +means it is not possible to travel through the city of Konigsberg, crossing every bridge exactly once, and returning to +your starting position. +12.5 • Euler Circuits +1313 + +Figure 12.113 Degrees of Vertices in Konigsberg Bridge Multigraph +VIDEO +Existence of Euler Circuits in Graph Theory (https://openstax.org/r/existence_Euler_circuits) +Chinese Postman Problem +At Camp Woebegone, campers travel the waterways in canoes. As part of the Camp Woebegone Olympics, you will hold +a canoeing race. You have placed a checkpoint on each of the 11 different streams. The competition requires each team +to travel each stream, pass through the checkpoints in any order, and return to the starting line, as shown in the Figure +12.114. +Figure 12.114 Map of Canoe Event Checkpoints +Since the teams want to go as fast as possible, they would like to find the shortest route through the course that visits +each checkpoint and returns to the starting line. If possible, they would also like to avoid backtracking. Let’s visualize the +course as a multigraph in which the vertices represent turns and the edges represent checkpoints as in Figure 12.115. +Figure 12.115 Multigraph of Canoe Event +The teams would like to find a closed walk that repeats as few edges as possible while still visiting every edge. If they +never repeat an edge, then they have found a closed trail, which is a circuit. That circuit must cover all edges; so, it would +be an Euler circuit. The task of finding a shortest circuit that visits every edge of a connected graph is often referred to as +the Chinese postman problem. The name Chinese postman problem was coined in honor of the Chinese +mathematician named Kwan Mei-Ko in 1960 who first studied the problem. +If a graph has an Euler circuit, that will always be the best solution to a Chinese postman problem. Let’s determine if the +multigraph of the course has an Euler circuit by looking at the degrees of the vertices in Figure 12.116. Since the degrees +of the vertices are all even, and the graph is connected, the graph is Eulerian. It is possible for a team to complete the +canoe course in such a way that they pass through each checkpoint exactly once and return to the starting line. +1314 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.116 Degrees of Vertices in Graph of Canoe Event +EXAMPLE 12.26 +Understanding Eulerian Graphs +A postal delivery person must deliver mail to every block on every street in a local subdivision. Figure 12.117 is a map of +the subdivision. Use the map to answer each question. +Figure 12.117 Map of Subdivision +1. +Draw a graph or multigraph to represent the subdivision in which the vertices represent the intersections, and the +edges represent streets. +2. +Is your graph connected? Explain how you know. +3. +Determine the degrees of the vertices in the graph. +4. +Is your graph an Eulerian graph? +5. +Is it possible for the postal delivery person to visit each block on each street exactly once? Justify your answer. +Solution +1. +The graph is in Figure 12.118. +Figure 12.118 Graph of Subdivision +2. +The graph is connected. It has only one component and there is a path between each pair of vertices. +3. +There are four corner vertices of degree 2, there are eight exterior vertices of degree 3, and there are four interior +vertices of degree 4. +4. +The graph is not Eulerian because it has vertices of odd degree. +5. +No. Since the graph is not Eulerian, there is no Euler circuit, which means that there is no route that would pass +through every edge exactly once. +YOUR TURN 12.26 +A pest control service has at least one customer on every block of every street or cul-de-sac in a neighborhood. Use +12.5 • Euler Circuits +1315 + +the map of the neighborhood to answer each question. +Map of Neighborhood +1. Draw a graph or multigraph of the neighborhood in which the vertices represent intersections, and the edges +represent the streets between them. +2. Is your graph connected? Explain how you know. +3. Determine the degrees of the vertices in the graph. +4. Is your graph an Eulerian graph? +5. Is it possible for the postal delivery person to visit each block on each street exactly once and start and end at +the same position? Justify your answer. +Identifying Euler Circuits +Solving the Chinese postman problem requires finding a shortest circuit through any graph or multigraph that visits +every edge. In the case of Eulerian graphs, this means finding an Euler circuit. The method we will use is to find any +circuit in the graph, then find a second circuit starting at a vertex from the first circuit that uses only edges that were not +in the first circuit, then find a third circuit starting at a vertex from either of the first two circuits that uses only edges that +were not in the first two circuits, and then continue this process until all edges have been used. In the end, you will be +able to link all the circuits together into one large Euler circuit. +Let’s find an Euler circuit in the map of the Camp Woebegone canoe race. In Figure 12.119, we have labeled the edges of +the multigraph so that the circuits can be named. In a multigraph it is necessary to name circuits using edges and +vertices because there can be more than one edge between adjacent vertices. +Figure 12.119 Multigraph of Canoe Race with Vertices and Edges Labeled +We will begin with vertex 1 because it represents the starting line in this application. In general, you can start at any +vertex you want. Find any circuit beginning and ending with vertex 1. Remember, a circuit is a trail, so it doesn’t pass +through any edge more than once. Figure 12.120 shows one possible circuit that we could use as the first circuit, 1 → A → +2 → B → 3 → C → 4 → G → 5 → J → 1. +1316 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.120 First Circuit +From the edges that remain, we can form two more circuits that each start at one of the vertices along the first circuit. +Starting at vertex 3 we can use 3 → H → 5 → I → 1 → K → 3 and starting at vertex 2 we can use 2 → D → 6 → E → 7 → F → 2, +as shown in Figure 12.121. +Figure 12.121 Second and Third Circuits +Now that each of the edges is included in one and only once circuit, we can create one large circuit by inserting the +second and third circuits into the first. We will insert them at their starting vertices 2 and 3 +becomes +Finally, we can name the circuit using vertices, 1 → 2 → 6 → 7 → 2 → 3 → 5 → 1 → 3 → 4 → 5 → 1, or edges, A → D → E → F → +B → H → I → K → C → G → J. +Let's review the steps we used to find this Eulerian Circuit. +Steps to Find an Euler Circuit in an Eulerian Graph +Step 1 - Find a circuit beginning and ending at any point on the graph. If the circuit crosses every edges of the graph, the +circuit you found is an Euler circuit. If not, move on to step 2. +Step 2 - Beginning at a vertex on a circuit you already found, find a circuit that only includes edges that have not +previously been crossed. If every edge has been crossed by one of the circuits you have found, move on to Step 3. +Otherwise, repeat Step 2. +Step 3 - Now that you have found circuits that cover all of the edges in the graph, combine them into an Euler circuit. You +can do this by inserting any of the circuits into another circuit with a common vertex repeatedly until there is one long +circuit. +EXAMPLE 12.27 +Finding an Euler Circuit +Use Figure 12.122 to answer each question. +12.5 • Euler Circuits +1317 + +Figure 12.122 Graph F +1. +Verify the Graph F is Eulerian. +2. +Find an Euler circuit that begins and ends at vertex c. +Solution +1. +The graph is connected because it has only one component. Also, each of the vertices in graph F has even degree as +shown in Figure 12.123. So, the graph is Eulerian. +Figure 12.123 Degrees of Vertices in Graph F +2. +Step 1: Beginning at vertex c, identify circuit c → e → b → c as shown in Figure 12.124. Since this circuit does not +cover every edge in the graph, move on to Step 2. +1318 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.124 First Circuit in Graph F +Step 2: Find another circuit beginning at one of the vertices in the first circuit, using only edges that have not been +used in the first circuit. It is possible to do this using either vertex c or vertex b. In Figure 12.125, we have used +vertex b as the starting and ending point for a second circuit, b → d → c → a → b. Since all edges have been crossed +by one of the two circuits, move on to Step 3. +Figure 12.125 Second Circuit in Graph F +Step 3: Combine the two circuits into one. Replace vertex b in the first circuit with the whole second circuit that +begins at vertex b. +. +An Euler circuit that begins at vertex c is c → e → b → d → c → a → b → c. +YOUR TURN 12.27 +Use Graphs X and Y to answer each question. +12.5 • Euler Circuits +1319 + +1. Which graph does not have an Euler circuit? Explain how you know. +2. Find an Euler circuit in the other graph. +Eulerization +The Chinese postman problem doesn’t only apply to Eulerian graphs. Recall the postal delivery person who needed to +deliver mail to every block of every street in a subdivision. We used the map in Figure 12.126 to create the graph in +Figure 12.127. +Figure 12.126 Map of Subdivision +Figure 12.127 Graph of Subdivision +Since the graph of the subdivision has vertices of odd degree, there is no Euler circuit. This means that there is no route +through the subdivision that visits every block of every street without repeating a block. What should our delivery person +do? They need to repeat as few blocks as possible. The technique we will use to find a closed walk that repeats as few +edges as possible is called eulerization. This method adds duplicate edges to a graph to create vertices of even degree +so that the graph will have an Euler circuit. +In Figure 12.128, the eight vertices of odd degree in the graph of the subdivision are circled in green. We have added +duplicate edges between the pairs of vertices, which changes the degrees of the vertices to even degrees so the +resulting multigraph has an Euler circuit. In other words, we have eulerized the graph. +Figure 12.128 An Eulerized Graph +1320 +12 • Graph Theory +Access for free at openstax.org + +The duplicate edges in the eulerized graph correspond to blocks that our delivery person would have to travel twice. By +keeping these duplicate edges to a minimum, we ensure the shortest possible route. It can be challenging to determine +the fewest duplicate edges needed to eulerize a graph, but you can never do better than half the number of odd +vertices. In the graph in Figure 12.128, we have found a way to fix the eight vertices of odd degree with only four +duplicate edges. Since four is half of eight, we will never do better. +IMPORTANT! The duplicate edges that we add indicate places where the route will pass twice. An entirely new edge +between two vertices that were not previously adjacent would indicate that our postal delivery person created a new +road through someone’s property! So, we can duplicate existing edges, but we cannot create new ones. +EXAMPLE 12.28 +Eulerizing Graphs +Use Graph A and multigraphs B, C, D, and E given in Figure 12.129 to answer the questions. +Figure 12.129 Graph A and Multigraphs B through E +1. +Which of the multigraphs are not eulerizations of Graph A? Explain your answer. +2. +Which eulerization of Graph A uses the fewest duplicate edges? How many does it use? +3. +Is it possible to eulerize Graph A using fewer duplicate edges than your answer to part 2? If so, give an example. If +not, explain why not. +Solution +1. +Multigraph B is not an eulerization of A because the edge N between vertices c and d is not a duplicate of an +existing edge. Multigraph E is not an eulerization of A because vertices b and e have odd degree. +2. +Multigraph C uses 3 duplicate edges while multigraph D only uses 2. So, D uses the fewest. +3. +Since there were four vertices in Graph A, the fewest number of edges that could possibly eulerize the graph is half +of four, which is two. So, it is not possible to eulerize Graph A using fewer edges. +YOUR TURN 12.28 +1. Create an eulerization using the fewest duplicate edges possible for Graph Z. +Graph Z +IMPORTANT! Since only duplicate edges can be added to eulerize a graph, it is not possible to eulerize a +disconnected graph. +12.5 • Euler Circuits +1321 + +WORK IT OUT +Figure 12.130 Map of the Magical Land of Oz (credit: “Map of Oz within the surrounding deserts” by L. Frank Baum/ +Wikimedia Commons, Public Domain) +In The Wonderful Wizard of Oz, written by L. Frank Baum and illustrated by W. W. Denslow, the region of the Emerald +City lies at the center of the magical land of Oz, with Gillikin Country to the north, Winkie Country to the east, +Munchkin Country to the west, and Quadling Country to the south. Munchkin Country and Winkie Country each +shares a border with Gillikin Country and Quadling Country. Let’s apply graph theory to Dorothy’s famous journey +through Oz. Draw a graph in which each vertex is one of the regions of Oz. Then answer each question: +1. +Is there an Euler trail circuit that Dorothy could follow, instead of the yellow brick road, to lead her from the land +of the Munchkins, through all the regions of Oz and back, passing over each border exactly once? If not, how +could the graph be Eulerized most efficiently? +2. +What is the chromatic number of the graph? Find a graph coloring that demonstrates your answer and use it to +draw and color a graph of Oz. +Check Your Understanding +For the following exercises, determine whether each statement is always true, sometimes true, or never true. +41. A disconnected graph has only one component. +42. A graph that has all vertices of even degree is connected. +43. There is a route through the city of Konigsberg that a person may pass over each bridge exactly once and +return to the starting point. +44. A graph with vertices of all even degree is Eulerian. +45. An Eulerian graph has all vertices of even degree. +46. An Euler circuit is a closed trail. +47. An Euler circuit is a closed path +48. To eulerize a graph, add new edges between previously nonadjacent vertices until no vertices have odd degree. +49. To eulerize a graph, add duplicate edges between adjacent vertices until no vertices have odd degree. +50. The number of duplicate edges required to eulerize a graph is half the number of vertices of odd degree. +SECTION 12.5 EXERCISES +Use Graphs A, B, C, D, E, F, G, H, and I to answer the following exercises. Identify any graph or graphs with the given +characteristics or indicate that none do. +1322 +12 • Graph Theory +Access for free at openstax.org + +1. Connected +2. Disconnected +3. Exactly two components +4. Exactly three components +5. Exactly four components +6. Exactly five components +7. At least one vertex of odd degree +8. All vertices of even degree +9. An Euler circuit +10. A path between vertex j and each other vertex on the same graph +Use the graphs in the previous exercise to answer the following exercises. For each exercise, list the set of vertices for +each component in the given graph. +11. Graph B +12. Graph E +13. Graph F +14. Multigraph I +Use the graphs in the initial exercise to answer the following exercises. For each exercise, a graph and a vertex on the +graph are given. Find a path between the given vertex and each other vertex on the graph. If this is not possible, +indicate that it is not. +15. Graph A, vertex c. +16. Graph B, vertex m. +17. Graph D, vertex x. +18. Graph F, vertex w. +19. Graph G, vertex a. +20. Graph H, vertex e. +Use the graphs in the initial exercise to answer the following exercises. For each exercise, a graph and a vertex on the +graph are given. Find an Euler circuit beginning and ending at the given vertex if one exists. If no Euler circuits exist, +explain how you know that they do not. +21. Graph A, vertex c. +22. Graph B, vertex k. +23. Graph D, vertex w. +24. Graph G, vertex b. +12.5 • Euler Circuits +1323 + +25. Graph H, vertex o. +26. Multigraph I, vertex p. +For the following exercises, use the connected graphs. In each exercise, a graph is indicated. Determine if the graph is +Eulerian or not and explain how you know. If it is Eulerian, give an example of an Euler circuit. If it is not, state which +edge or edges you would duplicate to eulerize the graph. +27. Graph J +28. Graph K +29. Multigraph L +30. Graph M +31. The figure shows a map of zoo exhibits A through P with walkways shown in gray. Draw a graph, or multigraph, to +represent the routes through the zoo in which the edges represent walkways and the vertices represent turns and +intersections, which are each marked with a star. Notice that there is exactly one exhibit between each pair of +adjacent vertices. Label the edges with the corresponding exhibit. Use it to determine if it is possible for a visitor to +begin at the entrance, view each exhibit exactly once, and end back at the entrance. If it is possible, give an +example of a circuit on the graph that would represent a route the visitor could take. If it is not possible, explain +why. +The figure shows the map of the exhibits at an indoor aquarium with hallways shown in gray. Turns and intersections +of hallways are marked with stars. +32. Use the Euler circuit theorem and a graph in which the edges represent hallways and the vertices represent +turns and intersections to explain why a visitor to the aquarium cannot start at the entrance, visit every exhibit +exactly once, and return to the entrance. +1324 +12 • Graph Theory +Access for free at openstax.org + +33. Use an eulerization of the graph you found to determine the least amount of backtracking necessary to allow a +visitor to begin at the entrance, see every exhibit at least once, and return to the entrance. How many hallways +must be covered twice? Explain your reasoning. +The map of the states of Imaginaria is given. +34. Use a graph to determine if it is possible to travel through Imaginaria crossing each the border between each +pair of states exactly once, and returning to the state in which you started. +35. Use an eulerization of the graph you found to determine the fewest borders that must be covered twice in +order to cross each border at least once and return to the state in which you started. Explain your reasoning. +12.5 • Euler Circuits +1325 + +12.6 Euler Trails +Figure 12.131 The Pony Express mail route spanned from California to Missouri. (credit: “Map of Pony Express” by +Nathan Hughes Hamiltonh/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Describe and identify Euler trails. +2. +Solve applications using Euler trails theorem. +3. +Identify bridges in a graph. +4. +Apply Fleury’s algorithm. +5. +Evaluate Euler trails in real-world applications. +We used Euler circuits to help us solve problems in which we needed a route that started and ended at the same place. +In many applications, it is not necessary for the route to end where it began. In other words, we may not be looking at +circuits, but trails, like the old Pony Express trail that led from Sacramento, California in the west to St. Joseph, Missouri +in the east, never backtracking. +Euler Trails +If we need a trail that visits every edge in a graph, this would be called an Euler trail. Since trails are walks that do not +repeat edges, an Euler trail visits every edge exactly once. +EXAMPLE 12.29 +Recognizing Euler Trails +Use Figure 12.132 to determine if each series of vertices represents a trail, an Euler trail, both, or neither. Explain your +reasoning. +Figure 12.132 Graph H +1. +a → b → e → g → f → c → d → e +2. +a → b → e → g → f → c → d → e → b → a → d → g +3. +g → d → a → b → e → d → c → f → g → e +1326 +12 • Graph Theory +Access for free at openstax.org + +Solution +1. +It is a trail only. It is a trail because it is a walk that doesn’t cover any edges twice, but it is not an Euler trail because +it didn’t cover edges ad or dg. +2. +It is neither. It is not a trail because it visits ab and be twice. Since it is not a trail, it cannot be an Euler trail. +3. +It is both. It is a trail because it is a walk that doesn’t cover any edges twice, and it is an Euler trail because it visits all +the edges. +YOUR TURN 12.29 +Use the figure to determine if each sequence of vertices represents an Euler trail or not. If not, explain why. +Graph I +1. e → f → b → a → d → c → g → h → e → d +2. d → a → b → f → e → h → g → c +3. d → a → b → e → f → b → e → d → c → g → h → e +The Five Rooms Puzzle +Just as Euler determined that only graphs with vertices of even degree have Euler circuits, he also realized that the only +vertices of odd degree in a graph with an Euler trail are the starting and ending vertices. For example, in Figure 12.132, +Graph H has exactly two vertices of odd degree, vertex g and vertex e. Notice the Euler trail we saw in Excercise 3 of +Example 12.29 began at vertex g and ended at vertex e. +This is consistent with what we learned about vertices off odd degree when we were studying Euler circuits. We saw that +a vertex of odd degree couldn't exist in an Euler circuit as depicted in Figure 12.133. If it was a starting vertex, at some +point we would leave the vertex and not be able to return without repeating an edge. If it was not a starting vertex, at +some point we would return and not be able to leave without repeating an edge. Since the starting and ending vertices +in an Euler trail are not the same, the start is a vertex we want to leave without returning, and the end is a vertex we +want to return to and never leave. Those two vertices must have odd degree, but the others cannot. +Figure 12.133 A Vertex of Degree 3 +Let’s use the Euler trail theorem to solve a puzzle so you can amaze your friends! This puzzle is called the “Five Rooms +Puzzle.” Suppose that you were in a house with five rooms and the exterior. There is a doorway in every shared wall +between any two rooms and between any room and the exterior as shown in Figure 12.134. Could you find a route +through the house that passes through each doorway exactly once? +12.6 • Euler Trails +1327 + +Figure 12.134 Five Rooms Puzzle +Let’s represent the puzzle with a graph in which vertices are rooms (or the exterior) and an edge indicates a door +between two rooms as shown in Figure 12.135. +Figure 12.135 Graph of Five Rooms Puzzle +To pass through each doorway exactly once means that we cross every edge in the graph exactly once. Since we have +not been asked to start and end at the same position, but to visit each edge exactly once, we are looking for an Euler +trail. Let’s check the degrees of the vertices. +Figure 12.136 Degrees of Vertices in Five Rooms Puzzle +Since there are more than two vertices of odd degree as shown in Figure 12.136, the graph of the five rooms puzzle +contains no Euler path. Now you can amaze and astonish your friends! +Bridges and Local Bridges +Now that we know which graphs have Euler trails, let’s work on a method to find them. The method we will use involves +identifying bridges in our graphs. A bridge is an edge which, if removed, increases the number of components in a +graph. Bridges are often referred to as cut-edges. In Figure 12.137, there are several examples of bridges. Notice that an +edge that is not part of a cycle is always a bridge, and an edge that is part of a cycle is never a bridge. +Figure 12.137 Graph with Bridges +Edges bf, cg, and dg are “bridges” +The graph in Figure 12.137 is connected, which means it has exactly one component. Each time we remove one of the +bridges from the graph the number of components increases by one as shown in Figure 12.138. If we remove all three, +the resulting graph in Figure 12.138 has four components. +1328 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.138 Removing a Bridge Increases Number of Components +In sociology, bridges are a key part of social network analysis. Sociologists study two kinds of bridges: local bridges and +regular bridges. Regular bridges are defined the same in sociology as in graph theory, but they are unusual when +studying a large social network because it is very unlikely a group of individuals in a large social network has only one +link to the rest of the network. On the other hand, a local bridge occurs much more frequently. A local bridge is a +friendship between two individuals who have no other friends in common. If they lose touch, there is no single individual +who can pass information between them. In graph theory, a local bridge is an edge between two vertices, which, when +removed, increases the length of the shortest path between its vertices to more than two edges. In Figure 12.139, a local +bridge between vertices b and e has been removed. As a result, the shortest path between b and e is b → i → j → k → e, +which is four edges. On the other hand, if edge ab were removed, then there are still paths between a and b that cover +only two edges, like a → i → b. +Figure 12.139 Removing a Local Bridge +The significance of a local bridge in sociology is that it is the shortest communication route between two groups of +people. If the local bridge is removed, the flow of information from one group to another becomes more difficult. Let’s +say that vertex b is Brielle and vertex e is Ella. Now, Brielle is less likely to hear about things like job opportunities, that +Ella many know about. This is likely to impact Brielle as well as the friends of Brielle. +EXAMPLE 12.30 +Identifying Bridges and Local Bridges +Use the graph of a social network in Figure 12.140 to answer each question. +Figure 12.140 Graph of a Social Network +1. +Identify any bridges. +2. +If all bridges were removed, how many components would there be in the resulting graph? +3. +Identify one local bridge. +4. +For the local bridge you identified in part 3, identify the shortest path between the vertices of the local bridge if the +local bridge were removed. +12.6 • Euler Trails +1329 + +Solution +1. +The edges ku, gh, and hi are bridges. +2. +If the bridges were all removed, there would be four components in the resulting graph, {i}, {h}, {u, v, w, x}, and {a, +b, c, d, e, f, g, j, k, m, n, o, p, q, r, s, t} as shown in Figure 12.141. +Figure 12.141 Graph of Social Network without Bridges +3. +Three local bridges are dn, ef, and qt, among others. +4. +If dn were removed, the shortest path between d and n would be d → e → f → j → o → m → n. +YOUR TURN 12.30 +1. How many bridges and local bridges are in a complete graph with three or more vertices? Explain your +reasoning. +VIDEO +Bridges and Local Bridges in Graph Theory (https://openstax.org/r/bridges_local_bridges) +Finding an Euler Trail with Fleury’s Algorithm +Now that we are familiar with bridges, we can use a technique called Fleury’s algorithm, which is a series of steps, or +algorithm, used to find an Euler trail in any graph that has exactly two vertices of odd degree. Here are the steps +involved in applying Fleury’s algorithm. +Here are the steps involved in applying Fleury’s algorithm. +Step 1: Begin at either of the two vertices of odd degree. +Step 2: Remove an edge between the vertex and any adjacent vertex that is NOT a bridge, unless there is no other +choice, making a note of the edge you removed. Repeat this step until all edges are removed. +Step 3: Write out the Euler trail using the sequence of vertices and edges that you found. For example, if you removed +ab, bc, cd, de, and ef, in that order, then the Euler trail is a → b → c → d → e → f. +Figure 12.142 shows the steps to find an Euler trail in a graph using Fleury’s algorithm. +1330 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.142 Using Fleury’s Algorithm To Find Euler Trail +The Euler trail that was found in Figure 12.142 is t → v → w → u → t → w → y → x → v. +EXAMPLE 12.31 +Finding an Euler Trail with Fleury’s Algorithm +Use Fleury’s Algorithm to find an Euler trail for Graph J in Figure 12.143. +Figure 12.143 Graph J +Solution +Step 1: Choose one of the two vertices of odd degree, c or f, as your starting vertex. We will choose c. +Step 2: Remove edge ca, cb, or cd. None of these are cut edges so we can select any of the three. We will choose cb as +shown in Figure 12.144 to be the first edge removed. +Figure 12.144 Graph J with cb Removed +Repeat Step 2 The next choice is to remove edge ba, bd, or bf as shown in Figure 12.144, but bf is not an option since it +is a bridge. We will choose ba as shown in Figure 12.145 to be the second edge removed. +Figure 12.145 Graph J with cb and ba Removed +Repeat Step 2 for the third, fourth, fifth, sixth, and seventh edges. As shown in Figure 12.145, until we get to the seventh +edge there is only one option each time, ac, cd, db, and bf in that order. For the seventh edge, we must choose between +12.6 • Euler Trails +1331 + +fe and fg. Neither of these are bridges. We choose fe. Figure 12.146 shows that ac, cd, db, bf, and fe have been removed. +Figure 12.146 Graph J with Seven Edges Removed +Repeat Step 2 for the eight, ninth, tenth, and eleventh edges. As shown in Figure 12.146, there is only one option for +each of these edges, eh, hi, ig, and gf, in that order. +Step 3: Write out the Euler trail using the vertices in the sequence that the edges were removed. We removed cb, ba, ac, +cd, db, bf, fe, eh, hi, ig, and gf, in that order. The Euler trail is c → b → a → c → d → b → f → e → h → i → g → f. +TIP! To avoid errors, count the number of edges in your graph and make sure thatyour Euler trail represents that +number of edges. +YOUR TURN 12.31 +Use Graph L to fill in the blanks to complete the steps in Fleury’s algorithm. +Graph L +1. The two vertices that can be used as the starting vertex are ____ and s. +2. If sq is the first edge removed, the three options for the second edge to be removed are qr, ___, and ___; +however, ___ cannot be chosen because it is a ________________. +3. If qr is the second edge removed, the next four edges to be removed must be ___, ___, ___, and ___, in that +order. +4. After qn is removed, the three options for the next edge to be removed are no, ___, and ___, however, ___ +cannot be chosen because it is a _____________. +5. If no is the next edge removed, the last four edges removed will be ___, ___, ___, and ___, in that order. +6. The final Euler trail using the answers to parts 1 through 5 is _________________________. +In the previous section, we found Euler circuits using an algorithm that involved joining circuits together into one large +circuit. You can also use Fleury’s algorithm to find Euler circuits in any graph with vertices of all even degree. In that case, +you can start at any vertex that you would like to use. +Step 1: Begin at any vertex. +Step 2: Remove an edge between the vertex and any adjacent vertex that is NOT a bridge, unless there is no other +choice, making a note of the edge you removed. Repeat this step until all edges are removed. +Step 3: Write out the Euler circuit using the sequence of vertices and edges that you found. For example, if you removed +ab, bc, cd, de, and ea, in that order, then the Euler circuit is a → b → c → d → e → a. +VIDEO +Fluery's Algorithm to Find an Euler Circuit (https://openstax.org/r/Fleurys_algorithm) +IMPORTANT! Since a circuit is a closed trail, every Euler circuit is also an Euler trail, but when we say Euler trail in this +1332 +12 • Graph Theory +Access for free at openstax.org + +chapter, we are referring to an open Euler trail that begins and ends at different vertices. +EXAMPLE 12.32 +Finding an Euler Circuit or Euler Trail Using Fleury’s Algorithm +Use Fleury’s algorithm to find either an Euler circuit or Euler trail in Graph G in Figure 12.147. +Figure 12.147 Graph G +Solution +Graph G has all vertices of even degree so it has an Euler circuit. +Step 1: Choose any vertex. We will choose vertex j. +Step 2: Remove one of the four edges that meet at vertex j. Since jn is a bridge, we must remove either jh, ji, or jk. We +remove ji as shown in Figure 12.148. +Figure 12.148 Graph G with 1 Edge Removed +Repeat Step 2: Since id is a bridge, we can remove either ih or ik next. We remove ih, and then the only option is to +remove hj as shown in Figure 12.149. +Figure 12.149 Graph G with 3 Edges Removed +Repeat Step 2: Since jn is a bridge, the next edge removed must be jk, and then the only option is to remove ki followed +by id as shown in Figure 12.149. Even though id is a bridge, it can be removed because it is the only option at this point. +Figure 12.150 shows Graph G with these additional edges removed. +Figure 12.150 Graph G with 6 Edges Removed +12.6 • Euler Trails +1333 + +Repeat Step 2: Choose any one of the edges db, dc, or de. We remove dc as shown in Figure 12.151. +Figure 12.151 Graph G with 7 Edges Removed +Repeat Step 2: Since co is a bridge, choose cb next. We remove cb, then bd, and then de as shown in Figure 12.152. +Figure 12.152 Graph G with 10 Edges Removed +Repeat Step 2: Next, remove ec and co. Then choose any of op, pn, or om. We remove on as shown in Figure 12.153. +Figure 12.153 Graph G with 13 Edges Removed +Repeat Step 2: Next, remove either nm, np, or nj, but nj is a So, we remove nm as shown in Figure 12.154. +Figure 12.154 Graph G with 14 Edges Removed +Repeat Step 2: Next, remove mo, op, pn, and nj. And we are done! +Step 3: Notice that the algorithm brought us back to the vertex where we started, forming an Euler circuit. Write out the +Euler circuit: +j → i → h → j → k → i → d → c → b → d → e → c → o → n → m → o → p → n → j +YOUR TURN 12.32 +1. Find an Euler circuit or trail through the graph using Fleury’s algorithm. +1334 +12 • Graph Theory +Access for free at openstax.org + +Graph T +WORK IT OUT +We have discussed a lot of subtle concepts in this section. Let’s make sure we are all on the same page. Work with a +partner to explain why each of the following facts about bridges are true. Support your explanations with definitions +and graphs. +1. +When a bridge is removed from a graph, the number of components increases. +2. +A bridge is never part of a circuit. +3. +An edge that is part of a triangle is never a local bridge. +Check Your Understanding +Fill in the blank to make the statement true. +51. An Euler trail is a trail that visits each ___________ exactly once. +52. __________ algorithm is a procedure for finding an Euler trail or circuit. +53. An Euler _____ always begins and ends at the same vertex, but an Euler _____ does not. +54. When a bridge is removed from a graph, the number of ________ is increased by one. +55. When a __________ is removed from a graph, the shortest path between its vertices will be greater than two. +56. When using Fleury’s algorithm to find an Euler trail, never remove a _________ unless it is the only option. +SECTION 12.6 EXERCISES +Use the figure to answer the following exercises. Identify the graph(s) with the given characteristics, if any. +1. Connected +2. All vertices of even degree +3. Exactly two vertices of odd degree +4. Has an Euler trail +5. Has an Euler circuit +6. Has neither an Euler trail nor an Euler circuit +7. ab is a bridge +8. ef is a bridge +9. ab is a local bridge +10. ef is a local bridge +Use the figure to answer the following exercises. In each exercise, a graph and a sequence of vertices are given. +Determine whether each sequence of vertices is an Euler trail, an Euler circuit, or neither for the graph. If it is neither, +explain why. +12.6 • Euler Trails +1335 + +11. Graph A, w → x → y → z → w → u → t → s → v → u +12. Graph A, u → v → s → t → u → w → z → y → x → w +13. Graph A, s → t → u → v → u → w → z → y → x → w +14. Graph A, w → x → y → z → w → v → u → t → s → v +15. Graph B, u → v → w → x → r → u → t → s → y → z → u +16. Graph B, v → w → x → r → u → z → y → s → t → u +17. Graph C, s → t → u → v → w → x → s +18. Graph C, t → u → x → w → u → s → t → v → w +19. Graph D, t → r → s → t → u → v → t → x → v → w → x → y → z → x +20. Graph D, x → v → w → x → y → z → x → t → r → s → t → u → v → t +Use the figure to answer the following exercises. For each graph, identify a bridge if one exists. If it does not, state so. If +it does, identify any components that are created when the bridge is removed. +21. Graph A +22. Graph B +23. Graph C +24. Graph D +Use the figuree to answer the following exercises. For each graph, identify a local bridge if one exists. If it does not, +state so. If it does, find a shortest path between the vertices of the local bridge if the local bridge is removed. +25. Graph A +26. Graph B +27. Graph C +28. Graph D +Use the graphs to answer the following exercises. In each exercise, a graph is given. Find two Euler trails in each graph +using Fleury’s algorithm. +29. Graph Q +1336 +12 • Graph Theory +Access for free at openstax.org + +30. Graph R +31. Graph S +32. In chess, a knight can move in any direction, but it must move two spaces then turn and move one more space. +The eight possible moves a knight can make from a given space are shown in the figure. +A graph in which each vertex represents a space on a five-by-six game board and each edge represents a move +a knight could make is shown in the figure. +A knight’s tour is a sequence of moves by a knight on a chessboard (of any size) such that the knight visits every +square exactly once. If the knight’s tour brings the knight back to its starting position on the board, it is called a +closed knight’s tour. Otherwise, it is called an open knight’s tour. Determine if the closed knight’s tour in the +figure is most accurately described as a trail, a circuit, an Euler trail, or an Euler circuit of the graph of all +possible knight moves. Explain your reasoning. +33. Determine if the open knight’s tour in the figure is most accurately described as a trail, a circuit, an Euler trail, or +an Euler circuit of the graph of all possible knight moves on a five-by-five game board. Explain your reasoning. +12.6 • Euler Trails +1337 + +34. The neighborhood of Pines West has three cul-de-sacs that meet at an intersection as shown in the figure. A postal +delivery person starts at the intersection and visits each house in a cul-de-sac once, returns to the intersection, +visits each house in the next cul-de-sac, and so on, returning to the intersection when finished. Describe how the +route can be represented as a graph. If there is no backtracking, in other words, the person never reverses +direction, is the route followed by the postal delivery person best described as a trail, a circuit, an Euler trail, or an +Euler circuit? Explain your reasoning. +35. Recall that the bridges of Konigsberg can be represented as a multigraph as shown in the figure. We have seen +that no route through Konigsberg passes over each bridge exactly once and returns to the starting point. Is there +a route that passes over each bridge exactly once but does not begin and end at the same point? Explain your +reasoning. +36. The figure shows the map of the exhibits at an indoor aquarium. Use a graph in which the edges represent +hallways and the vertices represent turns and intersections to explain why a visitor to the aquarium cannot start at +one of the turns or intersections, passes by every exhibit exactly once, and end at one of the turns or intersections. +1338 +12 • Graph Theory +Access for free at openstax.org + +37. The map of the states of Imaginaria is given. Use a graph to determine if it is possible to begin in one state, travel +through Imaginaria crossing the border between each pair of states exactly once, and end in a different state. If it +is possible, find such a route. If it is not, explain why. +12.6 • Euler Trails +1339 + +12.7 Hamilton Cycles +Figure 12.155 The symmetries of an icosahedron, with 30 edges, 20 faces, and 12 vertices, can be analyzed using graph +theory. (credit: "Really big icosahedron" by Clayton Shonkwiler/Wikimedia, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Describe and identify Hamilton cycles. +2. +Compute the number of Hamilton cycles in a complete graph. +3. +Apply and evaluate weighted graphs. +In Euler Circuits and Euler Trails, we looked for circuits and paths that visited each edge of a graph exactly once. In this +section, we will look for circuits that visit each vertex exactly once. Like many concepts in graph theory, the idea of a +circuit that visits each vertex once was inspired by games and puzzles. As early as the 9th century, Indian and Islamic +intellectuals wondered whether it was possible for a knight to visit every space on a chess board of a given size, which is +equivalent to visiting every vertex of a graph that represents the chess board. +In 1857, a mathematician named William Rowan Hamilton invented a puzzle in which players were asked to find a route +along the edges of a dodecahedron (see Figure 12.156), which visited every vertex exactly once. Let’s explore how graph +theory provides insight into these games as well as practical applications such as the Traveling Salesperson Problem. +Figure 12.156 A Dodecahedron +Hamilton’s Puzzle +Before we look at the solution to Hamilton's puzzle, let’s review some vocabulary we used in Figure 12.157. It will be +helpful to remember that directed cycle is a type of circuit that doesn’t repeat any edges or vertices. +1340 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.157 Closed Walks, Circuits, and Directed Cycles +The goal of Hamilton's puzzle was to find a route along the edges of the dodecahedron, which visits each vertex exactly +once. A dodecahedron is a three-dimensional space figure with faces that are all pentagons as we saw in Figure 12.156. +Since it is easier to visualize two dimensions rather than three, we will flatten out the dodecahedron and look at the +edges and vertices on a flat surface. Graph A in Figure 12.158 shows a two-dimensional graph of the edges and vertices, +and Graph B shows an untangled version of Graph A in which no edges are crossing. Graph B in Figure 12.158 is very +similar to the design of the game board that Hamilton invented for his puzzle. +Untangle graph +Figure 12.158 Graph of Edges and Vertices of Dodecahedron +We can see that this is a planar graph because it can be “untangled.” In order to solve Hamilton’s puzzle, we need to find +a circuit that visits every vertex once. A solution is shown in Figure 12.159. +In three Dimensions +> +Figure 12.159 A Solution to Hamilton’s Puzzle +A circuit that doesn’t repeat any vertices, like the one in Figure 12.159, is called a directed cycle. So, we can most +accurately say that Hamilton’s puzzle asks us to find a directed cycle that visits every vertex in a graph exactly once. +Because Hamilton created and solved this puzzle, these special circuits were named Hamilton cycles, or Hamilton +12.7 • Hamilton Cycles +1341 + +circuits. +WHO KNEW? +Icosian Game +When Hamilton invented his puzzle in the 19th century, it was originally called the Icosian game. This was a reference +to an icosahedron, not a dodecahedron. Why? Hamilton was actually interested in the symmetries of icosahedrons. It +turns out that these two space figures are related to each other. The dodecahedron has 30 edges, 20 faces, and 12 +vertices, while the icosahedron has 30 edges, 12 faces, and 20 vertices. By relating the vertices of the dodecahedron +to the faces of the icosahedron, Hamilton was able to make the mathematical connections necessary to use graph +theory and dodecahedrons to make discoveries about the symmetries of icosahedrons. +Hamilton Cycles vs. Euler Circuits +Let’s practice naming and identifying Hamilton cycles, as well as distinguishing them from Euler circuits. It is important +to remember that Euler circuits visit all edges without repetition, while Hamilton cycles visit all vertices without +repetition. Hamilton cycles are named by their vertices just like all circuits. An example is given in Figure 12.160. +Figure 12.160 Hamilton Cycle in Graph Z +Notice that the Hamilton cycle a → b → c → d for Graph Z in Figure 12.160 is NOT an Euler circuit, because it does not visit +edge +. Some Hamilton cycles are also Euler circuits while some are not, and some Euler circuits are Hamilton cycles +while some are not. +TIP! Sometimes students confuse Euler circuits with Hamilton cycles. To help you remember, think of the E in Euler +as standing for Edge. +EXAMPLE 12.33 +Differentiating between Hamilton Cycles or Euler Circuits +Use Figure 12.161 to determine whether the given circuit is a Hamilton cycle, an Euler circuit, both, or neither. +Figure 12.161 Graph Q +1. +a → b → c → e → h → g → f → d → a +2. +g → e → h → g → f → d → a → b → d → g +3. +a → b → c → e → h → g → f → d → b → e → g → d → a +1342 +12 • Graph Theory +Access for free at openstax.org + +Solution +1. +This circuit is a Hamilton cycle only. It visits each vertex exactly once, so, it is a Hamilton cycle. It is not an Euler +circuit because it doesn’t visit all of the edges. +2. +This circuit is neither Hamilton cycle nor an Euler circuit. It doesn’t visit vertex , so, it is not a Hamilton cycle. It also +doesn’t visit edges +and +, so, it is not an Euler circuit. +3. +This circuit is an Euler circuit only. It visits several vertices more than once; so, it is not a Hamilton cycle. It visits +every edge exactly once, so, it is an Euler circuit. +TIP! Euler circuits never skip a vertex, so, they only fail to be Hamilton cycles when they visit a vertex more than +once. Hamilton cycles never visit any edges twice, so, they only fail to be Euler circuits when they skip edges. +YOUR TURN 12.33 +1. Consider the figure shown. Is the circuit a → b → f → e → h → i → g → d → c → a an Euler circuit, a Hamilton cycle, +or both? +Graph T +Notice that the graph is a cycle. A cycle will always be Eulerian because all vertices are degree 2. Moreover, any circuit in +the graph will always be both an Euler circuit and a Hamilton cycle. It is not always as easy to determine if a graph has a +Hamilton cycle as it is to see that it has an Euler circuit, but there is a large group of graphs that we know will always +have Hamilton cycles, the complete graphs. Since all vertices in a complete graph are adjacent, we can always find a +directed cycle that visits all the vertices. For example, look at the directed six-cycle, n → o → p → q → r → s, in the +complete graph with six vertices in Figure 12.162. +Figure 12.162 Directed Cycle in Complete Graph +That is not the only directed six-cycle in the graph though. We could find another just be reversing the direction, and we +could find even more by using different edges. So, how many Hamilton cycles are in a complete graph with +vertices? +Before we tackle this problem, let’s look at a shorthand notation that we use in mathematics which will be helpful to us. +Factorials +In many areas of mathematics, we must calculate products like +or +, +products that involve multiplying all the counting numbers from a particular number down to 1. Imagine that the +product happened to be all the numbers from 100 down to 1. That’s a lot of writing! Instead of writing all of that out, +mathematicians came up with a shorthand notation. For example, instead of +, we write +, which is +read “7 factorial.” In other words, the product of all the counting numbers from +down to 1 is called +factorial and it is +12.7 • Hamilton Cycles +1343 + +written +EXAMPLE 12.34 +Evaluating Factorials +Evaluate +and +for +. +Solution +and +YOUR TURN 12.34 +1. Evaluate +and +for +. +A common use for factorials is counting the number of ways to arrange objects. Suppose that there were three students, +Aryana, Byron, and Carlos, who wanted to line up in a row. How many arrangements are possible? There are six +possibilities: ABC, ACB, BAC, BCA, CAB, or CBA. Notice that there were three students being arranged, and the number of +possible arrangements is three. +FORMULA +The number of ways to arrange +distinct objects is +. +EXAMPLE 12.35 +Counting Arrangements of Letters +Find the number of ways to arrange the letters a, b, c, and d. +Solution +YOUR TURN 12.35 +1. Find the number of ways to arrange the letters v, w, x, y, and z. +Counting Hamilton Cycles in Complete Graphs +Now, let’s get back to answering the question of how many Hamilton cycles are in a complete graph. In Table 12.8, we +have drawn all the four cycles in a complete graph with four vertices. Remember, cycles can be named starting with any +vertex in the cycle, but we will name them starting with vertex +. +1344 +12 • Graph Theory +Access for free at openstax.org + +Complete Graph +Cycle +Cycle +Cycle +Cycle Name Clockwise +(a, b, c, d) +(a, b, d, c) +(a, c, b, d) +Cycle Name Counterclockwise +(a, d, c, b) +(a, c, d, b) +(a, d, b, c) +Table 12.8 Four-Cycles in Complete Graph with Four Vertices +Table 12.8 shows that there are three unique four-cycles in a complete graph with four vertices. Notice that there were +two ways to name each cycle, one reading the vertices in a clockwise direction and one reading the vertices in a +counterclockwise direction. This is important to us because we are interested in Hamilton cycles, which are directed +cycles. Although the cycles (a, b, c, d) and (a, d, c, b) are the same cycle, the directed cycles, a → b → c → d → a and a → d +→ c → b → a, which travel the same route in reverse order are considered different directed cycles, as shown in Table +12.9. +Complete Graph +Cycle +Cycle +Cycle +Clockwise Hamilton Cycle +a → b → c → d → a +a → b → d → c → a +a → c → b → d → a +Counter-clockwise Hamilton Cycle +a → d → c → b → a +a → c → d → b → a +a → d → b → c → a +Table 12.9 Hamilton Cycles in a Complete Graph with Four Vertices +The six directed four-cycles in Table 12.9 are the only distinct Hamilton cycles in a complete graph with four vertices. Six +is also the number of ways to arrange the three letters b, c, and d. (Do you see why?) The number of ways to arrange +three letters is +. Similarly, the number of Hamilton cycles in a graph with five vertices is the number of +ways to arrange four letters, which is +. In general, to find the number of Hamilton cycles in a graph, +we take one less than the number of vertices and find its factorial. +12.7 • Hamilton Cycles +1345 + +FORMULA +The number of distinct Hamilton cycles in a complete graph with +vertices is +EXAMPLE 12.36 +Counting Hamilton Cycles in a Complete Graph +How many Hamilton cycles are in the complete graph in Figure 12.163? +Figure 12.163 Complete Graph L +Solution +There are five vertices in the graph. Using +, we have +Hamilton cycles. +YOUR TURN 12.36 +1. How many Hamilton cycles are in the graph? +Complete Graph R +Weighted Graphs +Suppose that an officer in the U.S. Air Force who is stationed at Vandenberg Air Force base must drive to visit three other +California Air Force bases before returning to Vandenberg. The officer needs to visit each base once. The vertices in the +graph in Figure 12.164 represent the four U.S. Air Force bases, Vandenberg, Edwards, Los Angeles, and Beale. The edges +are labeled to with the driving distance between each pair of cities. +Figure 12.164 Graph of Four California Air Force Bases +The graph in Figure 12.164 is called a weighted graph, because each edge has been assigned a value or weight. The +weights can represent quantities such as time, distance, money, or any quantity associated with the adjacent vertices +joined by the edges. The total weight of any walk, trail, or path is the sum of the weights of the edges it visits. +1346 +12 • Graph Theory +Access for free at openstax.org + +Notice that the officer’s trip can be represented as a Hamilton cycle, because each of the four vertices in the graph is +visited exactly once. +EXAMPLE 12.37 +Finding Hamilton Cycles of Lowest Weight +Use Figure 12.164 and the given Hamilton cycles to answer the following questions. +V → L → E → B → V +V → L → B → E → V +V → E → L → B → V +V → B → E → L → V +1. +Which of the Hamilton cycles (directed cycles) lie on the same cycle (undirected cycle) in the graph? +2. +Find the total weight of each cycle. +3. +Of the four, which of the Hamilton cycles describes the shortest trip for the officer? Describe the route. +Solution +1. +V → L → E → B → V and V → B → E → L → V follow the same edges in reverse order. +2. +Any Hamilton cycles that lie on the same cycle will have the same edges and the same total weight. +V → L → E → B → V and V → B → E → L → V each have total weight +. +V → L → B → E → V has a total weight +. +V → E → L → B → V has a total weight +. +3. +Hamilton cycles V → L → E → B → V and V → B → E → L → V each have the lowest total weight. The officer would take +the route from Vandenberg, to Los Angeles, to Edwards, to Beale, and back to Vandenberg, or reverse that route. +YOUR TURN 12.37 +1. Find the weight of the Hamilton cycle m → o → p → n → q → m in the given figure. +Weighted Complete Graph with Five Vertices +Check Your Understanding +For the following exercises, fill in the blank to make the statement true. +57. A Hamilton cycle is a circuit that visits each ___________ exactly once. +58. A __________ graph with +vertices has +Hamilton cycles. +For the following exercises, fill in the blank with is or is not to make the statement true. +59. A Hamilton cycle ________ a circuit. +60. A Hamilton cycle that visits every edge ________ an Euler circuit. +61. A Hamilton cycle ________ different from a Hamilton circuit. +62. An Euler circuit that visits every vertex ________ a Hamilton cycle. +63. The total weight of a trail _______ the sum of the weights of the edges visited by the trail. +64. A weighted graph _______ always a complete graph. +65. The number of ways to arrange n objects _______ +66. Every cycle ____ a circuit. +12.7 • Hamilton Cycles +1347 + +SECTION 12.7 EXERCISES +For the following exercises, use the figure to determine whether the sequence of vertices in the given graph is a +Hamilton cycle, an Euler circuit, both, or neither. +1. Graph A: f → b → g → e → d → c → f +2. Graph A: g → b → f → c → d → e → g +3. Graph A: d → e → g → d → f → b → g → f → c → d +4. Graph L: h → i → k → n → j → h +5. Graph L: n → i → h → j → m → k → n +6. Graph L: j → i → n → k → i → j → k → m → j +7. Graph U: v → w → r → s → t → o → q → v +8. Graph U: w → q → r → s → t → o → v → w +For the following exercises, use the figure to find a circuit that fits the description. +9. A Hamilton cycle in Graph P that begins at vertex c. +10. An Euler circuit in Graph P that begins at vertex c. +11. A directed cycle in Graph P that is NOT a Hamilton cycle, and explain why it is not a Hamilton cycle. +12. A directed circuit in Graph P that is NOT an Euler circuit, and explain why it is not an Euler circuit. +13. A Hamilton cycle in Graph Q that begins at vertex n. +14. An Euler circuit in Graph Q that begins at vertex n. +15. A directed cycle in Graph Q that is NOT a Hamilton cycle, and explain why it is not a Hamilton cycle. +16. A directed circuit in Graph Q that is NOT an Euler circuit, and explain why it is not an Euler circuit. +17. Use the results of Exercises 9–16 to make an observation about which tends to involve a longer sequence of +vertices, Hamilton cycles or Euler circuits. Explain why you think this is. +For the following exercises, evaluate the factorial expression for the given value of +. +18. +19. +20. +21. +For the following exercises, find the number of arrangements letters in the given word. +22. have +23. teamwork +24. anime +25. making +For the following exercises, find the number of Hamilton cycles in a complete graph with the given number of vertices. +26. 12 vertices +27. 13 vertices +28. 9 vertices +1348 +12 • Graph Theory +Access for free at openstax.org + +29. 8 vertices +30. +vertices +For the following exercises, given the number of Hamilton cycles in a complete graph, determine the number of +vertices. +31. +Hamilton cycles +32. +Hamilton cycles +33. +Hamilton cycles +34. +Hamilton cycles +For the following exercises, all the distinct Hamilton cycles for a complete graph are given. Indicate which pairs of +Hamilton cycles (directed cycles) lie on the same cycle (undirected cycle) in the graph. +35. +1. +b → a → c → d → b +2. +b → a → d → c → b +3. +b → c → a → d → b +4. +b → c → d → a → b +5. +b → d → a → c → b +6. +b → d → c → a → b +36. +1. +i → f → g → h → e → i +2. +i → f → g → e → h → i +3. +i → f → h → g → e → i +4. +i → f → h → e → g → i +5. +i → f → e → g → h → i +6. +i → f → e → h → g → i +7. +i → g → f → h → e → i +8. +i → g → f → e → h → i +9. +i → g → h → f → e → i +10. +i → g → h → e → f → i +11. +i → g → e → f → h → i +12. +i → g → e → h → f → i +13. +i → h → g → f → e → i +14. +i → h → g → e → f → i +15. +i → h → f → g → e → i +16. +i → h → f → e → g → i +17. +i → h → e → g → f → i +18. +i → h → e → f → g → i +19. +i → e → g → h → f → i +20. +i → e → g → f → h → i +21. +i → e → h → g → f → i +22. +i → e → h → f → g → i +23. +i → e → f → g → h → i +24. +i → e → f → h → g → i +For the following exercises, use the figure to find the weight of the given Hamilton cycle. +37. q → r → s → v → y → x → w → t → u → q +38. u → y → x → w → t → q → r → s → v → u +39. y → v → s → r → u → q → t → w → x → y +40. u → v → s → r → q → t → w → x → y → u +12.7 • Hamilton Cycles +1349 + +41. The neighborhood of Pines West has three cul-de-sacs that meet at an intersection as shown. A postal delivery +person starts at the intersection and visits each house in a cul-de-sac once, returns to the intersection, visits each +house in the next cul-de-sac, and so on, returning to the intersection when finished. Describe how the route can be +represented as a graph. If there is no backtracking, in other words, the person never reverses direction, is the +route followed by the postal delivery person best described as a trail, an Euler circuit, a Hamilton cycle, or neither? +Explain your reasoning. +42. In chess, a knight can move in any direction, but it must move two spaces then turn and move one more space. +The 8 possible moves a knight can make from a given space are shown in the figure. +A graph in which each vertex represents a space on a five-by-six game board and each edge represents a move a +knight could make is shown in the figure. +A knight’s tour is a sequence of moves by a knight on a chessboard (of any size) such that the knight visits every +square exactly once. If the knight’s tour brings the knight back to its starting position on the board, it is called a +closed knight’s tour. Otherwise, it is called an open knight’s tour. +1350 +12 • Graph Theory +Access for free at openstax.org + +Determine if the closed knight’s tour in Figure 12.239 is most accurately described as an Euler circuit or a Hamilton +cycle, or both, of the graph of all possible knight moves. Explain your reasoning. +12.8 Hamilton Paths +Figure 12.165 A school bus picks up children along a planned route. (credit: “Kids at School Bus Stop” by Ty Hatch/Flickr, +CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Describe and identify Hamilton paths. +2. +Evaluate Hamilton paths in real-world applications. +3. +Distinguish between Hamilton paths and Euler trails. +In the United States, school buses carry 25 million children between school and home every day. The total distance they +travel is around 6 billion kilometers per year. In the city of Boston, Massachusetts, the 2016 budget for running those +buses was $120 million dollars. In 2017, the city held a competition to find ways to cut costs and the Quantum Team +from the MIT Operations Research Center came to the rescue, using a computer algorithm to identify the most efficient +and least costly routes, which saved the city of Boston $5 million each year and even reduced daily CO2 emissions by +9,000 kilograms! (This U.S. city put an algorithm in charge of its school bus routes and saved $5 million, Sean Fleming, +World Economic Forum) +The problem the Quantum Team tackled involves graph theory. Imagine a graph in which vertices are the bus depot, the +school, and the bus stops along a particular route. The bus must start at the depot, visit every stop exactly once, and end +at the school. The route is a special kind of path that visits every vertex exactly once. Can you guess what those paths are +called? +12.8 • Hamilton Paths +1351 + +Hamilton Paths +Just as circuits that visit each vertex in a graph exactly once are called Hamilton cycles (or Hamilton circuits), paths that +visit each vertex on a graph exactly once are called Hamilton paths. As we explore Hamilton paths, you might find it +helpful to refresh your memory about the relationships between walks, trails, and paths by looking at Figure 12.166. We +know that paths are walks that don’t repeat any vertices or edges. So, a Hamilton path visits every vertex without +repeating any vertices or edges. Figure 12.167 shows a path from vertex A to vertex E and a Hamilton path from vertex A +to vertex E. +Figure 12.166 Walks, Trails, and Paths +Figure 12.167 Path or Hamilton Path? +EXAMPLE 12.38 +Identifying Hamilton Paths +Which of the following sequences of vertices is a Hamilton path for Graph Q in Figure 12.168? +Figure 12.168 Graph Q +1. +a → d → b → c → e → g → f +2. +c → b → e → h → g → f → d → a +3. +h → e → g → d → b → e → g → f → d → a → b → c +Solution +Sequence 1 is a path, because it is a walk that doesn’t repeat any vertices or edges, but not a Hamilton path because it +skips vertex h. Sequence 2 is a path that visits each vertex exactly once; so, it is a Hamilton path. Sequence 3 is a walk, +but it is not a path because it visits vertices g, e, and b each more than once; so, it cannot be a Hamilton path. So, we can +1352 +12 • Graph Theory +Access for free at openstax.org + +see that only sequence 2 is a Hamilton path. +YOUR TURN 12.38 +Use the graphs to determine if the given sequence of vertices represents a Hamilton path or not. +Graphs A and K +1. Graph A, b → c → d → e → f +2. Graph A, b → e → f → c → b → d +3. Graph K, n → q → o → p → m +TIP! Since a Hamilton path visits each vertex exactly once, it must have the same number of vertices listed as appear +in the graph. +Finding Hamilton Paths +Suppose you were visiting an aquarium with some friends. The map of the aquarium is given in Figure 12.169. The +letters represent the exhibits. +Figure 12.169 Map of Aquarium Exhibits +Figure 12.170 shows a graph of the aquarium in which each vertex represents an exhibit and each edge is a route +12.8 • Hamilton Paths +1353 + +between the pair of exhibits that doesn’t bypass another exhibit. +Figure 12.170 Graph of Aquarium +Let’s see if we can plan a tour of the exhibits that visits each exhibit exactly once, beginning at exhibit O and ending at +exhibit C. Suppose that, after exhibit O, we plan to visit exhibit Q and then exhibit M. After M, should we plan to visit N, L, +or R? Take a look at Figure 12.171. If R is not chosen next, that will cause a problem later on. Do you see what it is? +Figure 12.171 Choosing Vertex L, N, or R +If L or N is chosen next, the only way to get to R later will be to go from S to R, and then we will not be able to continue +without repeating a vertex. So, we will pick R next, and then the only option is S. After S we have another choice to make. +As shown in Figure 12.172, the next choice is between B and E. Keeping in mind that the goal is to end at C, which would +be the better choice? +Figure 12.172 Choosing Vertex B or E +If you said vertex B, you are right! Otherwise, we will not be able to visit B later. After B, the only option is E. Then we can +choose either D or G. Either will work fine. Let’s choose G as shown in Figure 12.173. After G, you must visit H, but should +you visit K or L after that? +1354 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.173 Choose Vertex L or K +If you said to go to vertex L next, you are right! Otherwise, it will be impossible to visit N without repeating a vertex. So, +next is L, then N, then K, and then at J you have another decision to make we can see in Figure 12.174. Should you +choose F, I, or P next? +Figure 12.174 Choose Vertex F, I, or P +If you said P, you are right! If you choose either of the other two vertices, you will not be able to visit P later without +passing through another vertex twice. Once P is chosen, vertex I must be next followed by F. Then you have to choose +between A and D as shown in Figure 12.175. +Figure 12.175 Choose Vertex A or D +In this case, we must go to D then to A so that we can visit C without backtracking. The complete Hamilton path is shown +in Figure 12.176. +12.8 • Hamilton Paths +1355 + +Figure 12.176 Complete Hamilton Path from O to C +So, one Hamilton path that begins at O and ends at C is O → Q → M → R → S → B → E → G → H → L → N → K → J → P → I → F +→ D → A → C. +There is no set sequence of steps that can be used to find a Hamilton path if it exists, but it does help to keep in mind +where we are headed and avoid choices that will make returning to a particular vertex impossible without repeating +vertices. Let’s practice finding Hamilton paths. +EXAMPLE 12.39 +Finding a Hamilton Path +Use Figure 12.177 to find a Hamilton path between vertices C and D. +Figure 12.177 Graph G +Solution +If we start at vertex C, A must be next. Then we must choose between B and F. If we choose F, we will have to backtrack +to get to include B; so, we must choose B. Once we choose B, we must choose F next. After F, we choose E, because we +want to end at D. So, a Hamilton path between C and D is C → A → B → F → E → D. +YOUR TURN 12.39 +1. Use Figure 12.253 to find a Hamilton path between vertices C and E. +Existence of a Hamilton Path +It turns out that there is no Hamilton path between vertices A and E in Graph G in Figure 12.177. To understand why, let’s +imagine there is a red apple tree on one side of a bridge and a green apple tree on the other side of the bridge. Now +suppose someone asked you to pick up all the fallen apples under each tree without crossing the bridge more than once, +and making sure that the first apple you pick up and the last apple you pick up are both red. You would say, that is +impossible! To have the first and last apple be red would either require leaving the green apples on the ground or +crossing the bridge twice. +Let’s see how this relates to finding a Hamilton path between A and E in Graph G. The edge AC is a bridge because, if it +were removed, the graph would become disconnected with two components, the component {C} and the component {A, +B, D, E, F}. So, we can think of the vertices A, B, D, E, and F as the red apples, vertex C as the green apple, and the edge +AC is the bridge between them as in Figure 12.178. +1356 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.178 Bridge between Red and Green Apples +The creation of a Hamilton path requires a visit to each vertex, just as picking up all the apples requires a visit to each +apple. A and E are both red apples; so, a path from A to E would both start and end at a red apple, just as you were asked +to do. And you wouldn’t be able to cross the bridge twice because that would mean visiting A twice, which is not allowed +in a Hamilton path. So, it is impossible to find a Hamilton path from A to E just as it was impossible to pick up all the +apples without crossing the bridge more than once. By the same reasoning, if a graph has a bridge, there will never be a +Hamilton path that begins and ends on the same side of that bridge, meaning beginning and ending at vertices that +would be in the same component if the bridge were removed from the graph. +EXAMPLE 12.40 +Finding a Hamilton Path If One Exists +Find a Hamilton path from vertex s to vertex v for each graph in Figure 12.179 or indicate that there is none. +Figure 12.179 Graphs A, B, C, and D +Solution +Graph A: Edge uw is a bridge connecting component {s, t, u, v} to the component {w, x, y, z}. There is no Hamilton path +from vertex s to vertex v because they would be part of the same component if the bridge uw were removed. +Graph B: There are no bridges in Graph B. The only method we have to determine if a Hamilton path from vertex s to +vertex v exists is to try every possibility. From vertex s, we can visit either vertex y or vertex t. We will try vertex y first and +then come back to see what happens with vertex t. After visiting y, we must visit z and then u, but then we have to +decide between vertices r, t, and v next as shown in Figure 12.180. +Figure 12.180 Choose Vertex r, t, or v +Vertex v is not an option since we want to end at v. Vertex t is not an option since that would force us to go to visit s a +second time. So, we must go to vertex r next. After vertex r, we must visit x, then w, then v, but we missed vertex t as +shown in Figure 12.181. +12.8 • Hamilton Paths +1357 + +Figure 12.181 Missed Vertex t +Let’s go back to the beginning and choose t instead of y. After t, we must go to u and then we have a choice to make +between r, v, and z as shown in Figure 12.182. +Figure 12.182 Choose Vertex v, r, or z +Vertex v is not an option since we want to end at v. Vertex z is not an option since that would force us to go y and then to +visit s a second time. So, we must go to vertex r next. After r, we must go to x then w then v, where we have to stop even +though we have missed vertices y and z, as shown in Figure 12.183. +Figure 12.183 Missed Vertices y and z +So, we have tried every possible route and there are no Hamilton paths between s and v in Graph B. +Graph C: In Graph C, there is a Hamilton path, s → t → u → x → w → v. +Graph D: In Graph D, there is a bridge, tx, which would form components {r, s, t, u, q} and {v, w, x, y, z} if it were +removed. Since s and v would be in different components, it is possible there is a Hamilton path between them. The only +way to know is to try all possibilities. If we begin at s, we can go to r then t, or we can go directly to t, either way, we have +a problem as you can see in Figure 12.184. +Figure 12.184 Vertices Visited Twice or Skipped +If we visit all the vertices in the component {r, s, t, u, q}, we will have to visit t a second time in order to cross the bridge. +If we visit t only once, we have to skip some of the vertices. So, there is no Hamilton path between s and v. +YOUR TURN 12.40 +Use Graph L to find a Hamilton path between each pair of vertices or indicate that there isn’t one. +1358 +12 • Graph Theory +Access for free at openstax.org + +Graph L +1. p to r +2. m to p +3. o to q +There is not a short way to determine if there is a Hamilton path between two vertices on a graph that works in every +situation. However, there are a few common situations that can help us to quickly determine that there is no Hamilton +path. Some of these are listed in Table 12.10. +Scenario +Diagram +Scenario 1 If an edge ab is a bridge, then there is no Hamilton path between a +pair of vertices that are on the same side of edge ab. We saw this in Graph A +of Example 12.40. +No Hamilton path between any two +vertices in the component +{a, c, d, f}. +No Hamilton path between any two +vertices in {b, e, h, g, i}. +Scenario 2 If an edge ab is a bridge with at least three components on each +side, then there is no Hamilton path beginning or ending at a or b. We saw +this in Graph D of Example 12.40. +No Hamilton path beginning or +ending at a or b. +Scenario 3 If a graph is composed of two cycles joined only at a single vertex +p, and v is any vertex that is NOT adjacent to p, then there are no Hamilton +paths beginning or ending at p. We saw this in Graph B of Example 12.40. +No Hamilton path can be formed +starting or ending at vertices, r, v, +or u because they are not adjacent +to p. +Table 12.10 Some Impossible Hamilton Paths +There are also many other situations in which a Hamilton path is not possible. These are just a few that you will +encounter. +Hamilton Path or Euler Trail? +We learned in Euler Trails that an Euler trail visits each edge exactly once, whereas a Hamilton path visits each vertex +exactly once. Let’s practice distinguishing between the two. +12.8 • Hamilton Paths +1359 + +EXAMPLE 12.41 +Distinguishing between Hamilton Path and Euler Trail +Use Figure 12.185 to determine if the given sequence of vertices is a Hamilton path, an Euler trail, both, or neither. +Figure 12.185 Graphs A, F, and K +1. +Graph A, e → b → a → e → d → c → b +2. +Graph F, f → g → j → h → i +3. +Graph K, k → l → m → n → o +Solution +1. +Since the sequence covers every edge once but visits vertices more than once, it is only an Euler trail. +2. +Since the sequence visits every vertex exactly once but skips some edges, it is only a Hamilton path. +3. +Since the sequence visits each edge and each vertex exactly once, it is both an Euler trail and a Hamilton path. +YOUR TURN 12.41 +Use Figure 12.262 to determine if the given sequence of vertices is a Hamilton path, an Euler trail, both, or neither. +1. Graph A, a → b → e → d → c +2. Graph F, g → h → i → j → h → f → g → j → f → i +3. Graph K, o → n → m → l +WORK IT OUT +As we saw in Figure 12.130, the Emerald City lies at the center of the Magical Land of Oz, with Gillikin Country to the +north, Winkie Country to the east, Munchkin Country to the west, and Quadling Country to the south. Munchkin +Country and Winkie Country each shares a border with Gillikin Country and Quadling Country. Let’s apply graph +theory to Dorothy’s famous journey through Oz one more time! +Draw a graph in which each vertex is one of the regions of Oz. Is there a Hamilton path that Dorothy could follow, +instead of the yellow brick road, to lead her from the land of the Munchkins, through all the regions of Oz exactly +once, and end in the Emerald City? If so, what might it be? Compare your results with those of a classmate. +Check Your Understanding +For the following exercises, fill in the blank with the same as or different from to make the statement true. +67. Unlike in a Hamilton cycle, the vertex where the Hamilton path begins is _________ the vertex where the +Hamilton path ends. +68. If a sequence of vertices represents a Hamilton path, the number of vertices listed should be _______ the +number of vertices in the whole graph. +69. To determine if a graph has a Hamilton path, use a method that is _________ the method used to determine if a +graph has an Euler trail. +70. If a graph with a bridge has a Hamilton path, the starting vertex should be on the side of the bridge that is +________ the side of the bridge with the ending vertex. +71. A path between two vertices of a graph that visits each vertex of the graph exactly once is called an Euler path. +a. +True +1360 +12 • Graph Theory +Access for free at openstax.org + +b. +False +72. Any graph that has exactly two vertices of odd degree has a Hamilton path. +a. +True +b. +False +73. If a graph is composed of two cycles joined only at a single vertex p, then no Hamilton path can be formed starting +or ending at any vertex that is adjacent to p. +a. +True +b. +False +74. If an edge ab is a bridge with at least three components on each side, then there is no Hamilton path between +vertex a and any vertex on the other side of edge ab. +a. +True +b. +False +SECTION 12.8 EXERCISES +For the following exercises, use the figure to determine whether the sequence of vertices in the given graph is a +Hamilton path, an Euler trail, both, or neither. +1. Graph G: f → b → g → e → d → c +2. Graph G: g → b → f → c → d → e +3. Graph G: f → b → g → d → f → c → d → e → g +4. Graph W: v → w → r → s → t → o → q +5. Graph W: s → r → w → v → q → o → t +6. Graph N: h → i → k → n → j → h +7. Graph N: n → i → h → j → m +8. Graph N: m → j → h → i → k → n → i → j → k +For the following exercises, use the figure to explain why the given sequence of vertices does not represent a Hamilton +path. +9. Graph A: t → s → v → u → x → w → y → z +10. Graph B: w → x → r → u → z → y → s → t → u → v +11. Graph C: s → u → w → v → t +12. Graph D: r → → t → q → u → t → x → v → w → x → z → y +For the following exercises, use the figure to find a path that fits the description or indicate which scenario from the +figure makes it impossible. +12.8 • Hamilton Paths +1361 + +13. A Hamilton path in Graph H that begins at vertex c and ends at vertex e. +14. A Hamilton path in Graph Q that begins at vertex n and ends at vertex h. +15. A Hamilton path in Graph H that begins at vertex c and ends at vertex g. +16. A Hamilton path in Graph Q that begins at vertex m and ends at vertex j. +17. A Hamilton path in Graph H that begins at vertex g. +18. A Hamilton path in Graph Q that begins at vertex i. +19. A path between n and j in Graph Q that is NOT a Hamilton path, and explain why it is not a Hamilton path. +20. A path between a and c in Graph H that is NOT a Hamilton path, and explain why it is not a Hamilton path. +21. In chess, a knight can move in any direction, but it must move two spaces then turn and move one more space. +The eight possible moves a knight can make from a given space are shown in the figure. +A knight’s tour is a sequence of moves by a knight on a chessboard (of any size) such that the knight visits every +square exactly once. If the knight’s tour brings the knight back to its starting position on the board, it is called a +closed knight’s tour. Otherwise, it is called an open knight’s tour. Determine if the Knight’s tour shown in the figure +is a Hamilton path, an Euler trail, or both, for the graph of all possible knight moves on an eight-by-eight chess +board in which the vertices are spaces on the board and the edges indicate that the knight can move directly from +one space to the other. Explain your reasoning. +Recall from the section Euler Circuits, as part of the Camp Woebegone Olympics, there is a canoeing race with a +1362 +12 • Graph Theory +Access for free at openstax.org + +checkpoint on each of the 11 different streams as shown in the figure. The contestants must visit each checkpoint. +22. Draw a graph in which the vertices represent checkpoints, and an edge indicates that it is possible to travel +from one checkpoint to the next without passing through another checkpoint. +23. Find a Hamilton path beginning at vertex A and ending at vertex E. +24. What does this Hamilton path represent in the context of the race? +The figure shows a map of zoo exhibits A through P. Use it to answer each question. +25. Draw a graph to represent the routes through the zoo in which the edges represent walkways and the vertices +represent exhibits. Two vertices are connected if a person can walk between the exhibits they represent without +passing another exhibit. +26. Use the graph you created to find a route that begins at exhibit M, ends at exhibit J, and visits each exhibit +exactly once. +12.8 • Hamilton Paths +1363 + +12.9 Traveling Salesperson Problem +Figure 12.186 Each door on the route of a traveling salesperson can be represented as a vertex on a graph. (credit: +"Three in a row, Heriot Row" by Jason Mason/Flickr, CC BY 2.1) +Learning Objectives +After completing this section, you should be able to: +1. +Distinguish between brute force algorithms and greedy algorithms. +2. +List all distinct Hamilton cycles of a complete graph. +3. +Apply brute force method to solve traveling salesperson applications. +4. +Apply nearest neighbor method to solve traveling salesperson applications. +We looked at Hamilton cycles and paths in the previous sections Hamilton Cycles and Hamilton Paths. In this section, we +will analyze Hamilton cycles in complete weighted graphs to find the shortest route to visit a number of locations and +return to the starting point. Besides the many routing applications in which we need the shortest distance, there are also +applications in which we search for the route that is least expensive or takes the least time. Here are a few less common +applications that you can read about on a website set up by the mathematics department at the University of Waterloo +in Ontario, Canada: (https://openstax.org/r/University_of_Waterloo) +• +Design of fiber optic networks +• +Minimizing fuel expenses for repositioning satellites +• +Development of semi-conductors for microchips +• +A technique for mapping mammalian chromosomes in genome sequencing +Before we look at approaches to solving applications like these, let's discuss the two types of algorithms we will use. +Brute Force and Greedy Algorithms +An algorithm is a sequence of steps that can be used to solve a particular problem. We have solved many problems in +this chapter, and the procedures that we used were different types of algorithms. In this section, we will use two +common types of algorithms, a brute force algorithm and a greedy algorithm. A brute force algorithm begins by +listing every possible solution and applying each one until the best solution is found. A greedy algorithm approaches a +problem in stages, making the apparent best choice at each stage, then linking the choices together into an overall +solution which may or may not be the best solution. +To understand the difference between these two algorithms, consider the tree diagram in Figure 12.187. Suppose we +want to find the path from left to right with the largest total sum. For example, branch A in the tree diagram has a sum +of +. +1364 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.187 Points Along Different Paths +To be certain that you pick the branch with greatest sum, you could list each sum from each of the different branches: +A: +B: +C: +D: +E: +F: +G: +H: +Then we know with certainty that branch E has the greatest sum. +Figure 12.188 Branch E +Now suppose that you wanted to find the branch with the highest value, but you only were shown the tree diagram in +phases, one step at a time. +12.9 • Traveling Salesperson Problem +1365 + +Figure 12.189 Tree Diagram Phase 1 +After phase 1, you would have chosen the branch with 10 and 7. So far, you are following the same branch. Let’s look at +the next phase. +Figure 12.190 Tree Diagram Phase 2 +After phase 2, based on the information you have, you will choose the branch with 10, 7 and 4. Now, you are following a +different branch than before, but it is the best choice based on the information you have. Let’s look at the last phase. +Figure 12.191 Tree Diagram Phase 3 +After phase 3, you will choose branch G which has a sum of 32. +1366 +12 • Graph Theory +Access for free at openstax.org + +The process of adding the values on each branch and selecting the highest sum is an example of a brute force algorithm +because all options were explored in detail. The process of choosing the branch in phases, based on the best choice at +each phase is a greedy algorithm. Although a brute force algorithm gives us the ideal solution, it can take a very long +time to implement. Imagine a tree diagram with thousands or even millions of branches. It might not be possible to +check all the sums. A greedy algorithm, on the other hand, can be completed in a relatively short time, and generally +leads to good solutions, but not necessarily the ideal solution. +EXAMPLE 12.42 +Distinguishing between Brute Force and Greedy Algorithms +A cashier rings up a sale for $4.63 cents in U.S. currency. The customer pays with a $5 bill. The cashier would like to give +the customer $0.37 in change using the fewest coins possible. The coins that can be used are quarters ($0.25), dimes +($0.10), nickels ($0.05), and pennies ($0.01). The cashier starts by selecting the coin of highest value less than or equal to +$0.37, which is a quarter. This leaves +. The cashier selects the coin of highest value less than or +equal to $0.12, which is a dime. This leaves +. The cashier selects the coin of highest value less than +or equal to $0.02, which is a penny. This leaves +. The cashier selects the coin of highest value less +than or equal to $0.01, which is a penny. This leaves no remainder. The cashier used one quarter, one dime, and two +pennies, which is four coins. Use this information to answer the following questions. +1. +Is the cashier’s approach an example of a greedy algorithm or a brute force algorithm? Explain how you know. +2. +The cashier’s solution is the best solution. In other words, four is the fewest number of coins possible. Is this +consistent with the results of an algorithm of this kind? Explain your reasoning. +Solution +1. +The approach the cashier used is an example of a greedy algorithm, because the problem was approached in +phases and the best choice was made at each phase. Also, it is not a brute force algorithm, because the cashier did +not attempt to list out all possible combinations of coins to reach this conclusion. +2. +Yes, it is consistent. A greedy algorithm does not always yield the best result, but sometimes it does. +YOUR TURN 12.42 +1. Suppose that you lost the combination to a combination lock that consisted of three digits, and each was +selected from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. You desperately need to open the lock without breaking it. You +decide to check all possible combinations methodically, 000, then 001, then 002, and so on until you find the +right combination. Is this an example of a brute force algorithm or a greedy algorithm? +The Traveling Salesperson Problem +Now let’s focus our attention on the graph theory application known as the traveling salesperson problem (TSP) in +which we must find the shortest route to visit a number of locations and return to the starting point. +Recall from Hamilton Cycles, the officer in the U.S. Air Force who is stationed at Vandenberg Air Force base and must +drive to visit three other California Air Force bases before returning to Vandenberg. The officer needed to visit each base +once. We looked at the weighted graph in Figure 12.192 representing the four U.S. Air Force bases: Vandenberg, +Edwards, Los Angeles, and Beal and the distances between them. +Figure 12.192 Graph of Four California Air Force Bases +12.9 • Traveling Salesperson Problem +1367 + +Any route that visits each base and returns to the start would be a Hamilton cycle on the graph. If the officer wants to +travel the shortest distance, this will correspond to a Hamilton cycle of lowest weight. We saw in Table 12.11 that there +are six distinct Hamilton cycles (directed cycles) in a complete graph with four vertices, but some lie on the same cycle +(undirected cycle) in the graph. +Complete Graph +Cycle +Cycle +Cycle +Clockwise Hamilton Cycle +a → b → c → d → a +a → b → d → c → a +a → c → b → d → a +Counterclockwise Hamilton Cycle +a → d → c → b → a +a → c → d → b → a +a → d → b → c → a +Table 12.11 Hamilton Cycles in a Complete Graph with Four Vertices +Since the distance between bases is the same in either direction, it does not matter if the officer travels clockwise or +counterclockwise. So, there are really only three possible distances as shown in Figure 12.193. +Figure 12.193 Three Possible Distances +The possible distances are: +1368 +12 • Graph Theory +Access for free at openstax.org + +So, a Hamilton cycle of least weight is V → B → E → L → V (or the reverse direction). The officer should travel from +Vandenberg to Beal to Edwards, to Los Angeles, and back to Vandenberg. +Finding Weights of All Hamilton Cycles in Complete Graphs +Notice that we listed all of the Hamilton cycles and found their weights when we solved the TSP about the officer from +Vandenberg. This is a skill you will need to practice. To make sure you don't miss any, you can calculate the number of +possible Hamilton cycles in a complete graph. It is also helpful to know that half of the directed cycles in a complete +graph are the same cycle in reverse direction, so, you only have to calculate half the number of possible weights, and the +rest are duplicates. +FORMULA +In a complete graph with +vertices, +• +The number of distinct Hamilton cycles is +. +• +There are at most +different weights of Hamilton cycles. +TIP! When listing all the distinct Hamilton cycles in a complete graph, you can start them all at any vertex you +choose. Remember, the cycle a → b → c → a is the same cycle as b → c → a → b so there is no need to list both. +EXAMPLE 12.43 +Calculating Possible Weights of Hamilton Cycles +Suppose you have a complete weighted graph with vertices N, M, O, and P. +1. +Use the formula +to calculate the number of distinct Hamilton cycles in the graph. +2. +Use the formula +to calculate the greatest number of different weights possible for the Hamilton cycles. +3. +Are all of the distinct Hamilton cycles listed here? How do you know? +Cycle 1: N → M → O → P → N +Cycle 2: N → M → P → O → N +Cycle 3: N → O → M → P → N +Cycle 4: N → O → P → M → N +Cycle 5: N → P → M → O → N +Cycle 6: N → P → O → M → N +4. +Which pairs of cycles must have the same weights? How do you know? +Solution +1. +There are 4 vertices; so, +. This means there are +distinct Hamilton cycles +beginning at any given vertex. +2. +Since +, there are +possible weights. +3. +Yes, they are all distinct cycles and there are 6 of them. +4. +Cycles 1 and 6 have the same weight, Cycles 2 and 4 have the same weight, and Cycles 3 and 5 have the same +weight, because these pairs follow the same route through the graph but in reverse. +TIP! When listing the possible cycles, ignore the vertex where the cycle begins and ends and focus on the ways to +arrange the letters that represent the vertices in the middle. Using a systematic approach is best; for example, if you +must arrange the letters M, O, and P, first list all those arrangements beginning with M, then beginning with O, and +then beginning with P, as we did in Example 12.42. +YOUR TURN 12.43 +1. A complete weighted graph has vertices V, W, X, Y, and Z. List all the distinct Hamilton cycles. Use V as the +starting vertex. Identify the pairs of cycles that have the same weight because they are reverses of each other. +12.9 • Traveling Salesperson Problem +1369 + +The Brute Force Method +The method we have been using to find a Hamilton cycle of least weight in a complete graph is a brute force algorithm, +so it is called the brute force method. The steps in the brute force method are: +Step 1: Calculate the number of distinct Hamilton cycles and the number of possible weights. +Step 2: List all possible Hamilton cycles. +Step 3: Find the weight of each cycle. +Step 4: Identify the Hamilton cycle of lowest weight. +EXAMPLE 12.44 +Applying the Brute Force Method +On the next assignment, the air force officer must leave from Travis Air Force base, visit Beal, Edwards, and Vandenberg +Air Force bases each exactly once and return to Travis Air Force base. There is no need to visit Los Angeles Air Force base. +Use Figure 12.194 to find the shortest route. +Figure 12.194 Distances between Five California Air Force Bases +Solution +Step 1: Since there are 4 vertices, there will be +cycles, but half of them will be the reverse of the others; +so, there will be +possible distances. +Step 2: List all the Hamilton cycles in the subgraph of the graph in Figure 12.195. +Figure 12.195 Subgraph with Cities B, E, T, and V +To find the 6 cycles, focus on the three vertices in the middle, B, E, and V. The arrangements of these vertices are BEV, +BVE, EBV, EVB, VBE, and VEB. These would correspond to the 6 cycles: +1: T → B → E → V → T +2: T → B → V → E → T +3: T → E → B → V → T +4: T → E → V → B → T +5: T → V → B → E → T +1370 +12 • Graph Theory +Access for free at openstax.org + +6: T → V → E → B → T +Step 3: Find the weight of each path. You can reduce your work by observing the cycles that are reverses of each other. +1: +2: +3: +4: Reverse of cycle 2, 1071 +5: Reverse of cycle 3, 1572 +6: Reverse of cycle 1, 1097 +Step 4: Identify a Hamilton cycle of least weight. +The second path, T → B → V → E → T, and its reverse, T → E → V → B → T, have the least weight. The solution is that the +officer should travel from Travis Air Force base to Beal Air Force Base, to Vandenberg Air Force base, to Edwards Air Force +base, and return to Travis Air Force base, or the same route in reverse. +YOUR TURN 12.44 +1. Suppose that the Air Force officer needed to leave from Travis Air Force base, visit each of Beal, Edwards, and Los +Angeles Air Force bases exactly once and return to Travis. Use Figure 12.278 to find the shortest route. +Now suppose that the officer needed a cycle that visited all 5 of the Air Force bases in Figure 12.194. There would be +different arrangements of vertices and +distances to compare +using the brute force method. If you consider 10 Air Force bases, there would be +different arrangements and +distances to consider. There must be another way! +The Nearest Neighbor Method +When the brute force method is impractical for solving a traveling salesperson problem, an alternative is a greedy +algorithm known as the nearest neighbor method, which always visit the closest or least costly place first. This method +finds a Hamilton cycle of relatively low weight in a complete graph in which, at each phase, the next vertex is chosen by +comparing the edges between the current vertex and the remaining vertices to find the lowest weight. Since the nearest +neighbor method is a greedy algorithm, it usually doesn’t give the best solution, but it usually gives a solution that is +"good enough." Most importantly, the number of steps will be the number of vertices. That’s right! A problem with 10 +vertices requires 10 steps, not 362,880. Let’s look at an example to see how it works. +Suppose that a candidate for governor wants to hold rallies around the state. They plan to leave their home in city A, visit +cities B, C, D, E, and F each once, and return home. The airfare between cities is indicated in the graph in Figure 12.196. +12.9 • Traveling Salesperson Problem +1371 + +Figure 12.196 Airfares between Cities A, B, C, D, E, and F +Let’s help the candidate keep costs of travel down by applying the nearest neighbor method to find a Hamilton cycle that +has a reasonably low weight. Begin by marking starting vertex as +for "visited 1st." Then to compare the weights of the +edges between A and vertices adjacent to A: $250, $210, $300, $200, and $100 as shown in Figure 12.197. The lowest of +these is $100, which is the edge between A and F. +Figure 12.197 Finding the Second Vertex +Mark F as +for "visited 2nd" then compare the weights of the edges between F and the remaining vertices adjacent to +F: $170, $330, $150 and $350 as shown in Figure 12.198. The lowest of these is $150, which is the edge between F and D. +Figure 12.198 Finding the Third Vertex +Mark D as +for "visited 3rd." Next, compare the weights of the edges between D and the remaining vertices adjacent to +1372 +12 • Graph Theory +Access for free at openstax.org + +D: $120, $310, and $270 as shown in Figure 12.199. The lowest of these is $120, which is the edge between D and B. +Figure 12.199 Finding the Fourth Vertex +So, mark B as +for "visited 4th." Finally, compare the weights of the edges between B and the remaining vertices +adjacent to B: $160 and $220 as shown in Figure 12.200. The lower amount is $160, which is the edge between B and E. +Figure 12.200 Finding the Fifth Vertex +Now you can mark E as +and mark the only remaining vertex, which is C, as +. This is shown in Figure 12.201. Make a +note of the weight of the edge from E to C, which is $180, and from C back to A, which is $210. +12.9 • Traveling Salesperson Problem +1373 + +Figure 12.201 Finding the Sixth Vertex +The Hamilton cycle we found is A → F → D → B → E → C → A. The weight of the circuit is +. This may or may not be the route with the lowest cost, but there is a +good chance it is very close since the weights are most of the lowest weights on the graph and we found it in six steps +instead of finding 120 different Hamilton cycles and calculating 60 weights. Let’s summarize the procedure that we used. +Step 1: Select the starting vertex and label +for "visited 1st." Identify the edge of lowest weight between +and the +remaining vertices. +Step 2: Label the vertex at the end of the edge of lowest weight that you found in previous step as +where the +subscript n indicates the order the vertex is visited. Identify the edge of lowest weight between +and the vertices that +remain to be visited. +Step 3: If vertices remain that have not been visited, repeat Step 2. Otherwise, a Hamilton cycle of low weight is +. +EXAMPLE 12.45 +Using the Nearest Neighbor Method +Suppose that the candidate for governor wants to hold rallies around the state but time before the election is very +limited. They would like to leave their home in city A, visit cities B, C, D, E, and F each once, and return home. The airfare +between cities is not as important as the time of travel, which is indicated in Figure 12.202. Use the nearest neighbor +method to find a route with relatively low travel time. What is the total travel time of the route that you found? +Figure 12.202 Travel Times between Cities A, B, C, D, E and F +Solution +Step 1: Label vertex A as +. The edge of lowest weight between A and the remaining vertices is 85 min between A and +1374 +12 • Graph Theory +Access for free at openstax.org + +D. +Step 2: Label vertex D as +. The edge of lowest weight between D and the vertices that remain to be visited, B, C, E, and +F, is 70 min between D and F. +Repeat Step 2: Label vertex F as +. The edge of lowest weight between F and the vertices that remain to be visited, B, C, +and E, is 75 min between F and C. +Repeat Step 2: Label vertex C as +. The edge of lowest weight between C and the vertices that remain to be visited, B +and E, is 100 min between C and B. +Repeat Step 2: Label vertex B as +. The only vertex that remains to be visited is E. The weight of the edge between B +and E is 95 min. +Step 3: A Hamilton cycle of low weight is A → D → F → C → B → E → A. So, a route of relatively low travel time is A to D to F +to C to B to E and back to A. The total travel time of this route is: +YOUR TURN 12.45 +1. Use the nearest neighbor method to find a Hamilton cycle of relatively low weight beginning and ending at +vertex D in Figure 12.240 and find its total weight. Give your answer in hours and minutes. +Check Your Understanding +75. The advantage of a greedy algorithm is that it is more efficient. +a. +True +b. +False +76. The disadvantage of a brute force algorithm is that it does not always give the ideal solution. +a. +True +b. +False +77. The nearest neighbor method is an example of a brute force algorithm. +a. +True +b. +False +78. The brute force method is an example of a greedy algorithm. +a. +True +b. +False +79. The brute force method is used to find a Hamilton cycle of least weight in a complete graph. +a. +True +b. +False +80. The nearest neighbor method is used to find the ideal solution to the traveling salesperson problem. +a. +True +b. +False +81. The traveling salesman problem involves finding the shortest route to travel between two points. +a. +True +b. +False +82. The traveling salesman problem can be represented as finding a Hamilton cycle of least weight on a weighted +graph. +a. +True +b. +False +12.9 • Traveling Salesperson Problem +1375 + +83. There is always more than one Hamilton cycle of least weight, a given Hamilton cycle and the reverse of that +Hamilton cycle. +a. +True +b. +False +84. The greatest possible number of distinct weights for the Hamilton cycles of a complete graph with +vertices is ( +-1)! +a. +True +b. +False +SECTION 12.9 EXERCISES +For the following exercises, determine whether the algorithm described is a greedy algorithm or a brute force +algorithm. +1. The algorithm for creating graph colorings in the section Navigating Graphs involved coloring the vertex of +highest degree first, coloring as many other vertices as possible each color from highest to lowest degree, then +repeating this process for the remaining vertices. +2. Pallets of goods are to be transported on 10 flatbed trucks which have weight limits. To determine which goods +will be shipped together, all the possible ways to divide the goods into 10 groups is listed and the total weight of +each group is calculated. +3. A wedding planner is creating a seating arrangement for the reception dinner. The couple has provided a list of +which guests must be seated together. The wedding planner prefers to use the fewest tables possible so that +there is more space to mingle at the reception. The planner creates a list of all possible seating arrangements +and selects one that meets these criteria. +4. Packages must be loaded into freight cars to be transported by train. It is preferred to use the fewest freight +cars possible to keep the costs down. As each freight car is packed, the package with the largest girth that will +fit in the freight car is loaded and this is repeated until the freight car can hold no more packages. When a +freight car can hold no more packages, the next freight car is loaded. +For the following exercises, use the figure to calculate the number of distinct Hamilton cycles beginning at the given +vertex in the given graph. How many of those could possibly result in a different weight? +5. Graph A, vertex a +6. Graph B, vertex e +7. Graph C, vertex k +8. Graph D, vertex o +For the following exercises, use the figure to list all the distinct Hamilton cycles beginning at the given vertex in the +given graph. Indicate which pairs of Hamilton cycles are reverses of each other. +1376 +12 • Graph Theory +Access for free at openstax.org + +9. Graph A, vertex a +10. Graph B, vertex e +11. Graph C, vertex k +12. Graph D, vertex o +For the following exercises, use the figure to find a Hamilton cycle of least weight for the given graph, beginning at the +given vertex, and using the brute force method. What is the weight of the cycle? +13. Graph A, vertex a +14. Graph B, vertex e +15. Graph C, vertex k +16. Graph D, vertex o +For the following exercises, use the figure to find a Hamilton cycle of low weight for the given graph, beginning at the +given vertex, and using the nearest neighbor method. What is the weight of the cycle? +17. Graph A, vertex a +18. Graph B, vertex e +19. Graph C, vertex k +20. Graph D, vertex o +12.9 • Traveling Salesperson Problem +1377 + +For the following exercises, use your solutions to Exercises 13–20 to compare the results of the brute force method to +the results of the nearest neighbor method for each graph. Indicate whether the Hamilton cycle was the same or +different and whether the weights were the same or different. If the weights are different, indicate which method gave +the lower weight. Are your observations consistent with the characteristics of brute force algorithms and greedy +algorithms? Explain your reasoning. +21. Graph A, vertex a +22. Graph B, vertex e +23. Graph C, vertex k +24. Graph D, vertex o +For the following exercises, use the table to create a complete weighted graph in which the vertices are the given cities, +and the weights are the distances between them. +Cities +U +V +W +X +Y +Z +U +0 +89 +37 +49 +54 +28 +V +89 +0 +76 +68 +92 +112 +W +37 +76 +0 +45 +52 +49 +X +49 +68 +45 +0 +66 +47 +Y +54 +92 +52 +66 +0 +29 +Z +28 +112 +49 +47 +29 +0 +Distances between Cities U, V, W, X, Y, and Z in +kilometers +25. U, V, W, X +26. U, W, Y, Z +27. U, X, Y, Z +28. U, V, W, X, Y +29. U, W, X, Y, Z +30. U, V, W, X, Y, Z +For the following exercises, use your solutions to Exercises 25–30 and the nearest neighbor method to find a Hamilton +cycle to solve the traveling salesperson problem of finding a reasonably short route to leave from city U, visit each of +the other cities listed and return to city U. Indicate the distance required to travel the route you found. +31. U, V, W, X +32. U, W, Y, Z +33. U, X, Y, Z +34. U, V, W, X, Y +35. U, W, X, Y, Z +36. U, V, W, X, Y, Z +For the following exercises, use your solutions to Exercises 25–30 and the brute force method to find a Hamilton cycle +of lowest weight to solve the traveling salesperson problem of finding a shortest route to leave from city U, visit each of +the other cities listed and return to city U. Indicate the distance required to travel the route you found. +37. U, V, W, X +38. U, W, Y, Z +39. U, X, Y, Z +40. U, V, W, X, Y +41. U, W, X, Y, Z +For the following exercises, use your solutions to the indicated exercises to compare the results of the brute force +method to the results of the nearest neighbor method for each traveling salesman problem of finding a reasonably +short route to leave from city U, visit each of the other cities listed and return to city U. Indicate whether the greedy +algorithm resulted in a Hamilton cycle of the same weight, lower weight, or higher weight. Is this consistent with the +1378 +12 • Graph Theory +Access for free at openstax.org + +characteristics of brute force algorithms and greedy algorithms? Explain your reasoning +42. Exercises 32 and 38: U, W, Y, Z +43. Exercises 31 and 37: U, V, W, X +44. Exercises 34 and 40: U, V, W, X, Y +45. Exercises 35 and 41: U, W, X, Y, Z +46. Exercises 33 and 39: U, X, Y, Z +The products at a particular factory are manufactured in phases. The same equipment is utilized for each phase, but it +must be formatted differently to accomplish different tasks. The transition time to convert between a format for one +task and another task varies. The times are given in Table 12.13. In the following exercises, use the table and the +nearest neighbor method to find an order in which to complete the tasks, which keeps the transition times down and +ends with the same set up as it began so that the factory is ready to start the next batch. Assume that there are no +restrictions on which tasks can be completed in which order. Hint: The nearest neighbor algorithm may give a different +result depending on which vertex is the starting vertex, so, you must check all possibilities. +Task +A +B +C +D +E +F +G +A +0 +75 +130 +45 +120 +70 +100 +B +75 +0 +140 +65 +115 +25 +60 +C +130 +140 +0 +35 +55 +20 +125 +D +45 +65 +35 +0 +50 +30 +40 +E +120 +115 +55 +50 +0 +95 +145 +F +70 +25 +20 +30 +95 +0 +15 +G +100 +60 +125 +40 +145 +15 +0 +Manufacturing Equipment Transition Times in Minutes +47. A, C, D, F, G +48. B, D, E, F, G +49. A, B, C, D, E, F +50. B, C, D, E, F, G +51. A, B, C, D, E, F, G +12.9 • Traveling Salesperson Problem +1379 + +12.10 Trees +Figure 12.203 In graph theory, graphs known as trees have structures in common with live trees. (credit: “Row of trees in +Roslev” by AKA CJ/Flickr, Public Domain) +Learning Objectives +After completing this section, you should be able to: +1. +Describe and identify trees. +2. +Determine a spanning tree for a connected graph. +3. +Find the minimum spanning tree for a weighted graph. +4. +Solve application problems involving trees. +We saved the best for last! In this last section, we will discuss arguably the most fun kinds of graphs, trees. Have you +every researched your family tree? Family trees are a perfect example of the kind of trees we study in graph theory. One +of the characteristics of a family tree graph is that it never loops back around, because no one is their own grandparent! +What Is A Tree? +Whether we are talking about a family tree or a tree in a forest, none of the branches ever loops back around and rejoins +the trunk. This means that a tree has no cyclic subgraphs, or is acyclic. A tree also has only one component. So, a tree is +a connected acyclic graph. Here are some graphs that have the same characteristic. Each of the graphs in Figure 12.204 +is a tree. +Figure 12.204 Graphs T, P, and S +Let’s practice determining whether a graph is a tree. To do this, check if a graph is connected and has no cycles. +1380 +12 • Graph Theory +Access for free at openstax.org + +EXAMPLE 12.46 +Identifying Trees +Identify any trees in Figure 12.205. If a graph is not a tree, explain how you know. +Figure 12.205 Graphs M, N, and P +Solution +• +Graph M is not a tree because it contains the cycle (b, c, f). +• +Graph N is not a tree because it is not connected. It has two components, one with vertices h, i, j, and another with +vertices k, l, m. +• +Graph P is a tree. It has no cycles and it is connected. +YOUR TURN 12.46 +1. There are some configurations that are commonly used when setting up computer networks. Several of them are +shown in the given figure. Which of the configurations in the figure appear to have the characteristics of a tree +graph? If a configuration does not appear to have the characteristics of a tree graph, explain how you know. +Common Network Configurations +Types of Trees +Mathematicians have had a lot of fun naming graphs that are trees or that contain trees. For example, the graph in +Figure 12.206 is not a tree, but it contains two components, one containing vertices a through d, and the other +containing vertices e through g, each of which would be a tree on its own. This type of structure is called a forest. There +12.10 • Trees +1381 + +are also interesting names for trees with certain characteristics. +• +A path graph or linear graph is a tree graph that has exactly two vertices of degree 1 such that the only other +vertices form a single path between them, which means that it can be drawn as a straight line. +• +A star tree is a tree that has exactly one vertex of degree greater than 1 called a root, and all other vertices are +adjacent to it. +• +A starlike tree is a tree that has a single root and several paths attached to it. +• +A caterpillar tree is a tree that has a central path that can have vertices of any degree, with each vertex not on the +central path being adjacent to a vertex on the central path and having a degree of one. +• +A lobster tree is a tree that has a central path that can have vertices of any degree, with paths consisting of either +one or two edges attached to the central path. +Examples of each of these types of structures are given in Figure 12.207. +Figure 12.206 Forest Graph F +Figure 12.207 Six Types of Trees +1382 +12 • Graph Theory +Access for free at openstax.org + +EXAMPLE 12.47 +Identifying Types of Trees +Each graph in Figure 12.208 is one of the special types of trees we have been discussing. Identify the type of tree. +Figure 12.208 Graphs U and V +Solution +Graph U has a central path a → b → d → f → i → l → o → q. Each vertex that is not on the path has degree 1 and is adjacent +to a vertex that is on the path. So, U is a caterpillar tree. +Graph V is a path graph because it is a single path connecting exactly two vertices of degree one, r → s → u → v → w. +YOUR TURN 12.47 +Of the network configurations from Figure 12.297, which, if any, has the characteristics of a +1. Star tree? +2. Caterpillar tree? +3. Path graph? +Characteristics of Trees +As we study trees, it is helpful to be familiar with some of their characteristics. For example, if you add an edge to a tree +graph between any two existing vertices, you will create a cycle, and the resulting graph is no longer a tree. Some +examples are shown in Figure 12.209. Adding edge bj to Graph T creates cycle (b, c, i, j). Adding edge rt to Graph P +creates cycle (r, s, t). Adding edge tv to Graph S creates cycle (t, u, v). +Figure 12.209 Adding Edges to Trees +It is also true that removing an edge from a tree graph will increase the number of components and the graph will no +longer be connected. In fact, you can see in Figure 12.210 that removing one or more edges can create a forest. +Removing edge qr from Graph P creates a graph with two components, one with vertices o, p and q, and the other with +12.10 • Trees +1383 + +vertices r, s, and t. Removing edge uw from Graph S creates two components, one with just vertex w and the other with +the rest of the vertices. When two edges were removed from Graph T, edge bf and edge cd, creates a graph with three +components as shown in Figure 12.210. +Figure 12.210 Removing Edges from Trees +A very useful characteristic of tree graphs is that the number of edges is always one less than the number of vertices. In +fact, any connected graph in which the number of edges is one less than the number of vertices is guaranteed to be a +tree. Some examples are given in Figure 12.211. +Figure 12.211 Number of Vertices and Edges in Trees vs. Other Graphs +FORMULA +The number of edges in a tree graph with +vertices is +. +A connected graph with n vertices and +edges is a tree graph. +EXAMPLE 12.48 +Exploring Characteristics of Trees +Use Graphs I and J in Figure 12.212 to answer each question. +1384 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.212 Graphs I and J +1. +Which vertices are in each of the components that remain when edge be is removed from Graph I? +2. +Determine the number of edges and the number of vertices in Graph J. Explain how this confirms that Graph J is a +tree. +3. +What kind of cycle is created if edge im is added to Graph J? +Solution +1. +When edge be is removed, there are two components that remain. One component includes vertices a, b, and c. The +other component includes vertices d, e, and f. +2. +There are seven vertices and six edges in Graph J. This confirms that Graph J is a tree because the number of edges +is one less than the number of vertices. +3. +The pentagon (i, h, j, l, m) is created when edge im is added to Graph J. +YOUR TURN 12.48 +Use Graphs I and J in Figure 12.301 to answer each question. +1. Which vertices are in each of the components that remain when edge jl is removed from Graph J? +2. Determine the number of edges and the number of vertices in Graph I. Explain how this confirms that Graph I +is a tree. +3. What kind of cycle is created if edge cf is added to Graph I? +WHO KNEW? +Graph Theory in the Movies +In the 1997 film Good Will Hunting, the main character, Will, played by Matt Damon, solves what is supposed to be an +exceptionally difficult graph theory problem, “Draw all the homeomorphically irreducible trees of size +.” That +sounds terrifying! But don’t panic. Watch this great Numberphile video to see why this is actually a problem you can +do at home! +VIDEO +The Problem in Good Will Hunting by Numberphile (https://openstax.org/r/Hunting_Numberphile) +Spanning Trees +Suppose that you planned to set up your own computer network with four devices. One option is to use a “mesh +topology” like the one in Figure 12.213, in which each device is connected directly to every other device in the network. +12.10 • Trees +1385 + +Figure 12.213 Common Network Configurations +The mesh topology for four devices could be represented by the complete Graph A1 in Figure 12.214 where the vertices +represent the devices, and the edges represent network connections. However, the devices could be networked using +fewer connections. Graphs A2, A3, and A4 of Figure 12.214 show configurations in which three of the six edges have been +removed. Each of the Graphs A2, A3 and A4 in Figure 12.214 is a tree because it is connected and contains no cycles. Since +Graphs A2, A3 and A4 are also subgraphs of Graph A1 that include every vertex of the original graph, they are also known +as spanning trees. +Figure 12.214 Network Configurations for Four Devices +By definition, spanning trees must span the whole graph by visiting all the vertices. Since spanning trees are subgraphs, +they may only have edges between vertices that were adjacent in the original graph. Since spanning trees are trees, they +are connected and they are acyclic. So, when deciding whether a graph is a spanning tree, check the following +characteristics: +• +All vertices are included. +• +No vertices are adjacent that were not adjacent in the original graph. +• +The graph is connected. +• +There are no cycles. +EXAMPLE 12.49 +Identifying Spanning Trees +Use Figure 12.215 to determine which of graphs M1, M2, M3, and M4, are spanning trees of Q. +1386 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.215 Graphs Q, M1, M2, M3, and M4 +Solution +1. +Graph M1 is not a spanning tree of Graph Q because it has a cycle (c, d, f, e). +2. +Graph M2 is a spanning tree of Graph Q because it has all the original vertices, no vertices are adjacent in M2 that +weren’t adjacent in Graph Q, Graph M2 is connected and it contains no cycles. +3. +Graph M3 is not a spanning tree of Graph Q because vertices a and f are adjacent in Graph M3 but not in Graph Q. +4. +Graph M4 is not a spanning tree of Graph Q because it is not connected. +So, only graph M2 is a spanning tree of Graph Q. +YOUR TURN 12.49 +Use the given figure for the following exercises. +1. Since sq is not an edge in Graph H, Graph N1 cannot be a spanning tree of H. +a. +True +b. +False +2. Graph N2 is a spanning tree of Graph H. +a. +True +b. +False +3. Graph N3 is a spanning tree of Graph H. +a. +True +b. +False +4. Since there is no path between p and t in Graph N4, it cannot be a spanning tree of any graph. +a. +True +b. +False +Constructing a Spanning Tree Using Paths +Suppose that you wanted to find a spanning tree within a graph. One approach is to find paths within the graph. You can +start at any vertex, go any direction, and create a path through the graph stopping only when you can’t continue without +backtracking as shown in Figure 12.216. +12.10 • Trees +1387 + +Figure 12.216 First Phase to Construct a Spanning Tree +Once you have stopped, pick a vertex along the path you drew as a starting point for another path. Make sure to visit +only vertices you have not visited before as shown in Figure 12.217. +Figure 12.217 Intermediate Phase to Construct a Spanning Tree +Repeat this process until all vertices have been visited as shown in Figure 12.218. +Figure 12.218 Final Phase to Construct a Spanning Tree +The end result is a tree that spans the entire graph as shown in Figure 12.219. +Figure 12.219 The Resulting Spanning Tree +Notice that this subgraph is a tree because it is connected and acyclic. It also visits every vertex of the original graph, so +it is a spanning tree. However, it is not the only spanning tree for this graph. By making different turns, we could create +any number of distinct spanning trees. +EXAMPLE 12.50 +Constructing Spanning Trees +Construct two distinct spanning trees for the graph in Figure 12.220. +1388 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.220 Graph L +Solution +Two possible solutions are given in Figure 12.221 and Figure 12.222. +Figure 12.221 First Spanning Tree for Graph L +Figure 12.222 Second Spanning Tree for Graph L +YOUR TURN 12.50 +1. Construct three distinct spanning trees for Graph J. +Graph J +Revealing Spanning Trees +Another approach to finding a spanning tree in a connected graph involves removing unwanted edges to reveal a +spanning tree. Consider Graph D in Figure 12.223. +12.10 • Trees +1389 + +Figure 12.223 Graph D +Graph D has 10 vertices. A spanning tree of Graph D must have 9 edges, because the number of edges is one less than +the number of vertices in any tree. Graph D has 13 edges so 4 need to be removed. To determine which 4 edges to +remove, remember that trees do not have cycles. There are four triangles in Graph D that we need to break up. We can +accomplish this by removing 1 edge from each of the triangles. There are many ways this can be done. Two of these +ways are shown in Figure 12.224. +Figure 12.224 Removing Four Edges from Graph D +VIDEO +Spanning Trees in Graph Theory (https://openstax.org/r/Spanning_Trees_in_Graph_Theory) +EXAMPLE 12.51 +Removing Edges to Find Spanning Trees +Use the graph in Figure 12.225 to answer each question. +Figure 12.225 Graph V +1. +Determine the number of edges that must be removed to reveal a spanning tree. +2. +Name all the undirected cycles in Graph V. +3. +Find two distinct spanning trees of Graph V. +Solution +1. +Graph V has nine vertices so a spanning tree for the graph must have 8 edges. Since Graph V has 11 edges, 3 edges +must be removed to reveal a spanning tree. +2. +(a, c, d), (a, c, f), (a, d, c, f), and (b, e, h, i, g) +3. +To find the first spanning tree, remove edge ac, which will break up both of the triangles, remove edge cf , which will +break up the quadrilateral, and remove be, which will break up the pentagon, to give us the spanning tree shown in +Figure 12.226. +1390 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.226 Spanning Tree Formed Removing ac, cf, and be +To find another spanning tree, remove ad, which will break up (a, c, d) and (a, d, c, f), remove af to break up (a, c, f), +and remove hi to break up (b, e, h, i, g). This will give us the spanning tree in Figure 12.227. +Figure 12.227 Spanning Tree Formed Removing ad, af, and hi +YOUR TURN 12.51 +1. Name three edges that you could remove from Graph V in Figure 12.316 to form a third spanning tree, different +from those in the solution to Example 12.50 Exercise 3. +WHO KNEW? +Chains of Affection +Here is a strange question to ask in a math class: Have you ever dated your ex’s new partner’s ex? Research suggests +that your answer is probably no. When researchers Peter S. Bearman, James Moody, and Katherine Stovel attempted +to compare the structure of heterosexual romantic networks at a typical midwestern high school to simulated +networks, they found something surprising. The actual social networks were more like spanning trees than other +possible models because there were very few short cycles. In particular, there were almost no four-cycles. +Figure 12.228 Chains of Affection +“…the prohibition against dating (from a female perspective) one’s old boyfriend’s current girlfriend’s old boyfriend – +accounts for the structure of the romantic network at [the highschool].” +In their article “Chains of Affection: The Structure of Adolescent Romantic and Sexual Networks,” the researchers went +on to explain the implications for the transmission of sexually transmitted diseases. In particular, social structures +based on tree graphs are less dense and more likely to fragment. This information can impact social policies on +disease prevention. (Peter S. Bearman, James Moody, and Katherine Stovel, “Chains of Affection: The Structure of +Adolescent Romantic and Sexual Networks,” American Journal of Sociology Volume 110, Number 1, pp. 44-91, 2004) +Kruskal’s Algorithm +In many applications of spanning trees, the graphs are weighted and we want to find the spanning tree of least possible +weight. For example, the graph might represent a computer network, and the weights might represent the cost involved +in connecting two devices. So, finding a spanning tree with the lowest possible total weight, or minimum spanning +tree, means saving money! The method that we will use to find a minimum spanning tree of a weighted graph is called +Kruskal’s algorithm. The steps for Kruskal’s algorithm are: +12.10 • Trees +1391 + +Step 1: Choose any edge with the minimum weight of all edges. +Step 2: Choose another edge of minimum weight from the remaining edges. The second edge does not have to be +connected to the first edge. +Step 3: Choose another edge of minimum weight from the remaining edges, but do not select any edge that creates a +cycle in the subgraph you are creating. +Step 4: Repeat step 3 until all the vertices of the original graph are included and you have a spanning tree. +VIDEO +Use Kruskal's Algorithm to Find Minimum Spanning Trees in Graph Theory (https://openstax.org/r/ +Trees_in_Graph_Theory) +EXAMPLE 12.52 +Using Kruskal’s Algorithm +A computer network will be set up with six devices. The vertices in the graph in Figure 12.229 represent the devices, and +the edges represent the cost of a connection. Find the network configuration that will cost the least. What is the total +cost? +Figure 12.229 Graph of Network Connection Costs +Solution +A minimum spanning tree will correspond to the network configuration of least cost. We will use Kruskal’s algorithm to +find one. Since the graph has six vertices, the spanning tree will have six vertices and five edges. +Step 1: Choose an edge of least weight. We have sorted the weights into numerical order. The least is $100. The only +edge of this weight is edge AF as shown in Figure 12.230. +1392 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.230 Step 1 Select Edge AF +Step 2: Choose the edge of least weight of the remaining edges, which is BD with $120. Notice that the two selected +edges do not need to be adjacent to each other as shown in Figure 12.231. +Figure 12.231 Step 2 Select Edge BD +Step 3: Select the lowest weight edge of the remaining edges, as long as it does not result in a cycle. We select DF with +$150 since it does not form a cycle as shown in Figure 12.232. +Figure 12.232 Step 3 Select Edge DF +12.10 • Trees +1393 + +Repeat Step 3: Select the lowest weight edge of the remaining edges, which is BE with $160 and it does not form a cycle +as shown in Figure 12.233. This gives us four edges so we only need to repeat step 3 once more to get the fifth edge. +Figure 12.233 Repeat Step 3 Select Edge DF +Repeat Step 3: The lowest weight of the remaining edges is $170. Both BF and CE have a weight of $170, but BF would +create cycle (b, d, f) and there cannot be a cycle in a spanning tree as shown in Figure 12.234. +Figure 12.234 Repeat Step 3 Do Not Select Edge BF +So, we will select CE, which will complete the spanning tree as shown in Figure 12.235. +1394 +12 • Graph Theory +Access for free at openstax.org + +Figure 12.235 Repeat Step 3 Select Edge CE +The minimum spanning tree is shown in Figure 12.236. This is the configuration of the network of least cost. The +spanning tree has a total weight of +, which is the total cost of this network +configuration. +Figure 12.236 Final Minimum Spanning Tree +YOUR TURN 12.52 +1. Find a minimum spanning tree for the weighted graph. Give its total weight. +Weighted Graph +Check Your Understanding +85. The number of cycles in a spanning tree is one less than the number of vertices. +a. +True +12.10 • Trees +1395 + +b. +False +86. A spanning tree contains no triangles. +a. +True +b. +False +87. A spanning tree includes every vertex of the original graph. +a. +True +b. +False +88. There is a unique path between each pair of vertices in a spanning tree. +a. +True +b. +False +89. A spanning tree must be connected. +a. +True +b. +False +90. Kruskal’s algorithm is a method for finding all the different spanning trees in a given graph. +a. +True +b. +False +91. Only graphs that are trees have spanning trees. +a. +True +b. +False +92. A minimum spanning tree of a given graph can be found using Kruskal’s algorithm. +a. +True +b. +False +93. A minimum spanning tree of a given graph is the subgraph, which is a tree, includes every vertex of the original +graph, and which has the least weight of all spanning trees. +a. +True +b. +False +94. If a graph contains any cut edges, they must be included in any spanning tree. +a. +True +b. +False +SECTION 12.10 EXERCISES +For the following exercises, refer to the figure shown. +1. Which graphs, if any, are trees? +2. Which graphs, if any, are not trees because they are not connected? +3. Which graphs, if any, are not trees because they contain a cycle? +For the following exercises, refer to the figure shown. Identify any graphs that fit the given description. +1396 +12 • Graph Theory +Access for free at openstax.org + +4. Tree graph +5. Star graph +6. Star like graph +7. Line graph (or path graph) +8. Lobster graph +9. Caterpillar graph +10. Forest graph +For the following exercises, use the figure shown to answer the questions. +11. Determine whether Graph H1 is a spanning tree of Graph H. If not, explain how you know. +12. Determine whether Graph H2 is a spanning tree of Graph H. If not, explain how you know. +13. Determine whether Graph H3 is a spanning tree of Graph H. If not, explain how you know. +14. Determine whether Graph Q1 is a spanning tree of Graph Q. If not, explain how you know. +12.10 • Trees +1397 + +15. Determine whether Graph Q2 is a spanning tree of Graph Q. If not, explain how you know. +16. Determine whether Graph Q3 is a spanning tree of Graph Q. If not, explain how you know. +For the following exercises, a student has been asked to construct a spanning tree for Graph O, as shown in the figure. +The dashed lines show the first step that the student took, creating a path from vertex h to vertex d. +17. How many more edges must be included with the dashed edges to create a spanning tree? +18. List three unused (solid) edges from Graph O that cannot be used to complete the spanning tree. +19. Give an example of a set of edges that do not have e as an endpoint, which would complete the spanning tree. +20. Give an example of a set of edges that do not have f as an endpoint, which would complete the spanning tree. +For the following exercises, a student has been asked to construct a spanning tree for Graph O, as shown in the figure. +The dashed lines show the first step that the student took, creating a path from vertex c to vertex h. +21. How many more edges must be included with the dashed edges to create a spanning tree? +22. List two unused edges from Graph O that cannot be used to complete the spanning tree. +23. Give an example of a set of edges that do not have f as an endpoint, which would complete the spanning tree. +24. Give an example of a set of edges that have neither c nor e as an endpoint, which would complete the spanning +tree. +For the following exercises, use Graphs A, B, and C. +25. How many edges must be removed from Graph A to create a spanning tree? +26. How many edges must be removed from Graph B to create a spanning tree? +1398 +12 • Graph Theory +Access for free at openstax.org + +27. How many edges must be removed from Graph C to create a spanning tree? +28. Identify all the distinct cyclic subgraphs of Graph A. +29. Identify all the cyclic subgraphs of Graph B. +30. Identify all the cyclic subgraphs of Graph C. +31. Draw four spanning trees of Graph A each of which includes edges vs, uv, wz and xy. +32. Draw four spanning trees of Graph B which includes edge ut, but not ur. +33. Draw four spanning trees of Graph C that each have only one edge with an endpoint at vertex u. +For the following exercises, use the figure shown. Draw a graph that fits the given description. +34. S1, S2, S3 and S4 are all spanning trees. +35. S1 and S2 are spanning trees, but S4 is not. +36. S3 and S4 are spanning trees but S1 is not. +37. S2 and S3 are spanning trees but S1 is not. +38. S2 and S3 are spanning trees but S4 is not. +39. S1, S2, and S3 are spanning trees but S4 is not. +40. S2, S3, and S4 are spanning trees but S1 is not. +41. S1 and S4 are spanning trees but S2 and S3 are not. +For the following exercises, use the figure shown to find the weight of the given spanning tree. +42. +43. +44. +45. Use Kruskal’s algorithm to draw a minimum spanning tree for Graph Z in the provided figure. Find its weight. +12.10 • Trees +1399 + +For the following exercises, draw a minimum spanning tree for the given graph, and calculate its weight. +46. Graph A +47. Graph C +48. Graph B +49. Graph D +For the following exercises, draw a weighted graph to represent the given information. Then use the graph to find a +minimum spanning tree and give its weight. Explain what the weight represents in the given scenario. +50. City planners are tasked with building roadways to connect locations A, B, C, and D. The cost to build the +roadways between any given pair of locations is given in the table. +A +B +C +D +A +- +125 +320 +275 +B +125 +- +110 +540 +C +320 +110 +- +1,010 +D +275 +540 +1,010 +- +Construction Costs in Thousands +between Locations +51. In a video game, the goal is to visit five different lands, V, W, X, Y and Z, without losing all your lives. The paths +between the lands are rated for danger, 1 being lowest and 10 being highest. Once a path has been traversed +successfully, it is free from danger. The ratings are given in the table. +V +W +X +Y +Z +V +- +2 +4 +9 +10 +W +2 +- +6 +8 +No Path +X +4 +6 +- +7 +No Path +Y +9 +8 +7 +- +5 +Z +10 +No Path +No Path +5 +- +Danger Ratings between Lands +1400 +12 • Graph Theory +Access for free at openstax.org + +Chapter Summary +Key Terms +12.1 Graph Basics +• +vertex +• +edge +• +loop +• +graph (simple graph) +• +multigraph +• +adjacent (neighboring) +• +degree +12.2 Graph Structures +• +complete +• +subgraph +• +cycle +• +cyclic subgraph +• +clique +12.3 Comparing Graphs +• +isomorphic +• +isomorphism +• +planar +• +nonplanar +• +complement +• +complementary +12.4 Navigating Graphs +• +walk (directed walk) +• +trail (directed trail) +• +path (directed path) +• +closed +• +open +• +closed walk +• +circuit (closed trail) +• +directed cycle (closed path) +• +coloring (graph coloring) +• +-coloring +• +chromatic number +12.5 Euler Circuits +• +connected +• +component +• +disconnected +• +Euler circuit +• +Eulerian graph +• +Chinese postman problem +• +Eulerization +12.6 Euler Trails +• +algorithm +• +Fleury’s algorithm +• +Euler trail +• +bridge +• +local bridge +12 • Chapter Summary +1401 + +12.7 Hamilton Cycles +• +Hamilton cycle, or Hamilton circuit +• +factorial +• +weighted graph +• +total weight +12.8 Hamilton Paths +• +Hamilton path +12.9 Traveling Salesperson Problem +• +brute force algorithm +• +greedy algorithm +• +traveling salesperson problem (TSP) +• +brute force method +• +nearest neighbor method +12.10 Trees +• +acyclic +• +tree +• +forest +• +path graph or linear graph +• +star tree +• +root +• +starlike tree +• +caterpillar tree +• +lobster tree +• +spanning tree +• +minimum spanning tree +Key Concepts +12.1 Graph Basics +• +Graphs and multigraphs represent objects as vertices and the relationships between the objects as edges. +• +The degree of a vertex is the number of edges that meet it and the degree can be zero. +• +An edge must have a vertex at each end. +• +Multigraphs may contain loops and double edges, but simple graphs may not. +12.2 Graph Structures +• +The sum of the degrees of the vertices in a graph is twice the number of edges. +• +In a complete graph every pair of vertices is adjacent. +• +A subgraph is part of a larger graph. +• +Cycles are a sequence of connected vertices that begin and end at the same vertex but never visit any vertex twice. +12.3 Comparing Graphs +• +Two graphs are isomorphic if they have the same structure. +• +When graphs are relatively small, we can use visual inspection to identify an isomorphism by transforming one +graph into another without breaking connections or adding new ones. +• +An isomorphism between two graphs preserves adjacency. +• +If two graphs differ in number of vertices, number of edges, degrees of vertices, or types of subgraphs, they cannot +be isomorphic. +• +When the complements of two graphs are isomorphic, so are the graphs themselves. +12.4 Navigating Graphs +• +Walks, trails, and paths are ways to navigate through a graph using a sequence of connected vertices and edges. +• +Closed walks, circuits, and directed cycles are ways to navigate from a vertex on a graph and return to the same +vertex. +• +Colorings are a way to organize the vertices of a graph into groups so that no two members of a group are adjacent. +1402 +12 • Chapter Summary +Access for free at openstax.org + +• +Maps can be represented with planar graphs, which can always be colored using four colors or fewer. +12.5 Euler Circuits +• +A connected graph has only one component. +• +The Euler circuit theorem states that an Euler circuit exists in every connected graph in which all vertices have even +degree, but not in disconnected graphs or any graph with one or more vertices of odd degree. +• +The Chinese postman problem asks how to find the shortest closed trail that visits all edges at least once. +• +If an Euler circuit exists, it is always the best solution to the Chinese postman problem. +• +Eulerization is the process of adding duplicate edges to a graph so that the new multigraph has an Euler circuit. +• +The minimum number of duplicated edges needed to eulerize a graph is half the number of odd vertices or more. +12.6 Euler Trails +• +An Euler trail exists whenever a graph has exactly two vertices of odd degree. +• +When a bridge is removed from a graph, the number of components increases. +• +A bridge is never part of a circuit. +• +When a local bridge is removed from a graph, the distance between vertices increases. +• +An edge that is part of a triangle is never a local bridge. +12.7 Hamilton Cycles +• +A Hamilton cycle is a directed cycle, or circuit, that visits each vertex exactly once. +• +Some Hamilton cycles are also Euler circuits, but some are not. +• +Hamilton cycles that follow the same undirected cycle in the same direction are considered the same cycle even if +they begin at a different vertex. +• +The number of unique Hamilton cycles in a complete graph with n vertices is the same as the number of ways to +arrange +distinct objects. +• +Weighted graphs have a value assigned to each edge, which can represent distance, time, money and other +quantities. +12.8 Hamilton Paths +• +A Hamilton path visits every vertex exactly once. +• +Some Hamilton paths are also Euler trails, but some are not. +12.9 Traveling Salesperson Problem +• +A brute force algorithm always finds the ideal solution but can be impractical whereas a greedy algorithm is efficient +but usually does not lead to the ideal solution. +• +A Hamilton cycle of lowest weight is a solution to the traveling salesperson problem. +• +The brute force method finds a Hamilton cycle of lowest weight in a complete graph. +• +The nearest neighbor method is a greedy algorithm that finds a Hamilton cycle of relatively low weight in a +complete graph. +12.10 Trees +• +A brute force algorithm always finds the ideal solution but can be impractical whereas a greedy algorithm is efficient +but usually does not lead to the ideal solution. +• +A Hamilton cycle of lowest weight is a solution to the traveling salesperson problem. +• +The brute force method finds a Hamilton cycle of lowest weight in a complete graph. +• +The nearest neighbor method is a greedy algorithm that finds a Hamilton cycle of relatively low weight in a +complete graph. +Videos +12.1 Graph Basics +• +Graph Theory: Create a Graph to Represent Common Boundaries on a Map (https://openstax.org/r/Graph_Theory) +12.2 Graph Structures +• +The Mathematical Secrets of Pascal's Triangle by Wajdi Mohamed Ratemi (https://openstax.org/r/ +Wajdi_Mohamed_Ratemi) +12 • Chapter Summary +1403 + +12.3 Comparing Graphs +• +Determine If Two Graphs Are Isomorphic and Identify the Isomorphism (https://openstax.org/r/Determine_If_Two) +12.4 Navigating Graphs +• +Walks, Trails, and Paths in Graph Theory (https://openstax.org/r/walks_trails_paths) +• +Closed Walks, Closed Trails (Circuits), and Closed Paths (Directed Cycles) in Graph Theory (https://openstax.org/r/ +closed_walks) +• +Coloring Graphs Part 1: Coloring and Identifying Chromatic Number (https://openstax.org/r/ +Coloring_and_Identifying) +• +The Four Color Map Theorem – Numberphile (https://openstax.org/r/The_Four) +• +Coloring Graphs Part 2: Coloring Maps and the Four Color Problem (https://openstax.org/r/Coloring_Maps) +• +Neil deGrasse Tyson Explains the Möbius Strip (https://openstax.org/r/Neil_de_Grasse) +12.5 Euler Circuits +• +Connected and Disconnected Graphs in Graph Theory (https://openstax.org/r/connected_disconnected_graphs) +• +Recognizing Euler Trails and Euler Circuits (https://openstax.org/r/Euler_trails_circuits) +• +Existence of Euler Circuits in Graph Theory (https://openstax.org/r/existence_Euler_circuits) +12.6 Euler Trails +• +Bridges and Local Bridges in Graph Theory (https://openstax.org/r/bridges_local_bridges) +• +Fluery's Algorithm to Find an Euler Circuit (https://openstax.org/r/Fleurys_algorithm) +12.10 Trees +• +The Problem in Good Will Hunting by Numberphile (https://openstax.org/r/Hunting_Numberphile) +• +Spanning Trees in Graph Theory (https://openstax.org/r/Spanning_Trees_in_Graph_Theory) +• +Use Kruskal's Algorithm to Find Minimum Spanning Trees in Graph Theory (https://openstax.org/r/ +Trees_in_Graph_Theory) +Formula Review +12.2 Graph Structures +For the Sum of Degrees Theorem, +or +The number of edges in a complete graph with +vertices is the sum of the whole numbers from 1 to +, +. +The number of edges in a complete graph with +vertices is +. +12.7 Hamilton Cycles +The number of ways to arrange +distinct objects is +. +The number of distinct Hamilton cycles in a complete graph with +vertices is +. +12.9 Traveling Salesperson Problem +• +In a complete graph with +vertices, the number of distinct Hamilton cycles is +. +• +In a complete graph with +vertices, there are at most +different weights of Hamilton cycles. +12.10 Trees +• +The number of edges in a tree graph with +vertices is +. A connected graph with n vertices and +edges is a +tree graph. +Projects +Everyone Gets a Turn! – Graph Colorings +Let’s put your knowledge of graph colorings to work! Your task is to plan a field day following these steps. +1. +Select between seven and ten activities for your field day. You can look online for ideas. +2. +Create a survey asking for the participants to select the three to five events in which they would most like to +participate. Survey between seven and ten people. +1404 +12 • Chapter Summary +Access for free at openstax.org + +3. +Use the results of your survey to create a graph in which each vertex represents one of the events. A pair of vertices +will be adjacent if there is at least one participant who would like to participate in both events. +4. +Find a minimum coloring for the graph. Explain how you found it and how you know the chromatic number of the +graph. +5. +Use your solution to part d to determine the minimum number of timeslots you must use to ensure that everyone +has the opportunity to participate in their top three events. +6. +Find the complement of the graph you created. Explain what the edges in this graph represent. +A Beautiful Day in the Neighborhood – Euler Circuits +Let’s apply what you have learned to the community in which you live. Using resources such as your county’s property +appraiser’s website, create a detailed graph of your neighborhood in which vertices represent turns and intersections. +Represent a large enough part of your community to include no fewer than 10 intersections or turns. Then use your +graph to answer the following questions. +1. +Label the edges of your graph. +2. +Determine if your graph is Eulerian. Explain how you know. If it is not, eulerize it. +3. +Find an Euler circuit for your graph. Give the sequence of vertices that you found. +4. +What does the Euler circuit you found in part c represent for your community? +5. +Describe an application for which this Euler circuit might be used. +Dream Vacation – Hamilton Cycles and Paths +Where in the world would you like to travel most: the Eiffel Tower in Paris, a Broadway musical in New York city, a bike +tour of Amsterdam, the Tenerife whale and dolphin cruises in the Canary Islands, the Giza Pyramid in Cairo, or maybe +the Jokhang Temple in Tibet? Let's plan your dream vacation! +1. +Which four destinations are at the top of your bucket list? +2. +Draw a complete weighted graph with five vertices representing the four destinations and your home city, and the +weights representing the cost of travel between cities. +3. +Use a website (such as Travelocity (https://openstax.org/r/travelocity)) to find the best airfare between each pair of +cities. List the airlines and flight numbers along with the prices. Include cost for ground transportation from the +nearest airport if there is no airport at the destination you want to visit. +4. +Use the nearest neighbor algorithm to find a Hamilton cycle of low weight beginning and ending in your hometown. +What is the weight of this circuit and what does it represent? +5. +Use the brute force method to find a Hamilton cycle of lowest weight beginning and ending in your hometown. +What is the weight of this circuit? Is it the same or different from the weight of the Hamilton cycle you found in +Exercise 4? +6. +Suppose that instead of returning home, you planned to move to your favorite location on the list, but you wanted +to stop at the other three destinations once along the way. Where would you move? List all Hamilton paths between +your hometown and your favorite location. +7. +Find the weights of all the Hamilton paths you found in Exercise 6. +12 • Chapter Summary +1405 + +Chapter Review +Graph Basics +For the following exercises, use the given figure. +1. Determine the number of vertices in Graph S. +2. Identify the graph with the fewest edges. +3. Name the vertices in Graph U. +4. Identify any pairs of vertices in Graph S that are not adjacent. +5. Which graphs only has vertices of degree 2? +6. Identify the graph in which the sum of the degrees of the vertices is 16. +7. Amazon.com has a network of warehouses that are used to move packages around the United States. Delivery +trucks from warehouse deliver packages to other locations. These mail trucks also pick up packages to bring back +to their home warehouse. Explain how a graph or multigraph might be drawn to model this scenario by identifying +the objects that could be represented by vertices and the connections that could be represented by edges. Indicate +whether a graph or a multigraph would be a better model. +8. Eduardo has two groups of four friends, group A and group B. Within each group, each of the members of the +group are friends with each other but not with those in the other group. The individuals in group A have no other +friends, but the individuals in group B each has their own group of four friends, and the individuals within those +groups are all friends with each other, but with no one outside their own group. Draw a graph to represent the +given scenario. +Graph Structures +For the following exercises, use the given figure. +9. Explain why Graph B is not a subgraph of Graph C. +10. Identify any graphs that have a quadrilateral cyclic subgraph and name the vertices. +11. Identify a clique in Graph C by listing its vertices. +12. Draw a graph with the following characteristics: largest clique has 4 vertices, a pentagonal cyclic subgraph, +exactly two vertices of degree 4. +13. How many edges are in a complete graph with 12 vertices? +14. How many triangles are in a complete graph with 11 vertices? +Comparing Graphs +15. Identify three differences between Graph 1 and Graph 2 that demonstrate the graphs are not isomorphic. +For the following exercises, use the given figure. +1406 +12 • Chapter Summary +Access for free at openstax.org + +16. Determine if Graph 3 is isomorphic to Graph 4. If so, identify a correspondence between the vertices which +demonstrates the isomorphism. If not, identify at least two characteristics that verifies this. +17. Determine if Graph 3 is isomorphic to Graph 5. If so, identify a correspondence between the vertices that +demonstrates the isomorphism. If not, identify a characteristic that verifies this. +18. Consider Graph 4 and Graph 5. Determine if one graph is a subgraph of the other. If so, give a correspondence +between vertices that demonstrates this relationship. If not, identify conflicting characteristics. +For the following exercises, use the given figure. +19. Draw the complements of Graphs 7 and 9. Determine whether the graphs you drew are isomorphic to each +other and explain how you know. Use this information to determine whether Graphs 7 and 8 are isomorphic. +20. Draw the complements of Graphs 7 and 10. Determine whether the graphs you drew are isomorphic to each +other and explain how you know. Use this information to determine whether Graphs 7 and 10 are isomorphic. +21. Draw the complements of Graphs 8 and 9. Determine whether the graphs you drew are isomorphic to each +other and explain how you know. Use this information to determine whether Graphs 8 and 9 are isomorphic. +Navigating Graphs +For the following exercises, use Graph A in the given figure. Consider each sequence of vertices. Determine if it is only a +walk, both a walk and a path, both a walk and a trail, all three, or none of these. +22. e → d → b → e → f +23. e → b → d → e → f → c → b → d +24. e → f → b → d +For the following exercises, use Graph K in Figure 12.352. Identify each sequence of vertices as a closed walk, circuit +(closed trail), and/or directed cycle (closed path). Indicate all that apply. +25. n → o → q → n → p → m → n +26. m → n → o → q → n → p → o → n → m +27. p → o → q → n → m → p +For the following exercises, use Graphs A and K in Figure 12.352. +28. Are Graphs A and K planar graphs? What does your answer tell you about the chromatic number of each graph? +29. How many vertices are in the largest complete subgraph of Graph A? What does this tell you about the +chromatic number of Graph A? +30. Create a coloring of Graph A, which uses exactly four colors, or explain why it is not possible. +31. Create a coloring of Graph K, which uses exactly two colors, or explain why it is not possible. +32. Determine the chromatic number of Graph A. Give a coloring that supports your conclusion. +33. Determine the chromatic number of Graph K. Give a coloring that supports your conclusion. +12 • Chapter Summary +1407 + +Euler Circuits +For the following exercises, use the graphs and multigraphs shown. Identify any graphs and/or multigraphs with the +given characteristics. If there are none, state so. +34. connected +35. disconnected +36. Eulerian +Euler Trails +For the following exercises, use the graphs and multigraphs in Figure 12.354. +37. List the set of vertices for each component in Graph 13. +38. Determine whether the sequence of edges represents an Euler circuit in Multigraph 15: +K → L → N → M → O → S → T → Q → U → P → R +39. Find an Euler circuit beginning and ending at vertex g in Graph 12 if one exists. Otherwise, explain how you +know such an Euler circuit does not exist. +40. Give an example of a pair of edges that could be duplicated to eulerize Multigraph 14. +For the following exercises, use the graphs and multigraphs in Figure 12.354. Identify any graphs and/or multigraphs +with the given characteristics. If there are none, state so. +41. Exactly two vertices of odd degree +42. Has an Euler trail +43. Has exactly one local bridge +For the following exercises, use the graphs and multigraphs in Figure 12.354. In each exercise a graph and a sequence +of vertices are given. +44. Determine whether the sequence of edges, A → B → C → H → G → D → F → E, is an Euler trail, an Euler circuit, or +neither for the graph. If it is neither, explain why. +45. Suppose that an edge were added to Graph 11 between vertices s and w. Determine if the graph would have an +Euler trail or an Euler circuit, and find one. +Hamilton Cycles +For the following exercises, refer to the graph in the given figure. +46. A student has been asked to use Fleury’s algorithm to construct an Euler trail in the given graph. The student +decides to begin the trail at vertex d. Is this a good choice, why or why not? +47. A student who is using Fleury’s algorithm to construct an Euler trail has decided to begin with f → d → a → b → +…. If the student is off to a good start, help the student by completing the Euler trail. If the student has made an +error, explain the error. +48. Use Fleury’s algorithm to construct an Euler trail for Graph 16 beginning at the vertex of your choice. +For the following exercises, use the graphs from the given figures to determine whether the sequence of vertices in the +given graph is a Hamilton cycle, an Euler circuit, both, or neither. +1408 +12 • Chapter Summary +Access for free at openstax.org + +49. Graph F. a → b → e → c → b → d → c → a +50. Graph K. m → n → q → o → p → m +51. Graph A. b → d → e → f → c → b → e +For the following exercises, evaluate the factorial expression for the given value of +. +52. +53. +54. Calculate the number of distinct Hamilton cycles in a complete graph with 13 vertices. +For the following exercises, use the figure shown to find the weight of each Hamilton cycle. +55. q → t → w → x → u → y → v → s → r → q +56. w → x → y → u → v → s → r → q → t → w +Hamilton Paths +For the following exercises, use the figure shown to determine whether the sequence of vertices in the given graph is a +Hamilton path, an Euler trail, both, or neither. +12 • Chapter Summary +1409 + +57. Graph A. e → b → c → f → e → b → e +58. Graph A. b → c → f → e → d → b → e +59. Graph K. n → q → o → p → m +60. Graph K. o → q → m → n → p +Recall the three common scenarios in which it is not possible to have a Hamilton between two vertices. +Scenario 1: If an edge ab is a bridge, then there is no Hamilton path between a pair of vertices that are on the same +side of edge ab. +Scenario 2: If an edge ab is a bridge with at least three components on each side, then there is no Hamilton path +beginning or ending at a or b. +Scenario 3: If a graph is composrd of two cycles joined only at a single vertex p, and v is any vertex that is not adjacent +to p, then there are no Hamilton paths beginning or ending at p. +For the following exercises, identify the scenario that guarantees there is no Hamilton path between the given pair of +vertices in the given graph. +61. Vertices a and e. +62. Vertices d and f. +The principal of an elementary school plans to visit each classroom exactly once before leaving for the day. The floor +plan of the school is given. Use this information for the following exercises. +63. Draw a graph to represent the floor plan for the elementary school in which each vertex is a room and an edge +between two vertices indicates that there is a path between the two rooms that does not pass the door to +another room. +64. Use your graph to find a route that the principal can take beginning at the Front Office in room A, visiting each +room exactly once (without passing by another room), and ending at room F, which is next to the exit. +65. Would the route you found be best to trace out a Hamilton path, an Euler trail, both, or neither in the graph you +drew? Explain how you know. +Traveling Salesperson Problem +For the following exercises, determine whether the algorithm described is a greedy algorithm or a brute force +algorithm. +66. A lumber distributor is loading pallets onto trucks with the intention of using the fewest trucks possible to send +a shipment. The distributor loads the pallets with the greatest length that will fit on the truck first and +continues loading until no more pallets will fit. Then the next truck is loaded using the same approach. +67. A traveler wants to visit five cities by airplane. The traveler lists all the possible orders in which the cities can be +visited then calculates the best airfare for each itinerary and selects the least expensive option. +For the following exercises, list all the distinct Hamilton cycles beginning at the given vertex in the given graph. Indicate +which pairs of Hamilton cycles are reverses of each other. +1410 +12 • Chapter Summary +Access for free at openstax.org + +68. Graph 17, vertex a +69. Graph 18, vertex m +For the following exercises, find a Hamilton cycle of least weight for the given graph in Figure 12.362, beginning at the +given vertex, and using the brute force method. What is the weight of the cycle? +70. Graph 18; vertex m +71. Graph 17; vertex a +For the following exercises, find a Hamilton cycle of low weight for the given graph in Figure 12.362, beginning at the +given vertex, and using the nearest neighbor method. What is the weight of the cycle? +72. Graph 18 vertex m +73. Graph 17, vertex a +74. The products at a particular factory are manufactured in phases. The same equipment is utilized for each phase, +but it must be formatted differently to accomplish different tasks. The transition time to convert between a format +for one task and another task varies. The times are given in the table below. +Task +A +B +C +D +E +F +G +A +B +C +D +E +F +G +Use the table to draw a graph in which the vertices represent tasks A, C, D, F, and G, and the weights of the edges +indicate the times in minutes to transition between pairs of tasks. Use the nearest neighbor method to find an +order in which to complete tasks A, C, D, F, and G, which keeps the transition times down and ends with the same +set up as it began so that the factory is ready start the next batch. Assume that there are no restrictions on which +tasks can be completed in which order. Hint: The nearest neighbor algorithm may give a different result depending +on which vertex is the starting vertex; so, you must check all possibilities. +Trees +For the following exercises, draw a graph with the given characteristics. +75. A tree with eight vertices, exactly two of degree three. +76. A connected graph with eight vertices, exactly two of degree 3, which is not a tree. +12 • Chapter Summary +1411 + +For the following exercises, identify each type of graph in the given figure: tree graph, star graph, starlike graph, line +graph, lobster graph, caterpillar graph, and/or forest graph. +77. Graph 19 +78. Graph 20 +For the following exercises, use the figure shown to draw three possible spanning trees that fit the given description. +79. Three spanning trees of Graph A, which include both edges be and de. +80. Three spanning trees of Graph K, which include edges mn and oq, but do not include no. +For the following exercises, use Kruskal’s Algorithm to find a minimum spanning tree of the given graph. Graph it and +calculate its weight. +81. Graph 17 +82. Graph 18 +Chapter Test +For the following Exercises, use the figure shown. +1. Name the edges in Graph T. +2. Identify the graph(s) with six edges. +For the following exercises, use the figure shown. +1412 +12 • Chapter Summary +Access for free at openstax.org + +3. Identify any of the graphs that is a subgraph of Graph D. +4. Identify any graphs with no cyclic subgraphs of any size. +5. Consider Graph 4 and Graph 6 in the given figure. Determine if one graph is a subgraph of the other. If so, give a +correspondence between vertices that demonstrates this relationship. If not, identify conflicting characteristics. +6. For the following exercise, use Graph A in the given figure. Consider each sequence of vertices. Determine if it is +only a walk, both a walk and a path, both a walk and a trail, all three, or none of these. +d → e → b → c → f +7. For the following exercise, use Graphs A and K in Figure 12.369. Determine the chromatic number of Graph K. Give +a coloring that supports your conclusion. +For the following exercise, use the graphs and multigraphs in the given figure. +8. Identify any graphs and/or multigraphs that are not Eulerian. If there are none, state so. +9. List the set of vertices for each component in Graph 11. +10. Find an Euler circuit beginning and ending at vertex b in Graph 12 if one exists. +11. Use Fleury’s algorithm to construct an Euler trail for the given graph beginning at vertex f of your choice. +12. Use the graphs shown to determine whether the sequence of vertices d → b → c → f → e → d is a Hamilton cycle, an +Euler circuit, both, or neither. +12 • Chapter Summary +1413 + +13. Calculate the number of distinct Hamilton cycles in a complete graph with 15 vertices. +14. Use the figure shown to find the weight of the given Hamilton cycle: +t → w → x → u → y → v → s → r → q → t +For the following exercises, use this information: For the Halloween celebration at an elementary school, the students +from Classroom I will visit every classroom once and return to their own classroom. The floor plan of the school is +given. +15. Draw a graph to represent the classrooms of the elementary school in which each vertex is a classroom and an +edge between two vertices indicates that there is a path between the two rooms that does not pass the door to +another classroom. +16. Use your graph to find a Hamilton circuit beginning and ending at I. +17. Explain what the Hamilton circuit you found represents for the students in Classroom I. +18. Find a Hamilton cycle of low weight for Graph 18, beginning at vertex q, and using the nearest neighborhood +method. What is the weight of the cycle? +1414 +12 • Chapter Summary +Access for free at openstax.org + +19. Use Kruskal’s Algorithm to find a minimum spanning tree for the below graph. Graph the tree and give its weight. +12 • Chapter Summary +1415 + +1416 +12 • Chapter Summary +Access for free at openstax.org + +Figure 13.1 Math can be found in many different areas and subjects. (credit: modification of work "Chemistry/Physics +Library" by University Libraries/Flickr, CC BY 2.0) +Chapter Outline +13.1 Math and Art +13.2 Math and the Environment +13.3 Math and Medicine +13.4 Math and Music +13.5 Math and Sports +Introduction +Where do we find math around us? Math can be found in areas that are expected and sometimes in areas that are +surprising. There are many ways that mathematical concepts, such as those in this text, are infused in the world around +us. In this chapter, we will explore a sampling of five distinct areas from everyday life where math’s impact plays a +meaningful role. +Math's impact on art can be found in numerical relationships that are known to create or enhance beauty. The Fibonacci +numbers are one mathematical example that can be found in nature such as in petal count of a rose. On a different note, +a mathematical exploration can aid in making a convincing argument on how we can positively impact our environment. +Whether looking at the choices of a single individual or the larger impact offered from a collaborative effort, there are +measurable responses to positively address climate change. Turning to medicine, which has been a topic of global +importance in recent years, we will explore how math is used to determine drug dosage rates and test the validity of a +vaccine. Switching back to items of aesthetic nature, we will examine some foundational components of music which, +like art, brings beauty and joy to our lives. Finally, we will explore some ways that math is used in sports to predict future +performance and analyze tournaments styles. +13 +MATH AND... +13 • Introduction +1417 + +13.1 Math and Art +Figure 13.2 Sunflower seeds appear in a pattern that involves the Fibonacci sequence. (credit: “Sunflower Surprise” by +frankieleon/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Identify and describe the golden ratio. +2. +Identify and describe the Fibonacci sequence and its application to nature. +3. +Apply the golden ratio and the Fibonacci sequence relationship. +4. +Identify and compute golden rectangles. +Art is the expression or application of human creative skill and imagination, typically in a visual form such as painting or +sculpture, producing works to be appreciated primarily for their beauty or emotional power. +Oxford Dictionary +Art, like other disciplines, is an area that combines talent and experience with education. While not everyone considers +themself skilled at creating art, there are mathematical relationships commonly found in artistic masterpieces that drive +what is considered attractive to the eye. Nature is full of examples of these mathematical relationships. +Enroll in a cake decorating class and, when you learn how to create flowers out of icing, you will likely be directed as to +the number of petals to use. Depending on the desired size of a rose flower, the recommendation for the number of +petals to use is commonly 5, 8, or 13 petals. If learning to draw portraits, you may be surprised to learn that eyes are +approximately halfway between the top of a person’s head and their chin. Studying architecture, we find examples of +buildings that contain golden rectangles and ratios that add to the beautifying of the design. The Parthenon (Figure +13.3), which was built around 400 BC, as well as modern-day structures such the Washington Monument are two +examples containing these relationships. These seemingly unrelated examples and many more highlight mathematical +relationships that we associate with beauty in artistic form. +1418 +13 • Math and... +Access for free at openstax.org + +Figure 13.3 The Parthenon in Greece demonstrates the golden ratio. (credit: “Parthénon” by Julien Maury/Flickr, Public +Domain Mark 1.0) +Golden Ratio +The golden ratio, also known as the golden proportion, is a ratio aspect that can be found in beauty from nature to +human anatomy as well as in golden rectangles that are commonly found in building structures. The golden ratio is +expressed in nature from plants to creatures such as the starfish, honeybees, seashells, and more. It is commonly noted +by the Greek letter ϕ (pronounced “fee”). +, which has a decimal value approximately equal to 1.618. +Consider Figure 13.3: Note how the building is balanced in dimension and has a natural shape. The overall structure +does not appear as if it is too wide or too tall in comparison to the other dimensions. +Figure 13.4 Vitruvian Man by Leonardo da Vinci (credit: "Vitruvian Man" by Leonardo da Vinci/Wikimedia Commons, +Public Domain) +The golden ratio has been used by artists through the years and can be found in art dating back to 3000 BC. Leonardo da +Vinci is considered one of the artists who mastered the mathematics of the golden ratio, which is prevalent in his artwork +such as Virtuvian Man (Figure 13.4). This famous masterpiece highlights the golden ratio in the proportions of an ideal +body shape. +The golden ratio is approximated in several physical measurements of the human body and parts exhibiting the golden +ratio are simply called golden. The ratio of a person’s height to the length from their belly button to the floor is ϕ or +approximately 1.618. The bones in our fingers (excluding the thumb), are golden as they form a ratio that approximates +ϕ. The human face also includes several ratios and those faces that are considered attractive commonly exhibit golden +ratios. +13.1 • Math and Art +1419 + +EXAMPLE 13.1 +Using Golden Ratio and a Person’s Height +If a person’s height is 5 ft 6 in, what is the approximate length from their belly button to the floor rounded to the nearest +inch, assuming the ratio is golden? +Solution +Step 1: Convert the height to inches +Step 2: Calculate the length from the belly button to the floor, +. +The length from the person’s belly button to the floor would be approximately 41 in. +YOUR TURN 13.1 +1. If a person’s height is 6 ft 2 in, what is the approximate length from their belly button to the floor rounded to the +nearest inch if the ratio is golden? +Fibonacci Sequence and Application to Nature +Figure 13.5 Rose petals appear in a Fibonacci spiral. (credit: “rilke4” by monchoohcnom/Flickr, Public Domain Mark 1.0) +The Fibonacci sequence can be found occurring naturally in a wide array of elements in our environment from the +number of petals on a rose flower to the spirals on a pine cone to the spines on a head of lettuce and more. The +Fibonacci sequence can be found in artistic renderings of nature to develop aesthetically pleasing and realistic artistic +creations such as in sculptures, paintings, landscape, building design, and more. It is the sequence of numbers +beginning with 1, 1, and each subsequent term is the sum of the previous two terms in the sequence (1, 1, 2, 3, 5, 8, 13, +…). +The petal counts on some flowers are represented in the Fibonacci sequence. A daisy is sometimes associated with +plucking petals to answer the question “They love me, they love me not.” Interestingly, a daisy found growing wild +typically contains 13, 21, or 34 petals and it is noted that these numbers are part of the Fibonacci sequence. The number +1420 +13 • Math and... +Access for free at openstax.org + +of petals aligns with the spirals in the flower family. +EXAMPLE 13.2 +Applying the Fibonacci Sequence to Rose Petals +Suppose you were creating a rose out of icing, assuming a Fibonacci sequence in the petals, how many petals would be +in the row following a row containing 13 petals? +Solution +The number of petals on a rose is often modeled with the numbers in the Fibonacci sequence, which is 1, 1, 2, 3, 5, 8, +13,…, where the next number in the sequence is the sum of +. There would be 21 petals on the next row of the +icing rose. +YOUR TURN 13.2 +1. If a circular row on a pinecone contains 21 scales and models the Fibonacci sequence, approximately how many +scales would be found on the next circular row? +Golden Ratio and the Fibonacci Sequence Relationship +Mathematicians for years have explored patterns and applications to the world around us and continue to do so today. +One such pattern can be found in ratios of two adjacent terms of the Fibonacci sequence. +Recall that the Fibonacci sequence = 1, 1, 3, 5, 8, 13,… with 5 and 8 being one example of adjacent terms. When +computing the ratio of the larger number to the preceding number such as 8/5 or 13/8, it is fascinating to find the +golden ratio emerge. As larger numbers from the Fibonacci sequence are utilized in the ratio, the value more closely +approaches ϕ, the golden ratio. +EXAMPLE 13.3 +Finding Golden Ratio in Adjacent Fibonacci Terms +The 24th Fibonacci number is 46,368 and the 25th is 75,025. Show that the ratio of the 25th and 24th Fibonacci numbers +is approximately ϕ. Round your answer to the nearest thousandth. +Solution +; The ratio of the 25th and 24th term is approximately equal to the value of ϕ rounded to the +nearest thousandth, 1.618. +YOUR TURN 13.3 +1. The 23rd Fibonacci number is 28,657 and the 24th is 46,368. Show that the ratio of the 24th and 23rd Fibonacci +numbers is approximately +. Round your answer to the nearest thousandth. +13.1 • Math and Art +1421 + +Figure 13.6 The pyramids of Giza in Egypt (credit: “Giza Pyramids” by Vincent Brown/Flickr, CC BY 2.0) +Golden Rectangles +Turning our attention to man-made elements, the golden ratio can be found in architecture and artwork dating back to +the ancient pyramids in Egypt (Figure 13.6) to modern-day buildings such as the UN headquarters. The ancient Greeks +used golden rectangles—any rectangles where the ratio of the length to the width is the golden ratio—to create +aesthetically pleasing as well as solid structures, with examples of the golden rectangle often being used multiple times +in the same building such as the Parthenon, which is shown in Figure 13.3. Golden rectangles can be found in twentieth- +century buildings as well, such as the Washington Monument. +Looking at another man-made element, artists paintings often contain golden rectangles. Well-known paintings such as +Leonardo da Vinci’s The Last Supper and the Vitruvian Man contain multiple golden rectangles as do many of da Vinci’s +masterpieces. +Whether framing a painting or designing a building, the golden rectangle has been widely utilized by artists and are +considered to be the most visually pleasing rectangles. +EXAMPLE 13.4 +Finding Golden Rectangle in Frames +A frame has dimensions of 8 in by 6 in. Calculate the ratio of the sides rounded to the nearest thousandth and determine +if the size approximates a golden rectangle. +Solution +8/6 = 1.333; A golden rectangle’s ratio is approximately 1.618. The frame dimensions are close to a golden rectangle. +YOUR TURN 13.4 +1. A frame has dimensions of 10 in by 8 in. Calculate the ratio of the sides rounded to the nearest thousandth and +determine if the size approximates a golden rectangle. +1422 +13 • Math and... +Access for free at openstax.org + +PEOPLE IN MATHEMATICS +M.C. Escher +Figure 13.7 M.C. Escher (credit: "M.C. Escher" by Hans Peters (ANEFO)/Dutch National Archives, CC0 1.0 Public +Domain) +Mauritis Cornelis Escher was a Dutch-born world-famous graphic artist and his work can be found in murals, stamps, +wallpaper designs, illustrations in books, and even carpets. Over his lifetime, M.C. Escher created hundreds of +lithographs and wood engravings as well as more than 2,000 sketches. +Escher’s work is characterized with the infusion of geometric designs that obey most of the mathematical rules. If you +study his work closely, you can see where he breaks a mathematical relationship to create famous illusions such as +soldiers marching around the top of a square turret where the soldiers appear to be always going uphill but are +contained on a single set of stairs in a square. Look closely and the golden ratio as well as golden rectangles abound +in Escher’s work. +Like many famous people, M.C. Escher did not find success in his early school years. Before finding success, Escher +failed his final school exam and quit a short stint in architecture. Finding a graphic arts teacher who recognized +Escher’s talent, Escher completed art school and enjoyed traveling through Italy, where he found much of his +inspiration for his work. +Check Your Understanding +1. What is the value of the golden ratio to the nearest thousandth? +2. What are the first 10 terms of the Fibonacci sequence? +3. What is a golden rectangle? +SECTION 13.1 EXERCISES +1. A person’s height is 5 ft 2 in. What is the approximate length from their belly button to the floor rounded to the +nearest inch? +2. A person’s height is 6 ft 3 in. What is the approximate length from their belly button to the floor rounded to the +nearest inch? +3. A person’s length from their belly button to the floor is 3 ft 11 in. What is the person’s approximate height rounded +13.1 • Math and Art +1423 + +to the nearest inch? +4. A person’s length from their belly button to the floor is 58 in. What is the person’s approximate height rounded to +the nearest inch? +5. The spikes on a pineapple mirror the Fibonacci sequence. If a row on a pineapple contains five spikes, +approximately how many spikes would be found on the next larger row of spikes? +6. The leaves on a plant mirror the Fibonacci sequence. If a set of leaves on the plant contains 5 leaves, how many +leaves would be found on the previous smaller set of leaves? +7. The spines on a head of lettuce mirror the Fibonacci sequence. If a head of lettuce contains 13 spines, +approximately how many spines would be found on the next inside layer? +8. The seeds on a sunflower mirror the Fibonacci sequence. If a circular layer on the sunflower contains 55 seeds, +approximately how many seeds would be found on the next larger circular layer? +9. The segments on a palm frond mirror the Fibonacci sequence. If a palm frond contains 89 segments, +approximately how many segments would be found on the next larger palm frond? +10. The 19th term of the Fibonacci sequence is 4,181 and the 20th term is 6,765. What is the 21st term of the +sequence? +11. The 23rd term of the Fibonacci sequence is 28,657 and the 24th term is 46,368. What is the 22nd term of the +sequence? +12. The 18th term of the Fibonacci sequence is 2,584 and the 20th term is 6,765. What is the 19th term of the +sequence? +13. The 25th term of the Fibonacci sequence is 75,025 and the 20th term is 6,765. What is the 24th term of the +sequence? +14. The 10th Fibonacci number is 55 and the 11th is 89. Show that the ratio of the 11th and 10th Fibonacci numbers is +approximately +. Round your answer to the nearest thousandth. +15. The 23rd Fibonacci number is 28,657 and the 24th is 46,368. Show that the ratio of the 24th and 23rd Fibonacci +numbers is approximately +. Round your answer to the nearest ten-thousandth. +16. The 22nd Fibonacci number is 17,711 and the 21st is 10,946. Show that the ratio of the 22nd and 21st Fibonacci +numbers is approximately +. Round your answer to the nearest ten-thousandth. +17. The 16th term of the Fibonacci sequence is 987. Use the approximate value of +of 1.618 to estimate the 15th +term. Round your answer to the nearest whole number. +18. The 26th term of the Fibonacci sequence is 121,393. Use the approximate value of +of 1.618 to estimate the 25th +term. Round your answer to the nearest whole number. +19. A frame has dimensions of 20 in by 24 in. Calculate the ratio of the sides rounded to the nearest tenth and +determine if the size approximates a golden rectangle. +20. A fence has dimensions of 75 in by 45 in. Calculate the ratio of the sides rounded to the nearest tenth and +determine if the size approximates a golden rectangle. +21. A frame has a length of 50 in. Calculate the width rounded to the nearest inch if the frame is to be a golden +rectangle. +1424 +13 • Math and... +Access for free at openstax.org + +13.2 Math and the Environment +Figure 13.8 Solar panels harness the sun's energy to power homes, businesses, and various methods of transportation. +(credit: modification of work “Craters of the Moon solar array” by NPS Climate Change Response/Flickr, Public Domain +Mark 1.0) +Learning Objectives +After completing this section, you should be able to: +1. +Compute how conserving water can positively impact climate change. +2. +Discuss the history of solar energy. +3. +Compute power needs for common devices in a home. +4. +Explore advantages of solar power as it applies to home use. +Climate change and emissions management have been debated topics in recent years. However, more and more people +are recognizing the impacts that have resulted in temperature changes and are seeking timely and effective action. The +World Meteorological Organization shared in a June 2021 publication that “2021 is a make-or-break year for climate +action, with the window to prevent the worst impacts of climate change—which include ever more frequent more +intense droughts, floods and storms—closing rapidly.” The problem no longer belongs to a few countries or regions but +rather is a worldwide concern measured with increasing temperatures leading to decreased glacier coverage and +resulting rise in sea levels. +The good news is, there are small steps that each of us can do that collectively can positively impact climate change. +Making a Positive Impact on Climate Change—Water Usage +Our use of water is one element that impacts climate change. Having access to clean, potable water is critical for not +only our health but also for the health of our ecosystem. About 1 out of 10 people on our planet do not have easy access +to clean water to drink. As each of us conserves water, we prolong the life span of fresh water from our lakes and rivers +and also reduce the impact on sewer systems and drainage in our communities. Additionally, as we conserve water, we +also conserve electricity that is used to bring water to and in our homes. So, what can we do to help conserve water? +EXAMPLE 13.5 +Brushing Your Teeth (One Person’s Contribution) +Brushing your teeth with the water running continually uses about 4 gal of water. Turning the faucet off when you are +not rinsing uses less than one-fourth of a gallon of water. Considering the recommendation to brush your teeth twice a +day, how much water would be saved in a week if the faucet was off when not rinsing? +Solution +Leaving the water running continually: +Step 1: Calculate gallons used not with water running continually: +Brushing twice a day for 7 days using 4 gal of water for each brushing +Step 2: Calculate gallons used turning the faucet off when you are not rinsing: +Brushing twice a day for 7 days using 0.25 gal of water for each brushing +13.2 • Math and the Environment +1425 + +Step 3: Calculate savings: +During one week, 52.5 gal of water would be saved if one person turned the faucet off except when rinsing when +brushing your teeth. +YOUR TURN 13.5 +1. Brushing your teeth with the water running continually uses about 4 gal of water. Turning the faucet off when +you are not rinsing uses less than one-fourth of a gallon of water. Consider the recommendation to brush your +teeth twice a day, how much water would be saved in a month (4 weeks) if the faucet was off when not rinsing? +EXAMPLE 13.6 +Brushing Your Teeth (Multiple People’s Contribution – Town) +Using the data in Example 13.5, how much water would be saved in a month if one-fifth of a town’s population of 15,000 +turned the faucet off when brushing their teeth except when rinsing? +Solution +Step 1: From Example 13.5, we found that 1 person saves 52.5 gal per week. +Step 2: Calculate the population to save water: +Step 3: One-fifth of a town’s population turning off the faucet when brushing their teeth for a month: +During one month, 630,000 gallons of water would be saved if one-fifth of a town of 15,000 people turned the faucet off +except when rinsing when brushing their teeth. +YOUR TURN 13.6 +1. Using the data in Your Turn 13.5, how much water would be saved in a month if one-third of a town’s population +of 15,000 turned the faucet off when brushing their teeth except when rinsing? +EXAMPLE 13.7 +Brushing Your Teeth (Multiple People’s Contribution – State) +Using the data in Example 13.5, how much water would be saved in a year if one-fourth of the population of the state of +Minnesota, which is approximately 5.6 million people, turned the faucet off when brushing their teeth except when +rinsing for a year (52 weeks)? +Solution +Step 1: From Example 13.5, we found that 1 person saves 52.5 gal per week. +Step 2: Calculate the population to save water: +Step 3: One-fourth of a town’s population turning off the faucet when brushing their teeth for a month: +1426 +13 • Math and... +Access for free at openstax.org + +During one year, 3,822 million gal of water would be saved if one-fourth of the state of Minnesota turned the faucet off +except when rinsing when brushing their teeth. +YOUR TURN 13.7 +1. Using the data in Example 13.5, how much water would be saved in a year if one-sixth of the population of the +state of Florida, which is approximately 21.6 million people, turned the faucet off when brushing their teeth +except when rinsing for a year (52 weeks)? +History of Solar Energy +In the mid-1800s, Willoughby Smith discovered photoconductive responsiveness in selenium. Shortly thereafter, William +Grylls Adams and Richard Evans Day discovery that selenium can produce electricity if exposed to the sun was a major +breakthrough. Less than 10 years later, Charles Fritts invented the first solar cells using selenium. Jumping a mere 100 +years later, Bell Labs in the United States produced the first practical photovoltaic cells in the mid-1950s and developed +versions used to power satellites in the same decade. +Solar panel use has exploded in recent decades and is now used by residences, organizations, businesses, and +government buildings such as the White House, space to power satellites, and various methods of transportation. One +reason for the expansion is a continuing drop in cost combined with an increase in performance and durability. In the +mid-1950s, the cost of a solar panel was around $300 per watt capability. Twenty years later, the cost was a third of the +1950s’ cost. Currently, solar panel cost has dropped to less than $1 per watt while decreasing in size as well as increasing +in longevity. The dropping price and improved performance has moved solar to a modest investment that can pay for +itself in less than half the time of systems from 15 years ago. +WHO KNEW? +Solar Power’s Age +The sun has been harnessed by humans for centuries. The earliest recorded use of tapping the sun’s energy for +power dates back to the seventh century BC when man focused the sun’s rays through a magnifying glass to create +fire. Four thousand years later, we find historical record of using mirrors to focus the sun and light torches, often for +ceremonial proceedings. Use of the sun to light torches continued through the centuries and has been recorded by +various cultures including the Chinese civilization in 20 AD and beyond. +In more recent years, the sun was harnessed to power ovens on ships traversing to oceans in the 1700s. At the same +time, the power of the sun was utilized to power steamboats through the 1800s. Mária Telkes, a Hungarian-born +American scientist, invented a widely deployed solar seawater distiller used on World War II life rafts. Soon after, she +partnered with architect Eleanor Raymond to design the first modern home to be completely heated by solar power. +Air warmed on rooftop collectors transferred heat to salts, which stored the heat for later use. +Although solar panels as we know them today are relatively new in history, use of the sun to harness power is much +older. +Compute Power Needs for Common Home Devices +A kilowatt (kW) is 1,000 watts (W). A kilowatt-hour (kWh) is a measurement of energy use, which is the amount of +energy used by a 1,000-watt device to run for an hour. Using the definition of a kilowatt-hour, to calculate how long it +would take to consume 1 kWh of power, we divide 1,000 by the watts use of a device. +FORMULA +For example, a 75 W bulb would take +to use 1 kW of power. +13.2 • Math and the Environment +1427 + +FORMULA +EXAMPLE 13.8 +Calculating the Kilowatt-Hours Needed to Run a Television +A 48 in plasma television uses about 200 W. How many kilowatt-hours are needed to run the television in a month if the +television is one for an average of 2.5 hours a day? +Solution +Step 1: +to use 1 kW +Step 2: +of use +Step 3: +The television will consume about 15 kW in a month. +YOUR TURN 13.8 +1. A 40 in plasma television uses about 175 W. How many kilowatt-hours are needed to run the television in a +month if the television is on for an average of 3 hours a day? +EXAMPLE 13.9 +Calculating the Cost to Run a Refrigerator +A medium-sized Energy Star–rated refrigerator uses about 575 W and runs for about 8 hours per day. What is the +monthly (30 days) cost of running the refrigerator if the electric rate is 12 cents per kilowatt-hour? +Solution +Step 1: Calculate the watts per day: +Step 2: Calculate the kilowatt-hours. +Step 3: Calculate the daily cost. +Step 4: Calculate the monthly cost. +It would cost about $16.50 to run the refrigerator for a month. +YOUR TURN 13.9 +1. A dorm-sized Energy Star–rated refrigerator uses about 375 W and runs for about 9 hours per day. What is the +monthly cost of running the refrigerator if the electric rate is 14 cents per kilowatt-hour? +1428 +13 • Math and... +Access for free at openstax.org + +EXAMPLE 13.10 +Calculating the Kilowatt-Hours to Run an Oven +An electric oven is labeled as 4,000 W. How much would it cost to bake a cake for 30 minutes if the electric rate is 14 +cents per kilowatt-hour? +Solution +Step 1: Determine the time it takes to use 1 kW of power: +For every 15 minutes, the oven uses 1 kW of power. +Step 2: Determine how many kilowatt-hours are needed to bake the cake for 30 minutes: +Step 3: Calculate the cost of the oven usage: +It would cost about 28 cents to bake the cake. +YOUR TURN 13.10 +1. A toaster oven is labeled as 2,000 W. How much would it cost to warm leftovers from a meal for 15 minutes if the +electric rate is 12 cents per kilowatt-hour? +Solar Advantages +There are multiple advantages that solar power can offer us today including reducing greenhouse gas and CO2 +emissions, powering vehicles, reducing water pollution, reducing strain on limited supply of other power options such as +fossil fuels. We will look further at reducing greenhouse gas and CO2 emissions. +Any gas that prevents infrared radiation from escaping Earth's atmosphere is a greenhouse gas. There are 24 currently +identified greenhouse gases of which carbon dioxide is one. When measuring the impact of any of the greenhouse +gases, the measurements are given in units of carbon dioxide emissions. For this reason, greenhouse gas and carbon +dioxide have become interchangeable in discussions. +PEOPLE IN MATHEMATICS +Charles Fritts and Mohammad M. Atalla +Charles Fritts, a New York inventor, is credited with creating the first solar cell, which he installed on a rooftop in New +York City in 1884. While the solar cell was not very efficient, having a rate of conversion between 1 to 2%, this was a +major step early in solar power energy. Today’s solar cells have an efficiency on average of 15 to 20%, which yields a +notably higher impact. Nonetheless, the work that Fritts successfully completed marked the start of solar energy +through the use of photovoltaic solar panels in the United States. +Mohamed M. Atalla was an Egyptian-born scientist who moved to the United States to complete his studies, and +undertook research and development at Bell Laboratories in New Jersey. Many of the early efficiency gains in solar +cells were due to his development of processes for using silicon within electronic devices. Atalla's work led to the +invention of silicon transistors and microchips (including his own invention of the MOSFET, the most widely used +transistor in the world), and quickly increased the efficiency of solar cells. +13.2 • Math and the Environment +1429 + +EXAMPLE 13.11 +Calculating the Solar Power for Average Home Use in Kilowatts +If a home uses approximately 30 kW of electricity per day, what size solar system would be needed to fuel 80% of a +home’s needs for a month (30 days)? +Solution +A solar system capable of producing 720 kW a month would be needed. +YOUR TURN 13.11 +1. A tiny home uses approximately 12 kW of electricity per day. What size solar system would be needed to fuel 80% +of a home’s needs for a month (30 days)? +Check Your Understanding +4. What is the relationship between 1 kW and watts? +5. An average bath uses about 35 gal of water. A water-saving showerhead uses approximately 2 gal of water per +minute. If a person typically takes one bath a week, how much water is saved by replacing the baths with showers +lasting 5 minutes over the course of a month? +6. How long does it take for a 30 W Google Nest audio to use 1 kW of power? +7. How long would it take for a 60 W bulb to consume 1 kW of power? Round your answer to the nearest hour. +SECTION 13.2 EXERCISES +1. A typical showerhead uses 5 gal of water per minute. A water-saving showerhead uses approximately 2 gal of +water per minute. How much water would one person save in a month if they take a 6-minute shower 4 times a +week? +2. An average toilet uses 5 gal of water per flush. A high-efficiency toilet uses about 1.25 gal per flush. How much +water would a household save in a week if the toilet was flushed 8 times a day? +3. When washing dishes, leaving the faucet running utilizes about 15 gal every 5 minutes, where filling the sink and +turning the faucet off except to rinse the dishes uses about 5 gal for washing a dishpan-sized load of dishes. How +much water is saved by not leaving the faucet running if it takes 10 minutes to wash a dishpan-sized load of dishes +each day for a month containing 30 days? +4. Leaving the water running when washing your hands consumes about 4 gal of water, whereas turning the water +off when lathering reduces the water used to 1 gal. How much water is saved in an apartment of 6 people in an +academic term of 16 weeks if each person averages washing their hands 3 times a day? +5. How long would it take for an energy-saving light bulb to consume 1 kW of power if the bulb is rated at 7.5 W? +Round your answer to the nearest hour. +6. How long would it take for a 120 W light bulb to consume 1 kW of power? Round your answer to the nearest hour. +7. A portable television uses about 80 W per hour. How many kilowatt-hours are needed to run the television during a +3-day trip if the television is run for an average of 5.5 hours a day? +8. A flat iron to straighten hair is rated at 331 W. If it is used for 15 minutes a day, 5 times a week, how much would it +cost a user over the course of a month if the electric rate is 13 cents per kilowatt-hour? Round your answer to the +nearest cent. +9. You purchase a window air conditioner for your apartment living room rated at 1,000 W. If you run the air +conditioner for an average of 3 hours a day for a week, how much would it cost if the electric rate is 12 cents per +kilowatt-hour? +10. A cabin uses approximately 25 kW of electricity per day. What size solar system would be needed to fuel 90% of the +1430 +13 • Math and... +Access for free at openstax.org + +cabin’s needs for a month if the cabin is used 2 days a week? +13.3 Math and Medicine +Figure 13.9 Shoppers wear masks during the Covid-19 pandemic. (credit: "True Covid Scene - Mask Buying" by Joey +Zanotti/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Compute the mathematical factors utilized in concentrations/dosages of drugs. +2. +Describe the history of validating effectiveness of a new drug. +3. +Describe how mathematical modeling is used to track the spread of a virus. +The pandemic that rocked the world starting in 2020 turned attention to finding a cure for the Covid-19 strain into a +world race and dominated conversations from major news channels to households around the globe. News reports +decreeing the number of new cases and deaths locally as well as around the world were part of the daily news for over a +year and progress on vaccines soon followed. How was a vaccine able to be found so quickly? Is the vaccine safe? Is the +vaccine effective? These and other questions have been raised through communities near and far and some remain +debatable. However, we can educate ourselves on the foundations of these discussions and be more equipped to analyze +new information related to these questions as it becomes available. +Concentrations and Dosages of Drugs +Consider any drug and the recommended dosage varies based on several factors such as age, weight, and degree of +illness of a person. Hospitals and medical dispensaries do not stock every possible needed concentration of medicines. +Drugs that are delivered in liquid form for intravenous (IV) methods in particular can be easily adjusted to meet the +needs of a patient. Whether administering anesthesia prior to an operation or administering a vaccine, calculation of the +concentration of a drug is needed to ensure the desired amount of medicine is delivered. +The formula to determine the volume needed of a drug in liquid form is a relatively simple formula. The volume needed +is calculated based on the required dosage of the drug with respect to the concentration of the drug. For drugs in liquid +form, the concentration is noted as the amount of the drug per the volume of the solution that the drug is suspended in +which is commonly measured in g/mL or mg/mL. +Suppose a doctor writes a prescription for 6 mg of a drug, which a nurse calculates when retrieving the needed +prescription from their secure pharmaceutical storage space. On the shelves, the drug is available in liquid form as 2 mg +per mL. This means that 1 mg of the drug is found in 0.5 mL of the solution. Multiplying 6 mg by 0.5 mL yields 3 mL, +which is the volume of the prescription per single dose. +FORMULA +. +A common calculation for the weight of a liquid drug is measured in grams of a drug per 100 mL of solution and is also +called the percentage weight by volume measurement and labeled as % w/v or simply w/v. +Note that the units for a desired dose of a drug and the units for a solution containing the drug or pill form of the +drug must be the same. If they are not the same, the units must first be converted to be measured in the same +13.3 • Math and Medicine +1431 + +units. +Suppose you visit your doctor with symptoms of an upset stomach and unrelenting heartburn. One possible recourse is +sodium bicarbonate, which aids in reducing stomach acid. +EXAMPLE 13.12 +Calculating the Quantity in a Mixture +How much sodium bicarbonate is there in a 250 mL solution of 1.58% w/v sodium bicarbonate? +Solution +sodium bicarbonate in 100 mL. If there is 250 mL of the solution, we have 2.5 times as much sodium +bicarbonate as in 100 mL. Thus, we multiply 1.58 by 2.5 to yield 3.95 g sodium bicarbonate in 250 mL solution. +YOUR TURN 13.12 +1. How many milligrams of sodium chloride are there in 200 mL of a 0.9% w/v normal saline solution? +EXAMPLE 13.13 +Calculating the Quantity of Pills Needed +A doctor prescribes 25.5 mg of a drug to take orally per day and pills are available in 8.5 mg. How many pills will be +needed each day? +Solution +The prescription and the pills are in the same units which means no conversions are needed. We can divide the units of +the drug prescribed by the units in each pill: +. So, 3 pills will be needed each day. +YOUR TURN 13.13 +1. How many pills would be needed for a patient who has been prescribed 25.5 mg of a drug if each pill contains +4.25 mg. +EXAMPLE 13.14 +Calculating the Drug Dose in Milligrams, Based on Patient Weight +A patient is prescribed 2 mg/kg of a drug to be delivered intramuscularly, divided into 3 doses per day. If the patient +weighs 45 kg, how many milligrams of the drug should be given per dose? +Solution +Step 1: Calculate the total daily dose of the drug based on the patient’s weight (measured in kilograms): +Step 2: Divide the total daily dose by the number of doses per day: +The patient should receive 30 mg of the drug in each dose. +YOUR TURN 13.14 +1. A patient is prescribed 1.4 mg/kg of a drug to be delivered intramuscularly, divided into 2 doses per day. If the +1432 +13 • Math and... +Access for free at openstax.org + +patient weighs 60 kg, how many milliliters of the drug should be given per dose? +Note that the units for a patient’s weight must be compatible with the units used in the medicine measurement. If +they are not the same, the units must first be converted to be measured in the same units. +EXAMPLE 13.15 +Calculating the Drug Dose in Milliliters, Based on Patient Weight +A patient is prescribed 2 mg/kg of a drug to be delivered intramuscularly, divided into 3 doses per day. If the drug is +available in 20 mg/mL and the patient weighs 60 kg, how many milliliters of the drug should be given per dose? +Solution +Step 1: Calculate the total daily dose of the drug (measured in milligrams) based on the patient’s weight (measured in +kilograms): +Step 2: Calculate the volume in each dose: +Step 3: Calculate the volume based on the strength of the stock: +The patient should receive 2 mL of the stock drug in each dose. +YOUR TURN 13.15 +1. A patient is prescribed 2.5 mg/kg of a drug to be delivered intramuscularly, divided into 2 doses per day. If the +drug is available in 5 mg/mL and the patient weighs 52 kg, how many milligrams of the drug should be given per +dose? +WHO KNEW? +Math Statistics from the CDC +The Centers for Disease Control and Prevention (CDC) states that about half the U.S. population in 2019 used at least +one prescription drug each month, and about 25% of people used three or more prescription drugs in a month. The +resulting overall collective impact of the pharmaceutical industry in the United States exceeded $1.3 trillion a year +prior to the 2020 pandemic. +Validating Effectiveness of a New Vaccine +The process to develop a new vaccine and be able to offer it to the public typically takes 10 to 15 years. In the United +States, the system typically involves both public and private participation in a process. During the 1900s, several vaccines +were successfully developed, including the following: polio vaccine in the 1950s and chickenpox vaccine in the 1990s. +Both of these vaccines took years to be developed, tested, and available to the public. Knowing the typical timeline for a +vaccine to move from development to administration, it is not surprising that some people wondered how a vaccine for +Covid-19 was released in less than a year’s time. +Lesser known is that research on coronavirus vaccines has been in process for approximately 10 years. Back in 2012, +concern over the Middle Eastern respiratory syndrome (MERS) broke out and scientists from all over the world began +working on researching coronaviruses and how to combat them. It was discovered that the foundation for the virus is a +spike protein, which, when delivered as part of a vaccine, causes the human body to generate antibodies and is the +13.3 • Math and Medicine +1433 + +platform for coronavirus vaccines. +When the Covid-19 pandemic broke out, Operation Warp Speed, fueled by the U.S. federal government and private +sector, poured unprecedented human resources into applying the previous 10 years of research and development into +targeting a specific vaccine for the Covid-19 strain. +PEOPLE IN MATHEMATICS +Shibo Jiang +Dr. Shibo Jiang, MD, PhD, is co-director the Center for Vaccine Development at the Texas Children’s Hospital and head +of a virology group at the New York Blood Center. Together with his colleagues, Jiang has been working on vaccines +and treatments for a range of viruses and infections including influenzas, HIV, Sars, HPV and more recently Covid-19. +His work has been recognized around the world and is marked with receiving grants amounting to over $20 million +from U.S. sources as well as the same from foundations in China, producing patents in the United States and China +for his antiviral products to combat world concerns. +Jiang has been a voice for caution in the search for a vaccine for Covid-19, emphasizing the need for caution to ensure +safety in the development and deployment of a vaccine. His work and that of his colleagues for over 10 years on other +coronaviruses paved the way for the vaccines that have been shared to combat the Covid-19 pandemic. +Mathematical Modeling to Track the Spread of a Vaccine +With a large number of people receiving a Covid-19 vaccine, the concern at this time is how to create an affordable +vaccine to reach people all over the world. If a world solution is not found, those without access to a vaccine will serve as +incubators to variants that might be resistant to the existing vaccines. +As we work to vaccinate the world, attention continues with tracking the spread of the Covid-19 and its multiple variants. +Mathematical modeling is the process of creating a representation of the behavior of a system using mathematical +language. Digital mathematical modeling plays a key role in analyzing the vast amounts of data reported from a variety +of sources such as hospitals and apps on cell phones. +When attempting to represent an observed quantitative data set, mathematical models can aid in finding patterns and +concentrations as well as aid in predicting growth or decline of the system. Mathematical models can also be useful to +determine strengths and vulnerabilities of a system, which can be helpful in arresting the spread of a virus. +The chapter on Graph Theory explores one such method of mathematical modeling using paths and circuits. Cell phones +have been helpful in tracking the spread of the Covid-19 virus using apps regulated by regional government public +health authorities to collect data on the network of people exposed to an individual who tests positive for the Covid-19 +virus. +1434 +13 • Math and... +Access for free at openstax.org + +PEOPLE IN MATHEMATICS +Gladys West +Figure 13.10 Gladys West (credit: "Dr. Gladys West Hall" by The US Air Force/Wikimedia Commons, Public Domain) +Dr. Gladys West is a mathematician and hidden figure with a rich résumé of accomplishments spanning Air Force +applications and work at NASA. Born in 1930, West rose and excelled both academically and in her professional life at +a time when Black women were not embraced in STEM positions. One of her many accomplishments is the Global +Positioning System (GPS) used on cell phones for driving directions. +West began work as a human computer, someone who computes mathematical computations by hand. Considering +the time and complexity of some calculations, she became involved in programming computers to crunch +computations. Eventually, West created a mathematical model of Earth with detail and precision that made GPS +possible, which is utilized in an array of devices from satellites to cell phones. The next time you tag a photo or obtain +driving directions, you are tapping into the mathematical modeling of Earth that West developed. +Consider the following graph (Figure 13.11): +Figure 13.11 Contact Tracing for Math 111 Section 1 +At the center of the graph, we find Alyssa, whom we will consider positive for a virus. Utilizing the technology of phone +apps voluntarily installed on each phone of the individuals in the graph, tracking of the spread of the virus among the 6 +individuals that Alyssa had direct contact with can be implemented, namely Suad, Rocio, Braeden, Soren, and Sandra. +Let’s look at José’s exposure risk as it relates to Alyssa. There are multiple paths connecting José with Alyssa. One path +includes the following individuals: José to Mikaela to Nate to Sandra to Alyssa. This path contains a length of 4 units, or +people, in the contact tracing line. There are 2 more paths connecting José to Alyssa. A second path of the same length +consists of José to Lucia to Rocio to Braeden to Alyssa. Path 3 is the shortest and consists of José to Lucia to Rocio to +13.3 • Math and Medicine +1435 + +Alyssa. Tracking the spread of positive cases in the line between Alyssa and José aids in monitoring the spread of the +infection. +Now consider the complexity of tracking a pandemic across the nation. Graphs such as the one above are not practical to +be drawn on paper but can be managed by computer programs capable of computing large volumes of data. In fact, a +computer-generated mathematical model of contact tracing would look more like a sphere with paths on the exterior as +well as on the interior. Mathematical modeling of contact tracing is complex and feasible through the use of technology. +EXAMPLE 13.16 +Using Mathematical Modeling +For the following exercises, use the sample contact tracing graph to identify paths (Figure 13.12). +Figure 13.12 Contact Tracing for ECON 250 Section 1 +1. +How many people have a path of length 2 from Jeffrey? +2. +Find 2 paths between Kayla and Rohan. +3. +Find the shortest path between Yara and Kalani. State the length and people in the path. +Solution +1. +5 (Lura, Naomi, Kalani, Vega, Yara) +2. +Answers will vary. Two possible answers are as follows: +a. +Kayla, Jeffrey, Rohan +b. +Kayla, Lura, Yara, Lev, Vega, Uma, Kalani, Rohan +3. +Length is 4. People in path = Yara, Lev, Vega, Uma, Kalani +YOUR TURN 13.16 +For the following exercises, use Figure 13.12. +1. List all the names of those who have a path length of three from Uma. +2. What is the length of the shortest path from Naomi to Vega? +Check Your Understanding +8. What two pieces of information are needed to calculate the volume of a prescription drug to be dispensed? +9. Research on coronavirus vaccines began in 2020. +a. +True +b. +False +10. What is mathematical modeling, and how is it used in the world of medicine with pandemics? +SECTION 13.3 EXERCISES +1. How many grams of sodium bicarbonate are contained in a 300 mL solution of 1.35% w/v sodium bicarbonate? +1436 +13 • Math and... +Access for free at openstax.org + +2. How many grams of sodium bicarbonate are contained in a 175 mL solution of 1.85% w/v sodium bicarbonate? +3. Using a saline solution that is 0.75% w/v, how many milligrams of sodium chloride are in 150 mL? +4. Using a saline solution that is 1.25% w/v, how many milligrams of sodium chloride are in 200 mL? +5. A prescription calls for a patient to receive 23 mg daily of a drug to be taken in pill form for 5 days. If the pills are +available in 5.75 mg, how many pills will the patient need for the full prescription run? +6. A prescription calls for a patient to receive 21 mg daily of a drug to be taken in pill form daily. If the pills are +available in 3.5 mg, how many pills will the patient need each day? +7. A patient is prescribed 4mg/kg of a drug to be delivered daily intramuscularly, divided into 2 doses. If the patient +weighs 30 kg, how many milligrams of the drug would be needed for each dose? +8. A patient is prescribed 1.5 mg/kg of a drug to be delivered intramuscularly, divided into 3 doses per day. If the +drug is available in 12.5 mg/mL and the patient weighs 50 kg, how many milliliters of the drug would be given per +dose? +9. A patient is prescribed 0.5 mg/kg of a drug to be delivered intramuscularly, divided into 2 doses per day. If the +drug is available in 2.5 mg/mL and the patient weighs 45 kg, how many milliliters of the drug would be given per +dose? +10. A patient is prescribed 1.5 mg/kg of a drug to be delivered intramuscularly, divided into 3 doses per day. If the +drug is available in 30 mg/mL and the patient weighs 54 kg, how many milliliters of the drug would be given per +dose? +For the following exercises, use the mathematical modeling graph showing contact tracing for students in a particular +class. +Contract Tracing for Students in a Class +11. List the people who have a length of 2 from Justin. +12. Find 2 paths of with a length of 3 from Emmet. +13. Find the shortest path from Aili to Kalina. +14. Which people does the model show as directly in contact with Nara? +15. Find the shortest path from Tai to Hani. +16. Of the 12 people in the model, how many have a path of 2 or less from Justin? +13.3 • Math and Medicine +1437 + +13.4 Math and Music +Figure 13.13 Friends sing music together around a campfire. (credit: modification of work “Fire is hot! (2)” by Chetan +Sarva/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Describe the basics of frequency related to sound. +2. +Describe the basics of pitch as it relates to music. +3. +Describe and evaluate musical notes, half-steps, whole steps, and octaves. +4. +Describe and find frequencies of octaves. +“The world’s most famous and popular language is music.” +Psy, South Korean singer, rapper, songwriter, and record producer +Imagine a world without music and many of us would struggle to fill the void. Music uplifts, inspires, heals, and generally +adds dimension to virtually every aspect of our lives. But what is music? For some it is a song; for others it may be the +sounds of birds or the rhythmic sound of drumming or a myriad of other sounds. Whatever you consider music, it is all +around us and is an integral part of our lives. “Music can raise someone’s mood, get them excited, or make them calm +and relaxed. Music also—and this is important—allows us to feel nearly or possibly all emotions that we experience in +our lives. The possibilities are endless” (Galindo, 2009). +What music you listen to can impact your mood and emotions. In similar fashion, the music we choose can often tell +those around us something about our current moods and emotions. Consider the music you may have been listening to +as you today or even as you are reading this text. What cues to your mood do your music selections share? Albert +Einstein is quoted as saying, “If I were not a physicist, I would probably be a musician. I often think in music. I live my +daydreams in music. I see my life in terms of music.” What clues do your recent music choices say about your mood or +how your day is going? +Basics of Frequency as It Relates to Sound +Every sound is created by an object vibrating and these vibrations travel in waves that are captured by our ears. Some +vibrations we may be able to see, such as a plucked guitar string moving, whereas other vibrations we may not be able +to see, such as the sound created when we hold our breath when accidentally dropping our cell phone on a hard floor. +We don’t see the vibrations of our cell phone hitting the floor; however, any audible sound created in the fall is the result +of vibrations in the form of sound waves, which can be pictured similarly to waves moving through the ocean. +The waves of sounds each have a frequency, or rate of vibration of sound waves, that measures the number of waves +completed in a single second and are measured in hertz (Hz; one Hz is one cycle per second). Louder sounds have +stronger vibrations or are created closer to our ear. The further an ear is from the source of the sound, the quieter the +sound will appear. +Sounds range in frequency from 16 Hz to ultrasonic values, with humans able to hear sounds in a frequency range of +about 20 Hz to 20,000 Hz. Adults lose the ability to hear the upper end of the range and typically top out in the ability to +hear in a frequency of 15,000–17,000 Hz. Sounds with a frequency above 17,000 Hz are less likely to be heard by adults +while still being audible to children. +While frequency plays a key role in audible sounds, so too does the sound level, which can be measured in decibels (dB), +which are the units of measure for the intensity of a sound or the degree of loudness. As a sound level increases, the +decibel level increases. +1438 +13 • Math and... +Access for free at openstax.org + +A person with average hearing can hear sounds down to 0 dB. Those with exceptionally good hearing can hear even +quieter sounds, down to approximately –5 dB. The following table includes sample sounds with their related decibel +values. +Sound +Decibels (dB) +Firecrackers +140 +Take-off of military jet from aircraft carrier +130 +Clap of thunder +120 +Auto horn standing next to the vehicle +110 +Outboard motor +100 +Motorcycle +90 +Noise inside a car in city traffic +80 +Typical washing machine +70 +Public conversation, such as at a restaurant +60 +Private conversation +50 +Hum of a computer with fan blowing +40 +Quiet whisper +30 +Swishing leaves +20 +Regular breathing +10 +Lowest typical sound audible by teenagers +0 +EXAMPLE 13.17 +Selecting Decibel Value of Sounds +Select the most representative decibel value for each of the following sounds: +1. +car wash: 25 dB, 55 dB, 85 dB +2. +vacuum cleaner: 15 dB, 70 dB, 90 dB +3. +ship’s engine room: 30 dB, 65 dB, 95 dB +4. +approaching subway car: 70 dB, 100 dB, 120 dB +Solution +1. +85 dB +2. +70 dB +3. +95 dB +4. +100 dB +13.4 • Math and Music +1439 + +YOUR TURN 13.17 +Match each of the following sounds with a corresponding decibel level: live outdoor concert, birds chirping, window +air conditioner, garbage disposal. +1. 40 dB +2. 60 dB +3. 80 dB +4. 100 dB +Basics of Pitch +When considering the various sound levels the human ear can hear, the ear perceives sound both from the frequency +level and the pitch of a sound. The quality of the sound is referred to as pitch, the tonal quality of a sound and how high +or low the tone. Sounds with a high frequency have a high pitch, such as 900 Hz, and sounds with a low pitch have a low +frequency, such as 50 Hz. +Let’s take a look at frequency and pitch using a string instrument such as a guitar or piano. When a string is plucked on a +guitar or a key is played on a piano, the related string vibrates at a frequency that is related to the length and thickness +of the string. The frequency is measurable and has a singular value. The pitch of the note played is open for +interpretation, as the pitch is a function of personal opinion. +WHO KNEW? +Graphing Calculators and Music +It may be interesting to note that a TI-84 and TI-Nspire graphing calculators can be utilized to tune a musical +instrument by measuring the frequency of a note using a small plug-in accessory that captures the sound waves from +a note and displays the corresponding frequency. Using the displayed frequency, an instrumentalist can then make +the needed adjustment to perfectly tune an instrument. This method of tuning an instrument can be helpful whether +a novice player or a seasoned instrumentalist because the instrument can be tuned precisely to the correct frequency. +Note Values, Half-Steps, Whole Steps, and Octaves +“There are not more than five musical notes, yet the combinations of these five give rise to more melodies than can ever +be heard.” +Sun Tzu, Chinese strategist +Figure 13.14 Piano Keys and Notes (credit: modification of work "Contemplate" by Walt Stoneburner/Flickr, CC BY 2.0) +Moving our exploration to note values, the frequency of all notes is well defined by a specific and unique frequency for +1440 +13 • Math and... +Access for free at openstax.org + +each note that is measurable. We will explore keys on a keyboard to discuss notes that have the same relationships with +any instrument or musical piece. +Let’s look at Figure 13.14. The white keys are labeled with the letters A–G and the photo begins with middle C, which can +be found in the middle of a keyboard. This labeling of the keys repeats across an entire keyboard and keys to the right +have a higher pitch and frequency than keys to the left. Each of the keys correlates to a musical note. +Movement up or down between any two consecutive keys (black and white) or notes constitutes a half-step. Movement +of one half-step sometimes involves a sharp (#) or a flat (♭) symbol. For example, D# is one half-step above D and D♭ is +one half-step below D. Note that this is not always true as one half-step above B is C, and one half-step below F is E. In +similar fashion, a whole step is movement up or down between any two half-steps on a keyboard. +EXAMPLE 13.18 +Identifying Half-Steps +Name which keys are one half-step up and one half-step down from the following: +1. +D +2. +E +3. +G# +Solution +1. +up D#, down D♭ +2. +up F, down E♭ +3. +up A, down G +YOUR TURN 13.18 +Name which keys are one half-step up and one half-step down from the following: +1. F# +2. B +3. G♭ +EXAMPLE 13.19 +Identifying Whole Steps +Name which keys are one whole step up and one whole step down from the following: +1. +F# +2. +E +3. +A♭ +Solution +1. +Up G#, down E +2. +Up F#, down D +3. +Up B♭, down G♭ +YOUR TURN 13.19 +Name which keys are one whole step up and one whole step down from the following: +1. D♭ +2. C# +3. E +You may have noticed that there are eight letters of the alphabet used to label notes. Selecting any one note and +counting up 12 half-steps you will find that the numbering for notes begins at the same value as you started from. This +13.4 • Math and Music +1441 + +collection of 12 consecutive half-notes is called an octave and is a basic foundational component in music theory. +EXAMPLE 13.20 +Listing All Notes in an Octave +List the 12 notes forming an octave, beginning with the note C. +Solution +C, C#, D, D#, E, F, F#, G, G#, A, A#, B +YOUR TURN 13.20 +1. List the 12 notes forming an octave, beginning with the note G. +Frequencies of Octaves +Notes that are one octave apart have the same name and are related in frequency values. Given the frequency of any +note, the frequency of same note one octave higher is doubled and this pattern continues as you move up and down the +notes on a keyboard or any other musical instrument. Song writers and singers use this knowledge to change the pitch +of a note up or down to align with a person’s vocal range. Regardless of which C is played or sung, the pitch is the same +and the frequency is related by a power or two. +Labeled keys on a keyboard are numbered for ease in identification. For example, middle C is labeled as C4 on a full +keyboard as it is the fourth C from the left in a set of eight notes. The frequency of C4 is 262 Hz, rounded to the nearest +whole number. +EXAMPLE 13.21 +Calculating the Frequency Values of Octaves +Given that the frequency of C4 is 262 Hz, find the approximate frequency of C6. +Solution +The frequency of each consecutive higher octave doubles. Given that the frequency of C4 is 262 Hz, the frequency of C5 is +found by doubling the frequency of C4, which is 524 Hz. In similar fashion, the frequency of C6 is found by doubling the +frequency of C5, which yields 1,048 Hz. +YOUR TURN 13.21 +1. Given that the frequency of E4 is 330 Hz, find the approximate frequency of E2 rounded to the nearest whole +number. +PEOPLE IN MATHEMATICS +David Cope +David Cope has a somewhat eclectic list of job titles ranging from author and music professor to scientist and artificial +intelligence researcher. Cope combined his interests when he developed software that can analyze a piece of music +and create a new and unique musical piece in the same style as the original. Some of his well-known products have +been based off of the classical music of Mozart, creating what has been called Mozart’s “42nd Symphony,” as well as +other genres including opera and a range of current music styles. Cope has also composed original musical pieces in +collaboration with a computer. +We have explored some basics components of frequency, pitch, note relationships, and octaves, which are building +blocks of music. It may be exciting to learn that the mathematical relationships found in music are vast and grow in +1442 +13 • Math and... +Access for free at openstax.org + +complexity beyond the math commonly studied in high school. +WHO KNEW? +Spotify Royalty Payments +Streaming services have grown exponentially in popularity thanks in large part to customized music listening through +cell phone use and devices such as Google as well as Amazon Echo and Alexa devices for home and vehicles, adding +to ways that artists are paid royalties. Spotify, which was launched in 2008, typically plays artists $0.06 per time a song +is streamed, with some artists receiving up to $0.84 per play amounting to over $9 billion in revenue for Spotify in +2020. Since 2014, Spotify’s revenue has grown over a billion dollars a year, with roughly half of their revenue being +paid out in royalties, which was good news for artists during the Covid-19 pandemic when in-person concerts and +shopping were hindered. +Check Your Understanding +11. What is the difference between frequency and pitch of a note? +12. Are higher frequencies associated with higher or lower pitch notes? +13. What frequency range can humans hear? +14. As a sound appears louder, does the decibel value increase or decrease? +15. What is the lowest decibel value that most people can hear? +16. What is the typical audible frequency range that adults can hear? +17. How does the frequency of a note change when increased by one octave? +SECTION 13.4 EXERCISES +For the following exercises, select the most representative decibel level for each sound. +1. Gas lawn mower: 70 dB, 100 dB, 120 dB +2. Radio playing loud enough to sing along to at home: 20 dB, 50 dB, 70 dB +3. Office noise on a floor of people working at desks: 40 dB, 60, dB, 80 dB +4. Conversation in a quiet room: 60 dB, 80 dB, 100 dB +5. Fast-moving train passing by when sitting in a car about 20 ft from the tracks with windows down: 40 dB, 60 dB, +80 dB +For the following exercises, state the requested note. +6. What is one half-step above G#? +7. What is one whole step down from G♭? +8. What is three half-steps above D? +9. What is two whole steps down from A#? +10. What is two half-steps down from E? +11. What is an octave above B4? +12. What is two octaves below F5? +13. What is three octaves above G2? +14. What is an octave below G2? +15. Given that the frequency of +, what is the frequency of A1? +16. Given that the approximate frequency of +, what is the approximate frequency of F5, rounded to the +nearest whole number? +17. Given that the approximate frequency of +, what is the approximate frequency of A6#, rounded to the +nearest whole number? +18. Given that the approximate frequency of +, what is the approximate frequency of D2, rounded to the +nearest whole number? +19. Given that the approximate frequency of +, what is the approximate frequency of G3#, rounded to the +13.4 • Math and Music +1443 + +nearest whole number? +20. Given that the frequency of +, what is the frequency of A5? +13.5 Math and Sports +Figure 13.15 Fans support their team by attending games and wearing team gear. (credit: “Cheering Touchdowns” by +Steven Miller/Flickr, CC BY 2.0) +Learning Objectives +After completing this section, you should be able to: +1. +Describe why data analytics (statistics) is crucial to advance a team’s success. +2. +Describe single round-robin method of tournaments. +3. +Describe single-elimination method of tournaments. +4. +Explore math in baseball, fantasy football, hockey, and soccer (projects at the end of the section). +Sports are big business and entertainment around the world. In the United States alone, the revenue from professional +sports is projected to bring in over $77 billion, which includes admission ticket costs, merchandise, media coverage +access rights, and advertising. So, whether or not you enjoy watching professional sports, you probably know someone +who does. Some celebrities compete to be part of half-time shows and large companies vie for commercial spots that are +costly but reach a staggering number of viewers, some who only watch the half-time shows and advertisements. +Data Analytics (Statistics) Is Crucial to Advance a Team’s Success +Analyzing the vast data that today’s world has amassed to find patterns and to make predictions for future results has +created a degree field for data analytics at many colleges, which is in high demand in places that might surprise you. +One such place is in sports, where being able to analyze the available data on your team’s players, potential recruits, +opposing team strategies, and opposing players can be paramount to your team’s success. +Hollywood turned the notion of using data analytics into a major motion picture back in 2011 with the release of +Moneyball, starring Brad Pitt, which grossed over $110 million. The critically acclaimed movie, based on a true story as +shared in a book by Michael Lewis, follows the story of a general manager for the Oakland Athletics who used data +analytics to take a team comprised of relatively unheard of players to ultimately win the American League West title in a +year’s time. The win caught the eye of other team managers and owners, which started an avalanche of other teams +digging into the data of players and teams. +In today’s world of sports, a team has multiple positions utilizing data analytics from road scouts who evaluate a +potential recruit’s skills and potential to the ultimate position of general manager who is typically the highest-paid (non- +player) employee with the exception of the coaches. Being able to understand and evaluate the available data is big +business and is a highly sought after skill set. In college and professional sports, it is no longer sufficient to have a strong +playbook and great players. The science to winning is in understanding the math of the data and using it to propel your +team to excelling. +1444 +13 • Math and... +Access for free at openstax.org + +Single Round-Robin Tournaments +Figure 13.16 Single-Round Robin Tournament +A common tournament style is single round-robin tournaments (Figure 13.16), where each team or opponent plays every +other team or opponent, and the champion is determined by the team that wins the most games. Ties are possible and +are resolved based on league rules. +An advantage of the round-robin tournament style is that no one team has the advantage of seeding, which eliminates +some teams from playing against each other based on rank of their prior performance. Rather, each team plays every +other team, providing equal opportunity to triumph over each team. In this sense, round-robin tournaments are deemed +the fairest tournament style. +One hindrance to employing a round-robin-style tournament is the potential for the number of games involved in +tournament play to determine a winner. Determining the number of games can be found easily using a formula which, +as we will see, can quickly grow in the number of games required for a single round-robin tournament. +FORMULA +The number of games in a single round-robin tournament with +teams is +. +EXAMPLE 13.22 +Calculating the Number of Games in Single Round-Robin Tournaments +Find the number of games in a single round-robin tournament for each of the following numbers of teams: +1. +4 teams +2. +8 teams +3. +20 teams +Solution +1. +Using the formula with 4 teams yields +tournament games. +2. +Using the formula with 8 teams yields +tournament games. +3. +Using the formula with 20 teams yields +tournament games. +YOUR TURN 13.22 +Find the number of games in a single round-robin tournament for each of the following numbers of teams: +13.5 • Math and Sports +1445 + +1. 5 teams +2. 12 teams +3. 25 teams +As the examples show, single round-robin tournament play can quickly grow in the number of games required to +determine a champion. As such, some tournaments elect to employ variations of single round-robin tournament play as +well as other tournament styles such as elimination tournaments. +Figure 13.17 Single-Elimination Tournament (credit: final TK) +Single-Elimination Tournaments +When desiring a more efficient tournament style to determine a champion, one option is single-elimination tournaments +(Figure 13.17), where teams are paired up and the winner advances to the next round of play. The losing team is +defeated from tournament play and does not advance in the tournament, although some leagues offer consolation +matches. +FORMULA +The number of games in a single-elimination tournament with +teams is +. +EXAMPLE 13.23 +Calculating the Number of Games in Single-Elimination Tournaments +Find the number of games in a single-elimination tournament for each of the following numbers of teams: +1. +4 teams +2. +8 teams +3. +20 teams +Solution +1. +Using the formula with 4 teams yields +tournament games. +2. +Using the formula with 8 teams yields +tournament games. +3. +Using the formula with 20 teams yields +tournament games. +YOUR TURN 13.23 +Find the number of games in a single-elimination tournament for each of the following numbers of teams: +1446 +13 • Math and... +Access for free at openstax.org + +1. 5 teams +2. 12 teams +3. 25 teams +A single-elimination tournament offers an advantage over single round-robin tournament style of play in the number of +games needed to complete the tournament. As you can see, in comparing the number of games in a single round-robin +tournament in Example 13.22 with the number of games in single-elimination tournament as shown in Example 13.23, +the number of games required for single round-robin can quickly become unmanageable to schedule. +There are modifications to both the round-robin and elimination tournament styles such as double round-robin and +double-elimination tournaments. Next time you observe a college or professional sporting event, see if you can +determine the tournament style of play. +WHO KNEW? +Sports Popularity Shifts +Sport has been a popular entertainment venue for hundreds of years and the popularity of various sports shifts over +time as well as in different regions of the world today. In today’s world, the most popular sport is, overwhelmingly, +soccer, with over 4 billion fans followed by cricket with 2.5 billion fans. It may surprise you to learn that American +football doesn’t rank in the top 10 most popular sports in the world today, yet table tennis ranks in sixth place and +golf fills the last spot in the top 10 world sports. +In the early 1930s, baseball ranked a close second, with basketball virtually tying for third place. By the mid-1940s, +hockey slightly led in first place over basketball. Ten years later, hockey remained in the number one sport, but cricket +pushed basketball to third place. In the 1960s, soccer dominated in popularity. Over the next 30 years, basketball +dropped in popularity and there was much movement in the popularity of sports. By the late 1990s, soccer swept to +first place, where it has since continued to grow in popularity. +Check Your Understanding +18. How is data analytics used in sports today? +19. What is the most popular sport in today’s world? +20. What is the formula to compute the number of games played in a single round-robin tournament with +teams +assuming single round? +21. What is the formula to compute the number of games played in a single-elimination tournament? +SECTION 13.5 EXERCISES +1. Briefly describe data analytics. +2. Name a sport’s career title that relies on analysis of data. +For the following exercises, compute how many games would be played in the style of tournament and number of +teams given in each question. Assume all tournaments are single round-robin or single-elimination. +3. How many games would be played with 4 teams using a round-robin tournament? +4. How many games would be played with 8 teams using a round-robin tournament? +5. How many games would be played with 10 teams using a round-robin tournament? +6. How many games would be played with 25 teams using a round-robin tournament? +7. How many games would be played with 12 teams using a round-robin tournament? +8. How many games would be played with 4 teams using a single-elimination tournament? +9. How many games would be played with 9 teams using a single-elimination tournament? +10. How many games would be played with 15 teams using a single-elimination tournament? +11. How many games would be played with 50 teams using a single-elimination tournament? +12. How many games would be played with 24 teams using a single-elimination tournament? +13.5 • Math and Sports +1447 + +Chapter Summary +Key Terms +13.1 Math and Art +• +golden ratio +• +ϕ +• +Fibonacci sequence +• +golden rectangle +13.2 Math and the Environment +• +greenhouse gas +• +CO2 emissions +• +watt +• +kilowatt (kW) +13.3 Math and Medicine +• +concentrations/dosages of drugs +• +mathematical modeling +13.4 Math and Music +• +frequency +• +pitch +• +Hertz +• +decibel +• +half-step +• +whole step +• +flat +• +sharp +• +octave +13.5 Math and Sports +• +data analytics +• +seeding +Key Concepts +13.1 Math and Art +• +The golden ratio, ϕ, can be found in nature, and the relationship is often associated with beauty and balance. +• +The Fibonacci sequence reflects a pattern of numbers that can be found in various places in nature. The sequence +can be used to predict other values that follow the Fibonacci pattern. +• +State some naturally occurring applications of the Fibonacci sequence. +• +State some naturally occurring applications of the golden ratio. +• +Determine if a rectangle is golden. +• +State some artistic applications of the golden rectangle. +13.2 Math and the Environment +• +Recognize how water conservation by one person, family, community, or nation can positively impact the world’s +freshwater supply. +• +Recall components from the history of solar use by mankind. +• +Calculate electrical demand given watts. +• +Recognize advantages of residential solar power. +13.3 Math and Medicine +• +Compute volumes of prescription drugs in liquid and pill form. +• +Validate the effectiveness of a new drug. +• +Mathematical modeling can be used to describe and track the spread of a virus. +1448 +13 • Chapter Summary +Access for free at openstax.org + +13.4 Math and Music +• +As frequency of a sound increases, the pitch of the sound increases. +• +Hertz is a unit of measurement for frequency. +• +Decibel is a unit of measurement for the intensity of sound. +• +Half-steps and whole-steps describe one type of movement between two notes on a keyboard. +• +Octaves are a collection of any 12 consecutive notes, which is a foundation in music theory. +13.5 Math and Sports +• +Describe how a round-robin tournament is organized. +• +Compute the number of games played in a round-robin tournament. +• +Describe how a single-elimination tournament is organized. +• +Compute the number of games played in a single-elimination tournament. +Formula Review +13.2 Math and the Environment +13.3 Math and Medicine +. +13.5 Math and Sports +Number of games in a single round-robin tournament with +teams is +. +Number of games in a single-elimination tournament with +teams is +. +Projects +Lucas Sequence and Fibonacci Sequence +The Lucas numbers bear some similarity to the Fibonacci numbers and exhibit a stronger link to the golden ratio. +Edouard Lucas is credited with naming the Fibonacci numbers and the Lucas numbers were so named in his honor. The +Lucas numbers play a role in finding prime numbers that are utilized in encrypting data for actions such as using your +debit card to obtain money at a cash machine or when making a credit card purchase for point of sale as well as when +shopping online. +Complete the following questions to explore numbers in the Lucas sequence as well as their relationships to the +numbers in the Fibonacci sequence. +1. +Conduct an Internet search to find out what a Lucas number is and how the Lucas numbers are related to the +Fibonacci numbers. +2. +What are the first two numbers in the Lucas sequence? +3. +Describe how the next number in the Lucas sequence is determined and compare this to how the next number is +determined in the Fibonacci sequence. +4. +Complete the following table listing the first 10 terms in the Fibonacci and Lucas sequence: +Term +Fibonacci Numbers +Lucas Numbers +5. +Interestingly, many patterns can be found in looking at the relationships in the Fibonacci and Lucas numbers. Look +closely at the chart in question 3 to discover one such pattern. Observe the Fibonacci numbers in the third and fifth +terms and compare with the Lucas number in the fourth term of the sequence. Describe the pattern found. Does +this pattern continue in the table? +6. +Research the Fibonacci or the Lucas numbers to find an application in our world distinct from what has been shared +in this project and section. Write a paragraph sharing the findings of your research. +13 • Chapter Summary +1449 + +Solar Array for a Residence +One of the first steps in adding solar to a residence is determining the size of a system to achieve the desired output. In +this project, we will explore the solar needs of a residence and estimate needs of a solar array to supply electrical output +to meet various percentages of electrical need. +Step 1: Obtain an electric bill from your apartment/home. Find the average monthly or yearly usage if listed or call the +electric company to inquire. If an electric bill is not available, use the Internet to find an average monthly or yearly +electric usage for your area. +Step 2: Determine a daily and hourly usage. Divide the average monthly usage by 30 or the yearly average by 365. Divide +again by 24 to calculate an average hourly electric usage, which will yield the average kilowatt-hours for how much +electrical power your is being utilized in an hour. +Step 3: Multiply your average hourly use (kilowatts) by 1,000 to convert to watts. +Step 4: Use the Internet to determine the average daily peak hours of sunlight where you live. +Step 5: Divide your average hourly watts (Step 3) by the average daily peak hours (Step 4) to calculate the average +energy needed for a solar array to produce every hour. +Step 6: Determine the average energy needed in a solar array per hour to meet each of the following: +a. +coverage of average energy (Step 5) +b. +coverage of average energy +c. +coverage of average energy +Step 7: Using the values computed in Step 6, compute the residential savings based on an average cost of 12 cents per +watt. +Vaccine Validation +Validation of vaccines is a topic that exploded in the news when the Covid-19 pandemic spread across the world. As +governments and organizations looked for a vaccine to curb the spread and minimize the severity of infection, concern +was expressed by some for what appeared to be a quick discovery for a Covid-19 vaccine. +Conduct an Internet search to explore the following questions. Pay special attention to the sources you select to ensure +that they are credible sources. +1. +Research the term efficacy rates. Express what efficacy means in your own words. +2. +Using a minimum of two sources, compose a well-developed paragraph describing what validation of a vaccine +means. +3. +Using a minimum of two sources, compose a well-developed paragraph sharing the steps in validating a vaccine. +4. +Using a minimum of two sources, compose a well-developed paragraph describing how a vaccine is determined to +be validated. +5. +Using your research for Questions 1–3, write a summary paragraph sharing your reflections on the validation of the +Covid-19 vaccine or on validating a vaccine in general. What key components did you learn? What would you like to +learn more about related to validating a vaccine? +Frequency and Ultrasonic Sounds +Ultrasonic sounds have been utilized for a variety of reasons, from purportedly repelling rodents and other animals as +well as a variety of other applications. Using an Internet search, complete the following questions to explore some of +these applications and examine the validity of various claims. +Repelling Insects, Rodents, and Small Animals +Some radio stations purport to play a high pitch sound dually with their music to aid in deterring insects and other +annoying bugs to aid in providing a bug-reduced listening environment. To deter small rodents, some products claim to +emit ultrasonic sounds that drive away mice and other similar pests. +1. +Research the science behind ultrasonic pest controls, paying attention to the source of the information that you +find. Compare the information found on advertisements, reviews, and scientific articles. +a. +What frequency ranges do ultrasonic pest deterrent devices utilize and how do these frequencies compare to +the audible range that humans can hear? +b. +Write a short paragraph comparing the claims in the advertisements with independent reviews and scientific +articles. +c. +What do you conclude about ultrasonic pest controls and why? +1450 +13 • Chapter Summary +Access for free at openstax.org + +Disbursing Teenagers +Some business owners and communities have turned to products such as the “mosquito” sonic deterrent device to +discourage groups of teenagers from loitering around storefronts and community landmarks, citing a public nuisance +issue and public safety concerns. +2. +Research the science behind ultrasonic deterrent devices as they apply to dispensing teenagers. Compare the +information found on advertisements, reviews, and scientific articles. +a. +What frequency ranges do ultrasonic teenager deterrent devices utilize and how do these frequencies compare +to the audible range that adults can hear? +b. +Write a short paragraph comparing the claims in advertisements with independent reviews and scientific +articles. +c. +What are some of the ethical debates surrounding the use of ultrasonic teenager deterrent devices? +d. +What do you conclude about business owners or communities using ultrasonic teenager deterrent devices and +why? +Jewelry Cleaner +Use of pastes and liquid chemicals to clean jewelry can be harsh on stones as well as metals. So how can we safely obtain +the sparkling clean look at home that jewelry stores provide? Some would say the answer is to use an ultrasonic jewelry +cleaner, but do these really work? +3. +Research the science behind ultrasonic jewelry cleaners, including the phrase “cavitation process” in your search. +Compare the information found on advertisements, reviews, and scientific articles. +a. +Write a short paragraph detailing how ultrasonic jewelry cleaners work and what role the cavitation process +plays in the claims for ultrasonic cleaning. +b. +Do ultrasonic jewelry cleaners utilize low or high frequencies? How do these frequencies compare to the +audible range that humans can hear? +c. +Write a short paragraph comparing the claims in the advertisements with independent reviews and scientific +articles. +d. +What do you conclude about ultrasonic jewelry cleaners and why? +Specialized Ringtones +As the use of cell phones has become commonplace and families grow towards each member having their own cell +phone, specialized ringtones have become popular and can aid in identifying who is calling just by the ringtone. +Ever hear of ringtones that can be heard by teens but often not their teachers? The banning of cell phone use by K–12 +students during class time as been implemented across a wide array of schools and some students have purportedly +found ways to get around teachers hearing a cell phone ring through the use of ultrasonic ringtones. +4. +Research the science behind ultrasonic ringtones. Compare the information found on advertisements, reviews, and +scientific articles. +a. +Do ultrasonic ringtones utilize low or high frequencies? +b. +Write a short paragraph detailing how ultrasonic ringtones work. How do these frequencies compare to the +audible range that adults can hear? Include ages that are purported to hear and not hear the ringtones as well +as the frequency ranges utilized. +c. +Write a short paragraph comparing the claims in the advertisements with independent reviews and scientific +articles. +d. +What do you conclude about ultrasonic ringtones and why? +Streaming Services and Math +With the ability to stream music virtually anywhere you are, it is not surprising that Google Play Music, Apple Music, and +a slew of other companies such as Spotify, Amazon Music, YouTube Music, Sound Cloud, Pandora, Deezer Music, Tidal, +Napster, and Bandcamp have invested heavily to bring streaming service to users worldwide. Streaming services have +expanded to offer virtually every genre of music with vast libraries to meet diverse user requests. +Considering all of the choices available for streaming music, there is a wide array of options for subscribing. Conduct an +Internet research to review your current streaming choices, if any, and evaluate competitors’ products. +1. +In this project, you will explore options for music streaming service subscriptions. Select a minimum of five +streaming services listed above that you are not currently utilizing and determine the below components. Format +your findings in an easy-to-read format such as a table similar to the one shown below. +a. +Monthly subscription cost +b. +Available features +13 • Chapter Summary +1451 + +Available Features +Service +Choice 1: +Service +Choice 2: +Service +Choice 3: +Service +Choice 4: +Service +Choice 5: +Able to stream unlimited +music +Able to purchase individual +songs +Able to purchase individual +whole albums +Ability to create personalized +play lists +Ability to select specific songs +to play +Ability to listen +Other = +2. +Premium cost per month, if available = ______ +a. +Feature(s) offered with premium monthly subscription = ______ +3. +What is (are) your current music streaming services, if any? +a. +What is your current monthly service charge(s)? +b. +What features does your current streaming service offer? +4. +Write a paragraph summarizing your findings and include if your current streaming service(s) meets your needs or +if research for this project has you considering changing streaming services. Support your rationale. +Math and Baseball +Baseball is known to have one of the largest pools of statistics related to the game and its players. Managers, coaches, +and pitchers study the statistics of the players on opposing teams to give their team an edge by knowing what pitches to +throw for the best probability to be missed by a batter. In similar fashion, batters study pitchers’ statistics to learn a +pitcher’s strength and how to predict what a pitcher will throw and how to best hit against a pitcher. +The three primary baseball statistics are batting average, home runs, and runs batted in (RBIs), which are the +components of the title of Triple Crown winner that is awarded to players who dominate in these three areas. However, +there is a wealth of other statistics to evaluate when studying the performance of a player. +Conduct an Internet search to research statistics and how they are calculated in the following categories: +Batting Statistics +1. +There are about 30 batting statistics. Select a minimum of 10 batting statistics. Compose an organized list including +the name of the statistic, abbreviation, explanation of what it represents, as well as how it is calculated. +As an example: AB/HR represents at bats per home run and is calculated by the number of times a player is at bat +divided by home runs. +Pitching Statistics +2. +There are about 40 pitching statistics. Select a minimum of 10 pitching statistics. Compose an organized list +including the name of the statistic, abbreviation, explanation of what it represents, as well as how it is calculated. +As an example: K/9 represents strikeouts per nine innings and is calculated by the number of strikeouts times nine +divided by the number of innings pitched. +Fielding Statistics +3. +There are around 10 fielding statistics. Select a minimum of five fielding statistics. Compose an organized list +including the name of the statistic, abbreviation, explanation of what it represents, as well as how it is calculated. +1452 +13 • Chapter Summary +Access for free at openstax.org + +As an example: FP represents fielding percentage is calculated by the number of total plays divided by the number +of total chances. +Overall +4. +Select a player from recent years to evaluate. The player can be one that you have followed, one from a favorite +team, or any current player. Find the statistics shared in Questions 1–3 to use in evaluating the potential strengths +and weaknesses of the player. Write a short paragraph analyzing your selected player, supported by the statistics +from the answers to Questions 1–3. +Math and Fantasy Football +Fantasy football offers spectators an added dimension to football season with a competitive math-based game where +the active components are real-life players in the current season. For clarity in this exercise, the actual fantasy football +players will be denoted as FFP and actual professional team members will be denoted as players. +While some fantasy football leagues have slightly different setups or scoring systems, most share some common +elements. +Often using a lottery system to determine who picks first, second, and so on, FFPs select 15 current players to comprise +their personal fantasy football team. The players selected can be from any professional teams and a FFP can utilize any +team recognized by their league. FFPs can elect to keep the same players on their team for the whole league play or +trade for any player not selected by another FFP in their league. +At the start of each week during football season, each fantasy football player selects their roster of actual players to +comprise their roster of starting players. Typically, a starting roster consists of the following players: +Number to Select +Position Abbreviation +Position Title +QB +Quarterback +K +Kicker +TE +Tight End +RB +Running Back +D/ST +Defense +WR +Wide receiver +RB or WR +Flex +As actual professional games are played, points are tallied based on your league’s scoring system. The points the team +members on your starting roster make during the week are computed and whichever FFP has the highest score for the +week wins that week. +The FFPs with the best records of wins versus losses enters fantasy football playoffs to determine the ultimate league +champion and collects the pot. +The above overview of fantasy football describes the basic game play. The fun comes in understanding and analyzing the +math behind the scoring. +1. +Talk to people you know who have played fantasy football in the past and interview them. Inquire about how they +select players and how league(s) they have played in have computed the weekly scoring. +a. +Was a league manager recruited to manage the record keeping of scores? +b. +Was an online scoring system utilized or did the fantasy football use a streamlined or specialized scoring +system? +c. +What strategies were used to select the team of 15 players and who would be on the weekly starting roster? +d. +Were any online resources utilized to aid in choosing which players to select? +e. +How was math utilized to select team members or who to place as a starter each week? Be as specific as +possible, citing examples when available. +13 • Chapter Summary +1453 + +2. +Conduct an Internet research to find two different resources that a FFP could utilize when selecting players for their +team or for their starting roster. Describe the math involved in the resources. +3. +Conduct an Internet research to learn more about the scoring utilized by fantasy football leagues. Write a short +paragraph sharing how the use of math and knowledge of football statistics is used and how they are calculated to +aid a FFP in selecting their team and starting roster. +Math and Hockey +Hockey is full of math from obvious components such as scoring and statistics to the shape of the rink and the angles +involved in puck movement. +Collegiate and professional hockey games are 60 minutes long and are divided into three periods of 20 (60/3) minutes +each. At any one time, there are five players and one goalie on the ice for each team. If a player is called on a penalty and +is placed in the penalty box, that team now has four players, which is 20% less players on the ice competing against five +opponents. In some instances, a team may have two players in the penalty box at one time, resulting in three players, or +40% less players on the ice compared to a full team. Being one player down for a 2-minute penalty or potentially 5 +minutes for a major penalty leads to an imbalance on the ice and calls for a quick change of offense and defense +strategy. +Rink Composition—North American +An ice rink is comprised of various geometrical shapes, each with precise dimensions. +1. +Research the dimensions of a hockey rink and draw an accurate scale model on graph paper. Be sure to include the +scale for your model and indicate all units. +2. +List the shapes and the numbers of each kind of hockey rink. You should find four different shapes, some with +multiple dimensions. +3. +Describe the shape and dimensions of a hockey puck using geometric vocabulary. +Statistics +Using an Internet search, select two top hockey players from the same league to answer the following questions: +4. +Write a paragraph sharing a minimum of five statistics for each player you have selected. Describe how each of the +statistics are calculated and what each statistic means. +5. +Write a paragraph comparing the two players and determine who you believe is the better player. Support your +choice. +Scoring +The basics of scoring in ice hockey is simple, the team with the most goals is the winner. But, how to score the most +goals involves much math! +6. +Research one of the following components involved in hockey puck movement on the ice and write a paragraph +summarizing your findings. Be specific and detailed in your summary. +a. +Angles +b. +Velocity vectors +c. +Angle of incident and angle of return +d. +Speed and acceleration (player as well as puck movement) +Math and Soccer +As the world’s most popular sport, you’ll be excited to confirm that soccer is full of mathematics ranging from scoring +and statistics to footwork, angles, and field shape. +Soccer requires understanding of mathematical concepts and equations as well as skill, fitness, and game knowledge. +One such example is angles, which you all will remember from your geometry class. While players are not carrying +protractors and measuring angles during play, mental calculation of angles is a constant in any successful player’s +thinking. A goalie is not physically able to cover the entire open net region and a player must calculate an angle to kick +the ball consistent with the net opening while predicting the ability of the goalie to stop the ball from entering the net. +1. +Research angles as they apply to soccer play. Provide two examples of different plays indicating angle of a player’s +body, angle of foot striking the ball, and angle to the net. Include relevant dimensions. Adding a diagram that may +aid in clarity is an option. +2. +Draw a scale model of a soccer field including dimensions with labels. +Using an Internet search, select two top soccer players from the same league to answer the following questions: +3. +Write a paragraph sharing a minimum of five statistics for each player you have selected. Describe how each of the +1454 +13 • Chapter Summary +Access for free at openstax.org + +statistics are calculated and what each statistic means. +4. +Write a paragraph comparing the two players and determine who you believe is the better player. Support your +choice. +13 • Chapter Summary +1455 + +Chapter Review +Section 13.1 Math and Art +1. A person’s length from their belly button to the floor is 25 in. What is the person’s approximate height rounded to +the nearest inch? +2. The spiral seeds on a fruit mirror the Fibonacci sequence. If a row of the spiral contains 34 leaves, how many leaves +would be found on the next larger spiral of seeds? +3. The seeds on a sunflower mirror the Fibonacci sequence. If a circular layer on the sunflower contains 89 seeds, +approximately how many seeds would be found on the next larger circular layer? +4. A term of the Fibonacci sequence is 17,711 and the previous term is 10,946. What is the following term of the +sequence? +5. The 35th term of the Fibonacci sequence is 9,227,465. Use the approximate value of +of 1.618 to estimate the +34th term. Round your answer to the nearest whole number. +6. A frame has dimensions of 16 in by 24 in. Calculate the ratio of the sides rounded to the nearest tenth and +determine if the size approximates a golden rectangle. +Section 13.2 Math and the Environment +7. An average toilet uses 5 gal of water per flush. A high-efficiency toilet uses about 1.25 gal per flush. How much +water would a household save in a month (30 days) if the toilet was flushed 6 times a day? +8. When washing dishes, leaving the faucet running utilizes about 15 gal every 5 minutes, whereas filling the sink and +turning the faucet off except to rinse the dishes uses about 6 gal for washing a dishpan-sized load of dishes. How +much water is saved by not leaving the faucet running if it takes 10 minutes to wash a dishpan-sized load of dishes +each day for a year containing 30 days in each month? +9. Leaving the water running when washing your hands consumes about 4 gal of water, whereas turning the water +off when lathering reduces the water used to 1 gal. How much water is saved in an apartment of 4 people in 6 +months (180 days) if each person averages washing their hands twice a day? +10. How long would it take for an energy-saving light bulb to consume 1 kW of power if the bulb is rated at 2.5 W? +Round your answer to the nearest hour. +11. How long would it take for a 60 W light bulb to consume 1 kW of power? Round your answer to the nearest hour. +12. A portable fan uses about 40 W. How many kilowatt-hours are needed to run the fan during a 10-day trip if the fan +is run for an average of 3.5 hours a day? +Section 13.3 Math and Medicine +13. In a 2% solution of lidocaine, how many milligrams are there per milliliter? +14. You are prescribed a nebulizer solution to aid with asthma. If the solution is 0.7% w/v, what volume is needed to +deliver a 7 mg dose? +15. A prescription calls for a patient to receive 17 mg of a drug to be taken in oral form. If the pills are available in 3.4 +mg, how many pills will the patient need daily? +16. A prescription calls for a patient to receive 9 mg daily of a drug to be taken daily for 10 days. If the pills are +available in 2.25 mg, how many pills are needed to fill the prescription? +17. A patient is prescribed 2 mg/kg of a drug to be delivered daily intramuscularly, divided into three doses. If the +patient weighs 30 kg, how many milligrams of the drug is needed for each dose? +18. A patient is prescribed 0.75 mg/kg of a drug to be delivered intramuscularly once per day. If the drug is available in +2.5 mg/mL and the patient weighs 63 kg, how many milliliters of the drug should be given per day? +Section 13.4 Math and Music +19. Select the most representative sound registering 50 dB: airport gate noise, light rain on a car, dirt bike racing. +20. Select the most representative sound registering 100 dB: city traffic when in a car with windows rolled up, children +playing on a school playground, professional sporting event such as football. +21. What frequency range can teenagers typically hear but adults over 20 are often not able to hear? +22. What is three whole steps up from G#? +1456 +13 • Chapter Summary +Access for free at openstax.org + +23. What is one half-step down from F? +24. What is an octave above D3? +25. Given that the frequency of +, what is the frequency of A6? +Section 13.5 Math and Sports +For the following exercises, compute how many games would be played in the style of tournament and number of +teams given in each question. Assume all tournaments are single round-round or single-elimination. +26. How many games would be played with 10 teams using a round-robin tournament? +27. How many games would be played with 25 teams using a round-robin tournament? +28. How many games would be played with 12 teams using a round-robin tournament? +29. How many games would be played with 4 teams using a single-elimination tournament? +30. How many games would be played with 9 teams using a single-elimination tournament? +Chapter Test +1. A person’s length from their belly button to the floor is 3 ft 8 in, assuming the ratio is golden. What is the person’s +approximate height rounded to the nearest inch? +2. The leaves on an invasive weed mirror the Fibonacci sequence. If a set of leaves on the plant contains 13 leaves, +how many leaves would be found on the previous smaller set of leaves? +3. The 14th term of the Fibonacci sequence is 377 and the 15th term is 610. What is the 13th term of the sequence? +4. A fence has dimensions of 40 in by 25 in. Calculate the ratio of the sides rounded to the nearest tenth and +determine if the size approximates a golden rectangle. +5. An average toilet uses 5 gal of water per flush. A high-efficiency toilet uses about 1.25 gal per flush. How much +water would a dorm save in an academic year (32 weeks) if the toilet was flushed 22 times a day? +6. When washing dishes, leaving the faucet running utilizes about 15 gal every 5 minutes, whereas filling the sink and +turning the faucet off except to rinse the dishes uses about 5 gal for washing a dishpan-sized load of dishes. How +much water is saved by not leaving the faucet running if it takes 10 minutes to wash a dishpan-sized load of dishes +each day for a week? +7. How long would it take for a 75 W light bulb to consume 1 kW of power? Round your answer to the nearest hour. +8. A phone charger uses about 20 W. How many kilowatt-hours are needed to use the charger during a month if the +charger is used for an average of 1.5 hours a day? +9. A pharmacy has a 1 L bottle of 0.9% solution of sodium chloride. How many grams of sodium chloride are +contained in the bottle? +10. A prescription calls for a patient to receive 5.1 mg of a drug to be taken in oral form. If the pills are available in 1.7 +mg, how many pills will the patient need daily? +11. A prescription calls for a patient to receive 7 mg daily of a drug to be taken daily for 7 days. If the pills are available +in 1.75 mg, how many pills are needed to fill the prescription? +12. A patient is prescribed 3.8 mg/kg of a drug to be delivered daily intramuscularly, divided into two doses. If the +patient weighs 20 kg, how many milligrams of the drug are needed for each dose? +13. Select the most representative decibel value for a college classroom with students working in groups: 10 dB, 30 dB, +50 dB. +14. What is typically the top frequency that adults over 20 can hear: 17 Hz, 170 Hz, 1,700 Hz, 17,000 Hz? +15. What note is three whole steps down from F#? +16. Given that the approximate frequency of +, what is the approximate frequency of C2# rounded to the +nearest whole number? +For the following exercises, compute how many games would be played in the style of tournament and number of +teams given in each question. Assume all tournaments are single round-robin or single-elimination. +17. How many games would be played with 22 teams using a round-robin tournament? +18. How many games would be played with 7 teams using a round-robin tournament? +19. How many games would be played with 8 teams using a single-elimination tournament? +13 • Chapter Summary +1457 + +20. How many games would be played with 17 teams using a single-elimination tournament? +1458 +13 • Chapter Summary +Access for free at openstax.org + +Nonnegative Integer Powers of 10 +The phrase nonnegative integers refers to the set containing 0, 1, 2, 3, … and so on. In the expression +, 10 is called +the base, and 5 is called the exponent, or power. The exponent 5 is telling us to multiply the base 10 by itself 5 times. +So, +. By definition, any number raised to the 0 power is 1. So, +. +In the following table, there are several nonnegative integer powers of 10 that have been written as a product. Notice +that higher exponents result in larger products. What do you notice about the number of zeros in the resulting product? +Exponential Form +Product +Number of Zeros in Product +That’s right! The number of zeros is the same as the power each time! +Negative Integer Powers of 10 +The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 10 is +. We use negative +exponents to indicate a reciprocal. For example, +. Similarly, any expression with a negative exponent +can be written with a positive exponent by taking the reciprocal. Several negative powers of 10 have been simplified in +the table that follows. What do you notice about the number of zeros in the denominator (bottom) of each fraction? +Exponential Form +Equivalent Simplified Expression +Number of Zeros in Denominator +That’s right! The number of zeros is the same as the positive version of the power each time. +In the following table, we will write the same powers of 10 as decimals. Count the number of decimal places to the right +of the decimal point. What do you notice? +CO-REQ APPENDIX: INTEGER POWERS OF 10 +A +A • Co-Req Appendix: Integer Powers of 10 +1459 + +Exponential Form +Equivalent Simplified Expression +Number of Decimal Places to Right of Decimal +That’s right! The number of decimal places to the right of the decimal point is the same as the positive version of the +power each time. +Multiplying Integers by Positive Powers of 10 +Did you know that the distance from the sun to Earth is over 90 million miles? This value can be represented as +90,000,000, or we can write it as a product: +, which is actually a more compact way of writing 90 +million. Notice that the power of 7 reflects the number of zeros in 90 million. Several products of positive integers and +powers of 10 are given in the table that follows. Notice that the number of zeros is the same as the exponent except in +one case. +Exponential Form +Product +Number of Zeros in Product +The only case in which the number of zeros didn’t equal the exponent was the last case. Why do you think that +happened? That’s right! We multiplied by 70 which also had a zero. So, the product had a zero from the 70 and 5 zeros +from +for a total of 6 zeros in 7,000,000. +Multiplying by Negative Powers of 10 +As we have seen, negative powers of 10 are decimals. Several products of positive integers and powers of 10 are given in +the table below. Notice that multiplying an integer by 10 raised to a negative integer power results in a smaller number +than you started with. Also, the number of decimal places to the right of the decimal point is the same as the exponent +except in one case. +Exponential Form +Product +Number of Decimal Places to Right of Decimal +1460 +A • Co-Req Appendix: Integer Powers of 10 +Access for free at openstax.org + +Exponential Form +Product +Number of Decimal Places to Right of Decimal +The only case in which the number of decimal places to the right of the decimal point didn’t equal the positive version of +the exponent was the last case. Why do you think that happened? That’s right! We multiplied by 70, which ended in zero. +Moving the Decimal Place +A helpful shortcut when multiplying a number by a power of 10 is to “move the decimal point.” The following table shows +several powers of 10, both positive and negative. Compare the location of the decimal point in the original number to +the location of the decimal point in the product. How has it changed? +Exponential +Form +Product +How the Position of the Decimal Point +Changed +Notice that multiplying by a positive power of 10 moves the decimal point to the right, making the value larger, while +multiplying by a negative power of 10 moves the decimal point to the left, making the value smaller. Also, the number of +decimal places that the decimal point moves is exactly the positive version of the exponent. +A • Co-Req Appendix: Integer Powers of 10 +1461 + +1462 +A • Co-Req Appendix: Integer Powers of 10 +Access for free at openstax.org + +Answer Key +Chapter 1 +Your Turn +1.1 +1. One possible solution: +. +1.2 +1. This is not a well-defined set. +2. This is a well-defined set. +1.3 +1. +1.4 +1. +1.5 +1. +1.6 +1. +1.7 +1. +1.8 +1. +2. +1.9 +1. finite +2. infinite +1.10 +1. Set +is equal to set +, +2. neither +3. Set +is equivalent to set +, +1.11 +1. +and +1.12 +1. A set with one member could contain any one of the following: +. +2. Any of the following combinations of three members would work: +, +, or +. +3. The empty set is represented as +or +. +1.13 +1. +1.14 +1. 512 +1.15 +1. +Answer Key +1463 + +1.16 +1. Serena also ordered a fish sandwich and chicken nuggets, because for the two sets to be equal they must contain +the exact same items: {fish sandwich, chicken nuggets} = {fish sandwich, chicken nuggets}. +1.17 +1. There are multiple possible solutions. Each set must contain two players, but both players cannot be the same, +otherwise the two sets would be equal, not equivalent. For example, {Maria, Shantelle} and {Angie, Maria}. +1.18 +1. The set of lions is a subset of the universal set of cats. In other words, the Venn diagram depicts the relationship +that all lions are cats. This is expressed symbolically as +. +1.19 +1. The set of eagles and the set of canaries are two disjoint subsets of the universal set of all birds. No eagle is a +canary, and no canary is an eagle. +1.20 +1. The universal set is the set of integers. Draw a rectangle and label it with +. Next, draw a circle in the +rectangle and label with Natural numbers. +Venn diagram with universal set, +, and subset +. +2. +Venn Diagram with universal set, +and subset +. +1.21 +1. +Venn Diagram with universal set, +Things that can fly with disjoint subsets Airplanes and Birds. +1.22 +1. +2. +or +1.23 +1. +1.24 +1. +1.25 +1464 +Answer Key +Access for free at openstax.org + +1. +1.26 +1. +1.27 +1. +1.28 +1. +. +1.29 +1. 33 +1.30 +1. 113 +1.31 +1. +or +2. +and +3. +or +4. ( +and +) and +1.32 +1. 127 +2. 50 +1.33 +1. +. +2. +. +3. +. +4. +. +1.34 +1. 40 +2. 0 +3. 27 +1.35 +1. +2. +3. +1.36 +Answer Key +1465 + +1. +Venn diagram – Attendees at a conference with sets: Soup, Sandwich, Salad – Complete Solution +1.37 +1. +2. +3. +1.38 +1. The left side of the equation is: +Venn diagram of intersection of two sets and its complement. +The right side of the equation is given by: +Venn diagram of union of the complement of two sets. +Check Your Understanding +1. set +2. cardinality +3. not a well-defined set +4. 12 +5. equivalent, but not equal +6. finite +7. Roster method: +and set builder notation: +8. subset +9. To be a subset of a set, every member of the subset must also be a member of the set. To be a proper subset, +there must be at least one member of the set that is not also in the subset. +10. empty +1466 +Answer Key +Access for free at openstax.org + +11. true +12. +13. equivalent +14. equal +15. relationship +16. universal +17. disjoint or non-overlapping +18. complement +19. disjoint +20. intersection +21. union +22. +23. +24. +25. +26. empty +27. +28. overlap +29. central +30. intersection of all three sets, +31. parentheses, complement +32. equation, true +Chapter 2 +Your Turn +2.1 +1. Logical statement, false. +2. Logical statement, true. +3. Not a logical statement, questions cannot be determined to be either true or false. +2.2 +1. +The movie Gandhi won the Academy Award for Best Picture in 1982. +2. +Soccer is the most popular sport in the world. +3. +All oranges are citrus fruits. +2.3 +1. Ted Cruz was born in Texas. +2. Adele does not have a beautiful voice. +3. Leaves do not convert carbon dioxide to oxygen during the process of photosynthesis. +2.4 +1. +2. +3. +2.5 +1. +2. Woody and Buzz Lightyear are not best friends. +2.6 +1. The sum of some consecutive integers results in a prime number. +2. No birds give live birth to their young. +3. All squares are parallelograms and have four sides. +2.7 +1. All apples are sweet. +2. Some triangles are squares. +Answer Key +1467 + +3. No vegetables are green. +2.8 +1. Negation; +. +2. Conjunction; +. +3. Biconditional; +. +2.9 +1. +2. +3. +4. +2.10 +1. If our friends did not come over to watch the game, then my roommates ordered pizza or I ordered wings. +2. If my roommates ordered pizza and I ordered wings, then our friends came over to watch the game. +3. It is not the case that my roommates ordered pizza or our friends came over to watch the game. +2.11 +1. +2. +3. +; this is another example of De Morgan’s Laws and it is always true. +2.12 +1. +, true +2. +: No houses are built with bricks; false +3. +: Abuja is not the capital of Nigeria; false +2.13 +1. True +2. False +3. True +2.14 +1. True +2. False +3. True +2.15 +1. +T +T +F +F +F +F +false +2. +T +T +F +T +T +T +true +3. +T +T +F +F +T +T +true +2.16 +1468 +Answer Key +Access for free at openstax.org + +1. +T +T +F +F +T +F +T +T +F +T +F +F +F +F +T +F +2. +T +T +T +F +T +F +T +F +F +T +T +F +F +F +F +T +3. +T +T +T +F +F +T +T +T +F +F +F +F +T +F +T +T +T +T +T +F +F +T +T +T +F +T +T +F +F +T +F +T +F +F +F +F +F +F +T +T +F +T +F +F +F +T +F +F +2.17 +1. Valid +T +F +T +F +T +T +2. Not valid +T +T +F +F +F +T +F +F +T +T +F +T +T +F +T +F +F +T +T +T +Answer Key +1469 + +2.18 +1. False +T +F +F +2. True +T +F +T +T +3. True +T +F +F +T +2.19 +1. Valid +T +T +F +T +T +T +F +F +F +T +F +T +T +T +T +F +F +T +T +T +2. Not valid +T +T +F +T +T +T +F +F +F +T +F +T +T +F +F +F +F +T +F +F +2.20 +1. False +T +F +F +2. True +T +F +T +T +3. True +1470 +Answer Key +Access for free at openstax.org + +T +F +F +T +2.21 +1. Valid +T +T +T +F +F +F +F +T +T +F +F +T +F +T +T +T +F +T +F +T +T +F +T +T +F +F +F +T +T +T +T +T +2. Not Valid +T +T +F +T +F +T +F +F +F +T +F +T +T +F +F +F +F +T +F +F +3. Valid +T +T +T +F +T +T +T +F +F +F +F +T +F +T +T +T +T +T +F +F +T +T +T +T +4. Valid +T +T +T +F +F +T +T +F +T +T +T +T +F +F +F +T +F +F +F +T +T +F +T +F +T +F +T +T +T +T +T +F +F +F +T +F +T +T +T +T +F +T +T +T +F +F +T +T +T +T +F +T +F +T +F +F +T +T +T +T +F +F +T +T +T +F +T +T +T +T +F +F +F +T +T +F +T +T +T +T +2.22 +Answer Key +1471 + +1. +is logically equivalent to +T +T +T +F +F +T +T +T +F +F +T +F +F +T +F +T +T +F +T +T +T +F +F +T +T +T +T +T +2. +is not logically equivalent to +T +T +T +F +T +T +T +F +F +T +T +F +F +T +T +F +F +F +F +F +T +T +T +T +2.23 +1. If Elvis Presley wore capes, then some superheroes wear capes. +2. If some superheroes wear capes, then Elvis Presley wore capes. +3. If Elvis Presley did not wear capes, then no superheroes wear capes. +4. If no superheroes wear capes, then Elvis Presley did not wear capes. +2.24 +1. +: Dora is an explorer. +2. +: Boots is a monkey. +3. Inverse +4. Converse +5. Converse +2.25 +1. If my friend does not live in California, then my friend lives in San Francisco. True. +2. If my friend does not live in San Francisco, then my friend lives in California. True. +3. If my friend lives in California, then my friend does not live in San Francisco. False. +2.26 +1. Jackie did not play softball and she did not run track. +2. Brandon did not study for his certification exam, or he did not pass his exam. +2.27 +1. Edna Mode made a new superhero costume, and it includes a cape. +2. I had pancakes for breakfast, and I did not use maple syrup. +2.28 +1. Some people like ice cream, but ice cream makers will not make a profit. +2. Raquel cannot play video games, but somebody will play video games. +2.29 +1. Eric needs to replace the light bulb, and Marcos did not leave the light bulb on all night, and Dan did not break the +light bulb. +2. Trenton went to school, and Regina went to work, and Merika did not clean the house. +1472 +Answer Key +Access for free at openstax.org + +2.30 +1. +T +T +T +F +F +F +F +T +T +F +T +F +F +T +F +T +F +T +T +F +T +F +F +T +F +F +F +T +T +T +T +T +2.31 +1. Some people like history. +2. Some people do not like reading. +3. The polygon is not an octagon. +2.32 +1. My classmate does not like history. +2. Homer likes to read. +3. The polygon does not have five sides. +2.33 +1. If my roommate does not go to work, then they will not be able to pay their bills. +2. If penguins cannot fly, then we will watch the news. +3. If Marcy goes to the movies, then she will buy water. +Check Your Understanding +1. logical statement +2. negation +3. +4. +5. premises +6. Inductive +7. quantifiers +8. Some giraffes are not tall. +9. compound statement +10. connective +11. biconditional, +12. Parentheses, +13. Conjunction, +; disjunction, +(in any order) +14. valid +15. true +16. truth table +17. four +18. two +19. one-way contract +20. conclusion +21. hypothesis +22. biconditional +23. biconditional +24. true +25. always true, valid, or a tautology. +26. conditional +27. logically equivalent +28. inverse +29. converse, inverse +30. +Answer Key +1473 + +31. +32. +33. De Morgan’s Laws +34. premise +35. valid +36. inductive +37. deductive +38. fallacy +39. sound +Chapter 3 +Your Turn +3.1 +1. Yes. When 54 is divided by 9, the result is 6 with no remainder. Also, 54 can be written as the product of 9 and 6. +3.2 +1. The last digit is 0, so 45,730 is divisible by 5, since the rule states that if the last digit is 0 or 5, the original number is +divisible by 5. +3.3 +1. The sum of the digits is 32. Since 32 is not divisible by 9, neither is 342,887. +3.4 +1. The last digit is even, so 2 divides 43,568. The sum of the digits is 26. Since 26 is not divisible by 3, neither is 43,568. +The rule for divisibility by 6 is that the number be divisible by both 2 and 3. Since 43,568 is not divisible by 3, it is not +divisible by 6. +3.5 +1. Since the last digit of 87,762 is not 0, it is not divisible by 10. +3.6 +1. The number formed by the last two digits of 43,568 is 68 and 68 is divisible by 4. Since the number formed by the +last two digits of 43,568 is divisible by 4, so is 43,568. +3.7 +1. Yes, 1,429 is prime. +3.8 +1. Yes, 859 is a prime number. +3.9 +1. 5,067,322 is a composite number. +3.10 +1. No, 1,477 is composite. +3.11 +1. +3.12 +1. +3.13 +1. +3.14 +1. The number 180 has three prime factors. +1474 +Answer Key +Access for free at openstax.org + +3.15 +1. The GCD is 9. +3.16 +1. The GCD of 36 and 128 is 4. +3.17 +1. 40 +3.18 +1. The largest square bricks that can be used are 20 cm by 20 cm. +3.19 +1. The largest team size that can be formed is 7 students. +3.20 +1. 60 +3.21 +1. 140 +3.22 +1. 92,400 +3.23 +1. 360 +3.24 +1. The sun, Venus, and Jupiter will line up again in 220,830 days. +3.25 +1. The first person to receive both giveaways would be the person who submits the 11,700th submission. +3.26 +1. integer +2. not an integer +3. not an integer +4. integer +5. integer +3.27 +1. +2. +3. +3.28 +1. +Answer Key +1475 + +and +3.29 +1. +and +3.30 +1. 101 is larger. +and +. +3.31 +1. 38 +3.32 +1. 81 +3.33 +1. −7. Since |−18| > |11|, the answer matches the sign of −18. +3.34 +1. −62. Since a larger positive number was subtracted from a smaller positive number, a negative result was expected. +3.35 +1. 71. Subtracting a negative number is the same as adding a positive number. +3.36 +1. −17. Since |19| < |−36|, the sign of the answer matches the sign of −36, which is negative. +3.37 +1. $89 +3.38 +1. 2,106. Since both numbers are positive, the product is positive. +3.39 +1. −234. Since the numbers have opposite signs, the product is negative. +3.40 +1. −29. The numbers have opposite signs, so the division will result in a negative number. +3.41 +1. 7. Since the signs of the numbers match, the division results in a positive number. +3.42 +1. The average daily balance was $529. +3.43 +1. 313 +3.44 +1. 72 +3.45 +1476 +Answer Key +Access for free at openstax.org + +1. 630 +3.46 +1. 2,701 +3.47 +1. −15 +3.48 +1. 1,516 +3.49 +1. 5 +3.50 +1. 403 +3.51 +1. 94 is not a perfect square. +2. 441 is a perfect square. +3.52 +1. not a rational number +2. rational number +3. rational number +4. rational number +5. rational number +3.53 +1. +and +. The fractions are not equivalent. +3.54 +1. +3.55 +1. +3.56 +1. +3.57 +1. +3.58 +1. +3.59 +1. +3.60 +1. +3.61 +1. +3.62 +Answer Key +1477 + +1. 5.108 +3.63 +1. 18.63 +3.64 +1. 1.6 +3.65 +1. +3.66 +1. +3.67 +1. 2.664 +3.68 +1. +2. +3.69 +1. +3.70 +1. +, or in decimal form, +3.71 +1. The process used above yields +. +3.72 +1. +Studying math: 5 hours +Studying history: 2.5 hours +Studying writing: 1.25 hours +Studying physics: 1.25 hours +3.73 +1. 720 calories of protein +3.74 +1. 321.868 km +3.75 +1. 23.656 liters +3.76 +1. +2. +3.77 +1. 0.14 +2. 0.07 +3.78 +1. 300 +1478 +Answer Key +Access for free at openstax.org + +2. 841.64 +3.79 +1. 120 +2. 800 +3.80 +1. 7% +2. 85% +3.81 +1. 440 calories of protein +3.82 +1. 12% of registered voters in the small town voted in the primaries. +3.83 +1. We want the original price of the item, which is the total. We know the percent, 40, and the percentage of the total, +$30. To find the original cost, use +, with +and +. Calculating with those values yields +. So, the original was $75. +3.84 +1. perfect square +2. not a perfect square +3. perfect square +4. not a perfect square +3.85 +1. rational +2. irrational +3. irrational +4. irrational +3.86 +1. +. The rational part is 5, and the irrational part is +. +3.87 +1. +. The rational part is 1, and the irrational part is +. +3.88 +1. +. The rational part is 11, and the irrational part is +. +3.89 +1. +3.90 +1. +3.91 +1. The two numbers being subtracted do not have the same irrational part, so the operation cannot be performed +without a calculator. +3.92 +1. +2. +Answer Key +1479 + +3.93 +1. +2. 19 +3.94 +1. +2. +3.95 +1. +3.96 +1. real +2. not real +3. real +3.97 +1. irrational number +2. integer, rational number +3. rational number +3.98 +1. Venn diagram showing ‒4, 13.863, 15, 871, +, and +3.99 +1. dstributive property +2. additive inverse property +3.100 +1. +. Using that, the problem can be changed to +. Change to +. Using the distributive +property, +. +3.101 +1. 93 = 9 (mod 12) +2. 387 = 3 (mod 12) +3.102 +1. 4:00 +3.103 +1. 9:00 +3.104 +1. 4 +1480 +Answer Key +Access for free at openstax.org + +3.105 +1. 5:00 +3.106 +1. Thursday +3.107 +1. +2. Since the bases are not the same (one is 3, the other 4), this cannot be simplified using the product rule for +exponents. +3.108 +1. +3.109 +1. +3.110 +1. +3.111 +1. +3.112 +1. +2. +3.113 +1. +2. +3.114 +1. +2. +3.115 +1. +2. +3.116 +1. +2. +3.117 +1. Is not written in scientific notation; 42.67 is not at least 1 and less than 10. +2. Is written in scientific notation +3. Is not written in scientific notation; The absolute value of –80.91 is not at least 1 and less than 10. +3.118 +1. +2. +3. +3.119 +Answer Key +1481 + +1. +3.120 +1. +3.121 +1. 1,020,000 +2. 0.0000409 +3.122 +1. +2. +3.123 +1. +2. +3.124 +1. +3.125 +1. +3.126 +1. +2. +3.127 +1. +2. +3.128 +1. The transistor is +m larger than the diameter of an atom. +3.129 +1. Neptune is +, or 89.3, times further from the sun that Mercury. +3.130 +1. +cubic meters +3.131 +1. A person exhales, on average, +, or 840 pounds of carbon dioxide per year. +3.132 +1. This is an arithmetic sequence. Every term is the previous term minus 2.2. +2. This is not an arithmetic sequence. The difference between terms 1 and 2 is 2, but between terms 3 and 4 the +difference is 4. The differences are not the same. +3. This is an (infinite) arithmetic sequence. Every term is the previous term plus 6. The ellipsis indicates the pattern +continues. +3.133 +1. +, +, +3.134 +1. +, +, and +3.135 +1. 12,675.5 +1482 +Answer Key +Access for free at openstax.org + +3.136 +1. Christina will save $265 in week 52. +3.137 +1. There are 2,520 seats in the theater. +3.138 +1. It is a geometric sequence; common ratio is 5. +2. It is not a common ratio; term 2 is the first term multiplied by −2, but the sixth term is the fifth term multiplied by 3. +3. It is a geometric sequence; common ratio is +. +3.139 +1. +2. 2,048 +3.140 +1. 84,652,645 +2. 40.444444 +3.141 +1. The amount in the account was $11,671.03 (rounded to two decimal places). +3.142 +1. There are +organisms after 20 hours. +3.143 +1. 0. 99996948242188 +Check Your Understanding +1. 31 and 701 are prime. 56, 213 and 48 are composite. +2. +3. 2 +4. 630 +5. The maximum number of bags that can be filled in this way is 10. +6. −4, 430 +7. +8. −13, −7, −2, 4, 10 +9. 7 +10. 13 +11. 36 +12. parentheses +13. exponents +14. −22 +15. parentheses +16. 49 +17. +are rational; +is not. +18. +19. +20. +21. +22. +23. +24. Using the process from the chapter, +, and there are other answers. +Answer Key +1483 + +25. +26. $110.25 +27. 228 +28. 7 new employees will be hired. +29. +30. +31. +32. +33. +34. +35. distributive property +36. 2 +37. 1 +38. 5 +39. Friday +40. +41. +or +42. +43. +44. +45. +46. +47. +48. +49. +50. +51. A pile of dollar bills that reaches the moon would contain +bills. +52. No. The difference from term 1 to term 2 is different than the difference from term 4 to term 5. +53. 8 +54. 613 +55. +, +56. 35050 +57. There will be 426 people in their survey group after 100 days. +58. Yes, each term is the previous term multiplied by 2. +59. The common ratio is −10. +60. 2,919.293 (rounded off to three decimal places) +61. 5.714 (rounded to three decimal places) +62. $30,188.57 (rounded off to two decimal places) +Chapter 4 +Your Turn +4.1 +1. 416 +2. 1,851 +3. 17,488 +4.2 +1. +2. +3. +1484 +Answer Key +Access for free at openstax.org + +4.3 +1. 621 +2. 3,203 +3. 40,630,891 +4.4 +1. 1269 +4.5 +1. 42,136 +4.6 +1. 6,105,643 +4.7 +1. 257 +4.8 +1. 6,054 +4.9 +1. 1,248,073 +4.10 +1. 77 +2. 240 +3. 3,447 +4.11 +1. XXVII +2. XLIX +3. DCCXXXIX +4. MMMDCXLVII +4.12 +1. 0, 1, 2, 3 +4.13 +1. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B +4.14 +1. 157 +4.15 +1. 2,014 +4.16 +1. 851 +4.17 +1. 27 +4.18 +1. 0, 1, 2, 3 +10, 11, 12, 13 +20, 21, 22, 23 +30, 31, 32, 33 +100 +Answer Key +1485 + +4.19 +1. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B +10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B +20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B +30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 3A, 3B +40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 4A, 4B +50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 5A, 5B +60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 6A, 6B +70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 7A, 7B +80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 8A, 8B +90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 9A, 9B +A0, A1, A2, A3, A4, A5, A6, A7, A8, A9, AA, AB +B0, B1, B2, B3, B4, B5, B6, B7, B8, B9, BA, BB +100 +4.20 +1. 0, 1, 2, 10, 11, 12, 20, 21, 22, 100 +4.21 +1. 20107 +4.22 +1. 554B12 +4.23 +1. 100010012 +4.24 +1. +2. +4.25 +1. The result has the digit 7 in it. In base 4, the 7 is an illegal symbol. +4.26 +1. The remainders include 10, which in base 6 is an illegal symbol. +4.27 +1. Since 12 is larger than 10, the base 10 number cannot have less digits than the base 12 number. Since it did, we +know an error has been made. +4.28 +1. 12426 +4.29 +1. ++ +0 +1 +2 +3 +0 +0 +1 +2 +3 +Base 4 Addition +Table +1486 +Answer Key +Access for free at openstax.org + +1 +1 +2 +3 +10 +2 +2 +3 +10 +11 +3 +3 +10 +11 +12 +Base 4 Addition +Table +4.30 +1. 6337 +4.31 +1. ++ +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +C +D +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +C +D +1 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +C +D +10 +2 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +C +D +10 +11 +3 +3 +4 +5 +6 +7 +8 +9 +A +B +C +D +10 +11 +12 +4 +4 +5 +6 +7 +8 +9 +A +B +C +D +10 +11 +12 +13 +5 +5 +6 +7 +8 +9 +A +B +C +D +10 +11 +12 +13 +14 +6 +6 +7 +8 +9 +A +B +C +D +10 +11 +12 +13 +14 +15 +7 +7 +8 +9 +A +B +C +D +10 +11 +12 +13 +14 +15 +16 +8 +8 +9 +A +B +C +D +10 +11 +12 +13 +14 +15 +16 +17 +9 +9 +A +B +C +D +10 +11 +12 +13 +14 +15 +16 +17 +18 +A +A +B +C +D +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +B +B +C +D +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +1A +C +C +D +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +1A +1B +D +D +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +1A +1B +1C +Base 14 Addition Table +4.32 +1. 13B912 +4.33 +1. 100100102 +4.34 +1. 326 +4.35 +1. 22712 +4.36 +1. The symbols 4 and 5 are not legal symbols in base 4. Careful use of the base 4 addition table would correct this +error. +Answer Key +1487 + +1 +3 +3 ++ +1 +1 +2 +3 +1 +1 +The correct answer is 3114. +4.37 +1. This is correct if the numbers are base 10 numbers, but these numbers are base 14 numbers. In base 14, 9 + 9 is not +18, but instead is 13. Careful use of the base 14 addition table generates the correct answer, +. +4.38 +1. +* +0 +1 +2 +3 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +2 +0 +2 +10 +12 +3 +0 +3 +12 +21 +Base 4 +Multiplication +Table +4.39 +1. +* +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +C +D +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +C +D +2 +0 +2 +4 +6 +8 +A +C +10 +12 +14 +16 +18 +1A +1C +3 +0 +3 +6 +9 +C +11 +14 +17 +1A +1D +22 +25 +28 +2B +4 +0 +4 +8 +C +12 +16 +1A +20 +24 +28 +2C +32 +36 +3A +5 +0 +5 +A +11 +16 +1B +22 +27 +2C +33 +38 +3D +44 +49 +6 +0 +6 +C +14 +1A +22 +28 +30 +36 +3C +44 +4A +52 +58 +7 +0 +7 +10 +17 +20 +27 +30 +37 +40 +47 +50 +57 +60 +67 +8 +0 +8 +12 +1A +24 +2C +36 +40 +48 +52 +5A +64 +6C +76 +9 +0 +9 +14 +1D +28 +33 +3C +47 +52 +5B +66 +71 +7A +85 +A +0 +A +16 +22 +2C +38 +44 +50 +5A +66 +72 +7C +88 +94 +B +0 +B +18 +25 +32 +3D +4A +57 +64 +71 +7C +89 +96 +A3 +C +0 +C +1A +28 +36 +44 +52 +60 +6C +7A +88 +96 +A4 +B2 +D +0 +D +1C +2B +3A +49 +58 +67 +76 +85 +94 +A3 +B2 +C1 +4.40 +1488 +Answer Key +Access for free at openstax.org + +1. 40006 +2. 10101112 +4.41 +1. 436912 +4.42 +1. +2. +4.43 +1. The symbols 4 and 5 are not legal symbols in base 4. Careful use of the base 4 multiplication table would correct this +error. The correct answer is 3334. +4.44 +1. This is correct if the numbers are base 10 numbers, but these numbers are base 14 numbers. In base 14, +is not 81, but instead is 5B. Careful use of the base 14 addition table (Table 4.9) generates the +correct answer, +. +Check Your Understanding +1. A system in which the position of a numeral determines the value associated with that numeral. +2. 279 +3. +4. 10 +5. A numeral is a symbol representing a number. A number is a quantity or amount. +6. 601,947 +7. 60 +8. 20 +9. Roman numerals do not use place value. +10. 341 +11. 209 +12. 247 +13. CDLXXIX +14. A base 25 system would require 25 symbols. +15. Since the 4 is the second digit, its place value is 181 times 4, or 72. +16. 329 +17. 409 +18. 5118 +19. 2,126 +20. In base 4, 5 is not a valid symbol. So, a mistake has been made. +21. ++ +0 +1 +2 +3 +4 +5 +6 +7 +0 +0 +1 +2 +3 +4 +5 +6 +7 +1 +1 +2 +3 +4 +5 +6 +7 +10 +2 +2 +3 +4 +5 +6 +7 +10 +11 +3 +3 +4 +5 +6 +7 +10 +11 +12 +4 +4 +5 +6 +7 +10 +11 +12 +13 +5 +5 +6 +7 +10 +11 +12 +13 +14 +6 +6 +7 +10 +11 +12 +13 +14 +15 +7 +7 +10 +11 +12 +13 +14 +15 +16 +22. 1216 +Answer Key +1489 + +23. 78 +24. 8214 +25. 512 +26. In base 8, 8 is not a valid symbol. So, a mistake has been made. +27. A common base 14 error is performing the operation in base 10. +28. the addition table +29. The process is the same, except the multiplication table for the base is used instead of the familiar base 10 rules. +30. the multiplication table for the base +31. 22406 +32. B14 +33. The 8 is not a symbol used in base 5. +34. A symbol that is not used in that base is present, or a base 10 rule is used. +Chapter 5 +Your Turn +5.1 +1. 18 plus 11 OR the sum of 18 and 11 +2. 27 times 9 OR the product of 27 and 9 +3. 84 divided by 7 OR the quotient of 84 and 7 +4. +minus +OR the difference of +and +5.2 +1. +2. +3. +4. +5.3 +1. +2. +OR +3. +4. +5.4 +1. +2. +3. +4. +5.5 +1. +2. +3. +4. +5.6 +1. +2. +5.7 +1. +5.8 +1. +5.9 +1. +1490 +Answer Key +Access for free at openstax.org + +2. +3. +4. +5. +5.10 +1. +5.11 +1. +5.12 +1. +5.13 +1. +5.14 +1. Henry has 42 books. +5.15 +1. Total cost is $48.85 +5.16 +1. Answers will vary. For example: You can rent a paddleboard for $25 per hour with a water shoe purchase of $75. If +you spent $200, how many hours did you rent the paddle board for? +You rented the paddle board for 5 hours. +5.17 +1. +, which is false; therefore, this is a false statement, and the equation has no solution. +5.18 +1. +, which is true; therefore, this is a true statement, and there are infinitely many solutions. +5.19 +1. +5.20 +1. +5.21 +1. +5.22 +1. +Graph: +5.23 +Answer Key +1491 + +1. +5.24 +1. +5.25 +1. Taleisha can send/receive 106 or fewer text messages and keep her monthly bill no more than $50. +5.26 +1. You could take up to 17 credit hours and stay under $2,000. +5.27 +1. Malik must tutor at least 23 hours. +5.28 +1. += 1 U.S. dollar, and += 1.21 Canadian dollars, the ratio is 1 to 1.21; or 1:1.21; or +. +5.29 +1. With += 170 pounds on Earth, and += 64 pounds on Mars, the ratio is 170 to 64; or 170:64; or +. +5.30 +1. $219.51 +5.31 +1. 501.6 pounds +5.32 +1. 720 cookies +5.33 +1. The constant of proportionality (centimeters divided by inches) is 2.54. This tells you that there are 2.54 centimeters +in one inch. +5.34 +1. 30.5 hours (or 30½ hours, or 30 hours and 30 minutes) +5.35 +1. $207.50 +5.36 +1. 125.9 miles +5.37 +1. The scale is +. The other borders would calculate as: eastern and western borders are 273.75 +miles, and northern border is 365 miles. +5.38 +1. 184 inches, or 15 feet, 4 inches. +5.39 +1492 +Answer Key +Access for free at openstax.org + +1. +5.40 +1. +I. +Yes +II. +Yes +2. +I. +No +II. +No +3. +I. +Yes +II. +Yes +4. +I. +Yes +II. +Yes +5.41 +1. +5.42 +Answer Key +1493 + +1. +Your friend will pay $1.65. +5.43 +1. Yes +2. Yes +3. No +4. No +5. No +5.44 +1. +5.45 +1. +5.46 +1. +2. +3. Answers will vary. +5.47 +1. +1494 +Answer Key +Access for free at openstax.org + +5.48 +1. +5.49 +1. +5.50 +1. +5.51 +1. +5.52 +1. +5.53 +1. +or +5.54 +1. +or +5.55 +1. +, +5.56 +1. +, +5.57 +1. +, +5.58 +1. +, +5.59 +1. 16, 15 and –16, –15 +5.60 +Answer Key +1495 + +1. +, +5.61 +1. 22 +2. 6 +3. +5.62 +1. There 205 unread emails after 7 days. +5.63 +1. This relation is a function. +2. This relation is not a function. +5.64 +1. Both George and Mike have two phone numbers. Each +-value is not matched with only one +-value. This relation is +not a function. +5.65 +1. function +2. not a function +3. function +5.66 +1. This graph represents a function. +5.67 +1. This graph does not represent a function. +5.68 +1. The domain is the set of all +-values of the relation: +. +2. The range is the set of all +-values of the relation: +. +5.69 +1. The ordered pairs of the relation are: +. +2. The domain is the set of all +-values of the relation: +. +3. The range is the set of all +-values of the relation: +. +5.70 +1. The graph crosses the +-axis at the point (2, 0). The +-intercept is (2, 0). The graph crosses the +-axis at the point (0, +−2). The +-intercept is (0, −2). +5.71 +1. The +-intercept is (4, 0) and the +-intercept is (0, 12). +1496 +Answer Key +Access for free at openstax.org + +5.72 +1. +5.73 +1. +5.74 +1. slope +and +2. slope +and +5.75 +Answer Key +1497 + +1. +5.76 +1. +5.77 +1498 +Answer Key +Access for free at openstax.org + +1. +5.78 +1. (0, 20) is the +and represents that there were 20 teachers at Jones High School in 1990. There is no +. +2. In the first 5 years the slope is 2; this means that on average, the school gained 2 teachers every year between 1990 +and 1995. Between 1995 and 2000, the slope is 4; on average the school gained 4 teachers every year. Then the +slope is 0 between 2000 and 2005 meaning the number of teachers remained the same. There was a decrease in +teachers between 2005 and 2010, represented by a slope of –2. Finally, the slope is 4 between 2010 and 2020, which +indicates that on average the school gained 4 teachers every year. +3. Answers will vary. Jones High School was founded in 1990 and hired 2 teachers per year until 1995, when they had +an increase in students and they hired 4 teachers per year for the next 5 years. Then there was a hiring freeze, and +no teachers were hired between 2000 and 2005. After the hiring freeze, the student population decreased, and they +lost 2 teachers per year until 2010. Another surge in student population meant Jones High School hired 4 new +teachers per year until 2020 when they had 80 teachers at the school. +5.79 +1. 50 inches +2. 66 inches +3. The slope, 2, means that the height +increases 2 inches when the shoe size(s) increases 1 size. +4. The +-intercept means that when the shoe size is 0, the woman’s height is 50 inches. +5.80 +1. $25 +Answer Key +1499 + +2. $85 +3. The slope, 4, means that the weekly cost, +, increases by $4 when the number of pizzas sold, +, increases by 1. The +-intercept means that when the number of pizzas sold is 0, the weekly cost is $25. +4. Graph: +5.81 +1. No +2. Yes +5.82 +1. No +2. No +5.83 +1. +5.84 +1. (3, 2) +5.85 +1. +1500 +Answer Key +Access for free at openstax.org + +5.86 +1. There is no solution to this system. +5.87 +1. There are infinitely many solutions to this system. +5.88 +1. Jenna burns 8.3 calories per minute circuit training and 11.2 calories per minute while on the elliptical trainer. +5.89 +1. not a solution +2. a solution +5.90 +1. The region containing (0, 0) is the solution to the system of linear inequalities. +5.91 +1. The solution is the darkest shaded region. +5.92 +1. The solution is the lighter shaded region. +Answer Key +1501 + +5.93 +1. No solution. +5.94 +1. The solution is the lighter shaded region. +1502 +Answer Key +Access for free at openstax.org + +5.95 +1. +2. +3. The point +is not in the solution region. Omar would not choose to eat 3 hamburgers and 2 cookies. +4. The point +is in the solution region. Omar might choose to eat 2 hamburgers and 4 cookies. +5.96 +1. With +the number of bags of apples sold, and +the number of bunches of bananas sold, the objective function +is +. +5.97 +1. +5.98 +1. The constraints are +and +. The summary is: +, +, and +. +5.99 +Answer Key +1503 + +1. The constraints are: +So the system is: +5.100 +1. The maximum value for the profit +occurs when +and +. This means that to maximize their profit, the +Robotics Club should sell 15 bags of apples and 5 bunches of bananas every day. +Check Your Understanding +1. +, +2. +, +3. +4. +5. +6. +7. multiplication +8. division +9. It is a correct solution strategy. +Let +10. It is a correct solution strategy. +Let +11. This is not a correct solution strategy. The negative sign is not distributed correctly in the second line of the solution +strategy. The second line should read +. +12. +13. $40.40 +14. +15. $40.20 +16. The Enjoyable Cab Company, because the cab fare will be $ 0.20 less than what it would cost to take a taxi from the +Nice Cab Company. +17. Luis is; there are infinitely many solutions. If this is solved using the general strategy, it simplifies to +. This is a +true statement, so there are infinitely many solutions. +18. d +19. b +, although d +is an equivalent formula. +20. a +21. b +22. d +23. b +24. c +25. a +1504 +Answer Key +Access for free at openstax.org + +26. e +27. b +28. c +29. c +30. b +31. d +32. a True +33. a True +34. a True +35. b False +36. a True +37. +38. +39. 324.00 British pounds (None of these.) +40. 10 inches +41. Yes he can, but barely. At 37 miles per gallon, Albert can drive +miles. While in theory he can make it, he +probably should fill up with gasoline somewhere along the way! +42. +43. d +44. +45. b +46. +47. b +48. d +49. b +50. c +51. d +52. b False +53. b False +54. a True +55. b False +Answer Key +1505 + +56. +57. a True +58. b False +59. b False +60. a True +61. b False +62. a True +63. b False +64. b False +65. a True +66. b False +67. Elimination +68. Substitution +69. Substitution +70. Elimination +71. Substitution +72. Elimination +73. Elimination +74. Substitution +75. c +76. e +77. d +78. b +79. a +80. c +81. a +82. b +83. d +84. a +85. c +86. d +87. +88. b +89. +Chapter 6 +Your Turn +6.1 +1. +2. +1506 +Answer Key +Access for free at openstax.org + +3. +4. +6.2 +1. 0.09 +2. 0.24 +3. 0.0218 +6.3 +1. 41% +2. 2% +3. 924.81% +6.4 +1. 338 +2. 2058.75 +6.5 +1. 250 +2. 6,000 +6.6 +1. 62.5% +2. 40% +6.7 +1. $529.50 +6.8 +1. 66.31% +6.9 +1. 80 bulbs. +6.10 +1. Discount is $496; sale price is $1,054.00 +2. Discount is $2.75; sale price is $24.75 +6.11 +1. 35% +2. 12% +6.12 +1. $13.00 +2. $220.00 +6.13 +1. Discount is $330; sale price of the bed is $220. +6.14 +1. 15% +6.15 +1. $59.00 +6.16 +1. Markup is $396; retail price is $2,196.00. +2. Markup is $1.05; retail price is $11.55. +Answer Key +1507 + +6.17 +1. 58.33% +2. 200% +6.18 +1. $35.00 +2. $8.99 +6.19 +1. The retail price of the bed is $1,260.00. +6.20 +1. 24% +6.21 +1. $38.00 +6.22 +1. Sales tax is $104.93; total price is $1,603.93. +2. Sales tax is $1.88; total price is $28.77. +6.23 +1. Sales tax rate = 5.25% +2. Sales tax rate = 6.75% +6.24 +1. $145.78 +2. $489.00 +6.25 +1. Sales tax is $111.22. The total price Daryl pays is $1,260.22. +6.26 +1. The sales tax is $9. +6.27 +1. Interest = $2,409.99, the loan payoff is $9,109.99 +2. Interest = $8,901, the loan payoff is $34,701.00 +6.28 +1. She will pay $878.75 in interest, total repayment is $10,378.75. +6.29 +1. He will pay $727.86 in interest, payoff is $9,127.86. +6.30 +1. The cost to borrow is $115.51 and their payoff is $3,815.51. +6.31 +1. Interest = $253.8, += $4,753.80 +2. Interest = $406.00, += $2,406.00 +3. Interest= $1019.18, += $121,019.18 +4. Interest = $253.89, += $4,933.89 +6.32 +1. Mia will receive $6,254.85. +6.33 +1508 +Answer Key +Access for free at openstax.org + +1. $846.58 +2. $172.91 +6.34 +1. $709.59 +2. $13,182.74 +6.35 +1. $2,313.74 +6.36 +1. $443.90 +6.37 +1. $14,285.72. $14,285.72 needs to be invested so that, after 10 years at 7.5% interest, the investment will be worth +$25,000. +2. $97,709.93. $97,709.93 needs to be invested so that, after 35 years at 6.5% interest, the investment will be worth +$320,000. +3. $48,813.56. $48,813.56 needs to be invested so that, after 270 months at 3.75% interest, the investment will be +worth $90,000. +6.38 +1. $10,549.46 +6.39 +1. The CD is worth $5,627.54 after 4 years. +6.40 +1. CD would have been worth $5,600. Oksana earned $27.54 more with compound interest. +6.41 +1. $11,443.81 +2. $26,551.23 +6.42 +1. $5,813.64 +6.43 +1. $3,555.88 +6.44 +1. $100,811.07 +2. $82,135.75 +6.45 +1. $96,271.23 +6.46 +1. A rate of 7% compounded quarterly is equivalent to a simple interest rate of 7.19%. +6.47 +1. The effective annual yield is 2.53%. +6.48 +1. The effective annual yields for the banks are 3.1158% for Smith Bank, 3.11% for Park Bank, 3.1381% for Town Bank, +and 3.144% for Community Bank. Community Bank has the best yield. +6.49 +Answer Key +1509 + +1. Mateo’s budget is below. +Income Source +Amount +Expense +Amount +Electrician +$3,375.00 +Mortgage +$987 +Side jobs +$300.00 +Truck payment +$589 +Truck insurance +$312 +Utilities +$167 +Clothing +$150 +Entertainment +$400 +Credit Card +$325 +Food +$470 +Gasoline +$375 +His total monthly income is $3,675.00, and his monthly expenses are $3,775. Mateo falls $100.00 short each month. +6.50 +1. Maddy’s budget is below. +Income Source +Amount +Expense +Amount +Engineer +$6,093.75 +Mortgage +$1,452.89 +Car payment +$627.38 +Car insurance +$179.00 +Health Insurance +$265.00 +Utilities +$320.00 +Clothing +$150.00 +Entertainment +$400.00 +Credit Card +$450.00 +Food +$370.00 +Gasoline +$175.00 +Her total monthly income is $6,093.75, and her monthly expenses are $4,389.27. Maddy has $1,704.48 in extra +income per month. This is her cushion in the budget. +6.51 +1. Heather’s budget is now +Income Source +Amount +Expense +Amount +Nursing +$3,765.40 +Mortgage +$1,240 +Part-time +$672.00 +Car Payment +$489 +Student Loan +$728 +1510 +Answer Key +Access for free at openstax.org + +Income Source +Amount +Expense +Amount +Car Insurance +$139 +Utilities +$295 +Clothing +$150 +Entertainment +$300 +Credit Card +$200 +Food +$400 +Gasoline +$250 +The changes have added $355.00 to her budget. As her extra income is $601.40, she can afford the changes. +6.52 +1. Heather’s total income is $4,437.40. +For the necessities, Heather should budget $2,218.70. +For her wants, she should budget $1,331.22. +For savings and extra debt service, she should budget $887.48. +Her necessities total $3,536.00, which exceeds the suggested budget amount of $2,218.70. +Her wants total $300.00, which is below the suggested budget amount of $1,331.22. +Her excess income is $601.40, which is below the suggested budget amount of $887.48. +Heather should make some changes. +6.53 +1. Elijah should budget $1,631.72 for needs, $979.03 for wants, and $652.69 for savings and debt service. When +choosing where to live, what to eat, and what to drive, he should make choices that keep those costs, combined with +debt service costs, gasoline, and utilities, below $1,631.72. This means he will have to make decisions about what his +priorities are. Elijah should then figure out his wants, and stay within the limits here, that is, keep those costs below +$979.03. Finally, he can begin to build his savings with the remaining $652.69. +6.54 +1. Her monthly income is $3,641.66. +Needs (50%): $1,820.83, Wants (30%): $1,092.50, Savings (20%): $728.33 +2. $1,061.83 +3. $1,092.50, $728.33 +4. She only has budgeted $1,061.83 for other necessities. It is difficult to imagine Fran being able to afford to change +jobs and move, unless she reallocates money that she would want to save or use for entertainment or takes on +another job. If Fran uses all the money that the 50-30-20 budget sets aside for savings, she then would have +$1,790.16 to spend on those other necessities. If she moves away from the 50-30-20 philosophy, she may be able to +afford the move. However, that means changing her priorities. +6.55 +1. $5,952.97 +6.56 +1. $3,741.46 +6.57 +1. $8,978.57 +6.58 +1. 17.31% +2. 13.2% +Answer Key +1511 + +6.59 +1. $42,499.63 +6.60 +1. $93,037.59 +6.61 +1. $1,227.30 +6.62 +1. $237.50 per year, $5,237.50 on maturity date. +6.63 +1. $108,501.30 +6.64 +1. 4.28% +2. 3.29% +6.65 +1. Looking at the table, the current price of a share is $30.68. +2. The high was $56.28, and the low was $30.05. +3. August 4, 2022 +4. 4.77% +5. $4.66 +6.66 +1. $336.00 +6.67 +1. $52 +2. 2.6% +6.68 +1. $108,352.43 +6.69 +1. $1,815.83 +6.70 +1. 23.75% +6.71 +1. 4.35% +6.72 +1. 371.74% +2. 13.8% +6.73 +1. $3,336.13 +6.74 +1. Traditional +6.75 +1. $5,460 +2. $10,920 +1512 +Answer Key +Access for free at openstax.org + +3. 100% return on the $5,460 deposit +6.76 +1. $813,128.60 +2. $457,384.83 +6.77 +1. $355.91 +2. $62.48 +6.78 +1. $213.07 +2. $209.96 +6.79 +1. $34,700, 3.79%, 6 years +2. $539.57 +3. $25,837.42 +4. $104.14 +5. $2,246.62 +6. The amount paid to principal increases each month +6.80 +1. $6,437.18 +6.81 +1. $9,745 +6.82 +1. Tiana can take out $12,500, the maximum in year 6. +6.83 +1. Gap = $10,850, Federal loans = $7,500, Remaining $3,350 +6.84 +1. $2,987.63 +6.85 +1. $82.70 +2. $137.42 +6.86 +1. $259.69 +6.87 +1. There is no impact on Ryann’s interest rate. The total Ryann pays back increases since the interest is extended to +more periods. +6.88 +1. $188.03 +6.89 +1. $5,310 +6.90 +1. $11,735 +2. $97.80 +Answer Key +1513 + +6.91 +1. Store-issued credit cards +2. Fees (both annual and penalty fees), reward programs, credit limits +3. Travel and entertainment cards +6.92 +1. April 01 2015 to April 30 2015 +2. $9,000.00 +3. $5.99 +6.93 +1. $567.32 +6.94 +1. $22.64 +2. $13.49 +3. $77.77 +6.95 +1. $566.44 +2. $3,267.17 +6.96 +1. $1,152.35 +6.97 +1. $47.47 +2. $84.00 +3. $19.90 +6.98 +1. $33,838.48 +6.99 +1. $32,777.75 +6.100 +1. $436.70 +2. $649.47 +6.101 +1. $1,516.05 +6.102 +1. $10,567.00 +6.103 +1. $401.05 +6.104 +1. 6.25% +2. +6.105 +1. $74.49 +2. $1,015.25 +6.106 +1514 +Answer Key +Access for free at openstax.org + +1. Purchase +2. Purchase +6.107 +1. Personal injury protection +2. Liability insurance +6.108 +1. $593.36 +6.109 +1. $42.02 +6.110 +1. $949.72 +6.111 +1. $319,620.60 +6.112 +1. $93,741.60 +6.113 +1. 6.1% +2. $1,383.61 +3. $730.38 +4. $723.66 +5. $133,513.05 +6. Payment 225 +6.114 +1. $1,521.20 +6.115 +1. $47,280 +6.116 +1. $80,300 +6.117 +1. $39,700 +6.118 +1. $25 +6.119 +1. Refund of $1,920 +6.120 +1. FICA is $244.80, SSI is $198.40 +6.121 +1. $9,114 +6.122 +1. $11,787 +6.123 +Answer Key +1515 + +1. His credits exceed his taxes owed, so he receives a refund, which is for $1,673.94. +Check Your Understanding +1. 100 +2. 0.387 +3. 190.4 +4. 834.15 (rounded to two decimal places) +5. 24.07% +6. 16 new drivers will be hired. +7. a reduction in price +8. the amount a retailer adds to the cost of goods to cover overhead and profit +9. $30.60 +10. $497.60 +11. Yes +12. $42.67 +13. Interest is money paid by a borrower to a lender for the privilege of borrowing money. +14. The amount borrowed is the principal. +15. $1,169.10 +16. $11,374.87 +17. $266.82 +18. $21,818.19 +19. interest that is earned on both the principal and the interest from previous periods +20. compound +21. $15,698.69 +22. $405.62 +23. $30,266.48 +24. Compare their effective annual yield. +25. 4.98% +26. A budget is an estimation of income and expenses over some period of time. +27. Expenses that represent basic living requirement and debt service. +28. +Income +Amount +Expenses +Amount +Job +$3,450.00 +Rent +$925.00 +Car payment +$178.54 +Car insurance +$129.49 +Credit card +$117.00 +Gas +$195.00 +Food +$290.00 +Amazon Prime +$21.99 +Internet +$49.99 +Going out +$400.00 +Total +$2,307.01 +29. $1,142.99 +30. David should allocate $1,735 to necessary expenses, $1,035 to expenses that are wants, and $690 to savings and +extra debt service. +31. David spends $1,835.03 on necessary expenses, which is fairly close to the 50-30-20 budget philosophy guidelines. +He spends $471.98 on expenses that are wants, which is well below the guideline of $1,035. He has $1,149.99 in +extra income, which is well above the savings and debt service guidelines. +32. A deposit account, held at a bank or other financial institution, which bears some interest on the deposited money. +1516 +Answer Key +Access for free at openstax.org + +33. Once money is deposited in a CD, the money cannot be withdrawn until the end of the CD period. +34. A money market account is more flexible. Transactions can be made after creating the money market account. +35. $8,549.93 +36. 25.6% +37. $21,441.03 +38. $2,030.65 +39. Stocks +40. Income must be below a threshold for a person to use a Roth IRA. +41. They invest in different investment vehicles so that no single investment can have a large impact on the value of the +mutual fund. +42. Each year, the holder of the bond receives $200. For 10 years, that’s $2,000 total earned with the bond. +43. Yes +44. $3,500, which is 4% of her income +45. $114.00 +46. 9.93% +47. $5,207.59 +48. Varies +49. Loan with a fixed period, and the borrower pays a fixed amount per period until the loan is paid off +50. More is applied to the principal. +51. $59.11 +52. $2,233.12 +53. $42,962.51 +54. $39,544.50 +55. Spring (March) before the start of the fall term +56. 9 years +57. The difference between the cost of college (including books, fees, room and board, etc.) and all non-loan financial +aid they receive +58. In subsidized loans, the interest does not accrue while the student is still in school. In unsubsidized student loans, +the interest begins accruing as soon as the money is disbursed. +59. $31,000 +60. $13,028.61 +61. 10 years +62. 10% +63. Bank issued, store issued, charge (Travel/entertainment) cards +64. Store issued +65. Due date, minimum payment due, and balance due +66. $19.30 +67. $4,104.27 +68. $696.21 +69. Cost of delivering the vehicle to the dealer +70. Registers your car with the state, gets the license plate, and assigns the title of the car to the lender +71. $628.89 +72. Interest rate higher loan term shorter +73. Title and registration, documentation fee, sales tax +74. $333.34 +75. 3.75% +76. Purchasing +77. Cover the damage caused to your car if involved in an accident with another vehicle +78. Buying +79. Renting +80. Buying +81. Mortgage +82. $663.32 +83. $129,802.20 +84. $554.06 +85. Escrow +86. Any and all income, including wages, salary, interest earned, gifts, and winnings +87. After deductions are subtracted +Answer Key +1517 + +88. $37,000 +89. After taxes are calculated, credits are subtracted from taxes owed. +90. No, only for up to $147,000 +91. $4,813 +92. $1,600 +Chapter 7 +Your Turn +7.1 +1. +7.2 +1. +7.3 +1. +7.4 +1. 120 +7.5 +1. 720 +2. 132 +3. 70 +7.6 +1. 30 +2. 24,024 +3. 5,814 +7.7 +1. +7.8 +1. combination +2. combination +7.9 +1. 15 +2. 45 +3. 2,002 +7.10 +1. 30,856 +2. 111,930 +7.11 +1. 290,004 +7.12 +1. +2. {A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K} +3. {1, 2, 3, 4} +4. {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} +7.13 +1. Independent +1518 +Answer Key +Access for free at openstax.org + +2. Dependent +7.14 +1. {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} +7.15 +1. {J +Q , J +K , Q +J , Q +K , K +J , K +Q } +7.16 +1. +2. +3. +7.17 +1. +2. +3. +7.18 +1. +7.19 +1. +2. +3. +7.20 +1. +7.21 +1. theoretical +2. subjective +3. empirical +7.22 +1. +7.23 +1. +7.24 +1. +2. +7.25 +1. +2. +7.26 +1. The odds for +are +and the odds against +are +. +7.27 +1. +2. +Answer Key +1519 + +7.28 +1. Mutually exclusive +2. Not mutually exclusive +3. Mutually exclusive +7.29 +1. +2. Not appropriate; the events are not mutually exclusive. +3. +7.30 +1. +2. +3. +7.31 +1. +2. +3. +7.32 +1. +2. +3. +7.33 +1. +Roll Probability +1 +2 +3 +4 +5 +6 +7.34 +1. Not binomial (more than two outcomes) +2. Not binomial (not independent) +3. Binomial +4. Not binomial (number of trials isn’t fixed) +7.35 +1. 0.146 +2. 0.190 +3. 0.060 +7.36 +1. 0.2972 +2. 0.6615 +1520 +Answer Key +Access for free at openstax.org + +3. 0.0919 +4. 0.5207 +5. 0.3643 +7.37 +1. +2. +3. += $6.25 +7.38 +1. If you roll the special die many times, the mean of the numbers showing will be around 3.33. +2. If you repeat the coin-flipping experiment many times, the mean of the number of heads you get will be around 1.5. +3. If you play this game many times, the mean of your winnings will be around $10. +7.39 +1. If the player bets on 7, the expected value is +. +2. If the player bets on 12, the expected value is +. +3. If the player bets on any craps, the expected value is +. +The best bet for the player is any craps; the best bet for the casino is the bet on 7. +Check Your Understanding +1. 540 +2. 14 +3. 1,024 +4. 800 +5. 25,920 +6. 120 +7. 120 +8. 1,320 +9. 1,680 +10. +11. permutations +12. combinations +13. 66 +14. 560 +15. 20 +16. 560 +17. {0, 1, 2, 3, 4, 5, 6} +18. +Crinkle Fries +Curly Fries +Onion Rings +8 Nuggets +8 nuggets with crinkle fries +8 nuggets with curly fries +8 nuggets with onion rings +12 Nuggets +12 nuggets with crinkle fries +12 nuggets with curly fries +12 nuggets with onion rings +Answer Key +1521 + +19. +20. {8 nuggets with crinkle fries, 8 nuggets with curly fries, 8 nuggets with onion rings, 12 nuggets with crinkle fries, 12 +nuggets with curly fries, 12 nuggets with onion rings} +21. {history with Anderson, history with Burr, English with Carter, sociology with Johnson, sociology with Kirk, sociology +with Lambert} +22. +23. +24. +25. 0 +26. theoretically +27. subjectively +28. empirically +29. +30. 89.9% +31. +32. +33. +34. +35. +36. +37. +38. +39. +40. +41. +42. +43. +44. +45. +46. +47. +48. +49. +50. +51. +52. +53. +1522 +Answer Key +Access for free at openstax.org + +54. +55. +56. No (more than two outcomes) +57. Yes +58. No (number of trials is not fixed) +59. 0.9453 +60. 0.1366 +61. 0.8230 +62. 4.1 +63. If you roll this die many times, the mean of the numbers rolled will be around 4.1. +64. $0.417 +65. If you play this game many times, the mean of the amount won/lost each time will be about 42 cents. +66. Yes; the expected value is positive. +Chapter 8 +Your Turn +8.1 +1. None of the above (there’s no sample being selected here; the entire population is being surveyed) +2. Stratified random sample (the strata are the different majors) +3. Simple random sample +8.2 +1. +Major +Frequency +Biology +6 +Education +1 +Political Science +3 +Sociology +2 +Undecided +4 +8.3 +1. +Number of people in the residence Frequency +1 +12 +2 +13 +3 +8 +4 +6 +5 +1 +8.4 +1. +Answer Key +1523 + +Age range Frequency +25-29 +2 +30-34 +6 +35-39 +2 +40-44 +4 +45-49 +1 +50-54 +2 +55-59 +2 +60-64 +0 +65-70 +1 +(Answers may vary depending on bin boundary decisions) +8.5 +1. +Major +Frequency +Proportion +Biology +6 +37.5% +Education +1 +6.3% +Political Science +3 +18.8% +Sociology +2 +12.5% +Undecided +4 +25% +Note that these percentages add up to 100.1%, due to the rounding. +8.6 +1. +8.7 +1. Southeast +2. Just over 10% +1524 +Answer Key +Access for free at openstax.org + +3. Outlying Areas and Rocky Mtns. +8.8 +1. +8.9 +1. Twenty-four +2. The longest commutes are 60, 50, and 36 miles; the shortest are 4, 6, and 7 miles. +3. 4, 6, 7, 10, 10, 10, 12, 12, 12, 14, 15, 18, 18, 20, 25, 25, 25, 30, 30, 35, 35, 36, 50, 60 +8.10 +1. +4 +7 +5 +9 7 4 +6 +9 8 7 +7 +8 7 5 2 2 1 0 +8 +9 6 5 4 4 1 +9 +7 7 6 3 3 1 +10 +7 6 3 1 +8.11 +1. +8.12 +1. Answers may vary based on bin choices. Here’s the result for bins of width 2,500: +Answer Key +1525 + +The data are strongly right-skewed. +8.13 +1. +8.14 +1. Top ten teams by wins: +1526 +Answer Key +Access for free at openstax.org + +Answer Key +1527 + +8.15 +1. There are two modes: 89 and 104, each of which appears three times. +8.16 +1. 2 +8.17 +1. 136 +8.18 +1. 82.5 +8.19 +1. 2 +8.20 +1. 12.556 +1528 +Answer Key +Access for free at openstax.org + +8.21 +1. +8.22 +1. Modes: 72, 84, 93, 97 +Median: 82.5 +Mean: 80.967 (rounded to three decimal places) +8.23 +1. Mode or median +2. Median or mean +3. Median or mean +8.24 +1. 67 +8.25 +1. +8.26 +1. +8.27 +1. $13,333.77. +8.28 +1. 15 +2. 70th percentile +8.29 +1. 1026 +2. 1318 +3. 82.7th +4. 99.2nd +8.30 +1. $3,120 +2. $26,465.20 +3. 32.3rd +4. 98.5th +8.31 +1. The red (leftmost) distribution has mean 11, the blue (middle) has mean 13, and the yellow (rightmost) has mean +14. +8.32 +1. 6 +8.33 +1. Mean: 150; standard deviation: 20 +8.34 +1. 99.7% +2. 95% +3. 68% +8.35 +1. 65 and 75 +Answer Key +1529 + +2. 26 and 54 +3. 110 and 290 +8.36 +1. 47.5% +2. 15.85% +3. 81.5% +8.37 +1. +8.38 +1. –44 +30 +6 +8.39 +1. 80th +9th +97th +8.40 +1. 3.9 +6.3 +2.9 +8.41 +1. 1.29 +2. 94.5th +3. An LSAT score of 161 is better +8.42 +1. 72 +2. 6 +3. 93rd +8.43 +1. Using NORM.INV: 1092.8 +Using PERCENTILE: 1085 +8.44 +1. Income +2. Either; neither one seems to directly influence the other (they’re both influenced by the student’s academic ability) +3. GPA +8.45 +1530 +Answer Key +Access for free at openstax.org + +1. +8.46 +1. +8.47 +1. +a. +No curved pattern +b. +Strong negative relationship, +2. +a. +Curved pattern +3. +a. +No curved pattern +b. +No apparent relationship, +4. +a. +No curved pattern +b. +Weak positive relationship, +8.48 +1. +1. +2. +Not appropriate +3. +4. +8.49 +Answer Key +1531 + +1. +8.50 +1. +8.51 +1. +2. 26.68 +3. Predicted: 28; actual: 18. The Phillies were caught around 10 fewer times than expected. +4. Every 10 additional steal attempts will result in getting caught about 1.7 times on average. +8.52 +1. +, where x is the proportion of made field goals and y is the proportion of made three-point +field goals +2. 0.347 +3. Predicted: 0.340; actual: 0.368. The Aces made about 2.8% more of their three-point shots than expected. +4. An increase of 1% in made field goal attempts will result in an expected increase of 0.55% in made three-point field +goal attempts. +Check Your Understanding +1. Randomization is being used; cluster random sample. +2. No randomization is being used. +3. Randomization is being used; stratified random sample. +4. +Genre +Frequency +Cooking +1 +Non-fiction +3 +Romance +4 +Thriller +3 +True Crime +3 +Young Adult +6 +5. +Number of classes Frequency +1 +1 +2 +3 +3 +16 +4 +8 +5 +4 +6. +Range of Cell Phone Subscriptions Per Hundred People Frequency +0.0 – 24.9 +1 +25.0 – 49.9 +3 +50.0 – 74.9 +1 +75.0 – 99.9 +6 +1532 +Answer Key +Access for free at openstax.org + +Range of Cell Phone Subscriptions Per Hundred People Frequency +100.0 – 124.9 +7 +125.0 – 149.9 +3 +150.0 – 174.9 +3 +175.0 – 199.9 +1 +(Note: Answers may vary based on choices made about bins.) +7. +8. +9. +12 +7 +13 +0 2 3 6 6 7 8 9 9 +14 +1 2 3 3 6 8 8 +15 +3 3 5 6 6 6 7 7 8 8 +16 +4 7 8 +Answer Key +1533 + +10. +11. +(data source:https://data.ed.gov/) +1534 +Answer Key +Access for free at openstax.org + +12. +(data source: https://data.ed.gov/) +(data source: https://data.ed.gov/) +13. Mode: 112 +Median: 113 +Mean: 112.64 +14. 3 +15. 3 +16. +17. 156 +18. 147 +19. 147.2 +Answer Key +1535 + +20. Mode: not useful; every value appears only once +Median: 0.322 +Mean: 0.3612 +21. Mode: not useful; every value appears only once +Median: $42,952 +Mean: $42924.78 +22. Mode: $13,380 +Median: $11,207 +Mean: $15,476.79 +23. Since the data are right skewed, the mean will be bigger than the mean. Thus, the workers would rather use the +median, while the management will prefer the mean. +24. 9 +25. +26. +27. 11.306 +28. Range: +29. Standard deviation: 0.170 +30. Range: +31. Standard deviation: $398.37 +32. 18 +33. 14 +34. 80th +35. 60th +36. 71.7 +37. 101.2 +38. 70.25 +39. 51.7th +40. 79.3rd +41. 13.8th +42. 95% +43. 34% +44. 84th +45. 0.583 +46. 71.2 +47. 72nd +48. 115.38 +49. 27th +50. 23.2 +51. 1300 on the SAT +52. +; +53. 96.6th +54. 57 +55. PERCENTILE gives 20.86; NORM.INV gives 21.11. +56. PERCENTILE gives 11.82; NORM.INV gives 11.68. +57. PERCENTRANK gives 92.6th; NORM.DIST gives 91.9th. +58. PERCENTRANK gives 77th; NORM.DIST gives 79.2nd. +59. +1536 +Answer Key +Access for free at openstax.org + +60. +a. +Yes +61. +a. +No +b. +Weak positive relationship; +62. 0.96 +63. +64. $7,475 +65. Less than expected by $1,495.68 +66. For every $1,000 increase in out-of-state tuition, we expect average monthly salary to increase by $161. +Chapter 9 +Your Turn +9.1 +1. gram (g) +2. meter (m) +3. liter (L) +9.2 +1. kilogram, hectogram, decagram +9.3 +1. 2.5 m +9.4 +1. 245 cm +9.5 +1. 0.76 daL +9.6 +1. 145.7 cg +9.7 +1. 1.25 L +9.8 +1. Yes, 352 mg of caffeine is equal to 0.352 g, which is less than 0.5 g. +9.9 +1. Since 3,810 m is 3.81 km, Kyrie’s skydive was at a greater altitude. +9.10 +1. km2 +9.11 +1. 1,800 cm2 +9.12 +1. Yes. The unit of value, cm2, is appropriate given the size of a sheet of paper. Because the total area of 2,412.89 cm2 +represents four sheets of paper, each sheet is approximately 600 cm2, which seems reasonable for a sheet of letter- +size paper. +9.13 +1. 153.29 m2 +Answer Key +1537 + +9.14 +1. 1,365,000 cm2 +9.15 +1. 1.06 m2 +9.16 +1. 34 m2 +9.17 +1. 1,250 m2 +9.18 +1. 70 m2 +9.19 +1. 8 m3 +9.20 +1. 40,000 cm2 +9.21 +1. Yes. The unit of value, mm3, may be appropriate given the size of the paperweight. The volume of 125,000 mm3 is +equivalent to a volume of 125 cm3, which would represent a cubic box with dimensions of +. +9.22 +1. 8,000 mm3 +9.23 +1. 37,854,000 cm3 +9.24 +1. 465 m3 +9.25 +1. 19 L +9.26 +1. 37,854 L +9.27 +1. 800 mL +9.28 +1. 100,000 boxes +9.29 +1. 6,000 boxes +9.30 +1. 18.75 L +9.31 +1. 2.5 g +9.32 +1. 1,300 kg +1538 +Answer Key +Access for free at openstax.org + +9.33 +1. A reasonable value for the weight of an Etruscan shrew is 2 g. A 2 L bottle of water weighs 2 kg, and since it is one of +the smallest mammals, the shrew must weigh less than that, so 2 g seems reasonable. +9.34 +1. 0.175 kg +9.35 +1. 4,869 g +9.36 +1. 1,230,000 mg +9.37 +1. 7.5 mL +9.38 +1. 5 kg +9.39 +1. 111,111 potatoes +9.40 +1. 250 °C +9.41 +1. 167 °F +9.42 +1. Since Toronto has a temperature of 14.1 °C, Madrid has the higher temperature by 2.6 °C. +9.43 +1. Recall that water boils at 100 °C. The more reasonable value for the temperature of the mixture is 148 °C. So, 48 °C is +not sufficient to bring the mixture to a boil. +9.44 +1. 37 °C +9.45 +1. 60 °C +9.46 +1. –2 °C +9.47 +1. 63 °C +9.48 +1. 180 minutes, or 1 hour and 30 minutes +Check Your Understanding +1. meter +2. grams +3. 12 hectoliters +4. 15.20 m or 15.2 m +5. 134 dam +6. 1.27 hg +7. 750,000,000 mm +Answer Key +1539 + +8. They are equal in size. +9. 12 m2 +10. 1,200 m2 +11. 2.5 m2 +12. 2 +13. 57,000 +14. 21,700 +16. 50,000 cm3 +17. 236 mL +18. 1,500 cm3 +19. 42.5 +20. 1,500 +21. 6,750 +22. 800 +23. 10.2 +24. 163.52 +25. 50 g +26. 180 kg +27. 624 g +28. 8.9 +29. 17,000 +30. 70 +31. 0.144 +32. 196,000 +33. 0.09 +34. 28.5 °F +35. 20 °C +36. 71.1 °C +37. 210 +38. 100 +39. 35 +40. 107 +41. 81 +42. 399 +Chapter 10 +Your Turn +10.1 +1. +is the ray that starts at point +and extends infinitely in the direction of point +. +2. +represents the line segment that starts at point +and ends at point +. +3. +represents the ray that starts at point +and extends infinitely in the direction of point +. +4. +represents a line that contains the points +and +. Notice the arrowheads on both ends of the line above +, +which means that the line continues infinitely in both directions. +10.2 +1. Answers will vary. One way +; Second way +. +10.3 +1. +10.4 +1. +2. +1540 +Answer Key +Access for free at openstax.org + +3. +10.5 +1. Point +is located at +; Point +is located at +; Point +is located at +; Point +is located at +; +Point +is located at +; Point +is located at +. +2. Points +and +are on the straight line +. +3. The line that begins at point +in the direction of point +is a ray, +. +4. The line from point +to point +is a line segment +. +5. Yes, this represents a plane. One reason is that the figure contains four points that are not on the line +. +10.6 +1. Plane +intersects with plane +, and plane +intersects with plane +. +10.7 +1. +Acute Angles Obtuse Angles +Right Angles +Straight Angles +10.8 +1. +10.9 +1. +∡ +, +∡ +, +∡ +10.10 +1. +∡ +, +∡ +, +∡ +10.11 +1. +10.12 +1. +∡ +, +∡ +, +∡ +10.13 +1. +∡ +, +∡ +10.14 +1. +∡ +∡ +∡ +∡ +∡ +10.15 +Answer Key +1541 + +1. +∡ +, +∡ +, +∡ +10.16 +1. +∡ +; +∡ +; +∡ +10.17 +1. Triangle +is congruent to triangle +. +10.18 +1. SAS +10.19 +1. ASA +10.20 +1. The triangles are similar. +10.21 +1. +, +10.22 +1. The tree is 86 feet high. +10.23 +1. 60 ft +10.24 +1. rectangle +2. pentagon +3. heptagon +4. parallelogram +10.25 +1. Shapes 1, 2, 4, and 6 are triangles; shape 3 is a pentagon; shape 5 is a parallelogram; and shape 7 is a rectangle. +10.26 +1. 120 in +10.27 +1. 22.4 in +10.28 +1. +The sum of the interior angles is +. +10.29 +1. We have the sum of interior angles is +. Then, +The other angles measure +, +, +. +10.30 +1. +10.31 +1. +10.32 +1. +1542 +Answer Key +Access for free at openstax.org + +10.33 +1. You need to buy 15.74 feet of trim. +10.34 +1. The translated hexagon has labels +, +, +, +, +, +. +10.35 +1. Rotate the triangle about the rotation point +to the right three times. +10.36 +1. This tessellation could be produced with a reflection of the triangle vertically, then each triangle is rotated +and +translated to the right. +10.37 +1. From the first square on the left, rotate the square +to the right, or +. Then, reflect the square over the +horizontal, or +. Next, reflect all three squares over the vertical line. The lavender triangles comprise +another pattern that tessellates, fits in with the squares, and fills the gaps. +10.38 +1. No +10.39 +1. Not by themselves, but by adding an equilateral triangle, the two regular polygons do tessellate the plane without +gaps. +10.40 +1. We made a tessellation with a regular decagon (10 sides) and an irregular hexagon. We see that the regular +decagon will not fill the plane by itself. The gap is filled, however, with an irregular hexagon. These two shapes +together will fill the plane. +10.41 +1. +Answer Key +1543 + +10.42 +1. 108 ft2 +10.43 +1. 45 boxes at a cost of $2,025.00 +10.44 +1. +10.45 +1. 13,671 square feet; cost is $19,872.95. +10.46 +1. +10.47 +1. +10.48 +1. 40 in +10.49 +1. +10.50 +1. +10.51 +1. +10.52 +1. the 15-inch pizza +10.53 +1. +10.54 +1. +10.55 +1. $300 +10.56 +1. +10.57 +1. +10.58 +1. +10.59 +1. +10.60 +1544 +Answer Key +Access for free at openstax.org + +1. +10.61 +1. approximately 8 1/3 cans of soup +10.62 +1. 6.25 ft by 6.25 ft +10.63 +1. +$95.26 +10.64 +1. +10.65 +1. 1,140 ft +10.66 +1. The slanted distance will be 120.4 inches. +10.67 +1. The side lengths are +10.68 +1. The ladder reaches 12 feet up the wall and sits +from the wall. +10.69 +1. Each side +equals +10.70 +1. +10.71 +1. +, +10.72 +1. 2,241 ft +10.73 +1. +, one angle is +, and the other angle is +. +10.74 +1. +10.75 +1. 46 ft +Check Your Understanding +1. The line containing point +and point +is a line segment from point +to point +, +, or from point +to point +, +. +2. The line containing points +and +is a straight line that extends infinitely in both directions and contains points +and +. +3. This is a ray that begins at point +, although it does not contain point +, and extends in the direction of point +. +4. +. The union of line segment +and the line segment +contains all points in each line segment +Answer Key +1545 + +combined. +5. +. The intersection of the ray +and the line segment +contains only the points common to each +set, +. +6. +. The union of the ray starting at point +and extending infinitely in the direction of +and the ray +starting at point +and extending infinitely in the direction of +is the straight line extending infinitely in both +directions containing points +, +, +, and +. +7. Two lines are parallel if the distance between the lines is constant implying that the lines cannot intersect. +8. Perpendicular lines intersect forming a +angle between them. +9. Yes, because it contains a line and a point not on the line. +10. straight +11. obtuse +12. right +13. acute +14. +∡ +by supplementary angles with ∡ +15. +∡ +by vertical angles with the angle measuring +16. +∡ +by corresponding angles with ∡ +17. +18. +19. +and +20. These are similar triangles, so we can solve using proportions. +Then, +and +. +21. Set up the proportions. +Thus, +and +22. pentagon +23. octagon +24. heptagon +25. +26. +27. +28. +29. +30. The patterns are repeated shapes that can be transformed in such a way as to fill the plane with no gaps or +overlaps. +31. Starting with the triangle with the point labeled +, the triangle is translated point by point 3 units to the right and 3 +1546 +Answer Key +Access for free at openstax.org + +units up to point +. Then, the triangle labeled +is translated 3 units to the right and 3 units up to point +. +32. The triangle is rotated about the rotation point +to vertex +. +33. The dark triangle is reflected about the vertical line showing the light back, and then reflected about the horizontal +line. The pattern is repeated leaving a white diamond between the shapes. +34. 3.3.3.3.3.3 +35. 7.5 cm2 +36. 25 ft +37. +38. +39. 201.1 in2 +40. +41. +42. +43. +44. +45. +46. +47. +48. +49. +50. +51. 24.2 ft +52. +53. +Chapter 11 +Your Turn +11.1 +1. Yes, Joe Biden won the majority. +11.2 +1. Tony Cambell won a plurality of votes in the Republican primary. +11.3 +1. Hearn and Lim tied. Yes, there must be a third election. +2. Kelly must be removed in the runoff. +11.4 +1. Option B. +11.5 +1. +2. None +3. 4 +11.6 +1. Using ranked-choice voting, Yoda is determined to be the winner. +11.7 +1. Blue received 78 points. +11.8 +1. Candidate B would be considered the winner using the ranked-choice voting method. +2. Candidate C would be considered the winner using the Borda count method. +Answer Key +1547 + +3. Different candidates won. It appears that the vote counts were so close that a small shift in either direction could +change the results of either method. +11.9 +1. +2. Bong Joon-ho won according to the pairwise comparison method. +3. Yes, the winner is a Condorcet candidate. +11.10 +1. Candidate P wins. Candidate P is not a Condorcet candidate because P lost to Candidate Q. +11.11 +1. Candidate C won 25 points. +2. The greatest number of points another candidate could win is 24 points. +3. No, Candidate C cannot lose or tie. +11.12 +1. Dough Boys Pizza is the winner for dinner. +11.13 +1. There is a tie between Rainbow China and Dough Boys Pizza. +11.14 +1. Al Gore is the winner. +11.15 +1. Animal Kingdom +2. Animal Kingdom +3. Magic Kingdom +11.16 +1. Using the plurality voting method, there would be a tie between Option B and Option C. +2. Using the ranked-choice voting method, Option B would be the winner. +3. No, there was not a majority candidate in Round 1. +1548 +Answer Key +Access for free at openstax.org + +11.17 +1. Using the pairwise comparison voting method, Option B wins. +2. Using the Borda count method, Option A and Option B tie. +3. Yes. The majority criterion fails for Borda count. +11.18 +1. Yes, because Option A is a Condorcet candidate. +11.19 +1. Option A. Yes, the Condorcet criterion is satisfied. +2. Option A. Yes, the Condorcet criterion is satisfied. +3. Option B. No, the Condorcet criterion is not satisfied. +11.20 +1. Standard Poodle. +2. Standard Poodle. +3. This election does not violate the monotonicity criterion. +4. Increasing the ranking for a winner of a Borda count election on a ballot will increase that candidate’s Borda score +while decreasing another candidate’s Borda score, but leaving the remaining candidates’ Borda score unchanged. +So, a Borda count election will never violate the monotonicity criterion. +11.21 +1. Labrador retriever wins the election. +2. Golden retriever wins the election. +3. This election violates the monotonicity criterion. +4. It is possible that the monotonicity criterion would be met in other ranked-choice election scenarios, but overall, the +ranked-choice voting method is said to fail the monotonicity criterion even if it failed in only one scenario. +11.22 +1. Chihuahua. +2. Yorkshire Terrier. +3. Yes, this election violates IIA. +11.23 +1. Jim wins. +2. Pam wins. +3. Yes, this is a violation of IIA. +11.24 +1. B +; 0.67, C +; 0.67, D +; 0.67, and E +; 0.67 +2. B +; 1.5, C +; 1.5, D +; 1.5, and E +; 1.5 +3. Yes, the constant ratio is +pencils per desk. +11.25 +1. 42 pencils would be allocated. +2. 42 pencils would be allocated. +3. 34 desks +11.26 +1. IL 700,000; OH 700,000; MI 700,000; GA 800,000; NC 700,000 +2. IL 0.0000014; OH 0.0000014; MI 0.0000014; GA 0.0000013; NC 0.0000014 +3. IL 0.000001; OH 0.000001; MI 0.000001; GA 0.000001; NC 0.000001 +4. The ratio of State Population to Representative Seats seems to be either 700,000 or 800,000 to 1. There does appear +to be a constant ratio of about 0.000001 to 1 of Representative Seats to State Population when rounding to six +decimal places. This is the same as the top five states. +11.27 +Answer Key +1549 + +1. 60,449.4 +11.28 +1. The states are the Hernandez family and the Higgins family. The seats are the pieces of candy. The house size is 313. +The state populations are three in the Hernandez family and four in the Higgins family. The total population is 7. +2. The standard divisor is the ratio of the number of children to the number of pieces of candy. 0.0224 children per +piece of candy. +11.29 +1. 0.98 +11.30 +1. +Family +Family’s Standard Quota +Hernandez +175.6667 candies +Higgins +234.2222 candies +Ho +117.1111 candies +Total +527 +The sum is very close to 527. +11.31 +1. 6, 5, 6, 7, 5, 6 +2. 35 +3. No. +11.32 +1. It is not possible to give a fractional part of a gift card. Also, traditional rounding to the nearest integer results in 4 +gift cards for each student, which leaves one extra gift card. +11.33 +1. 30, 18, 20 +2. 68 +3. 2 +11.34 +1. The final Hamilton apportionment is Fictionville as follows: 1, Pretendstead 3, Illusionham 5, and Mythbury 26. +11.35 +1. Method V violates the quota rule because State F receives 4 seats instead of 5 or 6. +11.36 +1. 53.33 +2. 480 +3. 96 +4. Answers may vary. With a modified divisor of 0.100, the modified quota is 120. +5. The modified quota from part 3 was the smallest, because the divisor was the largest of the three. Dividing the same +number by a larger value gives a smaller result. +11.37 +1. Each state would receive the following seats: Fictionville 1, Pretendstead 2, Illusionham 4, and Mythbury 28. +11.38 +1. Fictionville 2, Pretendstead 3, Illusionham 5, and Mythbury 25. +1550 +Answer Key +Access for free at openstax.org + +11.39 +1. The apportionment is Fictionville 2, Pretendstead 2, Illusionham 5, and Mythbury 26. +11.40 +1. The largest state is Mythbury. The citizens would likely favor the Jefferson method of apportionment most since they +received the most seats by that method. They would likely favor the Adams and Webster methods of apportionment +least because they received the least number of seats by those method. +2. As a group the other three states received nine seats by either the Hamilton method, seven seats by the Jefferson +Method, and ten seats by either the Adams method or the Webster method. They would likely favor the Adams +method and Webster method most and favor the Jefferson methods least. +11.41 +1. State A loses 0, seats State B loses 0, and State C loses 3. +2. State A loses 0 seats, State B loses 1, and State C loses 4. +3. State C, the largest state, loses the most representatives each time the divisor is increased. +11.42 +1. The standard divisor is 214,079.1236 citizens per seat. The standard quota for Colorado is 2.5210 seats. +2. The standard divisor is 213,479.4622 citizens per seat. The standard quota for Colorado is 2.5281 seats. +3. The standard quota increased. +4. It must have been the case that either the fractional part 0.5281 ranked lower amongst the other fractional parts of +the state quotas than the fractional part 0.5210 did, or there were fewer remaining seats to be distributed, or both. +11.43 +1. The final apportionment is: A 183, B 124, C 14, and D 2, which sums to 323. +2. The final apportionment is: A 184, B 125, C 13, and D 2, which sums to 324. +3. Yes, this demonstrates the Alabama paradox because State C receives 14 seats if the house size is 323, but only 13 +seats if the house size is 324. +11.44 +1. Hospital C lost a respirator while hospital A gained a seat, but hospital C has a higher growth rate than hospital A. +11.45 +1. The Hamilton reapportionment is: 19 for Mudston, 13 for WallaWalla, and 6 for Dilberta. This is an example of the +population paradox because WallaWalla lost a seat to Dilberta, even though WallaWalla’s population grew by 2.24 +percent while Dilberta’s only grew by 1.56 percent. +11.46 +1. 90 +2. 95 +3. The original state of Beaversdam lost a seat to the original state of Beruna when the new state of Chippingford was +added. +11.47 +1. The reapportionment gives 38 seats to Neverwood, 46 seats to Mermaids Lagoon, and 9 seats to Marooners Rock. +This is an example of the new-states paradox because the original state Mermaids Lagoon lost a seat to the original +state Neverwood when the new state was added to the union. +11.48 +1. +There are +new seats to be apportioned to New Mexico. +11.49 +1. Only Hamilton’s method violates the population paradox. +Answer Key +1551 + +Check Your Understanding +1. Answers may vary. Example: Ranked-choice, Borda count, and pairwise comparison. +2. True. +3. True +4. Borda count +5. Pairwise comparison +6. Hare +7. In two-round voting, only the top two candidates from Round 1 move on to Round 2, and there are only two rounds. +In ranked-choice voting, all candidates except those in last place move on to the next round, and there can be many +rounds of voting. +8. (Independence of) Irrelevant Alternatives Criterion +9. Pairwise comparison +10. Borda count method +11. Pairwise comparison +12. All of them: Plurality, ranked-choice, pairwise comparison, and Borda count +13. True, because a majority candidate is always the Condorcet candidate. +14. False, because the ranked-choice method violates the Condorcet criterion, but it doesn’t violate the majority +criterion. +15. No. Arrow’s Impossibility Theorem does not apply to approval voting because it is not a ranked voting system +16. False. Arrow’s Impossibility Theorem says that no ranked voting system is perfect and that voter profiles may arise +that will lead to a violation of one or more fairness criteria, but it does not guarantee that those voter profiles will +occur or are even likely to occur. +17. True. Candidates are rated as approved or not approved, and voters can give multiple candidates the same rating. +18. Student 1 is correct. +19. Both are correct. +20. Neither are correct. +21. Neither are correct. +22. Student 2 is correct. +23. Student 2 is correct. +24. Both are correct. +25. The Hamilton method. +26. The Hamilton method. +27. The Jefferson method. +28. The Adams method. +29. The Adams method. +30. The Hamilton and Jefferson methods. +31. Webster’s method. +32. Larger +33. The Alabama paradox +34. Population paradox +35. Quota rule violation +36. None of these +37. New-states paradox +38. The population paradox could occur. +39. The Alabama paradox could occur. +40. The new-state paradox could occur. +Chapter 12 +Your Turn +12.1 +1. The vertices are p, q, r, s, and t. The edges are pq, pr, pt, qr, qs, qt, rs, and st. +12.2 +1. The pairs that are not adjacent are p and s, p and t, and r and t. +12.3 +1552 +Answer Key +Access for free at openstax.org + +1. Vertex q +12.4 +1. Graph Representing Common Boundaries Between Regions of Oahu +12.5 +1. The objects that are represented with vertices are poker players. An edge between a pair of vertices would indicate +the two players competed against each other at table. This is best represented as a graph because players from the +same table never meet a second time. Also, a player cannot compete against themself; so, there is no need for a +loop. +12.6 +1. Graph E, 34 +2. Graph D, 18 +3. higher +12.7 +1. 3 +2. Answers may vary. Two possible graphs are shown. +Graph 1: +Graph 2: +12.8 +1. 124,750 introductions +12.9 +1. +12.10 +1. Triangle: (a, b, e or a, d, e) +Quadrilateral: (b, c, d, e or a, b, e, d) +Pentagon: (a, b, c, d, e) +12.11 +Answer Key +1553 + +1. (B, D, C, E, F) +12.12 +1. 186 +12.13 +1. Graph C1 has 6 edges, but Graph C2 has 8. +Graph C1 has 4 vertices and Graph C2 has 5. +Graph C1 has no vertex of degree 4, but Graph C2 has one vertex of degree 4. +They do not have the same cycles. For example, Graph C2 has a pentagon cycle, but Graph C1 does not. +12.14 +1. Answers may vary. +• +The total number of vertices in each graph is different. The Diamonds graph has 17 vertices while Dots graph has +only 16. +• +The degrees of vertices differ. The Diamonds graph has vertices of degrees 4 while the Dots graph does not. +• +The graphs have different sizes of cyclic subgraphs. The Diamonds graph has 4 squares (4-cycles), while the Dots +graph has 3 squares. Also, the Dots graph has 8-cycles while the Diamonds graph does not. +12.15 +1. Answers may vary. There are four possible isomorphisms: +• +a − q, d − s, c − p, and b − r. +• +a − p, d − s, c − q, and b − r. +• +a − q, d − r, c − p, and b − s. +• +a − p, d − r, c − q, and b − s. +12.16 +1. Caden is correct because Graph E could be untangled so that d is to the left of c and the edges bc and ad are no +longer crossed. Maubi is incorrect because (a, b, c, d) is a quadrilateral in Graph E. Javier is incorrect because the +portion of the graph he highlighted does not have 3 vertices and is not a triangle. +12.17 +1. +12.18 +1. The degrees are 0. There are no adjacent vertices in Graph N since all vertices are adjacent in Graph M. +12.19 +1. c Only C → V → G → B → B is a walk. +12.20 +1. walk, path, and trail +2. walk and a trail, but not a path +3. none of these +12.21 +1. To fly from Palm Beach to any other city, you must take the flight from PBI to TPA. To return, you must take the flight +from TPA to PBI. This means that the graph of any itinerary that begins and ends at PBI covers the same edge twice +and is not a circuit. +2. The degree of PBI is 1 meaning that there is only one edge connecting PBI to other parts of the graph. +1554 +Answer Key +Access for free at openstax.org + +12.22 +1. Yes. The chromatic number is 4 or less. +2. 3 +3. 3 or 4 +4. Graphs 1 and 3 +5. Answers may vary. Here is an example of a 3-coloring: +12.23 +1. The chromatic number is 3. Here is a 3-coloring of the graph: +12.24 +1. Graphs G, H, and I are connected; so, they each have a single component with the vertices {a, b, c, d}. Graph F is +disconnected with two components, {a, c, d} and {b}. Graph J is disconnected with two components, {a, d} and {b, c}. +Graph K is disconnected with three components, {a}, {b, d}, and {c}. +12.25 +1. Each pair of vertices would be connected by a path that was at most 6 edges long. So, the graph would be +connected. +12.26 +1. The multigraph of the neighborhood is shown: +2. Yes, the graph is connected. It has only one component. +3. There are four corner vertices of degree 2 and the remaining twelve vertices are all degree 4. +4. Yes +5. Yes, an Eulerian graph has an Euler circuit; so, it is possible. +12.27 +1. Graph X does not have an Euler circuit because it is disconnected. +2. a → b → g → d → f → c → g → e → a +12.28 +1. +Answer Key +1555 + +12.29 +1. Yes. +2. No. It doesn’t cover every edge. +3. No. It is not a walk, because the vertices b and e are not adjacent. +12.30 +1. In a complete graph with three or more vertices, any three vertices form a triangle which is a cycle. Any edge in a +graph that is part of a cycle cannot be a bridge so, there are no bridges in a complete graph. Also, any edge that is +part of a triangle cannot be a local bridge because the removal of any edge of a triangle will leave a path of only two +edges between the vertices of that edge. So, there are no bridges or local bridges in a complete graph. +12.31 +1. m +2. qn, qt; qn bridge +3. rs, st, tq, qn +4. nm, np; nm,, bridge +5. om, mp, pn, nm +6. s → q → r → s → t → q → n → o → m → p → n → m +12.32 +1. The graph has Euler trails. They must begin and end at vertices u and v. An example is +v → w → x → u → z → y → w → u +12.33 +1. both +12.34 +1. +and +12.35 +1. +12.36 +1. 120 +12.37 +1. 26 +12.38 +1. No +2. No +3. Yes +12.39 +1. C → A → B → F → D → E +12.40 +1. p → m → o → n → q → t → s → r +2. none +3. none +1556 +Answer Key +Access for free at openstax.org + +12.41 +1. Hamilton path +2. Euler trail +3. neither +12.42 +1. brute force algorithm +12.43 +1. +1. +V → W → X → Y → Z → V +2. +V → W → X → Z → Y → V +3. +V → W → Y → X → Z → V +4. +V → W → Y → Z → X → V +5. +V → W → Z → X → Y → V +6. +V → W → Z → Y → X → V +7. +V → X → W → Y → Z → V +8. +V → X → W → Z → Y → V +9. +V → X → Y → W → Z → V +10. +V → X → Y → Z → W → V (reverse of 6) +11. +V → X → Z → W → Y → V +12. +V → X → Z → Y → W → V (reverse of 4) +13. +V → Y → X → W → Z → V +14. +V → Y → X → Z → W → V (reverse of 5) +15. +V → Y → W → X → Z → V +16. +V → Y → W → Z → X → V (reverse of 11) +17. +V → Y → Z → X → W → V (reverse of 2) +18. +V → Y → Z → W → X → V (reverse of 8) +19. +V → Z → X → Y → W → V (reverse of 3) +20. +V → Z → X → W → Y → V (reverse of 15) +21. +V → Z → Y → X → W → V (reverse of 1) +22. +V → Z → Y → W → X → V (reverse of 7) +23. +V → Z → W → X → Y → V (reverse of 13) +24. +V → Z → W → Y → X → V (reverse of 9) +12.44 +1. The officer should travel from Travis Air Force Base to Beal Air Force Base, to Edwards Air Force Base, to Los Angeles +Air Force Base, and return to Travis. +12.45 +1. D → B → E → A → C → F → D; 550 min or 9 hrs 10 min. +12.46 +1. The star topology is a tree, and the tree topology is a tree. The ring topology and the mesh topology are not trees +because they contain cycles. +12.47 +1. star topology +2. tree topology +3. none +12.48 +1. Vertices k, l, and m are in one component, and vertices g, h, i, and j are in the other. +2. There are 6 vertices and 5 edges, which confirms Graph I is a tree, because the number of edges is one less than the +number of vertices. +3. a quadrilateral +12.49 +Answer Key +1557 + +1. a True +2. b False +3. a True +4. a True +12.50 +1. Answers may vary. Here are three possible spanning trees of Graph J: +12.51 +1. The three edges must include one edge from list A and one pair of edges from list B. +List A: be, eh, hi, gi, bg +List B: ac and ad, ac and af, ac and cd, ac and cf, ad and af, ad and cd, ad and cf, af and cd, af and cf, or cd and cf. +12.52 +1. There are two possible minimum spanning trees, each with a total weight of 174: +Check Your Understanding +1. a True +2. b False +3. a True +4. b False +5. b False +6. b False +7. a True +8. b False +9. a True +10. b +11. a True +12. The sum of the degrees is always double the number of edges, which means it has to be an even number, but 13 is +odd. +13. Yes. If n is the number of vertices in a complete graph, the number of edges is +which is half the +number of vertices times one less than the number of vertices. +14. always true +15. never true +16. sometimes true +17. never true +18. always true +19. sometimes true +20. sometimes true +21. never true +22. sometimes true +23. always true +1558 +Answer Key +Access for free at openstax.org + +24. always true +25. sometimes true +26. always true +27. sometimes true +28. always true +29. always true +30. sometimes true +31. always true +32. sometimes true +33. always true +34. always true +35. has no repeated egdes +36. is closed +37. has no repeated vertices +38. has no repeated edges +39. +40. at least +51. edge +52. Fleury’s +53. circuit, trail +54. components +55. local bridge +56. bridge +57. vertex +58. complete +59. is +60. is +61. is not +62. is +63. is +64. is not +65. is not +66. is not +67. different from +68. the same as +69. different from +70. different from +71. b False +72. b False +73. b False +74. a True +75. a True +76. b False +77. b False +78. b False +79. a True +80. b False +81. b False +82. a True +83. a True +84. b False +85. b False +86. a True +87. a True +88. a True +89. a True +90. b False +91. b False +Answer Key +1559 + +92. a True +93. a True +94. a True +Chapter 13 +Your Turn +13.1 +1. 46 in +13.2 +1. 34 scales +13.3 +1. 1.618 +13.4 +1. 1.25, which is close to a golden rectangle +13.5 +1. 210 gal +13.6 +1. 1,050,000 gal +13.7 +1. 9,828 million gal of water +13.8 +1. 15.8 kW +13.9 +1. $14.18 +13.10 +1. $0.06 +13.11 +1. 288 kW +13.12 +1. 1.8 mg in 200 mL +13.13 +1. 6 pills +13.14 +1. 42 mg +13.15 +1. 13 mL +13.16 +1. Yara, Jeffrey +2. 5 +13.17 +1. birds chirping +1560 +Answer Key +Access for free at openstax.org + +2. window air conditioner +3. garbage disposal +4. live outdoor concert +13.18 +1. up G, down F +2. up C, down B♭ +3. up G, down F +13.19 +1. up E♭, down B +2. up D#, down B +3. up F#, down D +13.20 +1. G, G#, A, A#, B, C, C#, D, D#, E, F, F# +13.21 +1. 83 Hz +13.22 +1. 10 games +2. 66 games +3. 300 games +13.23 +1. 4 games +2. 11 games +3. 24 games +Check Your Understanding +1. 1.618 +2. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 +3. A rectangle whose length divided by the width is approximately equal to +or 1.618. +4. +5. 100 gal +6. 33 1/3 hours +7. 17 hours +8. medicine dosage required, weight of drug by volume +9. b Research on coronavirus vaccines began approximately 10 years before the COVID-19 pandemic hit the world. +10. Mathematical modeling is a graphical representation of a behavior, such as exposure to a virus. It can be helpful in +mapping contract tracing with virus exposure during a pandemic. +11. Frequency is a static measured value, whereas pitch is subjective. +12. higher +13. 20–20,000 Hz +14. increase +15. 0 dB +16. 20–17,000Hz +17. It doubles. +18. Data analytics is used in sports to analyze data on team’s players, potential recruits, opposing teams, and opposing +players. This helps to find patterns and make predictions for future results. +19. Soccer (football) +20. +21. +Answer Key +1561 + +1562 +Answer Key +Access for free at openstax.org + +Index +Symbols +1099 form +602 +68-95-99.7 Rule +892 +A +is greater than +151 +absolute value of an integer +152 +acquisition fee +679 +acute angle +1008 +acyclic +1380 +additive system of numbers +273 +adjacent +1240 +adjusted gross income +698 +Alabama paradox +1212 +Algebraic expressions +334 +angle of depression +1111 +angle of elevation +1111 +annual percentage rate +566 +apportion +1179 +apportionment paradox +1212 +apportionment problem +1179 +approval voting ballot +1150 +approval voting system +1150 +Area +952 +arithmetic sequence +240 +Arrow’s Impossibility Theorem +1175 +average (or mean) of a set of +156 +B +bar chart +830 +base +218 +base 10 +288 +base 10 system +269 +is less than +151 +biconditional +71 +billing period +658 +bimodal +857 +binned frequency distribution +821 +binomial +404 +binomial experiments +790 +binomial trials +790 +Bonds +612 +bridges +1328 +brute force algorithm +1364 +brute force method +1370 +budget +590 +C +cardinal value +10 +Cardinal voting systems +1175 +cardinality +10 +cardinality of the union of two sets +32 +categorical frequency distribution +819 +chain rule for conditional +arguments +112 +chromatic number +1293 +circuit +1290 +clock arithmetic +211 +closed +1289 +cluster sample +817 +coefficient +339 +combinations +733 +combinatorics +724 +common ratio +246 +complement +24 +composite +131 +compound event +771 +compound interest +579 +compound statement +70 +conclusion +64 +conditional +71, 88 +conditional probabilities +777 +Condorcet criterion +1166 +Condorcet method +1166 +conjunction +70 +connective +70 +constant +334 +constant difference +240 +constant of proportionality +369 +constant sequence +247 +constraints +520 +contrapositive +98 +converse +98 +correlation coefficient +917 +countably infinite +10 +credit limit +657 +cubic meter +944 +cubic units +960 +cumulative distribution function +793 +cycle +1255 +D +data +816 +data analytics +1444 +De Morgan’s Law +103 +dealer preparation fee +676 +decibels +1438 +deductive arguments +109 +degree +1241 +degrees +1007 +degrees Celsius +944 +dependent +739 +dependent variable +908 +destination fee +676 +directed cycle +1290 +directed path +1287 +directed trail +1287 +directed walk +1284 +discounts +553 +disjoint sets +22 +disjunction +70 +disposition fees +679 +distribution +839 +Distributive Property +340 +dividend +294 +dividends +613 +division in a base +323 +divisor +294 +documentation fee +676 +domain +428 +dominance of connectives +74 +doubling time +246 +down payment +676 +down-rank +1169 +E +edge +1238 +Effective annual yield +585 +elements +6 +empirical probability +757 +empty set +7 +equal +11 +equal sign +337 +equation +337 +equivalent +11 +Euler circuits +1313 +Euler trail +1326 +evaluating the function +429 +event +751 +exemptions +698 +expanded form +269 +expected value +798 +experiment +739 +explanatory variable +908 +exponent +218 +exponential expression +268 +exponential notation +17 +exponents +134 +expressions +336 +extended warranties +676 +extrapolation +925 +F +factorial +729, 1343 +FAFSA +645 +fallacy +108 +Fibonacci sequence +244, 1420 +finite sets +8 +fixed expenses +590 +fixed interest rate +631 +flat +1441 +Fleury’s algorithm +1330 +forest +1381 +formula +352 +Index +1563 + +Four-Color Map Theorem +1299 +Four-Color Theorem +1299 +fractional form +544 +frequency +1438 +function +428 +G +geometric sequence +246 +geometric solids +1088 +glide reflection +1057 +golden ratio +1419 +golden rectangles +1422 +grams +944 +graph +1238 +graph colorings +1293 +greatest common divisor +138 +greedy algorithm +1364 +gross income +698 +H +half-step +1441 +Hamilton circuits +1341 +Hamilton cycles +1341 +Hamilton paths +1352 +Hare Method +1132 +hertz +1438 +Hindu-Arabic numeration system +269 +Histograms +841 +hypothesis +71 +I +Inclusion/Exclusion Principle +773 +independent +739 +independent variable +908 +inductive arguments +64 +infinite sets +8 +installment loan +631 +instant runoff voting (IRV) +1137 +integers +9, 149 +intercepts +447 +interest +566 +intersection of two sets +29 +inverse +98 +irrational numbers +191 +isomorphic +1266 +issue price +612 +K +kilowatt +1427 +kilowatt-hour +1427 +Kruskal’s algorithm +1391 +L +landlord +687 +law of denying the consequent +111 +law of detachment +109 +lease +687 +least common multiple +142 +line of best fit +921 +linear equation +346 +linear inequalities +361 +Linear programming +519 +liters +944 +Loan amortization +631 +local bridge +1329, 1329 +Logic +59 +logical statement +60 +loop +1238 +lower quota +1193 +M +majority +1130 +majority criterion +1162 +mapping +431 +mass +970 +Mathematical modeling +1434 +mean +864 +median +859 +meters +944 +metric system +943 +minimum spanning tree +1391 +mode +857 +modulo 12 +212 +modulo 7 +216 +money market account +603 +monomial +404 +mortgage +689 +multigraph +1238 +multiple +142 +multiplication and division in +bases +314 +multiplication principle +82 +Multiplication Rule for Counting +725 +mutually exclusive +771 +N +natural numbers +8 +nearest neighbor method +1371 +negation of a logical statement +62 +negative linear relationship +915 +neighboring +1240 +net worth +154 +new-state paradox +1216 +-coloring +1293 +nonplanar +1268 +normal distributions +886 +number +268 +numeral +268 +O +obtuse angle +1008 +octave +1442 +odds against +766 +odds for +766 +odds in favor +766 +open +1289 +order of operations +158 +ordinary annuity +605 +origination date +566 +P +part-to-part ratio +1180 +path +1287 +percent +544 +percentiles +879 +permutations +728 +perpendicular lines +997 +pie chart +835 +pitch +1440 +place values +269 +planar +1268 +polygon +1035 +polynomial +404 +population +816 +population growth rate +1214 +population paradox +1214 +positional system of numbers +273 +positive linear relationship +915 +Postulate 1 +994 +Postulate 2 +994 +preference ranking +1136 +premises +64 +prime +131 +prime factorization +134 +principal +566, 631 +probability +752 +probability density function +793 +proper subset +15 +properties of the real numbers +208 +proportional +1180 +proportions +367, 828 +Pythagorean Theorem +1098 +Q +Quadratic equations +408 +quadratic formula +415 +quadrilateral +1070 +quantifier +64 +quantiles +879 +quartile +879 +Quintile +879 +quotient +294 +R +radians +1007 +ranked ballot +1136 +ranked voting +1136 +ranked-choice voting (RCV) +1137 +ratio +366 +1564 +Index +Access for free at openstax.org + +rational number +165 +Real numbers +204 +reapportionment +1214 +reflection +1057 +regression line +921 +relation +428 +remainder +294 +replicated +739 +representative democracies +1182 +response variable +908 +return on investment +604 +Revolving credit +631 +right angle +1008 +right cylinder +1092 +right prism +1088 +right triangle +204 +rise +451 +run +451 +runoff election +1132 +S +sample space +739 +samples +816 +savings account +601 +scaling factor +1025 +scatter plot +908 +scientific notation +227 +security deposit +679 +seeding +1445 +sequence +239 +Sequences +17 +series +17 +set +5, 6 +set builder notation +9 +shares +613 +sharp +1441 +simple graph +1238 +simple interest +566 +simple random sample +816 +simplify an expression +338 +slope +451 +solutions of a system of equations +474 +sound +108 +spanning trees +1386 +spoiler +1153 +square meter +944 +Square Root Property +414 +square units +952 +standard divisor +1184 +standard quota +1185 +standardized score +898 +Stem-and-leaf plots +837 +Stocks +613 +straight angle +1008 +stratified sample +817 +subjective probability +758 +subset +15 +subsidized loans +646 +Subtraction in bases +310 +symbolic form +61 +system of linear inequalities +491 +systematic random sample +817 +T +tautology +97 +tax credits +699 +taxable income +698 +Temperature +977 +term +239 +terms +339 +tessellations +1052 +theoretical probability +752 +title and registration fee +676 +total weight +1346 +trail +1287 +translation +1054 +traveling salesperson problem +(TSP) +1367 +tree +1380 +tree diagram +742 +trials +790 +truth table +78 +truth values +60 +two-round system +1132 +tyranny of the majority +1165 +U +union of two sets +30 +units +816 +universal set +21 +unsubsidized loans +646 +up-rank +1169 +upper quota +1196 +V +valid +84 +variable +334 +variable expenses +590 +variable interest rate +631 +Venn diagram with three +intersecting sets +40 +Venn diagrams +21 +vertex +1007 +vertical line test +434 +vertices +1238 +Volume +960 +W +walk +1284 +weighted graph +1346 +well-defined sets +6 +whole step +1441 +Z +Zero Product Property +412 +Index +1565 +