A Survey of ALLEN R. ANGEL Monroe Community College CHRISTINE D. ABBOTT Monroe Community College DENNIS C. RUNDE State College of Florida, Manatee-Sarasota ANNOTATED INSTRUCTOR’S EDITION 12TH EDITION Mathematics with Applications
Content Development: Robert Carroll, Phillip Oslin Content Management: Brian Fisher, Christina French Content Production: Preeti Saini, Shrawan Joshi, Pallavi Pandit, Stephanie Woodward, Sandra Rodriguez Product Marketing: Alicia Wilson Rights and Permissions: Anjali Singh Please contact www.AskPearsonSupport.com with any queries on this content. Please contact us with concerns about any potential bias at https://www.pearson.com/ report-bias.html. You can learn more about Pearson’s commitment to accessibility at https://www.pearson.com/ us/accessibility.html. Cover Image by Flavio Coelho/Moment/Getty Images. Copyright © 2025, 2021, 2017 by Pearson Education, Inc. or its affiliates, 221 River Street, Hoboken, NJ 07030. All Rights Reserved. Manufactured in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information regarding permissions, request forms, and the appropriate contacts within the Pearson Education Global Rights and Permissions department, please visit www.pearsoned. com/permissions/. Acknowledgments of third-party content appear on the appropriate page within the text. PEARSON and MYLAB are exclusive trademarks owned by Pearson Education, Inc. or its affiliates in the U.S. and/or other countries. Unless otherwise indicated herein, any third-party trademarks, logos, or icons that may appear in this work are the property of their respective owners, and any references to third-party trademarks, logos, icons, or other trade dress are for demonstrative or descriptive purposes only. Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc., or its affiliates, authors, licensees, or distributors. Library of Congress Cataloging-in-Publication Data Names: Angel, Allen R., author. | Abbott, Christine D., author. | Runde, Dennis C., author. Title: A Survey of Mathematics with Applications / Allen R. Angel, Monroe Community College, Christine D. Abbott, Monroe Community College, Dennis C. Runde, State College of Florida, Manatee-Sarasota. Description: Twelfth edition. | Hoboken, NJ : Pearson, [2024] | Includes index. Identifiers: LCCN 2024000708 | ISBN 9780138306298 (print) | ISBN 9780138306274 (rental) | ISBN 9780138306342 (annotated instructor copy) Subjects: LCSH: Mathematics—Textbooks. Classification: LCC QA39.3 .A54 2024 | DDC 510—dc23/eng/20240201 LC record available at https://lccn.loc.gov/2024000708 Annotated Instructor’s Copy: ISBN-10: 0-13-830634-6 ISBN-13: 978-0-13-830634-2 Rental Edition: ISBN-10: 0-13-830627-3 ISBN-13: 978-0-13-830627-4 Print Offer: ISBN-10: 0-13-830629-X ISBN-13: 978-0-13-830629-8 $PrintCode
To my wife, Kathy Angel A. R. A. To my grandchildren, Mia and Owen Abbott C. D. A. To the memory of my parents, Bud and Tina Runde D. C. R.
iv To the Student ix Preface xi 1 Critical Thinking Skills 1 1. 1 Inductive and Deductive Reasoning 2 1. 2 Estimation Techniques 9 1. 3 Problem-Solving Procedures 20 Chapter Summary 35 Chapter Review Exercises 35 Chapter Test 39 2 Sets 41 2. 1 Set Concepts 42 2. 2 Subsets 52 2. 3 Venn Diagrams, Set Operations, and Data Representation 58 2. 4 Venn Diagrams with Three Sets and Verification of Equality of Sets 71 2. 5 Set Applications and Survey Data Analysis 80 2. 6 Infinite Sets 87 Chapter Summary 91 Chapter Review Exercises 92 Chapter Test 95 3 Logic 96 3. 1 Statements and Logical Connectives 97 3. 2 Truth Tables for Negation, Conjunction, and Disjunction 107 3. 3 Truth Tables for the Conditional and Biconditional 121 3. 4 Equivalent Statements 131 3. 5 Symbolic Arguments 144 3. 6 Euler Diagrams and Syllogistic Arguments 154 3. 7 Switching Circuits 160 Chapter Summary 166 Chapter Review Exercises 168 Chapter Test 170 Contents
CONTENTS v 4 Systems of Numeration 171 4. 1 Additive, Multiplicative, and Ciphered Systems of Numeration 172 4. 2 Place-Value or Positional-Value Numeration Systems 181 4. 3 Other Bases 188 4. 4 Perform Computations in Other Bases 195 4. 5 Early Computational Methods 203 Chapter Summary 207 Chapter Review Exercises 209 Chapter Test 211 5 Number Theory and the Real Number System 212 5. 1 Number Theory 213 5. 2 The Integers 224 5. 3 The Rational Numbers 235 5. 4 The Irrational Numbers 250 5. 5 Real Numbers and Their Properties 258 5. 6 Rules of Exponents and Scientific Notation 265 5. 7 Arithmetic and Geometric Sequences 275 5. 8 The Fibonacci Sequence and The Golden Ratio 282 Chapter Summary 289 Chapter Review Exercises 290 Chapter Test 293 6 Algebra, Graphs, and Functions 294 6. 1 Order of Operations and Solving Linear Equations 295 6. 2 Formulas and Modeling 306 6. 3 Applications of Algebra 312 6. 4 Variation 321 6. 5 Solving Linear Inequalities 330 6. 6 Graphing Linear Equations 338 6. 7 Solving Systems of Linear Equations 352 6. 8 Linear Inequalities in Two Variables and Systems of Linear Inequalities 366 6. 9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula 376
vi CONTENTS 6. 10 Functions and Their Graphs 387 Chapter Summary 404 Chapter Review Exercises 406 Chapter Test 409 7 The Metric System 410 7. 1 Basic Terms and Conversions Within the Metric System 411 7. 2 Length, Area, and Volume 418 7. 3 Mass and Temperature 429 7. 4 Dimensional Analysis and Conversions to and from the Metric System 436 Chapter Summary 448 Chapter Review Exercises 449 Chapter Test 450 8 Geometry 452 8. 1 Points, Lines, Planes, and Angles 453 8. 2 Polygons 464 8. 3 Perimeter and Area 475 8. 4 Volume and Surface Area 487 8. 5 Transformational Geometry, Symmetry, and Tessellations 501 8. 6 Topology 520 8. 7 Non-Euclidean Geometry and Fractal Geometry 527 Chapter Summary 534 Chapter Review Exercises 536 Chapter Test 539 9 Mathematical Systems 541 9. 1 Groups 542 9. 2 Finite Mathematical Systems 550 9. 3 Modular Arithmetic 560 9. 4 Matrices 569 Chapter Summary 581 Chapter Review Exercises 581 Chapter Test 583
CONTENTS vii 10 Consumer Mathematics 584 10. 1 Percent 585 10. 2 Personal Loans and Simple Interest 595 10. 3 Compound Interest 605 10. 4 Installment Buying 615 10. 5 Buying a House with a Mortgage 630 10. 6 Ordinary Annuities, Sinking Funds, and Retirement Investments 641 Chapter Summary 650 Chapter Review Exercises 651 Chapter Test 654 11 Probability 656 11. 1 Empirical and Theoretical Probabilities 657 11. 2 Odds 671 11. 3 Expected Value (Expectation) 679 11. 4 Tree Diagrams 691 11. 5 Or and And Problems 701 11. 6 Conditional Probability 712 11. 7 The Fundamental Counting Principle and Permutations 720 11. 8 Combinations 732 11. 9 Solving Probability Problems by Using Combinations 739 11. 10 Binomial Probability Formula 744 Chapter Summary 753 Chapter Review Exercises 755 Chapter Test 758 12 Statistics 760 12. 1 Sampling Techniques and Misuses of Statistics 761 12. 2 Frequency Distributions and Statistical Graphs 770 12. 3 Measures of Central Tendency and Position 784 12. 4 Measures of Dispersion 797 12. 5 The Normal Curve 806 12. 6 Linear Correlation and Regression 822 Chapter Summary 837 Chapter Review Exercises 838 Chapter Test 841
viii CONTENTS 13 Graph Theory 842 13. 1 Graphs, Paths, and Circuits 843 13. 2 Euler Paths and Euler Circuits 853 13. 3 Hamilton Paths and Hamilton Circuits 865 13. 4 Trees 877 Chapter Summary 888 Chapter Review Exercises 890 Chapter Test 893 14 Voting and Apportionment 895 14. 1 Voting Methods 896 14. 2 Flaws of the Voting Methods 913 14. 3 Apportionment Methods 928 14. 4 Flaws of the Apportionment Methods 945 Chapter Summary 953 Chapter Review Exercises 954 Chapter Test 957 ANSWERS A-1 INDEX OF APPLICATIONS I-1 INDEX I-9
ix To the Student Mathematics is an exciting, living study. Its applications shape the world around you and influence your everyday life. We hope that as you read this book you will realize just how important mathematics is and gain an appreciation of both its usefulness and its beauty. We also hope you learn some practical mathematics that you can use every day and that will prepare you for further mathematics courses. The primary purpose of this text is to provide material that you can read, understand, and enjoy. To this end, we have used straightforward language and tried to relate the mathematical concepts to everyday experiences. The concepts, definitions, and formulas that deserve special attention are in boxes or are set in boldface, italics, or color type. We have also provided many detailed examples for you to follow. Be sure to read the chapter summary, work the review exercises, and take the chapter test at the end of each chapter. The answers to the odd-numbered exercises, all review exercises, and all chapter test exercises appear in the answer section in the back of the text. You should, however, use the answers only to check your work. The answers to all Recreational Mathematics exercises are provided either in the Recreational Math boxes themselves or in the back of the book. It is difficult to learn mathematics without becoming involved. To be successful, we suggest that you read the text carefully and work each exercise in each assignment in detail, for it is in doing the math that you really learn and enjoy it. If you are using this text within MyLab Math, you’ll find a wealth of other learning aids available there, including tutorial videos and homework help. We welcome your suggestions and your comments. You may contact us at math@pearson.com. (Please use the subject line “Angel Survey of Math.”) Good luck with your adventure in mathematics! Allen R. Angel Christine D. Abbott Dennis C. Runde
xi Preface We present A Survey of Mathematics with Applications, Twelfth Edition, with the knowledge that we use mathematics every day. In this edition, we stress how mathematics is used in our daily lives and why it is important. Our primary goal is to give students a text they can read, understand, and enjoy while learning how mathematics affects the world around them. Numerous real-life applications are used to motivate topics. A variety of interesting and useful exercises demonstrate the real-life nature of mathematics and its importance in students’ lives. The text is intended for students who require a broad-based general overview of mathematics, especially those majoring in the liberal arts, elementary education, the social sciences, business, nursing, and allied health fields. It is particularly suitable for those courses that satisfy the minimum competency requirement in mathematics for graduation or transfer. New to This Edition New Within the Textbook 7 Now Try Exercises – After each example, students are guided to a corresponding exercise in the exercise set that closely resembles the example. This allows students to assess their comprehension by solving the similar exercise. In the exercise sets, these Now Try exercises are indicated with a green color and underline, such as 15. 7 Financial Literacy – For problems which involve financial applications, you’ll see the icon $ . We’ve increased the number of exercises relating to money and finance because of their importance in consumer education. 7 Downloadable Data Sets – For problems and examples in which students are expected to analyze a set of data, you’ll see the icon DS, which indicates that the data is available to download in *.txt and *.csv formats. We’ve greatly increased the number of exercises in the text that use these downloadable data sets. All of the data sets are housed in MyLab Math and also at https://bit.ly/48bwKME. 7 Data and Context Updates – We’ve updated time-sensitive data to the most current available or changed the context of a problem or narrative so that it is more relevant to students. 7 Instructor Resources Boxes – In the annotated instructor’s editions of the text, new boxes have been added before the exercise sets in each section which list the available resources available in MyLab. New Within MyLab Math 7 Projects – Brand new activities and projects have been created, written specifically to pair with this text. You can find two projects per chapter to be assigned for group or individual work, delving into real-world applications to showcase the practical side of mathematical concepts. Emphasizing engagement and creativity, these projects encourage active learning, problem solving, and critical thinking, providing an adaptable and engaging math experience. 7 Corequisite Support – Corequisite material in the MyLab Math course has received a complete overhaul. The corequisite content offers a wide array of topics to cover any that may be required prior to taking a liberal arts math course. The corequisite support resources provide all the content and assessment resources necessary for students and instructors by offering videos, worksheets, and exercises for each objective. 7 Integrated Review – Integrated Review narrows down the corequisite content to a smaller subset of objectives that could be used to help fill students' prerequisite
xii PREFACE gaps, or for a corequisite course. Integrated Review prebuilt assignments include a chapter-level Skills Check on relevant prerequisite skills, and follow-up personalized assignments and resources to learn. This content has been revised to include resources specifically made for a liberal arts math corequisite course. 7 Skills Check Quizzes by chapter assess the prerequisite skills students need for that chapter. 7 Skills Review Homework, again by chapter, is personalized (based on the results of the Skills Check Quiz) to provide students with help on the prerequisite skills they are lacking. Students receive just the help they need—no more, no less. 7 Exercise Labeling – When creating assignments in the Assignment Manager, you will now have the ability to filter the available questions using Question Source to more easily find the Now Try, Financial Literacy, and Data Set exercises. Financial Literacy exercises have also been tagged with a “-FL” in their exercise number to identify them more easily. Continuing and Revised Features 7 Chapter Openers – Interesting and motivational applications introduce each chapter, which includes the Why This Is Important section, and illustrate the realworld nature of the chapter topics. 7 Problem Solving – Beginning in Chapter 1, students are introduced to problem solving and critical thinking. We continue the theme of problem solving throughout the text and present special problem-solving exercises in the exercise sets. 7 Critical Thinking Skills – In addition to a focus on problem solving, this book also features sections on inductive and deductive reasoning, estimation, and dimensional analysis. 7 Profiles in Mathematics – Brief historical sketches and vignettes present stories of people who have advanced the discipline of mathematics. In this edition, we included more diversity among the mathematicians included. 7 Did You Know? – These colorful, engaging, and lively features highlight the connections of mathematics to history, the arts and sciences, technology, and a broad variety of disciplines. 7 Mathematics Today – These features discuss current real-life uses of the mathematical concepts in the chapter. Each box ends with Why This Is Important. 7 Recreational Math – In these features, students are invited to apply the math in puzzles, games, and brain teasers. Answers are given in upside-down type at the bottom of the feature or (for longer answers) at the back of the book. In addition, Recreational Mathematics problems appear in the exercise sets so that they can be assigned as homework. 7 Technology Tips – The material in these features explains how students can use technology including calculators, spreadsheets, and smartphone apps to explore various mathematical concepts and solve application problems. 7 Timely Tips – These easy-to-identify boxes offer helpful information to make the material under discussion more understandable. 7 Key-Idea Boxes – Important definitions, formulas, and procedures are boxed, making key information easy to identify for students. 7 Animations – Located throughout the narrative you’ll find Animation icons like the one pictured at the left. They are designed to facilitate active learning and visualization of key concepts. These Animations are housed within MyLab Math and are ideal for classroom use during lecture or by students independently. They were created in GeoGebra and are usable on any type of device. They are also editable. 7 StatCrunch – Located throughout the narrative you’ll find StatCrunch® icons like the one pictured at the left. Like the Animation features, they are designed to StatCrunch
PREFACE xiii facilitate active learning and exploration. They are also housed within MyLab Math. StatCrunch is Pearson’s online statistical software. 7 Learning Catalytics – Learning Catalytics is a “bring your own device” student polling and assessment system, available in MyLab Math. Each section of the Annotated Instructor’s Edition features keywords that can be entered into Learning Catalytics to bring you directly to questions for use in lecture for that section. Detailed instructions for using these keywords can be found at http:// bit.ly/485Yna8. 7 Chapter Summaries, Review Exercises, and Chapter Tests – The endof-chapter summary charts provide an easy study experience by directing students to the location in the text where specific concepts are discussed. Review Exercises and Chapter Tests also help students review material and prepare for exams. Student Resources MyLab Math is tightly integrated with each author team’s style, offering a learning and practice experience that gives students a consistent experience from text to MyLab. MyLab Math includes assignable algorithmic exercises for unlimited practice opportunities. The complete eText is available for learning in any environment. And a comprehensive gradebook gives instructors and students alike insight into how they are doing at any time. Additionally, the following resources are available to enrich student learning. 7 Section Lecture Videos – Lecture videos for each section, broken down to the objective-level as well, offer a modern, engaging approach that incorporates animations, applets, and StatCrunch. All videos have closed captioning available. Video assessment questions in the Assignment Manager check for student understanding of the video they just watched, making the videos assignable. 7 Interactive Concept Check Videos – These videos walk students through key concepts and pause to ask them questions, requiring student interaction throughout. For each incorrect answer, the video follows a different path focusing on the reason or misconception for selecting that particular incorrect answer. Students are then presented with another interactive question to gauge understanding. Assignable exercises are available for instructors to assess student understanding.
xiv PREFACE 7 Animations – Animations let students interact with the math in a visual, tangible way. These interactive figures, powered by GeoGebra, allow students to explore and manipulate the mathematical concepts, leading to more durable understanding. They can also be used by instructors in the classroom to enhance instruction. Look for the Animation icons in the margins. 7 StatCrunch – StatCrunch is a powerful web-based statistical software that allows users to collect, crunch, and communicate with data. Now integrated into this MyLab Math course, StatCrunch can be used to analyze and understand statistical concepts. DS icons in the text indicate that a data set is available to download online and analyze in statistical software, such as StatCrunch or Excel. StatCrunch applets are available in the MyLab Math course. 7 Mindset Videos – These videos and assignable, open-ended exercises foster a growth mindset in students. This material encourages them to maintain a positive attitude about learning, value their own ability to grow, and view mistakes as learning opportunities—which so often presents a hurdle for math students. 7 Integrated Review – Ideal for a corequisite course, or simply to get underprepared students up to speed, Integrated Review includes assessments, assignments, and resources on prerequisite topics. For each chapter, pre-made, assignable (and editable) quizzes assess students’ understanding of the prerequisite skills needed for that chapter. Personalized follow-up homework assignments and remediation resources, in the form of videos and worksheets, are available for any gaps in skills that are identified. 7 Student Solutions Manual – This manual provides detailed worked-out solutions to odd-numbered exercises, and to all Chapter Review and Test exercises.
PREFACE xv Instructor Resources Your course is unique. Whether you want to create custom assignments, teach multiple sections, or set prerequisites, MyLab Math provides the flexibility to effortlessly tailor a course that suits your requirements. 7 Annotated Instructor’s Edition – All answers are included. When possible, answers are on the page with the exercises. Longer answers are in the back of the book. The AIE now includes Instructor Resource boxes for each section to inform what is available in MyLab Math for each section. A version of the AIE is also available in the Instructor Resources section of the MyLab Math course. 7 Corequisite Support – Corequisite material in the MyLab Math course has received a complete overhaul. The corequisite content offers a wide array of topics to cover any that may be required prior to taking a liberal arts math course. The corequisite support resources provide all the content and assessment resources necessary for students and instructors by offering videos, worksheets, and exercises for each objective. 7 PowerPoint Lecture Slides – Fully editable slides correlated with the textbook are available. Accessible, screen reader-friendly versions of the slides are also available. 7 Learning Catalytics – Integrated into the MyLab course, Learning Catalytics uses students’ devices in the classroom for an engagement, assessment, and classroom intelligence system that gives instructors real-time feedback on student learning. To make it easy to integrate this learning tool into the classroom, keywords are available in the Annotated Instructor’s Edition at point of use. Detailed instructions for using these keywords can be found at http://bit.ly/485Yna8. 7 Instructor’s Solutions Manual – This manual includes fully worked solutions to all text exercises. 7 Instructor’s Testing Manual – This manual includes tests with answer keys for each chapter of the text. 7 TestGen® – TestGen® (www.pearson.com/testgen) enables instructors to build, edit, print, and administer tests using a computerized bank of questions developed to cover all the objectives of the text. 7 Accessibility – Pearson works continuously to ensure that our products are as accessible as possible to all students. Currently, we are working toward achieving WCAG 2.0 AA for our existing products (2.1 AA for future products) and Section 508 standards, as expressed in the Pearson Guidelines for Accessible Educational Web Media (wps.pearsoned.com/accessibility).
xvi PREFACE Acknowledgments We thank the reviewers from all editions of the book and all the students who have offered suggestions for improving it. A list of reviewers for all editions of this book follows, with reviewers of this edition noted with an asterisk (*). Thanks to you all for helping make A Survey of Mathematics with Applications one of the most successful liberal arts mathematics textbooks in the country. Kate Acks, University of Hawaii Maui College Marilyn Ahrens, Missouri Valley College David Allen, Iona College Mary Anne Anthony-Smith, Santa Ana College Frank Asta, College of DuPage Robin L. Ayers, Western Kentucky University Hughette Bach, California State University–Sacramento Tammy Barker, Hillsborough Community College Madeline Bates, Bronx Community College Rebecca Baum, Lincoln Land Community College Vivian Baxter, Fort Hays State University *John Becker, Indiana University Northwest *David Bell, Florida State College *Betty Berbari, SUNY Old Westbury Una Bray, Skidmore College David H. Buckley, Polk State College *Daniel Bueller, St. Petersburg College Robert C. Bueker, Western Kentucky University Carl Carlson, Moorhead State University Kent Carlson, St. Cloud State University Scott Carter, Palm Beach State College Yungchen Cheng, Missouri State University Joseph Cleary, Massasoit Community College Cash Clifton, Central New Mexico Community College Donald Cohen, SUNY College of Agriculture & Technology *Jung Colen, Chadron State College Celisa Counterman, Northampton Community College David Dean, Santa Fe College *Tuan Dean, Triton College *Scott Demsky, Keiser University John Diamantopoulos, Northeastern State University Darlene Diaz, Santiago Canyon College Greg Dietrich, Florida State College at Jacksonville *Elaine DiPerna, Community College of Allegheny County Charles Downey, University of Nebraska Ryan Downie, Eastern Washington University Jeffrey Downs, Western Nevada College Annie Droullard, Polk State College Patricia Dube, North Shore Community College Ruth Ediden, Morgan State University Lee Erker, Tri-County Community College Nancy Eschen, Florida State College at Jacksonville *Hope Essien, Malcolm X College Karen Estes, St. Petersburg College Mike Everett, Santa Ana College Robert H. Fay, St. Petersburg College *Jia Feng, James Sprunt Community College Teklay Fessanaye, Santa Fe College Kurtis Fink, Northwest Missouri State University Raymond Flagg, McPherson College Donna Fowler, Palm Beach Atlantic University Penelope Fowler, Tennessee Wesleyan College Gilberto Garza, El Paso Community College *Kathryn Gedamke, Keiser University Judith L. Gersting, Indiana University–Purdue University at Indianapolis *Kim D. Ghiselin, State College of Florida Rebecca Goad, Joliet Junior College Patricia Granfield, George Mason University Lucille Groenke, Mesa Community College Nick Haverhals, Avila University *Aziza Hina, Massasoit Community College Ryan Holbrook, University of Central Oklahoma Kaylinda Holton, Tallahassee Community College Terri Honeycutt, Lander University John Hornsby, University of New Orleans Mary Hoyt, University of North Texas at Dallas Judith Ink, Regent University Nancy Johnson, Broward College Phyllis H. Jore, Valencia College Burcu Karabina, Florida Atlantic University Heidi Kiley, Suffolk County Community College Daniel Kimborowicz, Massasoit Community College Reviewers for This and Previous Editions
PREFACE xvii Jennifer Kimrey, Alamance Community College Mary Lois King, Tallahassee Community College Harriet H. Kiser, Georgia Highlands College Nichole Klemmer, Washtenaw Community College Linda Kuroski, Erie Community College *Jennifer Lawhon, Valencia College Julia Ledet, Louisiana State University David Lehmann, Southwest Missouri State University Peter Lindstrom, North Lake College James Magliano, Union College Yash Manchanda, East Los Angeles College & Fullerton College Richard Marchand, Slippery Rock University Susan McCourt, Bristol Community College Robert McGuigan, Westfield State College Wallace H. Memmer, Brookdale Community College Maurice Monahan, South Dakota State University Julie Monte, Daytona State College Pedro J. Mora, Florida Gateway College Karen Mosely, Alabama Southern Community College Daniel Thomas Murphree, Great Basin College Kathleen Offenholley, Brookdale Community College *Brian Opoka, Montini Catholic High School Edwin Owens, Pennsylvania College of Technology *Cengiz Özgener, State College of Florida Wing Park, College of Lake County Bettye Parnham, Daytona State College Joanne Peeples, El Paso Community College *Joni B Pirnot, State College of Florida Traci M. Reed, St. John’s River State College *Kathy Renfro, Cuyahoga Community College Nelson Rich, Nazareth College Kenneth Ross, University of Oregon Timothy R. Ross, North Greenville University Ronald Ruemmler, Middlesex County College Rosa Rusinek, Queensborough Community College Len Ruth, Sinclair Community College John Samoylo, Delaware County Community College Sandra Savage, Orange Coast College *Michaela Schaben, Bellevue University Gerald Schultz, Southern Connecticut State University Richard Schwartz, College of Staten Island Kara Shavo, Mercer County Community College Minnie Shuler, Chipola College *Amanda Spencer-Barnes, Hazard Community and Technical College Paula R. Stickles, University of Southern Indiana *Deanne Stigliano, County College of Morris Kristin Stoley, Blinn College–Bryan *Alissa Sustarsic, Lake-Sumter State College Steve Sworder, Saddleback College *John Terrell, Troy University in Montgomery Shirley Thompson, Moorhead College Alvin D. Tinsley, Central Missouri State University Sherry Tornwall, University of Florida William Trotter, University of South Carolina Zia Uddin, Lock Haven University of Pennsylvania Michael Vislocky, University of Cincinnati Thomas G. Walker II, Western Kentucky Community College Richard Watkins, Dabney S Lancaster Community College *Carol Weideman, St. Petersburg College in Clearwater Sandra Welch, Stephen F. Austin State University Joyce Wellington, Southeastern Community College Sue Welsch, Sierra Nevada College Robert F. Wheeler, Northern Illinois University Susan Wirth, Indian River State College Judith B. Wood, College of Central Florida Jean Woody, Tulsa Community College James Wooland, Florida State University Kelly Young, Lander University Michael A. Zwick, Monroe Community College We thank Lauri Semarne for her conscientious job of checking the text and answers for accuracy. Finally, we thank our family members Kathy, Robert, and Steve Angel, Mathew and Jake Abbott, and Kris, Alex, Nick, and Max Runde for their support and encouragement throughout this project. We are grateful for their wonderful support and understanding while we worked on the book. They also gave us support and encouragement and were very understanding when we could not spend as much time with them as we wished because of book deadlines. Without the support and understanding of our families, this book would not be a reality. Allen R. Angel Christine D. Abbott Dennis C. Runde
1 b Should I buy a new car or repair my old car? If I buy a new car, is it worth spending the extra money to get a hybrid or an electric car? Critical thinking skills are needed to answer such questions we face in our daily lives. 1 Sections 1. 1 Inductive and Deductive Reasoning 1. 2 Estimation Techniques 1. 3 Problem-Solving Procedures Everyday life presents us with a wide range of problems that we must solve. For example, you may need to decide which classes to take in the next semester of college. Or you may need to decide whether it is more cost effective in the long run to repair your car or buy a new car. If you do buy a new car, you may need to decide between a gasoline-powered, hybrid, or electric-powered car. Decisions such as these can have long-term consequences for your financial future. The information you encounter in this chapter will help sharpen your critical thinking skills. These skills will help you make better decisions as you solve the problems that you encounter in your everyday life. Why This Is Important Critical Thinking Skills Halfpoint/123RF
2 CHAPTER 1 Critical Thinking Skills The science of biometrics involves the measurement and analysis of unique physical characteristics. Biometrics are usually used as a means of verifying personal identity. Fingerprints, iris patterns in eyes, facial recognition, DNA, and voice patterns can all be used for personal identification. Some smartphones can be unlocked by pressing a button that recognizes a unique fingerprint or by using facial recognition. Crime scene investigation often involves fingerprint and DNA evidence. Voice recognition software uses voice patterns of callers to help prevent fraud and to improve customer service. Inductive and Deductive Reasoning SECTION 1.1 LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand and use inductive reasoning to solve problems. 7 Understand and use deductive reasoning to solve problems. Why This Is Important Using biometrics for personal identification involves reasoning to a general conclusion through observation of specific cases. In this section, we will discuss how inductive and deductive reasoning are essential critical thinking skills used in biometrics and in many other applications. Inductive Reasoning Before looking at some examples of inductive reasoning and problem solving, let us first review a few facts about certain numbers. The natural numbers or counting numbers are the numbers 1, 2, 3, 4, 5, 6, 7, 8, .… The three dots, called an ellipsis, mean that 8 is not the last number but that the numbers continue in the same manner. A word that we sometimes use when discussing the counting numbers is “divisible.” If a b ÷ has a remainder of zero, then a is divisible by b. The counting numbers that are divisible by 2 are 2, 4, 6, 8, .… These numbers are called the even counting numbers. The counting numbers that are not divisible by 2 are 1, 3, 5, 7, 9, .… These numbers are the odd counting numbers. When we refer to odd numbers or even numbers, we mean odd or even counting numbers. Recognizing patterns is sometimes helpful in solving problems, as Examples 1 and 2 illustrate. Example 1 The Product of Two Even Numbers If two even numbers are multiplied together, will the product be an even number or an odd number? Solution To answer this question, we will examine the products of several pairs of even numbers to see if there is a pattern. 2 2 4 4 4 16 6 6 36 2 4 8 4 6 24 6 8 48 2 6 12 4 8 32 6 10 60 × = × = × = × = × = × = × = × = × = We see that all the products are even numbers. Thus, we might predict from these examples that the product of any two even numbers is always an even number. 7 Now try Exercise 25 Fernando Astasio Avila/123RF
1.1 Inductive and Deductive Reasoning 3 In Examples 1 and 2, we cannot conclude that the results are true for all counting numbers. From the patterns developed, however, we can make predictions. This type of reasoning process, arriving at a general conclusion from specific observations or examples, is called inductive reasoning, or induction. Example 2 The Sum of an Odd Number and an Even Number If an odd number and an even number are added, will the sum be an odd number or an even number? Solution Let’s look at a few examples in which one number is odd and the other number is even. 3 4 7 9 6 15 23 18 41 5 12 17 5 14 19 81 32 113 + = + = + = + = + = + = All these sums are odd numbers. Therefore, we might predict that the sum of an odd number and an even number is an odd number. 7 Now try Exercise 27 Induction often involves observing a pattern and from that pattern predicting a conclusion. Imagine an endless row of dominoes. You knock down the first, which knocks down the second, which knocks down the third, and so on. Assuming the pattern will continue uninterrupted, you conclude that any one domino that you select in the row will eventually fall, even though you may not witness the event. Inductive reasoning is often used by mathematicians and scientists to develop theories and predict answers to complicated problems. For this reason, inductive reasoning is part of the scientific method. When a scientist or mathematician makes a prediction based on specific observations, it is called a hypothesis or conjecture. After looking at the products in Example 1, we might conjecture that the product of two even numbers will be an even number. After looking at the sums in Example 2, we might conjecture that the sum of an odd number and an even number is an odd number. As described in the opening paragraph of this section, the science of biometrics is used for personal identification. By studying the biometrics of millions of people, scientists have never found two people who have the exact same fingerprints, iris patterns, DNA, or voice patterns. By induction, then, a conclusion can be reached that each of these biometrics provides a unique identification. A general conclusion is reached through the observation of specific cases. Therefore, the science of biometrics makes use of inductive reasoning. Examples 3 and 4 illustrate how we arrive at a conclusion using inductive reasoning. Example 3 Products Involving 5 a) Select some natural numbers and multiply the numbers by 5. b) Observe the ones digit (the rightmost digit) of the products from part (a). Use inductive reasoning and make a conjecture regarding products involving 5 and natural numbers. Definition: Inductive Reasoning Inductive reasoning is the process of reasoning to a general conclusion through observations of specific cases.
4 CHAPTER 1 Critical Thinking Skills Did You Know? Solution a)51 5 5210531554 205 5 25 56 3057 3558 4059 45510 50 × = × = × = × = × = × = × = × = × = × = b) Notice that the ones digit of the products are either 5 or 0. Based on these examples, we can make the conjecture that products involving 5 and natural numbers will have a ones digit of 5 or 0. As we will discuss in Section 5.1, this statement is indeed true. 7 Now try Exercise 29 Example 4 Use Inductive Reasoning to Make a Conjecture Pick any number and multiply the number by 4. Next, add 2 to the product and divide the sum by 2. Finally, subtract 1 from the quotient. a) What is the relationship between the number you started with and the final answer? b) Repeat this procedure for several different numbers. Note the original number and the final number. c) Use inductive reasoning to make a conjecture about the relationship between the original number and the final number. Solution a) We will go through the procedure step by step. Now try Exercise 37 Pick a number: say, 5 Multiply the number by 4: 5 4 20 × = Add 2 to the product: 20 2 22 + = Divide the sum by 2: 22 2 11 ÷ = Subtract 1 from the quotient: 11 1 10 − = Notice that we started with the number 5 and ended with the number 10. b) If we repeat the procedure, this time starting with the number 6, we will end with the number 12. If we start with the number 9, we will end with the number 18. If we start with the number 15, we will end with the number 30—verify these results for yourself. c) Based on the examples from parts a) and b), we can conjecture that when we follow the given procedure, the number you end with will always be twice the original number. 7 The result reached by inductive reasoning is often correct for the specific cases studied but not correct for all cases. History has shown that not all conclusions arrived at by inductive reasoning are correct. For example, Aristotle (384–322 b.c.) reasoned inductively that heavy objects fall at a faster rate than light objects. About 2000 years later, Galileo (1564 –1642) dropped two pieces of metal—one 10 times heavier than the other—from the Leaning Tower of Pisa in Italy. He found that both hit the ground at exactly the same moment, so they must have traveled at the same rate. See the Did You Know on the left for a similar experiment carried out on the moon by astronaut David Scott. An Experiment Revisited Apollo 15 astronaut David Scott used the moon as his laboratory to show that a heavy object (a hammer) does indeed fall at the same rate as a light object (a feather). Had Galileo dropped a hammer and feather from the Tower of Pisa, the hammer would have fallen more quickly to the ground and he still would have concluded that a heavy object falls faster than a lighter one. If it is not the object’s mass that is affecting the outcome, then what is it? The answer is air resistance or friction: Earth has an atmosphere that creates friction on falling objects. The moon does not have an atmosphere; therefore, no friction is created. m David Scott on the moon JSC PAO Web Team/NASA Learning Catalytics Keyword: Angel-SOM-1.1 (See Preface for additional details.)
1.1 Inductive and Deductive Reasoning 5 When forming a general conclusion using inductive reasoning, you should test it with several special cases to see whether the conclusion appears correct. If a special case is found that satisfies the conditions of the conjecture but produces a different result, such a case is called a counterexample . A counterexample proves that the conjecture is false because only one exception is needed to show that a conjecture is not valid. Galileo’s counterexample disproved Aristotle’s conjecture. If a counterexample cannot be found, the conjecture is neither proven nor disproven. Consider the statement “All birds fly.” A penguin is a bird that does not fly. Therefore, a penguin is a counterexample to the statement “All birds fly.” Deductive Reasoning A second type of reasoning process is called deductive reasoning , or deduction . Mathematicians use deductive reasoning to prove conjectures true or false. Definition: Deductive Reasoning Deductive reasoning is the process of reasoning to a specific conclusion from a general statement. Example 5 illustrates deductive reasoning. General Specific Deductive Reasoning Inductive Reasoning Timely Tip The following diagram helps explain the difference between inductive reasoning and deductive reasoning. Inductive reasoning is the process of reasoning to a general conclusion through observations of specific cases. Deductive reasoning is the process of reasoning to a specific conclusion from a general statement. In Example 4, we conjectured , using specific examples and inductive reasoning, that the result would be twice the original number selected. In Example 5, we proved , using deductive reasoning, that the result will always be twice the original number selected. Number Tricks Using Deductive Reasoning Instructor Resources for Section 1.1 in MyLab Math • Objective-Level Videos 1.1 • Animation: Number Tricks Using Deductive Reasoning • PowerPoint Lecture Slides 1.1 • MyLab Exercises and Assignments 1.1 Example 5 Use Deductive Reasoning to Prove a Conjecture Prove, using deductive reasoning, that the procedure given in Example 4 will always result in twice the original number selected. Solution To use deductive reasoning, we begin with the general case rather than specific examples. In Example 4, specific cases were used. Let’s select the letter n to represent any number . Pick any number: n Multiply the number by 4: n4 n4 means 4 times n. Add 2 to the product: n4 2 + Divide the sum by 2: + = + = + n n n 4 2 2 4 2 2 2 2 1 2 1 1 1 Subtract 1 from the quotient: n n 2 1 1 2 + − = Note that for any number n selected, the result is n2 , or twice the original number selected. Since n represented any number, we are beginning with the general case. Thus, this is deductive reasoning. 7 Now try Exercise 41
6 CHAPTER 1 Critical Thinking Skills Exercises Warm Up Exercises In Exercises 1–8, fill in the blank with an appropriate word, phrase, or symbol(s). 1. Another name for the counting numbers is the ________ numbers. Natural 2. If a b ÷ has a remainder of 0, then a is ________ by b. Divisible 3. A specific case that satisfies the conditions of a conjecture but shows the conjecture is false is called a ________. Counterexample 4. A belief based on specific observations that has not been proven or disproven is called a conjecture or ________. Hypothesis 5. The process of reasoning to a general conclusion through observation of specific cases is called ________ reasoning. Inductive 6. The process of reasoning to a specific conclusion from a general statement is called ________ reasoning. Deductive 7. The type of reasoning used to prove a conjecture is called ________ reasoning. Deductive 8. The type of reasoning generally used to arrive at a conjecture is called ________ reasoning. Inductive Practice the Skills In Exercises 9–12, use inductive reasoning to predict the next line in the pattern. 9. × = × = × = × = 5 1 5 5 2 10 5 3 15 5 4 20 5 5 25 × = 10. × = × = × = × = 15 10 150 16 10 160 17 10 170 18 10 180 19 10 190 × = 11. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 12. = = = = 10 10 100 10 1000 10 10,000 10 1 2 3 4 100,000 105 = In Exercises 13–16, draw the next figure in the pattern (or sequence). 13. , , , . . . * 14. , , , , , , . . . * 15. , , , , . . . * 16. , , , , , . . . * In Exercises 17–24, use inductive reasoning to predict the next three numbers in the pattern (or sequence). 17. … 1, 3, 5, 7, 9, 11, 13 18. … 4, 7, 10, 13, 16, 19, 22 19. − − … 1, 2, 4, 8, 16, 32, 64 − 20. 3, 9, 27, 81, − − … 243 − 21. … 1, 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , 1 7 22. 1 2 , 3 4 , 5 6 , 7 8 ,… 9 10 , 11 12 , 13 14 23. 1, 1, 2, 3, 5, 8, 13, 21,… 34, 55, 89 24. … 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199 Problem Solving In Exercises 25–28, choose several pairs of appropriate numbers to determine the sum or product indicated. Use these results to form a conjecture. See Example 1. 25. The Product of Two Odd Numbers If two odd numbers are multiplied together, is the product an even number or an odd number? The product of two odd numbers is an odd number. 26. The Product of an Even Number and an Odd Number If an even number is multiplied by an odd number, is the product an even number or an odd number? The product of an even number and an odd number is an even number. 27. The Sum of Two Even Numbers If two even numbers are added together, is the sum an even number or an odd number? The sum of two even numbers is an even number. 28. The Sum of Two Odd Numbers If two odd numbers are added together, is the sum an even number or an odd number? The sum of two odd numbers is an even number. 29. Products of 10 a) Select some natural numbers and multiply the numbers by 10. Answers will vary b) Observe the ones digit (the rightmost digit) of the products from part (a). Use inductive reasoning to make a conjecture regarding products involving 10 and natural numbers. Products involving 10 and natural numbers will have a ones digit 0. SECTION 1.1 R R R R R R *See Instructor Answer Appendix
1.1 Inductive and Deductive Reasoning 7 34. Triangles in a Triangle Four rows of a triangular figure are shown. a) If you added six additional rows to the bottom of this triangle, using the same pattern displayed, how many triangles would appear in the 10th row? 19 b) If the triangles in all 10 rows were added, how many triangles would appear in the entire figure? 100 35. Community College Tuition The following graph shows the annual tuition, to the nearest hundred dollars, at Clarence Community College for the years 2020–2024. a) Assuming this trend continues, use the graph to predict the annual tuition for the year 2025. $3700 b) Explain how you are using inductive reasoning to determine your answer. We are using specific cases to make a prediction. 36. Gym Membership Fees The following graph shows the annual membership fees for Crackle Fitness for the years 2020–2024. a) Assuming this trend continues, use the graph to predict the annual membership fee for the year 2025. $415 b) Explain how you are using inductive reasoning to determine your answer. We are using specific cases to make a prediction. $0 $50 $100 $150 $200 $250 $300 $350 $450 $400 2020 Annual Membership Fees Annual Membership Fees for Crackle Fitness 2021 2022 2023 2024 Year 30. Sum of the Digits a) Select a variety of one- and two-digit numbers between 1 and 99 and multiply each by 9. Record your results. Answers will vary. b) Determine the sum of the digits in each of your products in part (a). If the sum is not a one-digit number, determine the sum of the digits of the resulting sum again until you obtain a one-digit number. The sum is 9. c) Make a conjecture about the sum of the digits when a one- or two-digit number is multiplied by 9. The sum of the digits in the product when a one- or two-digit number is multiplied by 9 is 9. 31. A Square Pattern The ancient Greeks labeled certain numbers as square numbers. The numbers 1, 4, 9, 16, 25, and so on are square numbers. 1 4 9 16 25 a) Determine the next three square numbers. 36, 49, 64 b) Describe a procedure to determine the next five square numbers without drawing the figures. Square 6, 7, 8, 9 and 10 c) Is 72 a square number? Explain how you determined your answer. No, 72 is between 82 and 9 , 2 so it is not a square number. 32. A Triangular Pattern The ancient Greeks labeled certain numbers as triangular numbers. The numbers 1, 3, 6, 10, 15, 21, and so on are triangular numbers. 1 3 6 10 15 21 a) Can you determine the next two triangular numbers? 28, 36 b) Describe a procedure to determine the next five triangular numbers without drawing the figures. * c) Is 72 a triangular number? Explain how you determined your answer. No 33. Quilt Design The pattern shown is taken from a quilt design known as a triple Irish chain. Complete the color pattern by indicating the color assigned to each square. Blue: 1, 5, 7, 10, 12; Purple: 2, 4, 6, 9, 11; Yellow: 3, 8 1 8 9 10 11 4 5 6 2 3 7 12 *See Instructor Answer Appendix Annual Tuition at Clarence Community College $0 $500 $1000 $1500 $2000 $2500 $3000 $3500 $4000 2020 Annual Tuition 2021 2022 2023 2024 Year
8 CHAPTER 1 Critical Thinking Skills 44. Prove, using deductive reasoning, the conjecture you made in Exercise 40 part c). + + + = + + + = n n n n n n , 5, ( 5) 2 5,2 511 + + = + + − = n n n n n 2 16, 2 16 2 8, 8 8 In Exercises 45–50, determine a counterexample to show that each statement is incorrect. 45. The product of any two counting numbers is divisible by 2. 3 5 15, × = which is not divisible by 2. 46. The sum of any three two-digit numbers is a three-digit number. 10 11 12 33, + + = which is not a three-digit number. 47. When a counting number is added to 3 and the sum is divided by 2, the quotient will be an even number. (3 2)/2 5/2, + = which is not an even number. 48. The product of any two three-digit numbers is a five-digit number. 900 900 810,000, × = which is not a five-digit number. 49. The difference of any two counting numbers will be a counting number. 1 2 1, − = − which is not a counting number. 50. The sum of any two odd numbers is divisible by 4. 1 5 6, + = which is not divisible by 4. 51. Interior Angles of a Triangle a) Construct a triangle and measure the three interior angles with a protractor. What is the sum of the measures? The sum of the measures of the interior angles should be 180°. b) Construct three other triangles, measure the angles, and record the sums. Are your answers the same? Yes, the sum of the measures of the interior angles should be 180°. c) Make a conjecture about the sum of the measures of the three interior angles of a triangle. The sum of the measures of the interior angles of a triangle is 180°. 52. Interior Angles of a Quadrilateral a) Construct a quadrilateral (a four-sided figure) and measure the four interior angles with a protractor. What is the sum of the measures? The sum of the measures of the interior angles should be 360°. b) Construct three other quadrilaterals, measure the angles, and record the sums. Are your answers the same? Yes, the sum of the measures of the interior angles should be 360°. c) Make a conjecture about the sum of the measures of the four interior angles of a quadrilateral. The sum of the measures of the interior angles of a quadrilateral is 360°. Concept/Writing Exercises 53. Computer Log On While logging on to your computer, you type in your username followed by what you believe is your password. The computer indicates that a mistake has been made and asks you to try again. You retype your username and the same password. Again, the computer indicates a mistake has been made. You decide not to try again, reasoning you will get the same error message from the computer. What type of reasoning did you use? Explain. Inductive reasoning; a general conclusion is obtained from observation of specific cases. 37. Pick any number, multiply the number by 3, add 6 to the product, divide the sum by 3, and subtract 2 from the quotient. See Example 5. a) What is the relationship between the number you started with and the final number? You should obtain the original number. b) Arbitrarily select some different numbers and repeat the process, recording the original number and the result. You should obtain the original number. c) Make a conjecture about the relationship between the original number and the final number. The result is the original number. 38. Pick any number and multiply the number by 4. Add 6 to the product. Divide the sum by 2 and subtract 3 from the quotient. a) What is the relationship between the number you started with and the final answer? You should obtain twice the original number. b) Arbitrarily select some different numbers and repeat the process, recording the original number and the results. You should obtain twice the original number. c) Make a conjecture about the relationship between the original number and the final number. The result is always twice the original number. 39. Pick any number and add 1 to it. Determine the sum of the new number and the original number. Add 9 to the sum. Divide the new sum by 2 and subtract the original number from the quotient. a) What is the final number? 5 b) Arbitrarily select some different numbers and repeat the process. Record the results. You should obtain the number 5. c) Make a conjecture about the final number. The result is always the number 5. 40. Pick any number and add 5 to it. Determine the sum of the new number and the original number. Add 11 to the sum. Divide the new sum by 2 and subtract the original number from the quotient. a) What is the result? 8 b) Arbitrarily select some different numbers and repeat the process. Record the results. You should obtain the number 8. c) Make a conjecture about the final number. The result is always the number 8. 41. Prove, using deductive reasoning, the conjecture you made in Exercise 37 part c). + + = + + − = n n n n n n n , 3 , 3 6, 3 6 3 2, 2 2 42. Prove, using deductive reasoning, the conjecture you made in Exercise 38 part c). n n n n n n n , 4 , 4 6, 4 6 2 2 3, 2 3 3 2 + + = + + − = 43. Prove, using deductive reasoning, the conjecture you made in Exercise 39 part c). n n n n n n n , 1, ( 1)2 1,2 192 10, + + + = + + + = + + = + + − = n n n n 2 10 2 5, 5 5
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