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Using
& Understanding
Mathematics:
A
Quantitative Reasoning Approach (7th Edition) Jeffrey O. Bennett
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The Cosmic Perspective 9th Edition Jeffrey O. Bennett
An Introduction to Management Science: Quantitative Approach 15th Edition David R. Anderson & Dennis J. Sweeney & Thomas A. Williams & Jeffrey D. Camm & James J. Cochran
Mylab Math Online Course for Using & Understanding Mathematics, 7th edition, by
Bennett and Briggs
Mylab Math w ith
Integrated Review
In addition t o t he st andard Mylab"' Math course, Using & Understanding Mathematics offers a M yla b Math co u rse with In t egr at ed Revi ew, givi ng stu dent s th e su pport they need before completing their homewor k. A Skills Check is availa ble w it h each Integrat ed Review module t o ensure th at students ar e prepared for t he upcoming cha pter material.
Phones and Driving
Integrated Activit ies
Activit ies are heavi ly integrated
in to this edition of Using & Understanding Mathematics. Each chapt er o pens w it h an A ct iv i t y designed to spur d iscussion of some in terest ing face t of t he t opics cove red in the cha pter
NEW! Activity Ma nu al w ith In tegrated Review Worksheet s
The Ac t ivity M an u al provides additional activities for each chapt er in a workbook format. In add it ion t o th e activities, the workbook includes Integrated Review Worksheets t o su pport the Integrated Review version of the Mylab Math course.
Math for College, Career, and Life
We all use math i n our day-to-day lives. The goa l o f t his book is to imp rove students' mathematical lite racy so that t hey can use math mo re effective ly in everyday li f e. Mathematics can hel p students bette r u nde rsta nd a va riety of top ics and issues, making t hem mor e awa re of t he uses and abuses of numbe rs. The u lt imate goa l is to help t hem become bette r educated cit ize ns who are successfu l in t heir co llege experiences, the ir careers, and the ir lives.
I n Yo ur Wor ld boxes focu s on topics t hat students are likely to encounter in the world around t hem- in the news, in consumer decisions, or in polit ica l d iscussions. The conn ection to the rea l worl d is further enhanced by I n Yo ur Wo rl d exercises, designed to spur add it iona l research or discussion that w ill help students relate the unit's topics to t he t hemes of college, ca ree r, and li fe.
R WOR LD ..Fact Checking on the Web
Co ncepts are b rought to life in fi ve In You r Wor l d v ideos, assignable w it hi n Myl ab Math. These v ideos provide f urther connections to students' everyday lives i n an enterta ining and engaging way.
IN YOUR WORLD
45 Political Action This unit o\\tli.n«l m•mero us budge tary prob· lt:m!i facing rh<' 1L'i go\'crru nc-ru. a.c; 1hry a1 fh«' ti mr: tht text '''riltt:n in 20 17. thtn: been <&11)' >ign iricant political ac-don lO deal with any of d1ese problems? Learn whal , i l :r nyt hing, has t•hangcd sin('(" then write a one· page position paper ou tlining yo ur own 1-ccommc ndatiOI"li for the fu ture.
46. Debt Problem. How serious a problem is lhe g ross dcbl ? Find argument.-; o n hoth s ides or thls quesrion. Summarize the argumen t; and rour own opin ion.
NEW! I nt egrated Sta t Crunch
StatCru nch, powerfu l web -based statistica l softwa re, allows students to harness techno logy to pe rform data ana lysis. This ed it ion includes StatCrunch exercises in relevant exercise sets and some questions assignable in Mylab Math.
Why Should You Care About Quantitative Reasoning?
Quantitative reasoning is the ability to interpret and reason with information that involves numbers or mathematical ideas. It is a crucial aspect of literacy, and it is essential in making important decisions and understanding contemporary issues.
The topics covered in th is text wil l help you work with qua nt itative information and make crit ical decisions. For example:
• You should possess strong skills in critica l and logical thi nking so t hat you ca n make w ise personal decisions, navigate the media, and be an informed citizen. For example, do you know why you'd end up beh ind if you accepted a temporary 10% pay cut now and t hen received a 10% pay raise later? This particu lar question is covered in Un it 3A, but t hroughout t he book you'l l learn how to evaluate q uantitative q uestions on top ics ranging from personal decisions to major global issues .
• You should have a strong number sense and be proficient at estimation so that you can put numbers f rom the news into a context that makes them understandable For example, do you know how to make sense of t he more than $20 trill ion federal debt? Unit 38 discusses how you ca n put such huge nu mbers in perspective, and Unit 4F discusses how t he federal debt grew so large
• You should possess the mathematica l tools needed to make basic f in ancia l decisions. For example, do you enjoy a latte every morni ng before class? Sometimes two? Uni t 4A explores how such a seem ingly ha r mless habit can d rain more than $2400 from your wallet every year
• You should be able to read news reports of statistical stud ies in a way t hat wil l al low you to eva luate t hem critical ly and decide whether and how t hey should affect your persona l bel iefs. For example, how should you decide whether a new opin ion poll accu rately reflects t he views of Americans? Chapter 5 covers the basic concepts that lie behind t he statistical studies and graphics you'l l see in t he news, and d iscu sses how yo u can decide for yourself w hether you should beli eve a statistical study
• You should be famil iar with basic ideas of probab ility and r isk and be aware of how they affect your li fe. For examp le, wo ul d you pay $30,000 for a product that, over 20 years, wil l kill nearly as many people as live in San Francisco? In Un it 70, you'll see that t he answer is very likely yes- just one of ma ny surp r ises that you'll encoun ter as you study probab il ity in Chapter 7.
• You should understa nd how mathematics helps us study important socia l issues, such as globa l warm ing, t he g rowth of populations, t he depletion of resources, apportionment of congressional representatives, and methods of voti ng. For examp le, Un it 120 discusses t he nature of redistricting and how ger rymandering has made congressional elections less competit ive than t hey m ight otherw ise be.
In sum, this text w ill focus on understand ing and interpreti ng mathematical topics to help yo u develop t he q uantitative reasoni ng skills you will need for college, career, and life.
Jeffrey
Bennett
University of Colorado at Boulder
William
Briggs
University of Colorado at Denver
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library of Congress Catal oging-in-Pu blicatio n Da ta
Names: Bennett, jeffrey 0 , author. I Briggs, William L, author.
Title: Using and understanding mathematics: a q u antitative reasoning approach I jeiTTey Bennett and William Briggs
Description: 7th edition . I Boston: Pearson, 12019 1
Identifiers: LCCN 201 7048514 l lSBN 9780134705187 (alk. paper) I ISBN 0134705181 (a lk paper)
Subjects: LCSH: Mathematics-Textbooks.
Classification: LCC QA39.3 B46 20191 DOC 510-dc23
LC record ava ilable at hups://lccn loc.gov/2017048514
This book is dedicated to everyone who wants a better understanding of our world, and especially to those who have struggled with mathematics in the past. We hope this book will help you achieve your goals.
And it is dedicated to those who make our own lives brighter; especially Lisa, julie, Katie, Grant, and Brooke .
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CONTENTS
Preface x Acknowledgments xvii
Prologue: Literacy for the Modern World P l
PART ONE LOGIC AND PROBLEM SOLVING
TH I NK ING CRITICALLY
@ff ACTIVITY Bursting Bubble 4
1A Living in the Media Age 5
In Your World: Fact Checking on the Web 1 1
18 Propositions and Truth Values 14
1C Sets and Venn Diagrams 25
1D
1E
Brief Review: Sets of Numbers 29
Analyzing Arguments 4 0
Mathematical Insight: Deductive Proof of the Pythagorean Theorem 49
Critical Thinking in Everyday Life 54
In Your World: Beware of ·up to" Deals 60
• APPROACHES TO PROBLEM SOLVI NG
({$ ACTIVITY Global Melting 70
2A
Understand, So lve, and Explain 71
Brief Review: Common Fractions 74
Brief Review: Decimal Fractions 80
Using Technology: Currency Exchange Rates 82
In Your World: Changing Money in Foreign Countries 84 28 Extending Unit Analysis 8 8
In Your World: Gems and Gold Jewelry 90
Brief Review: Powers of 10 92
Using Technology: Metric Conversions 95
In Your World: Save Money and Save the Earth 99 2C Problem-Solv ing Hints 10 1
Mathematical Insight: Zeno's Paradox 1 10
PART TWO QUANTITATIVE INFORMATION IN EVERYDAY LIFE
NUMBERS I N THE REAL WORLD 120
@t ACTIVITY Big Numbers 122
3A Uses and Abuses of Percentages 122
Brief Review: Percentages 123
Brief Review: What Is a Ratio? 129
38 Putting Numbers in Perspective 139
Brief Review: Working with Scientific Notation 14 0
Using Technology: Scientific Notation 143
3C Dea ling with Uncertainty 155
Brief Review: Rounding 157
Using Technology: Rounding in Excel 162
30 Index Numbers: The CPI and Beyond 166
Using Technology: The Inflation Calculator 171
In Your World: The Chained CPI and the Federal Budget 173
3E How Numbers Can Deceive : Polygraphs, Mammograms, and More 178
MANAGING MONEY
@t ACTIVITY Student loans 192
4A
Taking Control of Your Finances 192
4B The Power of Compounding 206
Brief Review: Powers and Roots 207
Using Technology: Powers 208
Using Technology: The Compound Interest Formula 211
Using Technology: The Compound Interest Formula for Interest Paid
More Than Once a Year 2 15
Using Technology: APY in Excel 216
Using Technology: Powers of e 218
Brief Review: Four Basic Rules of Algebra 218
In Your World: Effects of Low Interest Rates 220
4C
Savings Plans and Investments 225
Mathematical Insight: Derivation of the Savings Plan Formula 227
Using Technology: The Savings Plan Formula 230
Using Technology: Fractional Powers (Roots) 232
In Your World: Building a Portfolio 237
40 Loan Payments, Credit Cards, and Mortgages 244
Using Technology: The Loan Payment Formula (Installment Loans) 247
Mathematical Insight: Derivation of the Loan Payment Formula 248
Using Technology: Principal and Interest Portions of Loan Payments 249
In Your World: Avoiding Credit Card Trouble 251
In Your World: Choosing or Refinancing a Loan 255
4E Income Taxes 262
4F Understanding the Federal Budget 275
PART THREE PROBABILITY A ND STATISTICS
STATISTICAL REASONING
@t ACTIVITY Cell Phones and Driving 294
SA Fundamentals of Statistics 294
Using Technology: Random Numbers 298
SB Should You Believe a Statistical Study? 309
In Your World: The Gun Debate : Defensive Gun Use 314
SC Statistical Tables and Graphs 319
Using Technology: Frequency Tables in Excel 321
Using Technology: Bar Graphs and Pie Charts in Excel 325
Using Technology: Line Charts in Excel 327
SO Graphics in the Media 333
Using Technology: Graphs with Multiple Data Sets 337
SE Correlation and Causa lity 354
Using Technology: Scatterplots in Excel 359
PUT T I NG STATISTICS TO WORK
({t ACTIVITY Are We Smarter Than Ou r Parents? 372
6A Ch aract erizing Dat a 373
Using Technology: Mean, Median, and Mode in Excel 376
68 M easures of Variation 386
Using Technology: Standard Deviation in Ex cel 392
6C The No rmal Distribution 397
6D
Using Technology: Standard Scores and Percentiles in Ex cel 405
Statistica l lnference 408
In Your World: Is Polling Reliable? 4 14
PROBABILITY: LIVING W ITH T HE ODDS
({t ACTIVITY l otteri es 426
7A Fundamental s o f Probability 426
Brief Revi ew: The Multiplication Principle 431
78 Comb ining Pro babi lities 441
7C The Law of Large Numbers 453
7D Assessing Risk 464
In Your World: Terrorism, Risk, and Human Psychology 465
7E Count ing and Probabil ity 474
Using Technology: Factorials 476
Brief Review: Factorials 476
Using Technology: Permutati ons 477
Using Technology: Combinations 4 79
PART FOUR MOD ELING
EXPONENTIAL ASTONIS
({t ACTIVITY Towers of Hanoi 490
H MENT
SA Growth: Linear vs. Exponentia l 491
88 Dou b lin g Time and Half-Life 499
Using Technology: Logarithms 505
Brief Revi ew: Logarithms 506
SC Real Po p ulation Growth 510
In Your World: Choosing Our Fate 517
SD Logarithmic Scales: Earthquakes, Sounds, and Acids 520
In Your World: Ocean Acidification 526
• MODE LI NG OUR WORLD
({t ACTIVITY Climate Modeling 534
9A Functions: The Bui ld in g Blocks o f Mat hemat ica l Models 536
Brief Review: The Coordinate Plane 539
98 Linear Modeling 546
Using Technology: Graphing Functions 551
In Your World: Algebra 's Baghdad Connection 554
9C Exponential Modeling 560
Brief Revi ew: Algebra with Logarithms 563
Mathematical Insight: Doubling Time and Half-Life Formulas 565
In Your World: Changing Rates of Change 570
424
488
532
MODELING WITH GEOMETRY
@§ ACTIVITY Eyes in the Sky 578
10A Fundamentals of Geometry 579 Mathemati cal Insi ght: Archi medes and Pi 583 In Your Worl d: Plato, Geometry, and Atlantis 587 108 Prob lem Solving with Geometry 593 10C Fractal Geometry 608
PART FIVE FURTHER APPLICATIONS
MATHEMATICS AND THE ARTS
@3 ACTIVITY Dig ital Music Files 622
11A Mathematics and Music 623 In Your World: Music Just for You 627
11 B Perspective and Symmetry 630 11C Proportion and the Golden Ratio 643
MATHEMATICS
AND POLITICS
@3 ACTIVITY Partisan Redistricting 656
12A Voting : Does the Majority Always Rule? 657 In Your World: Counting Votes-Not as Easy as It Sounds 663 128 Theory of Voting 675 In Your World: The Electoral College and the Presi dency 680 12C Apportionment: The House of Representatives and Beyond 686 12D Div iding the Political Pie 101
Credits C-1
Answers to Quick Quizzes and Odd-Numbered Exercises A-1 Index 1 1
ABOUT THE AUTHORS
jeffrey Bennett served as the fi rst d irector of the program "Quantitative Reasoning and Mathematical Skills " at the Univers ity of Colorado at Boulder, whe re he devel oped the ground b reak ing curriculum that became the bas is of this textbook. He holds a BA in biophysics (Univers ity of California , San D iego) and an MS and a PhD in astrophysics (University of Colorado) , and has focused his career on math and science education. In addition to co -author i ng this textbook, he is also the lead author of best-se ll ing college textbooks on statistical reasoning , astronomy, and astrobiology, and of more than a dozen books for ch il dren and adults . All six ofhischildren's books have been selec ted fo r NASA's "Story Time From Space " (storytimefromspace.com), a project in wh ich astronauts on the International Space Station read books a loud and videos a re posted that anyone in the world can watch for free His most recent books include I, Humanity for ch ildren and Mat/1 for Life and A Global Wanning Primer fo r the general public. Among his many other endeavors , Dr. Bennett proposed and coJed the development of the Voyage Scale Model Solar System, which is located outs ide the National Air and Space Museum on the National Mall in Washing ton , DC. Learn more about Dr Bennett and his work at www.jeffreybennett com
William Br iggs was on the mathematics [acu i ty at Clarkson Un iversity fo r 6 years and at the Univers ity o f Colo rado at Denve r for 23 years , whe re he taught both unde rgraduate and g raduate cou rses, w ith a spec ial interest in applied mathematics Dur ing much of that time, he des igned and taught courses i n quantitative reasoning. In addition to this book, he has co -authored textbooks on statistical reasoni ng and calculus , as well as monog raphs in computational mathematics . He recentl y comp l eted the book How America Got Its Guns ( Unive rsity of New Mexico P ress). D r Br iggs is a Unive rsity of Co l orado President's Teaching Scholar and the recipient o f a Fulb r ight Fellowsh ip to I reland; he ho l ds a BA degree from the University of Colo rado and an MS and a PhD from Ha rvard Univers ity.
PREFACE
Huma n his to ry
b eco m es more and mo r e a r ace b e twe e n e du catio n and ca t as tro ph e - H G Wells
The Outline of History, 1920
To the Student
There is no escaping the importance of mathematics in the modern world. However, for most people, the imp o rtance o f mathematics lies nol in its abs1ract ideas, but in its application to persona l a nd socia l issues. This textbook is designed with such practica l co n side rati o n s in m ind. In parti c u lar, this book has three speci fi c purposes:
• to prepare yo u fo r the mathemat ics yo u will e nco u nte r in other co ll ege co urses, part ic ular ly co re co urses i n social and natural scie nces;
• to deve lop your ability to reason with quanti tative i nformation in a way that will h elp yo u achieve s u ccess in yo ur career ; and
• to provid e you with the critica l thinking and quantitative reasoning skills you need to understand maj or issues in yo ur Life
We hope this book will be useful 10 everyone, but it is designed primarily for those who are not planning to majo r in a field that requires advanced mathematical skills. In particular, if you've eve r felt any fear o r anxiety about mathematics, we've writte n this book wilh you in mind. We hope that, through this book, you will disco,•er Ihat mathematics is much more important and relevant to you r life than you thought and not as difficult as you previously imagined.
\Vhatever your interesLS-SOcial sciences, envi ro nme ntal issues, politics, business and economics, art and music, o r any of many other topics-you will find many relevant and up-to-date examples in this book. But 1he most important idea to take away from th is book is th a i math emati cs can help you und e rstand a variety of lopics a n d issues, making yo u a more aware and better educa1ed citize n O nce you have co mpleted your s tudy o f this book, you s h o uld be prepared to unde rs tand mos t quantitative issu es 1ha1you will encounter
To the Instructor
Whether you've taught this cou rse many tim es o r are teaching it for the first time, you a re undoubtedly aware that mathematics courses fo r non-majors present c halle nges that differ fro m those presented by more traditional courses. First and fo remost, there isn 't even a clear conse nsus on what exactly should be taugh t in these courses. While the res little debate
about what mathematical content is necessary for sc ience , technology, engineering, and mathematics (STEM) studentSfor example, these students all need 10 learn algebra and calculus-there's g reat debate about whal we should leach n on -STEM stude n ts, especially the large majorily who will not make use of fo rmal mathematics in their careers or daily lives.
As a re s ult of this debate, core mathemalics cou rses for non -STEM st ud e nts re presen t a broad a nd diverse range. Some schools re quire these stud e nts to take a trad iti o nal , calcu l us -track co u rse, s u ch as co ll ege algebra. O t hers have i nst it u ted cou rses focu se d o n so me of 1he h idden ways in wh ich conte mporary math e matics contr ibutes 10 soc ie ty, a nd still others have devel o ped co urses devoted almos1 excl us i vely to fi nancial literacy. Each of th e different co urse types has its me rits , but we believe there is a bcue r op tio n , largely because of the fo llow i ng fac t: The vast majori t y ( t yp ica ll y 95%) o f non-STEM students will never take a n ot h er co ll ege mathematics co u rse after comp leting their co re requireme nt.
Given this fact, we believe it is essentia l to teach these students the mathematical ideas that Ihey will need for the ir remainin g college course work, their careers , and their daily lives. In other words , we must emphasize those IOpics that are t rul y important to the future success of these students , and we must cove r a broad range of such lopics. The focus of this approach is less on formal calcu lati onthough some is certainly required-and more on leaching students how to think critically with numerical or mathematical information. In the terminology adop1ed by MAA, AMATYC, and other mathematical organizalions, swdems need to learn quantitative reasoning and to become quantitatively literate. The re's been a recent rise in 1he popularity of quantitativ e reasoning courses for non-STEM s wd e n ts. This book has been integral to the quanl it a live reaso ning movement for yea rs and co ntinu es to be a1 the forefront as a n established resource designed to h e l p yo u s u cceed i n teach ing qua n t itative re aso nin g to yo ur stud e n ts.
The Key to Success: A Context-Driven Approach
Broad ly speaking, approaches to teaching malh e matics ca n be d ivided into two ca tegor ies:
• A co ntent-driven approach is organized by mathematical ideas. After each mathematical topic is presen ted, exa mples o f its applications are shown.
• A context-driven approach is organized by practical contexts. Applications d rive the course , and mathematical ideas are p resented as needed to support the applications
The same conten t can be cove red th rough e i ther approach , but the context-d r iven approach has an enormous advantage : It mot ivates students by showing them directly how re l evant mathemat ics is to the i r lives. I n cont rast , the content-dr iven approach tends to come across as " learn this content because it's good fo r you ," caus i ng many students to tune out before reaching the p ractical applications. Fo r more details , see our a r ticle "General Education Mathematics: New Approaches for a New M i llenn i um " (AMATYC Revin11 Fall 1999) or the d iscuss ion i n the Epilogue of the book Math for Life by Jeffrey Bennett (Big Kid Sc ience , 20 14).
The Challenge: Winning Over Your Students
Perhaps the greatest challenge in teach ing mathematics lies in w inn i ng students ove r- that is , conv in cing them that you have someth ing useful to teach them . Th is challenge ar ises because by the time they reach college , many students dislike o r fear mathematics Indeed, the vast majo r ity of students in gene ra l educat ion mathemat ics cou rses a re the re not by choice, but because such courses are requ ired for graduat ion. Reach ing yo ur students the refore requi res that you teach w i th enthus iasm and conv i nce them that mathematics is useful and enjoyable.
We've bu il t this book around two impo r tant st rategies that are des igned to help you win students over:
• Confront negative att it u des about mathematics head on, show i ng students that thei r fear o r loath ing is ungrounded and that mathematics is relevant to thei r lives. This strategy is embodied in the Prologue of this boo k (pages P1- P13), wh ich we urge you to emphas ize in class. I t contin u es imp li c itl y th roughout the rest o f the text.
• Focus on goals that are meaningful to students--namely, on the goals of learning mathematics fo r college, career, and life. Your students will then Jearn mathematics because they will see how it affects their lives This strategy fonns the backbone of th is book, as we have tr ied to build every unit around topics relevant to college, caree r, and life.
Modular Structure of the Book
Although we have written this book so that it can be read as a nar rative from beginn ing to end, we recognize that many instructors might wish to teach mate r ial in a different o rde r than we have chosen or to cove r only selected port ions of the text , as time allows , for classes o f d iffe rent l ength o r fo r
students at diffe rent l eve ls. We have therefore organized the book w i th a modular structure that allows instructors to create a custom ized course. The 12 chapte rs a re o rgan ized broadly by contextual areas. Each chapter, in turn, is divided into a set of self- contained tmits that focus on particular concepts o r app li cations. In most cases, you can cover chapters in any order or skip units that are l owe r pr ior i ty fo r your pa r ticula r course. The following outline desc r ibes the flow of each chapte r:
Chapter Overview
Each chapte r begins with a twopage ove rview consisting of an int roductory parag raph and a multiple- cho ice question designed to illustrate an i mportant way in which the chapter content connects w ith the book themes of college, career, and life. The overview also includes a motivational quote and a unit-by- un it listi ng o f key content; the Iaue r is designed to show students how the chapte r is organized and to help instructors decide which units to cove r in class.
Chapt er Activity
Each chapte r next offe rs an activity des igned to spu r student discussion of some interesting facet of the top ics covered in the chapter The activities may be done e ither individually or in small groups. A new Activity Manual conta ining additional activities is available with this seventh ed ition in p ri nt form and also i n Mylab Math .
Num bered Units
Each chapter cons ists of numbe red units (e .g ., Un it 1A, Un i t 1B, ... ). Each un it begins w ith a short i ntroduction and includes the following key features:
• Headings to Ide n tify Key Topi cs. In keep ing with the modularity, each subtop ic wi thin a unit is clearly identified so that students understand what they will be learning.
• S u mmary Boxes. Key definitions and concepts a re highlighted in summa ry boxes for easy refe rence
• Examples and Case Studies. Numbered examp les a re des igned to build understand ing and to offer p ractice with the types o f questions that appea r in the exe rcises Each example is accompanied by a " Now try " suggestion that re l a tes the examp l e to specific similar exe rcises. Occasional case stud ies go into more depth than the numbered examp l es.
• Exe r cises. Each unit concludes w ith a set of exerc ises , subdivided i nto the following categor ies :
• Q u ick Quiz. This ten -question quiz appea rs at the end o f each unit and allows students to check whether they understand key concepts before s ta r ti ng the exerc ise set. Note that students are asked not onl y to choose the co r rect multiple-choice answer but a lso to w r ite a br ief explanation of the reasoni ng behind their choice. Answers a re included in the back of the text.
• Revi ew Questi o ns. Designed p rima r ily for selfstu dy, these questi ons ask stu dents to s u mma r ize the important ideas cove red in the u n it and generally can be answered s imply by revi ew i ng the tex t.
• Does It Make Se n se? These questions ask students to dete r mine whethe r a short statement makes sense , and ex plai n why o r why not These exe rcises a re genera ll y easy once st udents u n de rstand a particular concept , b u t d iffic u lt otherwise; they are the refore an excellent probe of comprehension.
• Ba s i c S kill s & Co n ce p ts. These q u estions offe r practice w i th the concepts covered i n the unit. They can be used for homework ass ig n ments o r fo r se l f- study (a nswe rs to most odd - numbered exe rc ises appear i n the back of the book ) . These questions a re refe renced by the " Now try " suggestions i n the u nit.
• Fur t h e r Appli cat i o ns. Th rough additional applications , these exerc ises extend the ideas and tech niques covered i n the un it.
• I n Yo ur vVo rld . These quest io ns are des igned to spur add itional research or discuss ion that will he l p stu dents relate the u n it content to the book themes of college, ca reer, and li fe.
• Tec hno l ogy Exe rcises. For u nits that include one or mo re Usi ng Tech no l ogy featu res, these exercises give stu dentS an opportun ity to p ractice calcu l a tor or software skills that have been introduced. Some of these exerc ises are des igned to be completed with StatCrunch co mes with the Mylab Math co u rse . Applicatio ns u si ng StatCrunch , powe rfu l Web-based statistical software that allows u sers to collect data, perfo rm anal yses , and generate compelling res u lts , a re i ncl u ded in th is edition fo r the fi rst lime.
Ch apter Summ ary Appear ing at the end of each chapte r, the Chapte r Su mmary offers a b r ief outline of the chapte r's content, includ i ng page nu mbers, that s t udents can use as a study g u ide.
Additio nal Pedagog ica l Feature s In addition tO the standard features of all chapte rs listed above, seve ral other pedagog ical feat u res occur throughout the text:
• Th ink Ab o u t It These features pose sho r t conceptual q u estions des ig n ed to help st u dents reflect on i mportant new ideas. They also serve as e xcelle n t starti ng pointS for class room d iscussions and, in some cases , ca n be used as a basis for cl icke r q uestions.
• Bri ef Rev i ew. This feature appears whe n a key mathematica l skill is first n eeded; topics i nclude fractio ns, powe rs and roots, basic algebraic ope rations , a n d mo re. The word " rev iew " i nd icates that most
students will have lea rn ed these skills previous l y, b u t many will n eed review and practice. Practice is ava ilab l e i n the exerc ise sets , with relevant exercises ident ifi ed by a " Now try ." sugges tion at the end of the Brief Review.
• I n Your Wo rl d . These features focus on topics that students are li ke l y to encounter in the world a round them , whethe r i n the news, in consume r dec isions , or i n po li tical disc u ssions. Exa mp l es i n cl u de how to u nderstand jewe l r y purchases , how to invest money i n a se ns ible way, and how to eval u ate the reliability of pre- e lect ion polls. (Note: These feat ures a re not n ecessar il y con n ected directly to the I n Your Wo rl d e x ercises, but both have direct re leva n ce to st u dentS' wo rl d )
• Usi ng Techn ology. These feat u res give st udents clea r i nst ru cti ons in the use o f various technolog ies for co mp u tation , i ncl u di ng scienti fic calculators, Mic rosoft Exce l , and online technologies such as those bu il t into Google. Book-spec ific TI Tech Tips conta in ing instructions for perfo r ming comp u tations with a g raphing calcu l ator, s u ch as the TI -83 or TI-84 , are available in the Tools for Success section o f Mylab Math.
• Cau t io n ! New to the seventh edition , these sho r t n o tes , integrated into examples or text, h igh light co mmon erro rs that st u dents should be careful to avoid.
• Math e ma tic a l Insigh t. This featu re, which occ u rs less frequently than the others , b u ilds on mathematical ideas i n the mai n narrative but goes so mewhat beyo nd the level of other material i n the book. Exa mples of the topics cove red a re proof of the Pythagorean theorem, Zenos paradox , and de r ivations of the financial fo r mu l as used fo r sav i ngs p l ans and mo r tgage loans.
• Margin Fe atures. The ma rgins contain severa l typ es of short fea t ures: By t h e Way, which offers i n teresting n o tes and as ides relevant to the top ic at hand; Histo r ica l No te, which g ives historical context to the top ic at hand; and Tec hnical No te, which offers details that are im po rtant mathematically, b u t generally do not affect studentS' unde rstand i ng of the mater ial. The margins also contai n occasio n al quotations
Prerequ isite Mathematical Background
Because of itS modula r structure and the inclusio n of the Br ief Rev iew features , th is book can be used by st u dents w i th a w ide range of mathematical backgrounds. Many of the un i ts req u ire nothing more tha n arith met ic and a will i ngness to think about quantitative iss u es in new ways On l y a few units u se techn iques o f algebra or geomet r y, and those skills are reviewed as they ar ise. Th is book should therefore be access ible to any student who has co mpleted two o r more years of high schoo l math -
ematics However, this booh is not remedial: Although much of the book relies on mathematical techn i ques from secondary school, the techniques arise in applicat ions that students generally are not taught in high school and that require students to demonst rate thei r cr i tica l thinking skills.
For courses in which students do requi re more extens ive prereq u isite review, we have created a version o f the Using & Understanding Mathematics Mylab Math course called Using & Understanding Mathematics with Integrated Review that includes just- in-time review of selected prerequisite topics .
Note on "Developmental Math" We are o ften asked whether this te x t can be used by students for whom placement tests suggest that they bel ong in developmental mathematics cou rses . In most cases, we believe the answer to be a resound i ng " yes. " Our ex pe ri ence suggests that many students who do poorly on mathemati cs p lacement tests are not rea ll y as weak as these tests may suggest. Most s t udents did learn basic mathematical skills at one lime, and if the skills ar ise with context (as they do in this book ), we've fo u nd that students can quickly relea rn them . Th is is especially true if you provide the students with a little bit of e x tra practice as offered in ou r Br ief Rev iew featu res or by the resources i n My Lab Math or Mylab Math w ith Integrated Review. Indeed, we believe that most stu dents in th is s ituation will lea rn basic mathematical skills better by taking a quantitative reason i ng cou rse based on this tex tbook than they will by taking a developmen ta l cou rse.
Changes in the Seventh Edition
We've been pleased by the positi ve responses from so many use rs of previous editions of this text. Neverthe l ess, a book that relies heavily on facts and data always req u i res a major updating effort to keep it current, and we a re always looking fo r ways to improve cla r ity and pedagogy. As a result, use rs o f pr ior editions wi ll find many sections of this book to have been substantia ll y revised o r rewritten. The changes are too many to list here , but some of the more significant changes are the following.
Chapter 1 We s ign ificantly revised Units l A a nd lE with the particular goal of helping students eval uate media info r mation and recognize " fake news "
Chapter 2 We reorganized and s ignificantly rewrote th is entire chapte r to introduce a basic problem -so l ving strategy in Un it 2A . Mo reover, we mod ified the fo u r-step strategy
presented in previous ed itions to create a simple r threestep st rategy called " Unde rsta n d -Solve- Explain ." We have found that this strategy is easier for students to remembe r and the refo re easier for them to put into p racti ce.
Ch apters 3 and 4 These two chapters conta i n seve ral units that revolve a rou n d econom ic data s u ch as demog raph ic data , the Consume r Pr ice I nde x , inte rest rates, ta x es , and the fede ra l b u dget. These data obvio u sly requi red major updates given the changes that have occ ur red in the U.S economy in the four years s ince the last ed i tion. In add i tion, we've added basic ideas abo u t hea l th i nsu ra n ce to our discussion of pe rsonal finances i n Unit 4A.
Chapters 5 and 6 These chapters foc u s on statistical data, wh ich means we u pdated o r replaced large sections of the chapter content to include more current data.
Chapter 7 We significantl y revised Section 70 on r isk, both fo r greater clarity and to u pdate data .
Chapters 8 and 9 Un i ts 8B, 8C , and 9C all rel y heavily on population data, which means we revised s ign ificant po r ti ons of these u n its to reflect the latest gl obal demograph ic data
Chapt er 12 The 2016 election p rovided nume rous new examples for ou r discuss ion of the electoral college in Un i t 12A Other recent examples o f the intersecti on of mathematics and politics a lso p rovide interesting new examp les and exercises throughout this chapte r.
In Your World We've added seven new In Your World fea tu res, so every chapter now has at least one, fu r the r showcasing math fo r college, ca reer, and life.
Caution! These short notes highlighting common e rrors are new to this ed ition.
Exercis e Sets We've thoroughly revised the exerc ise sets: Over 30% of the exerc ises are changed or new.
StatCrun ch StatC ru nch has been n ewl y integ rated into the MyLab Math cou rse and relevant Technology Exerc ises.
Video Progr am The seventh edition is accompanied by an all - new video program cons isti ng of both fam iliar lecture- style v ideos fo r eve ry example and innovative concept v ideos.
Resources for Success
Mylab Ma th Online Course for Using & Understanding Mathematics: A Quantitative Reasoning Approach, 7t h edi tion
by Jeffrey Bennett and Wi lliam Br iggs
Mylabm Ma t h is ava ilab le to accompany Pearson's market-lead ing t ext offer ings
To give students a consistent t one, voice, and teach ing method, each t ext's f lavor and app roach are tightly integra t ed th roughout the accompanying Mylab Math course, making learning t he mater ial as seamless as poss ible
NEW! Lecture Videos
Brand -n ew lect ure v ideos for ever y example are f r esh and modern and are accompa nied by assessment q uestio ns that give the instructor the ability to not just assign the v ideos but gauge student under stand ing.
At the end of 5 years. you have earned $100 in simple interest and have a total of $300 Invested In the account
NEW! Concept Videos
Dynam ic light board videos focus on some of th e most interesting and challenging concepts so students can better grasp them. Excit ing visuals are used to exp lain concepts such as comparisons of quanti t ies, st udent loans, and percentages in the world around us.
NEW! StatCrunc h Integration
StatCrunch is powerful web -based st atistica l softwa re that allows users to collect data, pe rform ana lyses, and generat e com pelling results. For t his seventh edition, St atCrunch questions have been added to relevant Techn ology Exe rcises and access to t he sof tware has been int egrat ed into the Mylab Math course.
Resources for Success
Instructor Resources
Mylab Math with Integrated Review
This Mylab Math course option can be used in co-requisite courses, or simply to help students who enter the quantitative reasoning course lacking prerequisite skills or a full understanding of prerequisite concepts.
For relevant chapters, students begin with a Skills Check assignment to pinpoint which prerequisite developmental topics, if any, they need to review.
Those who requ ire add itional review proceed to a personalized homework assignment that focuses on the specific prerequisite topics on which they need remediation.
Students can also review the relevant prereq· uisite concepts using videos and Integrated Review Worksheets in Mylab Math. The lnte· grated Review Worksheets are also available in printed form as part of the Activity Manual with Integrated Review Worksheets.
Specific to the Using & Understanding Mathematics
Mylab Math cour se:
NEW! Completely new lecture video pr ogra m with corresponding assessment
NEW! Dynamic concept videos
NEW! Interactive concept videos with corresponding assessment
NEW! Animations with corresponding assessment
NEW! Integration of StatCrunch in the left-hand navigation of the Mylab Math course makes it easy to access the software for completion of the Technology Exercises that use StatCrunch. Bonus unit on mathematics and business, including assessment
Instructor's Edition
(ISBNs: 0·13-470522-X /978·0·13·470522 4)
The Instructor's Edition of the text includes answers to all of the exercises and Quick Quizzes in the back of the book.
The following resources are ONLINE ONLY and are available for download from the Pearson Higher Education catalog at www.pearson.com/uslsign -in .html or within your MyLab Math course.
Instructor's Solution Manual
james Lapp
This manual i ncludes answers to all of the text's Think About It features, Quick Quizzes, Review Questions, and Does It Make Sense? questions and detailed. worked -out solutions to all of the Basic Skills & Concepts, Further Applications, and Technology Exercises (includ ing StatCrunch exercises).
Instructor's Testing Manual
Dawn Dabney
The Testing Manual provides four alternative tests per chapter, includ ing answer keys.
TestGen
TestGen® (www.pearsoned.com/testgen) enables instructors to bui ld, edit, print, and administer tests using a computerized bank of quest ions developed to cover all the objectives of t he text. TestGen is algorithmically based, allowing inst ructors to ere· ate multiple but equ ivalent ver sions of the same question or test with the click of a button. lnstruc· tors can also modify test bank questions or add new questions. The software and test bank can be downloaded from Pearson's Instructor Resource Center.
PowerPoint Lecture Presentation
These editable slides present key concepts and definitions from the text. Instructo r s can add art from the text located in t he Image Resource Library in Mylab Math or add slides t hey have created. PowerPoint slides are fu lly accessible.
Image Resource Library
This resource in the Mylab Math course contains all the art from the text for instructors to use in their own presentations and handouts.
Student Resources
Student's Study Guide and Solutions Manua l
(ISBNs: 0·13·470524-6/978·0·13-470524·8)
james Lapp
This manual contains answers to all Quick Quiz ques· tions and to odd-numbered Review Questions and Does It Make Sense? quest ions, as well as worked-out solutions to odd-numbered Basic Skills & Concepts, Further Applications, and Technology Exercises (including StatCrunch exercises).
NEW! Activ ity Manual with Pearson
My
lab Integrated Review Worksheets
(ISB Ns: 0-13-477664-X /978·0·13-477664·4)
Compiled by Donna Kirk. The College ofSt Scholastica More than 30 act ivities correlated to the textbook give students hands-on experiences t hat reinforce the course content. Activities can be completed indiv id ually or in a g roup. Each activity includes an overview, estimated time of completion, obj ec· t i ves, guidelines for group size, and list of materi· als needed. Add itionally, the manual provides t he worksheets for the Integrated Review version of the Mylab Math course.
Acknowledgments
A textboo k may carry its authors' names, but it is the result of hard work by hundred s o f commi tted individuals. This book has been under developm ent for m o re t han 30 years, and eve n its begi n n ings were a group ciTo n , as one of the authors was a me m ber of a commiuec at the Un ive rsi ty o f Co lo rado tha t wo rked to es t ablish one of t h e nat ion's first co urses in quanti tative reasoning. Si n ce that beginning, the book has bendi ted f rom input and feedback fro m many faculty m e mbers and students.
First and fo rem os t, we extend our thanks to Bill Poole a nd El ka Block, whose faith in this project from the beginning allowed it to grow f rom class notes into a true textbook. We 'd also like to thank other past and present members o f o ur outstanding publishing team at Pearso n Educatio n , including Greg Tobin, An n e Kelly, Mamie Gree nhut , Patty Bergin, Barbara Atkinson, Kyle DiGiannantonio, Stacey Miller, Hannah Laffe rty, and Nick Swee ny. We thank Rhea Meye rh o l tz and Pau l Lorcza k for a n exce ll e n t job o n a cc uracy c h ecking, and Shan e Goo dw i n o f BYUldaho for h is he lp in p reparing th e Us in g Technology boxes (and for man y other suggestiorts he has mad e as well).
We 'd like t o thank the following people for their help with one or m o re editions of this book . Those who assisted with this seventh edition have an asterisk be fo re their names.
*Merri Jill Ayers, Ge01gia Gwi mum College
Lou Barnes, Premier Mortgage Gro up
Ca ro l Bellisio, Monmoutl1 Uni versity
Bob Be rn har dt , East Carolina University
Terence R. Blows, Northent Ari zona Univers ity
•Lo i Booher, University of Central Arlwnsas
W. Wayne Bosche,jr., Dalto n College
Kristina Bowers, Uni ver sity of So uth Florida
Michael Bradshaw, Caldwell Com mtm ity College and Technical Instihtte
Shane Brewer, Utalt State Campus
W. E. Briggs , University of Colorado, Boulder
Annette Burden, Youngstown State University
Ov idiu Calin, Eastem Michigan Univers ity
Su sa n Ca rr, Oral Roberts Univer-sity
• He nry Chango, Community Co ll ege of Rlwd e Island
Margaret Cibes, Trinity College
WalLer C zarnec , Framingham State College
Adrian Daigle, Uni versity ofColonrdo, Boulder
Andrew J. Dane , Angelo State University
•or. A mit Dave, Georgia Gwinneu College
Jill DeWiu , Baker College of Muslu:gon
Greg Dietrich, Florida Comnmni ty College at jacksonville
Marsha J Driskill, Aims Comrmmity College
john Emert, Ball State University
Kath y Eppl er, Sa lt L.alle Comm w rity College
Kelli e Evans, Yorh College of Pen nsylva nia
Fred Fel d o n , Co a stline Community College
Anne F in e, East Central University
David E. Fles ner, Gettysburg College
Pat Foard, South Plains College
Brian Gaines , University of Illinois
*Jose Gimenez, Temple Unhnsity
Shane Goodwin, Br·igham Yotmg Uni vtrsity-ldalro
Barbara Grover, Sa lt L.alte Community College
Lo u ise Hai nlin e, Brooltlyn College
Ward He ilman, Bridgewater State University
Peg Hovd e, Gross mont College
Andrew Hug ine, South Carolina State University
Lynn R Hun, Dixie College
Hal Huntsman , Uni versity of Colorado, Boulder
joel Irish , Uni\•ersity of South ern Maine
Davidjabon , DePaul University
Melvin F. Janowitz , University of Massachusetts, Amherst
Craig johnson, Brigham Young University-Idaho
Vijay S. Joshi, Virginia Intermont College
Anton Kau l , Univers ity of So uth Florida
Bonni e Ke ll y, University of Soutlt Carolina
William Kil ey, George Mason University
*Donna Kirk , T he College of Saint Scholastica
J im Koehler, University of Colorado, Denver
*Ch arlotte Koleti , Georgia Gwinnett College
Robert Kuenzi , Unive rsity of \Visconsirt, Oshkosh
Erin Lee , Cent ral Washington Uni versity
R. Warren Le m e ric h , Laramie Cowrty Community College
Deann Leoni, Edmonds Community Co ll ege
Linda Les te r, Wright State University
Paul Lorcza k , Mat lt Soft, Inc.
jay Malms 1ro m, Oltlahoma City Community Co ll ege
*Howard Mand elbaum ,Johnjay Co ll ege
Erich McAiisler, Unive rsity of Colorado, Boulder
*Meghan Mcintyre , Wake Technical Commurr ity College
Judith McKnew, Clemsorr Uni ver sity
Lisa McMillen , Baker College
Patricia McNicholas, Robert Morris College
Phyllis Mellinger, Hollins University
Elaine Spendlove Merrill, Briglwm Young UniversityHawaii
*Dillon Miller, San jacinto College
*Mehdi Mirfauah , Long Beach City College
Carrie Muir, Unhersity of Colorado, Boulder
Coiro Mulcahy, Spelman College
*Bette Nelson , Alvin Community College
Stephen Nicoloff, Paradise Valley Community College
Paul O'Heron, Broome Community College
L. Taylor Ollmann , Austin Community College
*Diane Overturf, M.S , Viterbo University
A. Dean Palmer, Pima Community College
Mary K. PaLLon, University of Illin ois at Springfield
Frank Pecchioni,jefferson Co mmunity Co ll ege
*M ichae l Polley, Sout heastern Comm unity Co ll ege
jonathan Preweu, Ut tiversity of Wyoming
Evelyn Pupplo-Cody, Marshall University
Scou Reed, College of Lal:e County
Frederick A. Reese, Borough of Manlwttan Community College
Nancy Rivers , Waite Tecltnical Community College
Anne Roberts , Unh ersity of Utah
*Michelle Robinson , Fayeueville Technical Community College
Sylvester Roebuck,jr., Olh-e Han ey College
*Sheri Rogers, Linn-Belllon Community College
Lori Rosenthal , Austill Community College
Hugo Rossi, University ofUtalt
*Robin Rufatto, Ball State University
*Joana Sandra , Olive Han ey College
Doris Schraeder, Mclennan Community College
Dee Dee Shaulis, University of Colorado, Boulder
judith Silver, Marshall University
Laura Smallwood, Chandler-Gilbert Community College
Sybil Smith-Darlington, Middlesex Coumy College
Alu Srinivasan, Temple Unh ttsity
john Supra, University of Colorado, Boulder
Scou Surgem , Arizona State Unh·ersity
Tunothy C. Swyter, Fl'!'derick Comnnmity College
Louis A. Talman, Metwpolitan State College of Dem·er
David Theobald, Unh·ersity of Colorado, Boulder
Robert Thompson, Hunter College (CUNY)
Terry Tolle, Soutlnvestem Community College
Kathy Turrisi, Centenary College
*Claudio Valenzuela, Southwest Texas junior College
Christina Venullo, Marist Coll ege
Pam Wahl, Middlesex Com munity Co llege
Ian C. Walters,jr., D'Youville Co llege
Thomas Wangler, Benedictine University
Richard Watkins, Tidewater Com munity College
Charles D. Watson, University of Central Kansas
*Dr. Gale Watson, East Georgia State College
*D r. Beverly Watts, McDowell Tecltnical Community College
Emily Whaley, DeKalb College
*john Williamson, Sandhills Commu11ity College
David Wilson , Unh ersity of Colorado, Boulder
Robert Woods, Broome Community College
Fred Worth, Henderson State Unh-ersity
Yoder, Eastem Kentucky University
Zabdawi , Gorclon College
Fredric Zerla, University of South Florida
Donald]. Zielke , Concordia Lutheran College
LITERACY FOR THE MODERN WORLD
Equations are j ust the boring par t of mathematics . - Stephen Hawk ing, physicist
If you're like most students enrolled in a course using this text, you may think that your interests have relatively little to do with mathematics. But as you will see, nearly every ca reer today requires the use and understandi ng of some mathematics. Furthermore, the abil ity to reason quantitatively is crucial for the decisions that we face da ily as citizens in a modern technological society. In this Prologue, we'll discuss why mathematics is so important, why you may be better at it than you think, and how this course can provide you w ith the quantitative ski lls needed for you r college courses, your career, and your life.
Imag ine t ha t yo u're at a pa rty and you've j ust st ruck up a conversat io n with a dynam ic, su ccessfu l lawyer. W hich o f th e f o llowing a re yo u most li kely to hea r
her say d uring yo u r co nversa tio n ?
0 "I really don't k now how to read very well."
0 "I can't w rite a grammatically correct sentence."
$ "I'm awful at dea ling with people."
(!) "I've never been able to think logically."
0 "I'm bad at math."
We all know that the answer is E, because we've heard it so many times. Not j ust from lawyers, b ut from businesspeople, actors and athletes, construction workers and sales clerks, and sometimes even teachers and CEOs It would be difficult to imagine these same people admitting to any of choices
A through D, b ut many people consider it socially acceptable to say that they are "bad at math " Unfortunately, this social acceptability comes with some very negative social consequences. (See the d iscussion about Misconception Seven o n page P-7 .)
Job Satisfaction
Each ch apter in t h is textboo k begin s with an activity, whic h yo u may do ind ivid ually or in groups . For this Pro logue, t h e opening activity will h elp you examine the ro le of ma t hematics in careers Add i tional activities are available online in Myl ab Ma t h.
To11 20 Jobs for job Satisfaction
1. Mathematician
2. Actuary (works with insu rance statistics)
3. Statistician
4. Biologist
5. Software engineer
6. Computer systems analyst
7. Historian
8. Sociologist
9. Industrial designer
10. Accountant
11. Economist
12 Philosopher
13. Physicist
14. Parole officer
15. Meteorologist
16. Medical laboratory technician
17. Paralegal assistant
18. Computer programmer
19. Motion picture editor
20. Astronomer
Source Job:sRatcd.com.
Everyone wants to find a career path that will bring lifelong job satisfaction, but what careers are most likely to do that? A recent survey evaluated 200 different jobs according to five criteria: salary, long-term employment outlook, work environment, physical demands, and stress. The table to the leh shows the top 20 jobs according to this survey. Notice that most of the top 20 jobs require mathematical skills, and all of them require an ability to reason with quantitative information
You and your classmates can conduct your own smaller study of job satisfaction. There are many ways to do this, but here is one procedure you might try:
O Each of you should identify at least three people with full-time jobs to interview briefly. You may choose parents, friends, acquaintances, or just someone whose job interests you.
E) Identify an appropriate job category for each interviewee (similar to the categories in the table to the left). Ask each interviewee to rate his or her job on a scale of 1 (worst) to 5 (best) on each of the five criteria: salary, long-term employment outlook, work environment, physical demands, and stress. You can then add the ratings for the five criteria to come up with a total job satisfaction rating for each job.
0 Working together as a class, compile the data to rank all the jobs Show the final results in a table that ranks the jobs in order of job satisfaction
0 Discuss the results Are they consistent with the survey results shown in the table? Do they sur· prise you in any way? Will they have any effect on your own career plans?
What Is Quantitative Reasoning?
Literacy is the ab ility to read and w ri te, and it comes in va rying degrees. Some peop le can recogniz e on l y a few words and w ri te only the ir names ; othe rs read and w r ite in man y l anguages. A pr imary goal of ou r educational sys te m is to provide citizens w ith a level of lite racy s ufficient to read , w ri te , and reason about the impo r tant issues of our lime . Today, the abilities to inte rpret and reason with q uanti tative informatio n-i nfo r mation that involves mathematical ideas or numbers-are c ruc ial aspectS of lite racy These abilities, often called q ua n t i tative reason ing or q u an ti tative lite r acy , are esse ntial tO understanding issues that appear in the news every day. The purpose of this textbook is to help you gain skills in quantitative reasoning as it appli es to issues you will encounter in
• yo u r s ubsequent cou rsewo rk,
• you r career, and
• you r daily l ife.
Quanti tative Reason ing and Culture
Quantitative reasoning enriches the apprec i ation of both a n c ient a nd mode rn c u lt u re. The h istor ical reco rd shows that nearly all cult u res devoted substantial energy to mathematics and to sc ience (or to obse rvational stu dies that p redated modern science). Without a sense of how quanti tative conceptS are u sed in a r t, arch itectu re , and sc ience , you cannot fu ll y apprec i ate the inc redible ach ievementS o f the Mayans i n Cent ral America, the builders of the great city of Zimbabwe in Afr ica, the ancient Egyptians and Greeks, the ea rl y Po l ynesian sailo rs, and many others.
Simila rl y, q u anti tati ve conceptS can he l p you understand and appreciate the works of the great artists. Mathematical conceptS p l ay a major ro l e in eve ryth ing fro m the work of Rena issance artistS li ke Leona rdo da Vinc i and Michelangelo to the pop culture o f te levis ion shows like The Big Bang Tl1eory Othe r ties between mathematics and the arts can be found in both modern and class ical music, as well as i n the d igital production of m u s ic. I ndeed , it is ha rd to find pop u lar works of a r t, fil m, o r li teratu re that do not rely on mathematics in so me way.
Quantitative Reasoning in the Work Force
Quantitative reasoni ng is important in the work force. A l ack of q u antitative skills p u ts many o f the most challenging and highest- paying jobs o u t of reach. Tabl e P. l defines skill levels in l ang u age and mathematics on a scale of 1 to 6, and Table P.2 (on the next page) shows the typical l evels needed in ma n y j obs
Note that the occ u pati ons requ ir ing high skill levels are generally the most p restigious and highest pay i ng . No te a lso that most o f those occupatio ns ca ll for high skill levels in both l anguage and math, refu ting the myth that if you 're good at language, yo u do n't have to be good at mathematics, a n d v ice versa.
Mt.):j!jijl
Sk ill Leve ls
Reads signs and basic news reports; writes and speaks
Mathematics knows no races or geographi c boundarie s; f or math emati cs, th e cultural world is one country
Addition and subtraction; simple calculations with money, volume, length, and
2 Can read short stories and instruction manuals; writes Arithmetic; can compute ratios, rates, and percentages; can compound sentences with proper grammar and draw and interpret bar graphs
3 Reads novels and magazines; writes reports with proper Basic geometry and algebra; can calculate discounts, interest.
4 Reads novels, poems, and newspapers; prepares business Has true quantitative reasoning abilities: understands logic, letters, summaries, and reports; participates in panel problem solving, ideas of statistics and probability, and discussions
modeling
5 Reads literature, scientific and technical journals, Calculus and statistics f inancial reports, and legal documents; can write editorials, speeches, and critiques
Source: Adapt ed from levels des<ribed in t he Wall Stree t Journal
We are all mathematicians . . . [your] forte lies in navigating the comp lexities of social networks, weighing passions agains t histories, calculating reactions, and generally managing a system of information that, when all laid ou t, would boggle a computer.
- A. K. Dewd ney, 200% of Notlting
Misconceptions About Mathematics
Do you consider yourself to have " math phobia " (fear of mathematics ) o r " math loathing " (dis li ke of mathemat ics)) We hope not- but if you do, you aren ' t alone. Many adults ha rbo r fear or loath i ng of mathematics, and unfortunate l y, these attitudes are often reinforced by classes that present mathematics as an obscure and ster il e subject.
In reality, mathematics is not nea rly so dry as it sometimes seems in school. Indeed, attitudes toward mathematics often are directed not at what mathematics really is but at some common m isconceptions about mathematics Let's investigate a few of these m isconceptions and the reality behind them
Misconception One : Math Requires a Specia l Brain
One of the most pervas ive misconceptions is that some people just a ren ' t good at mathematics because l earn i ng mathematics requ ires special or rare abilities. The reality is that nearly eve r yone can do mathematics. All i t takes is self- confidence and hard work- the same qua li ties needed to learn to read , to master a mus ical instrument , or to become sk ill ed at a sport. Indeed, the belief that mathematics requ ires special talent found in a few elite peop l e is peculia r to the United States. I n most othe r countries, part icu larly in Europe and As ia, all studenLS are expected to become proficient in mathematics.
Of cou rse , d ifferent people learn mathematics at d ifferent rates and in d i fferent ways. For examp l e , some peop l e l earn by concentrating on concrete p roblems, others by th ink ing visua lly, and still others by thinking abst ractl y No matter what type of th inking style you prefe r, you can succeed in mathematics.
Misconception Two: The Math in Modern Issues Is Too Complex
Some people cla i m that the advanced mathematical concepts unde rlying ma n y modern issues a re too comp l ex for the average person to understa n d. It is true that only a few people rece ive the tra i n i ng needed to work with or discover advanced mathemati cal concepts Howeve r, most people are capable o f understand ing enough about the mathematical bas is of i mportant issues to develop i nformed and reasoned opin ions.
The situ ati on is s im ila r fo r othe r fields. For examp l e , years o f study and practice are required to become a proficient professional w ri ter, but most people can read a book. It takes ha rd work and a l aw degree LO become a lawyer, but most people can understand how the law affects them. And tho ugh few have the musical talent o f Mo zart , anyone can learn to appreciate his mus ic Mathematics is no di fferent. I f yo u 've made it th is fa r in school, yo u can unde rstand eno u gh mathematics to succeed as an i ndividual and a concerned citizen.
Misconception Three: Math Makes You less Sens itive
Some people believe that learning mathematics will somehow make them less sensitive to the romantic and aesthetic aspects of life I n fact, unde rstand i ng the mathematics that ex plains the colo rs of a sunset o r the geomet ri c beauty in a work of art can only enhance ou r aesthetic apprec iation of these th ings . Furthermore , many peop le find beauty and elegance in mathematics itself. It's no accident that peop l e tr ained in mathematics have made i mpo rtant cont r ibuti ons to art, music, and many other fields.
Misconception Four: Math Makes No Allowance for Creativity
The "turn the crank " natu re of the problems in many textbooks may give the i mpression that mathematics sti Oes cr eativity. Some o f the facts, formalisms, and ski lls requi red fo r mathematical pro fi ci ency a re fai rly cut and dr ied , but using these mathematical too ls takes creati vity Consider design i ng and b u ild ing a home. The task of construction requires speci fi c skills to lay the foundation, frame in the stru cture, i nstall p l umb ing and wiring, and paint walls But the full process involves much mo re : Creativity is needed to devel op the arch itectu ral des ign, respond to on - the- spot problems d u r ing const r uction, and factor in constra ints based on budgets and build i ng codes The mathematical ski lls you've lea rn ed in school a re like the s kills of ca rpent ry or plumb i ng App l ying mathematics is like the creative process of build ing a home.
Misconception Five: Math Provides Exact Answers
A mathematical fo r mul a 'viii y iel d a spec ific result, a nd i n school that result may be marked right or w rong. Bu t when yo u use mathematics i n real -life s ituat ions , answers a re never so clea r cut For example:
A bani! offers simple interest of 3%, paid at the end of 1 year (tl1at is, after 1 year the bani! pays you 3% of your account balance) If yo u deposit $1000 today and mahe no further deposits or withdrawals , l10w much will yo u have in yo ur account after 1 yearl
A straight mathematical calc u lat ion seems s i mple eno ugh: 3% o f $1000 is $30; so yo u sho uld have $1030 at the end of a yea r But will you? How will you r balance be affected by se rvice charges or taxes on inte rest earned? What if the ban k fails? What if the bank is located in a count ry in which the cu rrency collapses during the yea r ? Choosing a ba nk in which to invest your money is a real mathemat ics problem that doesn't necessarily have a simple or defin i tive solution.
Skills are to math ematics what scales are to music or spelling is t o writing. The objective of learning is to write, t o play music, or to solve problems- not just to master skills.
- from Eve•)•body C ounts, a repon o f the Na t io nal Research Co uncil
It is impossible to be a math emati cian without being a poet in th e soul.
-Sophia Kova levskaya (1850- 1891), Russian rnalhematician
Tell me, and I will f orget. Sh ow me, and I may remember. Involve me, and I will understand
-Con fu ciUs (c 55 1-4 79 B.C.E )
Pro bably th e most harmful misconce pti on is that math ematics is esse ntially a matter of computati on Believ ing this is roughly equi valent to believing that writing es says is th e sa me as ty ping th em.
- John Al1en Pa ul os, rnalhematician
People Who Studied Mathematics
The critical thinking skills developed through the study of math· ematics are valuable in many careers The following list represents a small sample of people who studied mathematics but are more famous for work in other fields (Many of the names come from "Famous Nonmathematicians," a list compiled by Steven G Buyske, Rutgers University )
Ralph Abernathy , civil rights leader, BS in mathematics, Alabama State University
Tammy Baldwin , U S. Senator (Wisconsin), BA in math· ematics, Smith College
Sergey Brin , co-founder of Google, BA in mathematics, University of Maryland
Mayim Bialik, actress on The Big Bang Theory, studied mathematics in working toward her PhD in neuroscience
Har ry Blackmun , former Supreme Court j ustice, summa cum laude in mathematics, Harvard University
James Came ron , film di rector, studied physics before leav· ing college, works in oceanic and space research
Lewis carroll (Charles Dodgson), mathematician and au· thor of Alice in Wonderland
Fe li cia Day, actress, BA in mathematics, University of Texas
David D inkins, former mayor of New York City, BA in mathematics, Howard Un iversity
A lberto Fujimo ri, former president of Peru, MS in math· ematics, Un iversity of Wisconsin
Art Garfun kel , musician, MA in mathematics, Columbia University
Reed Hastings, founder and CEO of Netflix, BA in math· ematics, Bowdoi n College
G race Hopper, computer pioneer and fi rst woman rear admiral in the U.S Navy, PhD in mathematics, Yale Un iversity
Mae Jemison, first African-American woman in space, studied mathematics in working toward her BS in chemical engineering, Stanford University
John Maynard Keynes, economist, MA in mathematics, Cambridge University
Hedy Lama rr, actress, invented and patented the math· ematical techn ique called " frequency hopping"
Lee Hsien Loong , Prime Minister of Singapore, BA in math· ematics, Cambridge University
Brian May , lead guitarist for the band Queen, completed his PhD in astrophysics in 2007, Imperial College
Danica McKella r, actress, BA with highest honors in math· ematics, UCLA, and co-discoverer of the Chayes-McKellar· Winn t heorem
Andrea Merke l, Chancellor of Germany, studied math· ematics in working toward her PhD in physics, University of Leipzig
Harvey M il k, politician and activist for gay rights, BA in mathematics, State University of New York
Edwin Moses, three-time Olympic champion in the 400-meter hu rdles, studied mathematics in working toward his BS in physics, Morehouse College
Florence Nightingale, pioneer in nursing, studied math· ematics and applied it to her work
Natali e Portman, Oscar-winning actress, semifinalist in Intel Science Talent Search and co-author of two published scientific papers
Sally Ride, f irst American woman in space, studied math· ematics in working toward her PhD in physics, Stanford Un iversity
Dav id Rob inson , basketball star, BA in mathematics, U.S Naval Academy
Alexander Solzhen itsyn , Nobel prize-winn ing Russian author, degrees in mathematics and physics, University of Rostov
Bram Stoker , author of Dracula, BA in mathematics, Trinity University, Dublin
Laurence Tribe , Harvard law professor, summa cum laude in mathematics, Harvard University
John U rschel , NFL offensive lineman (Baltimore Ravens) who retired at age 26 to pursue a PhD in mathematics, MIT
Vi rg inia Wade , Wimbledon champion, BA in mathematics, Sussex University
W h at Is Mat h ematics?
Misconception Six: Math Is Irrelevant to My Life
No matter what your path i n college, ca reer, and life , you will find mathematics involved in many ways. A majo r goal of this text is to show you hund reds of exampl es in wh ich mathematics app li es LO everyone's life We hope you will find that mathematics is not on l y relevant but a lso inte resting and enjoyable.
Misconception Seven: It's OK to Be "Bad at Math "
Fo r ou r final misconception , let's return to the multiple-choice question on the open ing page of th is Pro l ogue. You' ll not on l y hea r many otherwise i ntelligent people say " I'm bad at math," but i t's sometimes sa id a l most as a point of p r ide, with no hint o f embar rassment. Yet the statement often isn't even true . A success fu l l awye r, for example, a l most certa i nl y did well in all subjects in school , i nclud i ng math, and so is more likely ex press ing an attitude than a reality.
Unfortunately, th is type of atti tude can cause a lot of damage. Mathematics underl ies nearly every thing in modern society, from the daily financial dec is ions that all of us must make to the way in which we understand and approach global issues invo l ving the economy, politics, and sc ience We cannot poss ibly hope to act wisely if we approach mathematical ideas with a poor attitude. Moreover, it's an attitude that can easily spread to others . After all , i f a child hears a respec ted adult saying that he or she is "bad at math ," the child may be less insp i red to do well
So before you begin you r cou rsewo rk, think about your own atti t udes toward mathematics. The re's no reason why anyone shou l d be " bad at math " and every reason to develop skills of mathematical th inking. W i th a good attitude and some ha rd work, by the end of this cou rse you' ll not on l y be bette r at math, but you' ll be he l ping futu re generations by making i t soc i a ll y unacceptable fo r anyone to be " bad at math."
What Is Mathematics?
I n d iscuss i ng m isconcept ions, we identified what mathematics is not. Now l et's l ook at what mathematics is The word mathematics is derived from the Greek wo rd mathematilws, wh ich means " i ncl i ned to Jearn." Literally speaking, to be mathematica l is to be curious, open- m i nded, and always inte rested in learning more! Today, we tend to look at mathematics in three diffe rent ways : as the sum of its branches, as a way to model the wo rld , and as a language.
Mathematics as the Sum of Its Branches
As you prog ressed through school, you probably learned to assoc iate mathemat ics w i th some of its b ranches. Among the better- k n own branches of mathematics are these:
• l og ic- the study of pr inc iples o f reason i ng;
• a r i t hmet i c- methods fo r operati ng on numbers;
• a lgeb ra - methods for wo rking with unknown quantities;
• geo m etr y- the study of size and shape;
Neglect of mathematics works to all knowl edge - Roger Baco n (1214-1294), Engl ish p hiloso phe r
You must be th e change you wish to see in t he world - Mahatma Ga nd hi ( 1869- 1948)
Th e Book of Natu re is wr itten in th e language of ma thematics
-Galileo
• trigonometry- the study of triangl es and their uses;
• probabili ty- the study of chance ;
• statistics-methods for anal yz ing data; and
• calcu lu s-the study of quantiti es that change.
One can view mathematics as the sum of its branches , but i n th is text we'll focus on how different branches of mathematics support the more general goals of quantitative th ink ing and critical reason ing.
Mathematics as a Way to Model the World
Mathematics also may be v iewed as a tool for creating models, or representations of real phenomena Modeling is not unique to mathematics For examp le , a road map is a mode l that represents the roads in some region . Ma t hema t ica l mode ls can be as s imple as a s i ngle equation that predicts how the money in you r bank account will g row o r as complex as a set of thousands of inter related equations and pa rameters used to rep resent the gl obal climate. By studying models, we gain insight into otherwise unmanageab l e probl ems A global cl imate model, fo r examp l e , can he l p us unde rstand weathe r systems and ask " what i f " questions about how human activity may affect the climate. When a model is used to make a prediction that does not come t r ue , that fa il u re points out areas where further resea rch is needed. Today, mathematical modeling is used in nearl y every field of study. F igu re P. l shows some of the many d isciplines that use mathematical modeling tO solve prob lems .
Mathematics as a Language
A th i rd way to look at mathemat ics is as a language with its own vocabulary and grammar Indeed, mathematics often is called " the language of nature" because it is so useful fo r modeling the natural world. As with any language, d ifferent degrees of fluency are poss ible From this po int of view, quant itative lite racy is the level of fluency requ ired fo r success in today's world
The idea of mathematics as a language is also useful in thinking about how to leant mathematics. Table P.3 compares l earning mathematics to learning a language and learning about art.
