Common sense mathematics

365 Pages·2021·37.874 MB·English

by  Ethan D. Bolker, Maura B. Mast

AMS / MAA TEXTBOOKS VOL 63
Common Sense Mathematics
VOL
Second Edition
AMS / MAA TEXTBOOKS 63
Ethan D. Bolker and Maura B. Mast
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students, like infl ation, credit card debt, and loans. We use search engines, calculators, and lk a
spreadsheet programs as tools to reduce drudgery, explore patterns, and get information. e t
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4-Color Process 398 pages on 50lb stock • Softcover • Spine 13/16"

Common Sense
Mathematics
Second Edition

AMS/MAA TEXTBOOKS
VOL 63
Common Sense
Mathematics
Second Edition
Ethan D. Bolker
Maura B. Mast
MAATextbooksEditorialBoard
SuzanneLynneLarson,Co-Editor
WilliamR.Green,Co-Editor
DebraS.Carney VirginiaA.Noonburg RonTaylor
HughN.Howards ThomasC.Ratliff ElizabethThoren
WilliamJohnston PameaRichardson RuthVanderpool
MichaelJ.McAsey JeffreyL.Stuart ElizabethWilcox
2010MathematicsSubjectClassification.Primary00-01,00A05,00A06.
Foradditionalinformationandupdatesonthisbook,visit
www.ams.org/bookpages/text-63
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Names:Bolker,EthanD.,author.|Mast,MauraB.,author.
Title:Commonsensemathematics/EthanD.Bolker,MauraB.Mast.
Description: Secondedition. |Providence,RhodeIsland: AmericanMathematicalSociety,2020. |Series:
AMS/MAAtextbooks,2577-1205;volume63|Includesbibliographicalreferencesandindex.
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Contents
PrefacetotheSecondEdition xi
PrefacetotheFirstEdition xiii
1 CalculatingontheBackofanEnvelope 1
In this first chapter we learn how to think about questions that need only good
enoughanswers. Wefindthoseanswerswithquickestimatesthatstartwithrea-
sonable assumptions and information you have at your fingertips. To make the
arithmeticeasyweroundnumbersdrasticallyandcountzeroeswhenwehaveto
multiply.
1.1 Hailingaride 2
1.2 Howmanyseconds? 4
1.3 Heartbeats 5
1.4 Calculators 6
1.5 Millionsoftrees? 7
1.6 Carbonfootprints 8
1.7 Kilo,mega,giga 9
1.8 Exercises 10
2 UnitsandUnitConversions 27
In real life there are few naked numbers. Numbers usually measure something
likecost,population,time,speed,distance,weight,energyorpower. Oftenwhat’s
measuredisarate,likemilesperhour,gallonspermile,milespergallon,dollars
pergallon,dollarspereuroorcentimetersperinch.
2.1 Ratetimestimeequalsdistance 27
2.2 TheMPGillusion 29
2.3 Convertingcurrency 31
2.4 Unitpricingandcrimerates 33
2.5 Themetricsystem 34
2.6 Workingontherailroad 36
2.7 Scientificnotation,milliandmicro 37
2.8 Carpetingandpaint 39
2.9 Exercises 41
v
vi Contents
3 Percentages,SalesTaxandDiscounts 65
Thefocusofthischapteristhestudyofrelativechange,oftenexpressedasaper-
cent. Weaugmentanoftenmuchneededreviewintwoways—stressingquick
paperlessestimationforapproximateanswersand,forprecision,anewtechnique:
multiplyingby1+(percentchange).
3.1 Thefederalbudget 65
3.2 RedSoxticketprices 68
3.3 The1+trick 68
3.4 Exploitingthe1+trick 70
3.5 Largeandsmallpercentages 71
3.6 Percentagepoints 73
3.7 Percentiles 73
3.8 Exercises 74
4 Inflation 101
Weminetheinternetfordataaboutinflationandusethe1+techniquefromChap-
ter3tounderstandthatdata.
4.1 RedSoxticketprices 101
4.2 Inflationisarate 102
4.3 TheConsumerPriceIndex 104
4.4 Morethan100% 104
4.5 Howmuchisyourraiseworth? 105
4.6 Theminimumwage 105
4.7 Inflationhistory 107
4.8 Exercises 108
5 AverageValues 117
Westartbyrememberingthattocomputeanaverageyouaddthevaluesanddivide
bythecount. Wequicklymoveontoweightedaverages,whicharemorecommon
andmoreuseful. They’realittlehardertounderstand,butworththeeffort. They
helpexplainsomeinterestingapparentparadoxes.
5.1 Averagetestscore 117
5.2 Gradepointaverage 118
5.3 Improvingaverages 119
5.4 TheConsumerPriceIndex 120
5.5 Newcarpricesfall... 122
5.6 Anaveragingparadox 123
5.7 Exercises 123
6 IncomeDistribution—Spreadsheets,ChartsandStatistics 131
Thischaptercoversalotofground—twonewkindsofaverage(medianandmode)
andwaystounderstandnumberswhentheycomeinlargequantitiesratherthan
justafewatatime: barcharts,histograms,percentilesandthebellcurve. Todo
thatweintroducespreadsheetsasatool.
Contents vii
6.1 SalariesatWingAero 132
6.2 Whatif? 136
6.3 Usingsoftware 136
6.4 Median 137
6.5 Barcharts 138
6.6 Piecharts 141
6.7 Histograms 142
6.8 Mean,median,mode 144
6.9 Computingaveragesfromhistograms 145
6.10 Thebellcurve 146
6.11 Marginoferror 149
6.12 Exercises 150
7 ElectricityBillsandIncomeTaxes—LinearFunctions 169
Weuseanelectricitybillasahookonwhichtohanganintroductiontofunctions
ingeneralandlinearfunctionsinparticular,inalgebraandinspreadsheets. Then
weapplywhatwe’velearnedtostudytaxes—sales,incomeandSocialSecurity.
You’llalsofindhereageneraldiscussionofenergyandpower.
7.1 Rates 169
7.2 Readingyourelectricitybill 170
7.3 Linearfunctions 172
7.4 Linearfunctionsinaspreadsheet 173
7.5 Whichtrucktorent? 177
7.6 Energyandpower 178
7.7 Federalpayrolltaxes 180
7.8 Exercises 185
8 ClimateChange—LinearModels 199
Complicatedphysicalandsocialphenomenararelybehavelinearly,butsometimes
datapointslieclosetoastraightline. Whenthathappensyoucanuseaspread-
sheettoconstructalinearapproximation. Sometimesthat’susefulandinforma-
tive. Sometimesit’smisleading. Commonsensecanhelpyouunderstandwhich.
8.1 Climatechange 199
8.2 Thegreenhouseeffect 203
8.3 Howgoodisthelinearmodel? 205
8.4 Regressionnonsense 206
8.5 Exercises 209
9 CompoundInterest—ExponentialGrowth 217
Inthischapterweexplorehowinvestmentsandpopulationsgrowandhowradio-
activitydecays—exponentially.
9.1 Moneyearnsmoney 217
9.2 Exploringexponentialgrowthwithaspreadsheet 219
9.3 Depreciation 221
9.4 Doublingtimesandhalf-lives 222
9.5 Exponentialmodels 225
viii Contents
9.6 “Exponentially” 227
9.7 Exercises 228
10 BorrowingandSaving 243
Whenyouborrowmoney—onyourcreditcard,fortuition,foramortgage—you
payitbackininstallments. Otherwisewhatyouowewouldgrowexponentially.
Inthischapterweexplorethemathematicsthatdescribespayingoffyourdebt.
10.1 Debitandcreditcards 243
10.2 Canyouaffordamortgage? 247
10.3 Savingforcollegeorretirement 249
10.4 Effectiveinterestrate 250
10.5 Instantaneouscompounding 251
10.6 Exercises 251
11 Probability—Counting,Betting,Insurance 257
PierredeFermatandBlaisePascalinventedthemathematicsofprobabilitytoan-
swergamblingquestionsposedbyaFrenchnoblemanintheseventeenthcentury.
We follow history by starting this chapter with simple examples involving cards
anddice. Thenwediscussrafflesandlotteries,fairpayoffsandthehouseadvan-
tage,insurance,andriskswherequantitativereasoningdoesn’thelpatall.
11.1 Equallylikely 257
11.2 Odds 258
11.3 Raffles 259
11.4 Statelotteries 260
11.5 Thehouseadvantage 263
11.6 One-timeevents 265
11.7 Insurance 266
11.8 Sometimesthenumbersdon’thelpatall 267
11.9 Exercises 268
12 BreaktheBank—IndependentEvents 275
Unlikelythingshappen—justrarely! Herewecalculateprobabilitiesforcombi-
nationslikerunsofheadsandtails. Thenwethinkaboutluckandcoincidences.
12.1 Acoinandadie 275
12.2 Repeatedcoinflips 277
12.3 Doubleyourbet? 279
12.4 Cancerclusters 280
12.5 Thehundredyearflood 281
12.6 Improbablethingshappenallthetime 283
12.7 Exercises 284
13 HowGoodIsThatTest? 291
InChapter12welookedatprobabilitiesofindependentevents—thingsthathad
nothingtodowithoneanother. Herewethinkaboutprobabilitiesinsituations
whereweexpecttoseeconnections,suchasinscreeningtestsfordiseasesorDNA
evidenceforguiltinacriminaltrial.