Math In Society NMC Version-202002 PDF

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Math in Society

NMC Edition 2020

David&Lippman,&Pierce&College&Ft&Steilacoom& Table&of&Contents& Chapter&1&–&Problem&Solving&.........................................................&1& 1.1 – Percent.................................................................................................1 1.2 – Proportions and Rates .........................................................................9 1.3 – Dimensional Analysis .......................................................................12 1.4 – Geometry...........................................................................................16 1.5 – Problem Solving and Estimating ......................................................20 Try It Now Answers...................................................................................25 Exercises ....................................................................................................28 Solutions to Selected Chapter 1 Exercises .................................................41

Chapter&2&–&Finance&.......................................................................&43& 2.1 – Simple Interest and Personal Budget ................................................43 2.2 –Compound Interest .............................................................................46 2.3 – Annuities ...........................................................................................53 2.4 – Loans .................................................................................................56 2.5 – Consumer Price Index, Inflation .......................................................64 Exponents on your calculator.....................................................................65 A note on rounding ....................................................................................65 Calculator Solver, old operating system ....................................................66 Try It Now Answers...................................................................................68 Exercises ....................................................................................................71 2.6 – Taxes – Honor’s Section ...................................................................75 Solutions to Selected Chapter 2 Exercises .................................................83

Chapter&3&–&Growth&Models.........................................................&85& & 3.1 – Linear Growth ...................................................................................85 3.2 – Doubling and Halving .......................................................................93 3.3 – Exponential Growth ..........................................................................99 3.4 – Regression Equations and Correlation ............................................109 Try It Now Answers.................................................................................115 Exercises ..................................................................................................117 Solutions to Selected Chapter 3 Exercises ...............................................122

Chapter&4&–&Statistics&and&Probability...................................& & 125& 4.1 – Populations, Samples, Measures of Central Tendency ...................126 4.2 – Measures of Variation .....................................................................133 4.3 – Basic Probability and Counting ......................................................141 4.4 – Compound Probability and Expected Value ...................................147 Try It Now Answers.................................................................................155 Exercises ..................................................................................................157 Solutions to Selected Chapter 4 Exercises ...............................................166

Copyright&©&2013&David&Lippman& & This&book&was&edited&by&David&Lippman,&Pierce&College&Ft&Steilacoom& Development&of&this&book&was&supported,&in&part,&by&the&Transition&Math&Project&and& the&Open&Course&Library&Project.& & Statistics,&Describing&Data,&and&Probability&contain&portions&derived&from&works&by:& Jeff&Eldridge,&Edmonds&Community&College&(used&under&CC-BY-SA&license)& www.onlinestatbook.com&(used&under&public&domain&declaration)& & Apportionment&is&largely&based&on&work&by:& &&Mike&Kenyon,&Green&River&Community&College&&(used&under&CC-BY-SA&license)& & Historical&Counting&Systems&derived&from&work&by:& Lawrence&Morales,&Seattle&Central&Community&College&(used&under&CC-BY-SA&license)& & Cryptography&contains&portions&taken&from&Precalculus:&An&investigation&of&functions& by:& &&&David&Lippman&and&Melonie&Rasmussen&(used&under&CC-BY-SA&license)& & Front&cover&photo:&& Jan&Tik,&http://www.flickr.com/photos/jantik/,&CC-BY&2.0& & & This&text&is&licensed&under&a&Creative&Commons&Attribution-Share&Alike&3.0& United&States&License.&& & To&view&a©&of&this&license,&visit&http://creativecommons.org/licenses/by-sa/3.0/us/&or&send&a&letter& to&Creative&Commons,&171&Second&Street,&Suite&300,&San&Francisco,&California,&94105,&USA.& & You&are&free:& to&Share&—&to©,&distribute,&display,&and&perform&the&work& to&Remix&—&to&make&derivative&works& & Under&the&following&conditions:& Attribution.&You&must&attribute&the&work&in&the&manner&specified&by&the&author&or&licensor&(but¬& in&any&way&that&suggests&that&they&endorse&you&or&your&use&of&the&work).& Share&Alike.&If&you&alter,&transform,&or&build&upon&this&work,&you&may&distribute&the&resulting&work& only&under&the&same,&similar&or&a&compatible&license.& & With&the&understanding&that:& Waiver.&&Any&of&the&above&conditions&can&be&waived&if&you&get&permission&from&the©right&holder.&& Other&Rights.&&In&no&way&are&any&of&the&following&rights&affected&by&the&license:&& • Your&fair&dealing&or&fair&use&rights;& • Apart&from&the&remix&rights&granted&under&this&license,&the&author's&moral&rights;& • Rights&other&persons&may&have&either&in&the&work&itself&or&in&how&the&work&is&used,&such&as& publicity&or&privacy&rights.& • Notice&—&For&any&reuse&or&distribution,&you&must&make&clear&to&others&the&license&terms&of& this&work.&The&best&way&to&do&this&is&with&a&link&to&this&web&page:& http://creativecommons.org/licenses/by-sa/3.0/us/&

About&the&Author/Editor& & David&Lippman&received&his&master’s°ree&in&mathematics&from& Western&Washington&University&and&has&been&teaching&at&Pierce& College&since&Fall&2000.&&& & David&has&been&a&long&time&advocate&of&open&learning,&open& materials,&and&basically&any&idea&that&will&reduce&the&cost&of& education&for&students.&&It&started&by&supporting&the&college’s& calculator&rental&program,&and&running&a&book&loan&scholarship& program.&&Eventually&the&frustration&with&the&escalating&costs&of& commercial&text&books&and&the&online&homework&systems&that& charged&for&access&led&to&action.& & First,&David&developed&IMathAS,&open&source&online&math&homework&software&that&runs& WAMAP.org&and&MyOpenMath.com.&&Through&this&platform,&he&became&an&integral&part& of&a&vibrant&sharing&and&learning&community&of&teachers&from&around&Washington&State& that&support&and&contribute&to&WAMAP.&&These&pioneering&efforts,&supported&by&dozens& of&other&dedicated&faculty&and&financial&support&from&the&Transition&Math&Project,&have& led&to&a&system&used&by&thousands&of&students&every&quarter,&saving&hundreds&of& thousands&of&dollars&over&comparable&commercial&offerings.& & David&continued&further&and&wrote&the&first&edition&of&this&textbook,&Math%in%Society,& after&being&frustrated&by&students&having&to&pay&$100+&for&a&textbook&for&a&terminal& course.%%Together&with&Melonie&Rasmussen,&he&co-authored&PreCalculus:%An% Investigation%of%Functions%in&2010.& &

Acknowledgements& & David&would&like&to&thank&the&following&for&their&generous&support&and&feedback.&&& & • Jeff&Eldridge,&Lawrence&Morales,&and&Mike&Kenyon,&who&were&kind&enough&to& license&me&use&of&their&works.& & • The&community&of&WAMAP&users&and&developers&for&creating&some&of&the& homework&content&used&in&the&online&homework&sets.& & • Pierce&College&students&in&David’s&online&Math&107&classes&for&helping&correct& typos&and&identifying&portions&of&the&text&that&needed&improving,&along&with& other&users&of&the&text.& & • The&Open&Course&Library&Project&for&providing&the&support&needed&to&produce&a& full&course&package&for&this&book.&

&

Preface! & The&traditional&high&school&and&college&mathematics&sequence&leading&from&algebra&up& through&calculus&could&leave&one&with&the&impression&that&mathematics&is&all&about& algebraic&manipulations.&&This&book&is&an&exploration&of&the&wide&world&of&mathematics,& of&which&algebra&is&only&one&small&piece.&&The&topics&were&chosen&because&they&provide& glimpses&into&other&ways&of&thinking&mathematically,&and&because&they&have&interesting& applications&to&everyday&life.&&Together,&they&highlight&algorithmic,&graphical,&algebraic,& statistical,&and&analytic&approaches&to&solving&problems.& & This&book&is&available&online&for&free,&in&both&Word&and&PDF&format.&&You&are&free&to& change&the&wording,&add&materials&and§ions&or&take&them&away.&&I&welcome& feedback,&comments&and&suggestions&for&future&development.&If&you&add&a§ion,& chapter&or&problems,&I&would&love&to&hear&from&you&and&possibly&add&your&materials&so& everyone&can&benefit.&&& & The&course&shell&was&built&for&the&IMathAS&online&homework&platform,&and&is&available& for&Washington&State&faculty&at&www.wamap.org&and&mirrored&for&others&at& www.myopenmath.com.&

Problem Solving 1

Chapter 1: Problem Solving In#previous#math#courses,#you’ve#no#doubt#run#into#the#infamous#“word#problems.”## Unfortunately,#these#problems#rarely#resemble#the#type#of#problems#we#actually# encounter#in#everyday#life.##In#math#books,#you#usually#are#told#exactly#which#formula# or#procedure#to#use#and#are#given#exactly#the#information#you#need#to#answer#the# question.##In#real#life,#problem#solving#requires#identifying#an#appropriate#formula#or# procedure,#and#determining#what#information#you#will#need#(and#won’t#need)#to# answer#the#question.### # In#this#chapter,#we#will#review#several#basic#but#powerful#algebraic#ideas:##percent,# rates,#proportions,#dimensional#analysis,#geometry,#and#estimating.##We#will#then#focus# on#the#problem-solving#process#and#explore#how#to#use#these#ideas#to#solve#problems# where#we#don’t#have#perfect#information.#

1.1 – Percent In#the#2004#vice-presidential#debates,#Edwards's#claimed#that#US#forces#have#suffered# "90%#of#the#coalition#casualties"#in#Iraq.##Cheney#disputed#this,#saying#that#in#fact#Iraqi# security#forces#and#coalition#allies#"have#taken#almost#50#percent"#of#the#casualties 1.## Who#is#correct?##How#can#we#make#sense#of#these#numbers?# ! Percent!literally#means#“per#100,”#or#“parts#per#hundred.”##When#we#write#40%,#this#is# 40 equivalent#to#the#fraction# #or#the#decimal#0.40.##Notice#that#80#out#of#200#and#10#out# 100 80 10 40 of#25#are#also#40%,#since# .### = = 200 25 100 # To#calculate#a#percent#you#must#calculate#what#part#of#the#whole#you#have.#You#must# know#what#the#whole#and#part#are.#This#calculation#will#give#you#a#decimal#which#can# then#be#converted#into#parts#per#hundred.## !"#$ # Percent!=!%&'()! ! Example#1# Write#each#number#as#a#percent:##a)#0.4# b)#0.02# c)#2.35# d)#0.0005# # *, * / a)#0.4#=#+, = . +,,#=#40%# b)#0.02#=#+,,#=#2%# # # # # 1 +,, ,.,1 /01 1 c)#2.35#=# #=#235%# d)#0.0005#=# #=#0.05%# =. .÷. =. +,,

+,,,,,

+,,,,

+,,

+,,

# # Notice#that#the#percent#can#be#found#from#the#equivalent#decimal#by#moving#the# decimal#point#two#places#to#the#right# 1

http://www.factcheck.org/cheney_edwards_mangle_facts.html

© David Lippman

Creative Commons BY-SA

Example#2# Write#each#percent#as#a#number:##a)#42%# b)#0.6%# c)#100%# d)#2000%# # */ ,.5 a)#42%#=# #=#0.42# b)#0.6%#=# #=#0.006## # # +,, +,, # +,, /,,, c)#100%#=# #=#1# d)#2000%#=# #=#20# +,, +,, # # Notice#that#the#decimal#can#be#found#from#the#equivalent#percent#by#moving#the# decimal#point#two#places#to#the#left.# # # Example#3# 243#people#out#of#400#state#that#they#like#dogs.##What#percent#of#these#people#like#dogs?# # The#whole#is#the#400#people#and#the#part#is#the#243#people.# # 243 60.75 = 0.6075 = Percent#that#like#dogs#=# .### 400 100 # 60.75%#of#these#people#like#dogs.# Remember!to!write!a!sentence!answer!that! clearly!answers!the!question!asked.# # # Percent! If#we#have#a#part#that#is#some#percent#of#a#base*(or*whole),#then## !"#$

percent!=!6"7) !!!or#equivalently,## part!=!percent!·!base! # To#do#the#calculations,#we#write#the#percent#as#a#decimal.# # A#key#word#you#must#be#able#to#spot#and#interpret#is#the#word#of.#The!word!of!tells!you! to!multiply.! # Example#4# The#sales#tax#in#a#town#is#9.4%.##How#much#tax#will#you#pay#on#a#$140#purchase?# # Here,#$140#is#the#base,#and#we#want#to#find#9.4%#of#$140.##We#will#need#to#write#the# percent#as#a#number.##We#can#then#compute:# Tax# =# 9.4%#of#$140# Correctly written mathematics is clearly labeled, 8.* Tax# =# +,,#·#140# works down the page, and does only one calculation per line of work. Tax#=#13.16# I#will#pay#taxes#of#$13.16.##

of means multiply

2

Problem Solving 3 Example#5# In#the#news,#you#hear#“tuition#is#expected#to#increase#by#7%#next#year.”#(Tuition#will#be# 7%#more*than#current#tuition.)##If#tuition#this#year#was#$97.32#per#credit#hour,#what#will# it#be#next#year?# # The#tuition#next#year#will#be#the#current#tuition#plus#an#additional#7%#of#the#current# tuition.# New#tuition# =# current#tuition#+#7%#of#current#tuition# # =# $97.32#+#7%#of#$97.32# #

9

=# 97.32#+#+,,#·#97.32#

# =# 104.1324# Next#year’s#tuition#will#be#$104.13#per#credit#hour.### #

Note, This is money so it must be rounded to hundredths.

# # # A#shortcut#that#we#will#want#to#learn#to#use#is#a#shortcut#for#percent#increase#or#percent# decrease.#Let’s#look#back#at#example#5#where#we#are#asked#to#calculate#a#7%#increase.# Calculating#a#7%#increase#of##a#number#is#the#same#as#calculating#107%#of#the#number.# We#will#use#algebra#to#prove#this.# Let#n#=#the#number# Increase#the#number#by#7%#=# the#number#+#7%#of#the#number# # =# n#+#7%#of#n# Use#n#for#the#number# 9

#

=# n#+#+,,#·#n#

Change#7%#to#a#number#

#

=# n#+#0.07n#

Write#+,,#as#a#decimal#

# # #

=# n(1#+#0.07)# =# n(1.07)# =# 1.07n#

Factor#out#n# Add#1#+#0.07# Rewrite#so#the#number#factor#is# first#

#

=#

#

+,9 +,,

#n#

=# 107%#of#n#

9

Rewrite#1.07#as#a#fraction#with# 100#as#the#denominator# +,9

Write#+,,#as#a#percent.#

So,#a#7%#increase#of#a#number#is#the#same#as#107%#of#the#number.#Knowing#this# shortcut#saves#several#lines#of#work.#This#is#true#for#all#percent#increases.#For#example,# a#30%#increase#of#a#number#is#130%#of#the#number.# Try!it!Now!1! Convert#each#percent#increase#of#a#number#to#a#percent#of#the#number:# a)#Increase#of#10%# b)#Increase#of#200%# #

4 This#shortcut#also#works#for#a#percent#decrease.#Let’s#look#at#decreasing#a#number#by# 12%.# Let#n#=#the#number# Decrease#the#number#by#12%#=# the#number#-#12%#of#the#number# Use#n#for#the#number# # =# n#-#12%#of#n# +/

#

=# n#-#+,, #·#n#

Change#12%#to#a#number#

#

=# n#-#0.12n#

Write#

# # #

=# n(1#-#0.12)# =# n(0.88)# =# 0.88n#

Factor#out#n# Subtract#1#-#0.12# Rewrite#so#the#number#factor#is# first#

#

=#

#

::

#n#

+,,

=# 88%#of#n#

+/

+,,

#as#a#decimal#

Rewrite#0.88#as#a#fraction#with# 100#as#the#denominator# ::

Write#+,,#as#a#percent.#

So,#a#12%#decrease#of#a#numner#is#the#same#as#88%#of#the#number.#This#is#true#for#all# percent#decreases.## One#note#here#–#You#can#increase#a#number#by#more#than#100%#but#you#can#not# decrease#a#number#by#more#than#100%.# # Try!it!Now!2! Convert#each#percent#decrease#of#a#number#to#a#percent#of#the#number:# a)#Decrease#of#30%# b)#Decrease#of#1%# c)#Decrease#of#100%# #

# # Try!it!Now!3! A#TV#originally#priced#at#$799#is#on#sale#for#30%#off.##There#is#also#a#9.2%#sales#tax.## Find#the#final#cost#of#the#TV.# # Often,#we#are#interested#in#a#change#that#has#taken#place,#either#an#increase#or#a# decrease.#Usually#just#the#change#itself#is#not#good#information#because#there#is#no#way# to#judge#this#number.#A#better#calculation#is#a#percent#change.#A#percent#change#is#a# better#number#because#it#relates#the#change#to#the#base#amount.#Another#name#for# percent#change#is#relative#change.# # #

#

Problem Solving 5 Absolute!and!Relative!Change! Given#two#quantities,# Absolute#change#=#ending*value*–*starting*value# ;[email protected];EFB Relative#change#=# # =A;GAHEF.I;?@B

# Absolute#change#has#the#same#units#as#the#original#quantity.### Relative#change#gives#a#percent#change#and#is#unitless.# The#starting#quantity#is#called#the#base#of#the#percent#change.# #

# Example#6# The#value#of#a#car#dropped#from#$7400#to#$6800#over#the#last#year.##What#percent# decrease#is#this?# # To#compute#the#percent#change,#we#first#need#to#find#the#dollar#value#change,#or# absolute#change#=#$6800#-#$7400#=#-$600.## Since#we#are#computing#the#decrease#relative#to#the#starting#value,#we#compute#the# percent#change#out#of#$7400:# J$5,,

###Percent#change#=#$9*,, ## #

=#-0.081#

#

=# +,,#

#

=#-8.1%#

J:.+

There#is#an#8.1%#decrease#in#value.# This#is#called#a#relative!change.# #

# The#base#of#a#percent#is#very#important.##For#example,#while#Nixon#was#president,#it#was# argued#that#marijuana#was#a#“gateway”#drug,#claiming#that#80%#of#marijuana#smokers# went#on#to#use#harder#drugs#like#cocaine.##The#problem#is,#this#isn’t#true.##The#true#claim# is#that#80%#of#harder#drug#users#first#smoked#marijuana.##The#difference#is#one#of#base:## 80%#of#marijuana#smokers#using#hard#drugs#vs.#80%#of#hard#drug#users#having#smoked# marijuana.##These#numbers#are#not#equivalent.##As#it#turns#out,#only#one#in#2,400# marijuana#users#actually#go#on#to#use#harder#drugs2 .### # # #

2

http://tvtropes.org/pmwiki/pmwiki.php/Main/LiesDamnedLiesAndStatistics

6 Example#7#–#Caution:#Know#the#base# There#are#about#75#QFC#supermarkets#in#the#U.S.##Albertsons#has#about#215#stores.## Compare#the#size#of#the#two#companies.# When#we#make#comparisons,#we#must#ask#first#whether#an#absolute#or#relative# comparison.##The#absolute#difference#is#215#–#75#=#140.##From#this,#we#could#say# “Albertsons#has#140#more#stores#than#QFC.”##However,#if#you#wrote#this#in#an#article#or# paper,#that#number#does#not#mean#much.##The#relative#difference#may#be#more# meaningful.##There#are#two#different#relative#changes#we#could#calculate,#depending#on# which#store#we#use#as#the#base:# +*,

Using#QFC#as#the#base,# Percent#change#=# # 91 # # =#1.867# # # =#186.7%# This#tells#us#Albertsons#is#186.7%#larger#than#QFC.# J+*, Using#Albertsons#as#the#base,# Percent#change#=# /+1 # # # =#-0.651# # # =#-65.1%# This#tells#us#QFC#is#65.1%#smaller#than#Albertsons.# #

Notice#both#of#these#are#showing#percent#differences.### /+1 We#could#also#calculate#the#size#of#Albertsons#relative#to#QFC:##Size#comparison# =# # 91 # # # =#2.867# # # # =#286.7%# This#tells#us#that#Albertsons#is#286.7%#of#the#size#of#QFC.##Likewise,#we#could#calculate#the# 91 size#of#QFC#relative#to#Albertsons:# Size#comparison# =#/+1# # # =#0.349# # # =#34.9%# This#tells#us#that#QFC#is#34.9%#of#the#size#of#Albertsons.###

To#answer#this#question,#suppose#the#value#started#at#$100.##After#one#week,#the#value# dropped#by#60%:# # New#Value#after#drop# =#$100#-#$100(0.60)# We could also ...