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AllanG.Bluman

Secondedition

Copyright©2011,2005byMcGraw-Hill.Allrightsreserved.ExceptaspermittedundertheUnitedStatesCopyrightActof1976,nopartofthispublicationmaybereproducedordistributedinanyformorbyanymeans,orstoredinadatabaseorretrievalsystem,withoutthepriorwrittenpermissionofthepublisher.ISBN:978-0-07-176385-1MHID:0-07-176385-6ThematerialinthiseBookalsoappearsintheprintversionofthistitle:ISBN:978-0-07-176386-8,MHID:0-07-176386-4.Alltrademarksaretrademarksoftheirrespectiveowners.Ratherthanputatrademarksymbolaftereveryoccurrenceofatrademarkedname,weusenamesinaneditorialfashiononly,andtothebenefitofthetrademarkowner,withnointentionofinfringementofthetrademark.Wheresuchdesignationsappearinthisbook,theyhavebeenprintedwithinitialcaps.McGraw-HilleBooksareavailableatspecialquantitydiscountstouseaspremiumsandsalespromotions,orforuseincorporatetrainingprograms.Tocontactarepresentativepleasee-mailusatbulksales@mcgraw-hill.com.Trademarks:McGraw-Hill,theMcGraw-HillPublishinglogo,Demystified,andrelatedtradedressaretrademarksorregisteredtrademarksofTheMcGraw-HillCompaniesand/oritsaffiliatesintheUnitedStatesandothercountriesandmaynotbeusedwithoutwrittenpermission.Allothertrademarksarethepropertyoftheirrespectiveowners.TheMcGraw-HillCompaniesisnotassociatedwithanyproductorvendormentionedinthisbook.InformationcontainedinthisworkhasbeenobtainedbyTheMcGraw-HillCompanies,Inc.(“McGraw-Hill”)fromsourcesbelievedtobereliable.However,neitherMcGraw-Hillnoritsauthorsguaranteetheaccuracyorcompletenessofanyinformationpublishedherein,andneitherMcGraw-Hillnoritsauthorsshallberesponsibleforanyerrors,omissions,ordamagesarisingoutofuseofthisinformation.ThisworkispublishedwiththeunderstandingthatMcGraw-Hillanditsauthorsaresupplyinginformationbutarenotattemptingtorenderengineeringorotherprofessionalservices.Ifsuchservicesarerequired,theassistanceofanappropriateprofessionalshouldbesought.TERMSOFUSEThisisacopyrightedworkandTheMcGraw-HillCompanies,Inc.(“McGraw-Hill”)anditslicensorsreserveallrightsinandtothework.Useofthisworkissubjecttotheseterms.ExceptaspermittedundertheCopyrightActof1976andtherighttostoreandretrieveonecopyofthework,youmaynotdecompile,disassemble,reverseengineer,reproduce,modify,createderivativeworksbasedupon,transmit,distribute,disseminate,sell,publishorsublicensetheworkoranypartofitwithoutMcGraw-Hill’spriorconsent.Youmayusetheworkforyourownnoncommercialandpersonaluse;anyotheruseoftheworkisstrictlyprohibited.Yourrighttousetheworkmaybeterminatedifyoufailtocomplywiththeseterms.THEWORKISPROVIDED“ASIS.”McGRAW-HILLANDITSLICENSORSMAKENOGUARANTEESORWARRANTIESASTOTHEACCURACY,ADEQUACYORCOMPLETENESSOFORRESULTSTOBEOBTAINEDFROMUSINGTHEWORK,INCLUDINGANYINFORMATIONTHATCANBEACCESSEDTHROUGHTHEWORKVIAHYPERLINKOROTHERWISE,ANDEXPRESSLYDISCLAIMANYWARRANTY,EXPRESSORIMPLIED,INCLUDINGBUTNOTLIMITEDTOIMPLIEDWARRANTIESOFMERCHANTABILITYORFITNESSFORAPARTICULARPURPOSE.McGraw-Hillanditslicensorsdonotwarrantorguaranteethatthefunctionscontainedintheworkwillmeetyourrequirementsorthatitsoperationwillbeuninterruptedorerrorfree.NeitherMcGraw-Hillnoritslicensorsshallbeliabletoyouoranyoneelseforanyinaccuracy,errororomission,regardlessofcause,intheworkorforanydamagesresultingtherefrom.McGraw-Hillhasnoresponsibilityforthecontentofanyinformationaccessedthroughthework.UndernocircumstancesshallMcGraw-Hilland/oritslicensorsbeliableforanyindirect,incidental,special,punitive,consequentialorsimilardamagesthatresultfromtheuseoforinabilitytousethework,evenifanyofthemhasbeenadvisedofthepossibilityofsuchdamages.Thislimitationofliabilityshallapplytoanyclaimorcausewhatsoeverwhethersuchclaimorcausearisesincontract,tortorotherwise.

ToBettyClaire,Allan,Mark,andallmystudentswhohavemademyteachingcareeranenjoyableexperience

AbouttheAuthor

AllanG.Blumantaughtmathematicsandstatisticsinhighschool,college,andgraduateschoolfor39years.HereceivedhisdoctoratefromtheUniversityofPittsburgh.HehaswrittenthreemathematicstextbookspublishedbyMcGraw-Hill.HeisalsotheauthorofthreeothermathematicsbooksintheMcGraw-HillDeMYSTiFieDseries:Pre-AlgebraDeMysTifieD,ProbabilityDeMysTifieD,andBusinessMathDeMysTifieD.Heistherecipientof“AnApplefortheTeacher”awardforbringingexcellencetothelearningenvironmentandtwo“MostSuccessfulRevisionofaTextbook”awardsfromMcGraw-Hill.HisbiographicalrecordappearsinWho’sWhoinAmericanEducation,5thedition.HehasbeeninductedintotheMcKeesportHighSchoolAlumniHallofFame.

Contents

Introduction

Acknowledgments

CHAPTER1IntroductiontoProblemSolving

Four-StepMethodProblem-SolvingStrategiesSummaryQuiz

CHAPTER2SolvingDecimalandFractionProblems

OperationsRefresherI:DecimalsSolvingWordProblemsUsingDecimalsRefresherII:FractionsSolvingWordProblemsUsingFractionsSummaryQuiz

CHAPTER3SolvingPercentProblems

RefresherIII:PercentsSolvingPercentWordProblemsSummaryQuiz

CHAPTER4SolvingProportionandFormulaProblems

RatiosProportionsFormulasSummaryQuiz

CHAPTER5EquationsandAlgebraicRepresentation

RefresherIV:EquationsAlgebraicRepresentationSummaryQuiz

CHAPTER6SolvingNumberandDigitProblems

NumberProblemsDigitProblemsSummaryQuiz

CHAPTER7SolvingCoinandAgeProblems

CoinProblemsAgeProblemsSummaryQuiz

CHAPTER8SolvingDistanceandMixtureProblems

DistanceProblemsMixtureProblemsSummaryQuiz

CHAPTER9SolvingFinance,Lever,andWorkProblems

FinanceProblemsLeverProblemsWorkProblemsSummaryQuiz

CHAPTER10SolvingWordProblemsUsingTwoEquations

RefresherV:SystemsofEquationsSolvingWordProblemsUsingTwoEquationsSummaryQuiz

CHAPTER11SolvingWordProblemsUsingQuadraticEquations

RefresherVI:SolvingQuadraticEquationsbyFactoringSolvingWordProblemsUsingQuadraticEquationsSummaryQuiz

CHAPTER12SolvingWordProblemsinGeometry,Probability,andStatistics

SolvingGeometryProblemsSolvingProbabilityProblemsSolvingStatisticsProblemsSummaryQuiz

FinalExam

AnswerstoQuizzesandFinalExam

SuggestionsforSuccessinMathematics

Index

Introduction

Whatdidonemathematicsbooksaytoanotherone?“Boy,dowehaveproblems!”

Allmathematicsbookshaveproblems,andmostofthemhavewordproblems.Manystudentshavedifficultieswhenattemptingtosolvewordproblems.Onereasonisthattheydonothaveaspecificplanofaction.Amathematician,GeorgePolya(1887–1985),wroteabookentitledHowtosolveIt,explainingafour-stepprocessthatcanbeusedtosolvewordproblems.ThisprocessisexplainedinChapter1ofthisbookandisusedthroughoutthebook.Thisprocessprovidesaplanofactionthatcanbeusedtosolvewordproblemsfoundinallmathematicscourses.

Thisbookisdividedinto12chapters.Chapters1,2,3,and4explainhowtousethefour-stepprocesstosolvewordproblemsinarithmeticorpre-algebra.Chapter5reviewsequationsandexplainsalgebraicrepresentation.Chapters6through11explainhowtousetheprocesstosolveproblemsinalgebra,andthesechapterscoverallofthebasictypesofproblems(coin,mixture,finance,etc.)foundinanalgebracourse.Chapter12explainshowtosolvewordproblemsingeometry,probability,andstatistics.Thisbookalsocontainssix“Refreshers.”Theseareintendedtoprovideareviewoftopicsneededtosolvethewordproblemsthatfollowthem.Theyarenotintendedtoteachthetopicsfromscratch.Youshouldrefertoappropriatetextbooksifyouneedadditionalhelpwiththerefreshertopics.Thisbookcanbeusedeitherasaself-studybookorasasupplementtoyourtextbook.Youcanselectthechaptersthatare

appropriateforyourneeds.

CurriculumGuide

TheDeMysTified®booksarecloselylinkedtothestandardhighschoolandcollegecurricula,sotheCurriculumGuideontheinsidebackcoverisprovidedforyoutohaveaclearpathtomeetyourmathematicalgoals.Whatmanystudentsdonotknowisthatmathematicsisahierarchicalsubject.Whatthismeansisthatbeforeyoucanbesuccessfulinalgebra,youneedtoknowbasicarithmetic,sincetheconceptsofarithmetic(pre-algebra)areusedinalgebra.Beforeyoucanbesuccessfulintrigonometry,youneedtohaveabasicunderstandingofalgebraandgeometry,sincetrigonometryusesconceptsfromthesetwocourses.YoucanusethisGuideinyourmathematicalstudiestolearnwhichcoursesarenecessarybeforetakingthenextones.

HowtoUseThisBook

Asyouknow,inordertobuildatallbuilding,youneedtostartwithastrongfoundation.Thesameistruewhenmasteringmathematics.Thisbookpresentsthebasictypesofmathematicalwordproblemsandhowtosolvetheminalogical,easy-to-readformat.Thisbookcanbeusedasanindependentstudycourseorasasupplementtoothermathematicalcourses.

Tolearnhowtosolvewordproblems,youmustknowthebasicproceduresandbeabletoapplytheseprocedurestomathematicalwordproblems.Thisbookiswritteninastylethatwillhelpyouwithlearning.Asstatedpreviously,itfollowsthebasicproblem-solvingstrategystatedbyGeorgePolya.Italsocontainssixmathematicalrefresherstohelpyoureviewtopicsthatareusedinword-problemsolving.Basicfactsandhelpfulsuggestionscanbefoundinthe“StillStruggling”boxes.Eachsectionhasseveralworked-outexamplesshowingyouhowtousetherulesandprocedures.Eachsectionalsocontainsseveralpracticeproblemsforyoutoworkouttoseeifyouunderstandtheconcepts.Thecorrectanswersareprovidedimmediatelyaftertheproblemssothatyoucanseeifyouhavesolvedthemcorrectly.Attheendofeachchapter,thereisamultiple-choicequiz.Ifyouanswermostoftheproblemscorrectly,youcanmoveontothenextchapter.Ifnot,youcanrepeatthechapter.Makesurethatyoudonotlookattheanswerbeforeyouhaveattemptedtosolvetheproblem.Evenifyouknowsomeorallofthematerialinthechapter,itisbesttoworkthroughthechapterinordertoreviewthe

material.Thelittleextraeffortwillbeagreathelpwhenyouencounterthemoredifficultmateriallater.Afteryoucompletetheentirebook,youcantakethe50-questionfinalexamanddetermineyourlevelofcompetence.Itissuggestedthatyouuseacalculatortohelpyouwiththecomputations.Iwouldliketoanswertheage-oldquestion,“WhydoIhavetolearnthisstuff?”Thereareseveralreasons.First,

mathematicsisusedinmanyacademicfields.Ifyoucannotdomathematics,youseverelylimityourchoicesofanacademicmajor.Second,youmayberequiredtotakeastandardizedtestforajob,degree,orgraduateschool.Mostofthesetestshaveamathematicalsection.Third,aworkingknowledgeofwordproblemswillgoalongwaytohelpyousolvemathematicalproblemsthatyouencounterineverydaylife.Ihopethisbookwillhelpyoulearnmathematics.Forthesecondedition,mostoftheexamplesandexerciseshavebeenchanged.Also,atthebeginningofeachchapter,the

basicobjectiveshavebeenstatedandabriefsummaryappearsattheendofthechapter.Inaddition,the“StillStruggling”explanationboxeshavebeenadded.Thesectiononmixtureproblemshasbeenrewrittentoexplaintheideasmoreclearly.Inthesectiononprobabilityproblems,thesamplespaceforcardshasbeenadded,andthefourbasicrulesforprobabilityhavebeenincluded.Bestwishesonyoursuccess.

AllanG.Bluman

Acknowledgments

Iwouldliketothankmywife,BettyClaire,forhelpingmewiththisproject,andIwishtoexpressmygratitudetomyeditor,JudyBass,andtoCarrieGreenforhersuggestionsanderrorchecking.

chapter1IntroductiontoProblemSolving

Thischapterexplainsthebasicfour-stepproblem-solvingtechniquedevelopedbyGeorgePolya.Inaddition,somebasicproblem-solvingstrategiessuchasdrawingapicture,makingalist,etc.,areexplained.

CHAPTEROBJECTIVES

Inthischapter,youwilllearnhowto

•Usethefour-stepproblem-solvingmethod

•Solvewordproblemsusinggeneralproblem-solvingstrategies

Four-StepMethod

Ineveryareaofmathematics,youwillencounter“word”problems.Somestudentsareverygoodatsolvingwordproblemswhileothersarenot.Whenteachingwordproblemsinpre-algebraandalgebra,Ioftenhear,“Idon’tknowwheretobegin”or“Ihaveneverbeenabletosolvewordproblems.”Agreatdealhasbeenwrittenaboutsolvingwordproblems.AHungarianmathematician,GeorgePolya,didmuchintheareaofproblemsolving.Hisbook,entitledHowtoSolveIt,hasbeentranslatedintoatleast17languages,anditexplainsthebasicstepsofproblemsolving.Thesestepsareexplainednext.Step1:UnderstandtheproblemFirstreadtheproblemcarefullyseveraltimes.Underlineorwritedownanyinformationgivenintheproblem.Next,decidewhatyouarebeingaskedtofind.Thiswillbecalledthegoal.Step2:SelectastrategytosolvetheproblemTherearemanywaystosolvewordproblems.Youmaybeabletouseoneofthebasicoperationssuchasaddition,subtraction,multiplication,ordivision.Youmaybeabletouseanequationorformula.Youmayevenbeabletosolveagivenproblembytrialanderror.Thisstepwillbecalledstrategy.Step3:CarryoutthestrategyPerformtheoperation,solvetheequation,etc.,andgetthesolution.Ifonestrategydoesn’twork,tryadifferentone.Thisstepwillbecalledimplementation.Step4:EvaluatetheanswerThismeanstocheckyouranswerifpossible.Anotherwaytoevaluateyouransweristoseeifitisreasonable.Finally,youcanuseestimationasawaytocheckyouranswer.Thisstepwillbecalledevaluation.

Whenyouthinkaboutthefoursteps,theyapplytomanysituationsthatyoumayencounterinlife.Forexample,supposethatyouplaybasketball.Thegoalistogetthebasketballintothehoop.Thestrategyistoselectawaytomakeabasket.Youcanuseanyoneofseveralmethodssuchasajumpshot,alayup,aone-handedpushshot,oraslamdunk.Thestrategyyouusewilldependonthesituation.Afteryoudecideonthetypeofshottotry,youimplementtheshot.Finally,youevaluatetheaction.Didyoumakethebasket?Goodforyou!Didyoumissit?Whatwentwrong?Canyouimproveonthenextshot?Nowlet’sseehowthisprocedureappliestoamathematicalproblem.

EXAMPLE

Findthenexttwonumbersinthesequence

512815111814__________

SOLUTION

Goal:Youareaskedtofindthenexttwonumbersinthesequence.

Strategy:Hereyoucanuseastrategycalled“findapattern.”Askyourself,“What’sbeingdonetoonenumbertogetthenextnumberinthesequence?”Inthiscase,togetfrom5to12,youcanadd7.Buttogetfrom12to8,youneedtosubtract4.Soperhapsitisnecessarytodotwodifferentthings.

Implementation:Add7to14toget21.Subtract4from21toget17.Hence,thenexttwonumbersshouldbe21and17.

Evaluation:Inordertochecktheanswers,youneedtoseeifthe“add7,subtract4”solutionworksforallthenumbersinthesequence,sostartwith5.

Voilà!Youhavefoundthesolution!Nowlet’stryanotherone.

EXAMPLE

Findthenexttwonumbersinthesequence

13713213143__________

SOLUTION

Goal:Youareaskedtofindthenexttwonumbersinthesequence.

Strategy:Againwewilluse“findapattern.”Askyourself,“Whatisbeingdonetothefirstnumbertogetthesecondone?”Hereweareadding2.Doesadding2tothesecondnumber3giveusthethirdnumber7?No.Youmustadd4tothesecondnumbertogetthethirdnumber7.Howdowegetfromthethirdnumbertothefourthnumber?Add6.Let’sapplythestrategy.

Implementation:

1+2=33+4=77+6=1313+8=2121+10=3131+12=4343+14=5757+16=73

Hence,thenexttwonumbersinthesequenceare57and73.

Evaluation:Sincethepatternworksforthefirstsevennumbersinthesequence,wecanextendittothenexttwonumbers,whichthenmakestheanswerscorrect.

EXAMPLE

Findthenexttwolettersinthesequence

AZCYEXGW__________

SOLUTION

Goal:Youareaskedtofindthenexttwolettersinthesequence.

Strategy:Again,youcanusethe“findapattern”strategy.Noticethatthesequencestartswiththefirstletterofthealphabet,A,andthengoestothelastletter,Z,thenbacktoC,andsoon.Soitlooksliketherearetwosequences.

Implementation:ThefirstsequenceisACEG,andthesecondsequenceisZYXW.Hence,thenexttwolettersareIandV.

Evaluation:Puttingthetwosequencestogether,yougetAZCYEXGWIV.Nowyoucantryafew

problemstoseeifyouunderstandtheproblem-solvingprocedure.Besuretouseallfoursteps.

TRYTHESE

Findthenexttwonumbersorlettersineachsequence.

1.515144241123122__________

2.16362161,2967,776__________

3.8040442226__________

4.149162536__________

5.A6B13C20D27__________

SOLUTION

1.366and365.Multiplythefirstnumberby3togetthesecondnumber;subtract1fromthesecondnumbertogetthethirdnumber.Continue.

2.46,656and279,936.Multiplyeachnumberby6togetthenextnumber.

3.13and17.Dividethefirstnumberby2togetthesecondnumber,thenadd4togetthenextnumber.Repeattheprocess.

4.49and64.Squarethenumbersinthesequence:1,2,3,4,…

5.Eand34.Usethealphabetandadd7toeachnumber.

Well,howdidyoudo?Youhavejusthadanintroductiontosystematicproblemsolving.Theremainderofthisbookisdividedintothreeparts.Chapters2–5explainhowtosolvewordproblemsinarithmeticandpre-algebra.Chapters6–11explainhowtosolvewordproblemsinintroductoryandintermediatealgebra.Chapter12explainshowtosolvewordproblemsingeometry,probability,andstatistics.Aftersuccessfullycompletingthisbook,youwillbewellalongthewaytobecomingacompetentmathematicalwordproblemsolver.

Problem-SolvingStrategies

Therearesomegeneralproblem-solvingstrategiesyoucanusetosolvereal-worldproblemsandhelpyoucheckyouranswerswhenyouusethestrategiespresentedlaterinthisbook.Thesestrategiescanhelpyouwithproblemsfoundonstandardizedtests,inothersubjects,andineverydaylife.

Thesestrategiesare

1.Makeanorganizedlist

2.Guessandtest

3.Drawapicture

4.Findapattern

5.Solveasimplerproblem

6.Workbackwards

MakeanOrganizedListWhenyouusethisstrategy,youmakeanorganizedlistofpossiblesolutionsandthensystematicallyworkouteachoneuntilthecorrectanswerisfound.Sometimesithelpstomakethelistinatableformat.

EXAMPLE

Apersonhassevenbillsconsistingof$5billsand$10bills.Ifthetotalamountofthemoneyis$50,findthenumberof$5billsand$10billshehas.

SOLUTION

Goal:Youarebeingaskedtofindthenumberof$5billsand$10billsthepersonhas.

Strategy:Thisproblemcanbesolvedbymakinganorganizedlistandfindingthetotalamountofmoneyyouhaveasshown:

One$5billandsix$10billsmakesevenbillswithavalueof1×$5+6×$10=$65.Thisisincorrect,sotrytwo$5billsandfive$10billsandkeepgoinguntilasumof$50isreached.

Implementation:Finishthelist.

Hencefour$5billsandthree$10billsareneededtoget$50.

Evaluation:Four$5billsandthree$10billsmakesevenbillswhosetotalvalueis$50.

EXAMPLE

Inabarnyardthereareeightanimals,chickensandcows.Chickenshavetwolegsandcowshavefourlegs,ofcourse.Ifthetotalnumberoflegsis22,howmanychickensandcowsarethere?

SOLUTION

Goal:Youarebeingaskedtofindhowmanychickensandhowmanycowsareinthebarnyard.

Strategy:Youcanmakeanorganizedlist,asshown.

Thenumberofchickensandcowsmustsumto8andthatgivesatotalof30legs:

1×2+7×4=2+28=30

Implementation:Continuethetableuntilthecorrectanswer(22legs)isfound.

Hence,therearefivechickensandthreecowsinthebarnyard.

Evaluation:Fivechickenshave5×2=10legs,andthreecowshave3×4=12legs,10+12=22legs.

GuessandTestThisstrategyissimilartothepreviousoneexceptyoudonotneedtomakealist.Yousimplytakeaneducatedguessatthesolutionandthentryitouttoseeifitiscorrect.Ifnot,tryanotherguess;thentestit.

EXAMPLE

Thesumofthedigitsofatwo-digitnumberis9.Ifthedigitsarereversed,thenewnumberisninemorethantheoriginalnumber.

SOLUTION

Goal:Youarebeingaskedtofindatwo-digitnumber.

Strategy:Youcanusetheguessandteststrategy.Firstguesssometwo-digitnumberssuchthatthesumofthedigitsis9.Forexample,18,27,36,45,etc.,meetthispartofthesolution.Thenseeiftheymeettheotherconditionoftheproblem.

Implementation:

Guess:27;reversethedigits:72;subtract:72−27=45

Guess:36;reversethedigits:63;subtract:63−36=27

Guess:45;reversethedigits:54;subtract:54−45=9.Thisisthecorrectsolution;hence,thenumberis45.

Evaluation:Thesumofthedigits4+5=9,andthedifference54−45=9.

EXAMPLE

ThelettersXandWeachrepresentadigitfrom0through9.Findthevalueofeachlettersothatthefollowingistrue:

SOLUTION

Goal:YouarebeingaskedtofindwhatdigitsXandWrepresent.

Strategy:Useguessandtest.

Implementation:GuessafewdigitsforXandseewhatworks:

HenceX=5andW=1isthecorrectanswer.

Evaluation:Noticethatallthedigitsinthecolumnarethesame;thatis,theyareallthesamenumber.Youmustaddthreesingle-digitnumbersandgetthesamenumberastheone’sdigitofthesolution.Thereareonlytwopossibilities:0and5.Sincetheanswerhastwodigits,0isdisregarded.

DrawaPictureManytimesaproblemcanbesolvedusingapicture,figure,ordiagram.Also,drawingapicturecanhelpyoutodeterminewhichotherstrategycanbeusedtosolveaproblem.

EXAMPLE

Tentreesareplantedinarowatthree-footintervals.Howfarisitfromthefirsttreetothelasttree?

SOLUTION

Goal:Youarebeingaskedtofindthedistancefromthefirsttreetothelasttree.

Strategy:Drawafigureandcounttheintervalsbetweenthem;thenmultiplytheanswerby3.

Implementation:Solvetheproblem.SeeFigure1-1.

FIGURE1-1

Sincetherearenineintervals,thedistancebetweenthefirstandlastoneis9×3=27feet.

Evaluation:Thefigureshowsthat27feetisthecorrectanswer.

EXAMPLE

Afamilyhasthreechildren.Listthenumberofwaysaccordingtogenderthatthebirthscanoccur.

SOLUTION

Goal:Youarebeingaskedtolistthetotalnumberofwaysthreechildrencanbeborn.

Strategy:Drawadiagramshowingthewaythechildrencanbeborn.

FIGURE1-2

Implementation:Eachchildcouldbebornasamaleorafemale.SeeFigure1-2.Hencethereareeightdifferentpossibilities:

Evaluation:Sincetherearetwowaysforeachchildtobeborn,thereare2×2×2=8differentwaysthatthebirthscanoccur.

FindaPatternManyproblemscanbesolvedbyrecognizingthatthereisapatterntothesolution.Oncethepatternisrecognized,thesolutioncanbeobtainedbygeneralizingfromthepattern.

EXAMPLE

Awealthypersondecidedtopayanemployee$1forthefirstday’swork,$2forthesecondday’swork,and$4forthethirdday’swork,etc.Howmuchdidtheemployeeearnfor15daysofwork?

SOLUTION

Goal:Youarebeingaskedtofindtheamounttheemployeeearnedforatotalof15daysofwork.

Strategy:Youcanmakeatablestartingwiththefirstdayandcontinuinguntilyouseeapattern.

Implementation:

Noticethattheamountearnedeachdayisgivenby2n−1wherenisthenumberoftheday.Forexample,onthe6thday,thepersonearns26−1=25=$32.Soonthe15thday,apersonearns215−1or214=$16,384.Thetotalamountthepersonearnsisgivenbydoublingtheamountearnedthatdayandsubtractingone.Sothetotalamountearnedattheendofthe15daysis$16,384×2−1=$32,767.

Evaluation:Youcouldcheckyouranswerbycontinuingthepatternfor15days.

EXAMPLE

Findtheanswerto12345678×9+9usingapattern.

1×9+2=1112×9+3=111

123×9+4=1111

SOLUTION

Goal:Youarebeingaskedtofindtheanswerto12345678×9+9usingapattern.

Strategy:Makeatablestartingwith1×9+2,12×9+3,123×9+4,etc.Findtheanswerstotheseproblemsandseeifyoucanfindapattern.

Implementation:

1×9+2=1112×9+3=111

123×9+4=1111

Thepatternshowsthatyougetananswerthathasthesamenumberof1sasthelastdigitthatisadded.Sotheanswertotheproblemwouldbeanumberwhichhas91s,thatis,111,111,111.

Evaluation:Performtheoperationsonacalculatorandseeiftheansweriscorrect.

SolveaSimplerProblemTousethisstrategy,youshouldsimplifytheproblemormakeupashorter,similarproblemandfigureouthowtosolveit.Thenusethesamestrategytosolvethegivenproblem.

EXAMPLE

Ifthereare10peopleatatenniscourtandeachpersonplaysasinglestennismatchwithanotherperson,howmanydifferentmatchescanoccur?

SOLUTION

Goal:Youarebeingaskedtofindthetotalnumberofdifferentmatchesplayedifeverybodyplayseverybodyelseonetime.

Strategy:Simplifytheproblemusing4people,andthentrytosolveitwith10people.

Implementation:Assumethe4peopleareA,B,C,andD.Thenwritethedifferentgamesthatwouldoccur.

AB,AC,AD,BC,BD,CD

Hence,with4people,therewouldbe6differentgames.

Nowcallthe10peopleA,B,C,D,E,F,G,H,I,andJ.

Therewouldbe45differentgames.

Evaluation:Youcansolvetheproblemusingadifferentstrategyandseeifyougetthesameanswer.

WorkBackwardsSomeproblemscanbesolvedbystartingattheendandworkingbackwardstothebeginning.

EXAMPLE

Tinawentshoppingandspent$3forparkingandone-halfoftheremainderofhermoneyinadepartmentstore.Thenshespent$5forlunch.Arrivingbackhome,shefoundthatshehad$2left.Howmuchmoneydidshestartwith?

SOLUTION

Goal:YouarebeingaskedtofindhowmuchmoneyTinastartedwith.

Strategy:Workbackwards.

Implementation:Workforwardfirstandthenworkbackwards.

1.Spent$3onparking.Subtract$3.

2.Spent oftheremainderinthedepartmentstore.Divideby2.

3.Spent$5onlunch.Subtract$5.

4.Has$2left.

Reversingtheprocess:

Hence,shestartedoutwith$17.

Evaluation:Worktheproblemforwardstartingwith$17andseeifyouendupwith$2.

Manytimesthereisnosinglebeststrategytosolveaproblem.Youshouldrememberthatproblemscanbesolvedusingdifferentmethodsoracombinationofmethods.

TRYTHESE

Useoneormoreofthestrategiesshowninthelessontosolveeachproblem.

1.Howmanycutsareneededtocutalogintoeightpieces?

2.Eachletterstandsforadigit.Allidenticallettersrepresentthesamedigit.Findthesolution.

3.Thesumofthedigitsofatwo-digitnumberis8.If36issubtractedfromthenumber,theanswerwillbetheoriginalnumberwiththedigitsreversed.

4.Apersonpurchasedsevencandybarsthatcosttwodifferentprices,$0.89and$0.99.Howmanyofeachkinddidthepersonpurchaseifthetotalcostis$6.43?

5.An20-inchpieceofpipeiscutintotwopiecessuchthatonepieceisthreetimesaslongastheother.Findthelengthofeachpiece.

6.Howmanywayscanacommitteeoffourpeoplebeselectedfromsixpeople?

7.Frankwantstoshapeupforbasketball.Hedecidestocutbackbyeatingtwofewercookieseachdayforfivedays.Duringthefivedays,heateatotalof40cookies.Howmanydidheeatonthefirstday?

8.Amotherisfourtimesasoldasherdaughter.In16years,shewillbetwiceasoldasherdaughter.Findtheirpresentages.

9.Howmanywayscanfourdifferentbooksbelinedupinarowonashelf?

10.FindthetallestpersonifBettyistallerthanJan,SueisshorterthanBetty,andJanistallerthanSue.

SOLUTIONS

1.Strategy:Drawapicture:Sevencutsareneeded.SeeFigure1-3.

2.Strategy:Guessandtest:89+9=98

3.Strategy:Guessandtest:62−36=26

4.Strategy:Makeanorganizedlist:5candybarsat$0.89and2at$0.99.

5.Strategy:Guessandtest:5inchesand15inches

6.Strategy:Makeanorganizedlist:15ways

FIGURE1-3

7.Strategy:Guessandtest:12cookies

8.Strategy:Makeanorganizedlistorguessandtest:Mother’sageis32;daughter’sageis8.

9.Strategy:Solveasimplerproblem:24ways

10.Strategy:Drawapicture:Betty

Summary

Inthischapter,youhavelearnedthebasicwordproblem-solvingprocedurethatwasdevelopedbyGeorgePolya.Thisfour-stepprocedurewillbeusedthroughoutthisbook.

Also,youlearnedsomeotherwaystosolvewordproblems.Thesewaysincludemakinganorganizedlist,guessandtest,drawingapicture,findingapattern,solvingasimplerproblemusingthesamestrategyonamoredifficultproblem,andworkingbackwards.Thereareotherstrategiesthatcanalsobeused.Theycanbefoundinbooksonproblemsolving.

QUIZ

1.Thenextnumberinthesequence386119141217is

A.22

B.15

C.21

D.14


A.17

B.18

C.20

D.21


A.8

B.12

C.15

D.20

4.Thelargestnumberthatwilldivideevenlyinto180and600is

A.12

B.20

C.30

D.60

5.Marywentshoppingandboughtsomepencils($1each),notebooks($2each),andpens($3each).Ifshespentatotalof$12,howmanyofeachitemdidshepurchase?

A.3pencils,2notebooks,and2pens

B.2pencils,3notebooks,and1pen

C.4pencils,1notebook,and2pens

D.1pencil,3notebooks,and2pens

6.Fourhorsesranarace.Thebrownhorsefinishedaheadofthegrayhorsebutbehindtheblackhorse.Thewhitehorsefinishedbehindthebrownhorsebutaheadofthegrayhorse.Thewhitehorsefinishedexactlyonehorseaheadofthegrayhorse.Whatwasthefinishingorderofthehorses?

A.black,brown,white,gray

B.gray,white,brown,black

C.brown,white,gray,black

D.black,white,gray,brown

7.Fourstudentsaretossingabaseballtoeachother.Iftheballistossedbetweeneachoftheotherplayersonetime,howmanytossesweremade?

A.5

B.6

C.12

D.30

8.Foraparty,apersonsetsupsixcardtablesandpushesthemtogetherinarow.Howmanypeoplecanbeseatedatthearrangement?(Note:Onlyonepersoncansitoneachsideofacardtable.)

A.6

B.12

C.14

D.24

9.Allwholenumbershavefactors.Thefactorsof10are1,2,5,and10.Thesearenumbersthatdivideevenlyinto10.Thenumbers1,2,and5arecalledproperfactorsof10.Thenumber6iscalledaperfectnumbersinceitsproperfactorsaddupto6.(1+2+3=6).Whatisthenextperfectnumber?

A.8

B.12

C.24

D.28

10.Arubberballbouncesuphalfthepreviousheightitfell.Ifarubberballisdroppedfromaheightof20feet,howfardidittravelbythetimeithitsthegroundthreetimes?

A.35feet

B.30feet

C.50feet

D.40feet

chapter2SolvingDecimalandFractionProblems

Thischapterexplainshowtodeterminewhichoperation(addition,subtraction,multiplication,ordivision)youcanusetosolveproblemsinarithmeticorpre-algebra.Also,operationswithdecimalsandfractionsarereviewedintworefreshers.Finally,wordproblemsusingdecimalsandfractionsareexplained.

CHAPTEROBJECTIVES


•Solvewordproblemsusingwholenumbers

•Usetherulesforadding,subtracting,multiplying,anddividingdecimals

•Solvewordproblemsusingdecimals

•Add,subtract,multiply,anddividefractionsandmixednumbers,changefractionstodecimals,andchangedecimalstofractions

•solvewordproblemsusingfractions

Operations

Mostwordproblemsinarithmeticandpre-algebracanbesolvedbyusingoneormoreofthebasicoperations.Thebasicoperationsareaddition,subtraction,multiplication,anddivision.Sometimesstudentshaveaproblemdecidingwhichoperationtouse.Thecorrectoperationcanbedeterminedbythewordsintheproblem.

Useadditionwhenyouarebeingaskedtofind

thetotal,

thesum,

howmanyinall,

howmanyaltogether,

etc.,

andwhenalltheitemsintheproblemarethesametypeorhavethesameunits.

EXAMPLE

Fortheyears2000–2009,thenumberofspacelaunchesforeachcountryisUnitedStates,201;Russia,237;China,49;Japan,17;andothercountries,79.Findthetotalnumberofspacelaunchesforthe10-yearperiod.

SOLUTION

Goal:Youarebeingaskedtofindthetotalnumberofspacelaunchesthatwereconductedfrom2000to2009.

Strategy:Useadditionsinceyouneedtofindatotalandalltheitemsintheproblemarethesame(i.e.,spacelaunches).

Implementation:201+237+49+17+79=583.

Evaluation:Thetotalnumberofspacelaunchesis583.Thiscanbecheckedbyestimation.Roundeachvalueandthenfindthesum:200+240+50+20+80=590.Sincetheestimatedsumisclosetotheactualsum,youcanconcludethattheanswerisprobablycorrect.(Note:Whenusingestimation,youcannotbe100percentsureyouransweriscorrectsinceyouhaveusedroundednumbers.)

Usesubtractionwhenyouareaskedtofind

howmuchmore,howmuchless,howmuchlarger,howmuchsmaller,howmanymore,howmanyfewer,thedifference,thebalance,howmuchisleft,howfarabove,howfarbelow,howmuchfurther,etc.,andwhenalltheitemsintheproblemarethesameorhavethesameunits.

EXAMPLE

IfthehighesttemperaturerecordedinAfricawas136°F,andthehighesttemperaturerecordedinSouthAmericawas120°F,howmuchhigherwasthehighesttemperatureinAfricacomparedtoSouthAmerica?

SOLUTION

Goal:YouarebeingaskedtofindhowmuchhigheristhehighesttemperatureinAfricacomparedtothehighesttemperatureinSouthAmerica.

Strategy:Sinceyouarebeingasked“howmuchhigher”andbothitemsarethesame(degrees),youusesubtraction.

Implementation:136°F−120°F=16°.HencethehighesttemperatureinAfricawas16°higherthanthehighesttemperaturerecordedinSouthAmerica.

Evaluation:Youcancheckthesolutionbyadding:120°+16°=136°.

Usemultiplicationwhenyouarebeingaskedtofindtheproduct,thetotal,howmanyinall,howmanyaltogether,etc.,andwhenyouhavegroupsofindividualitems.

EXAMPLE

Findthetotalcostof15digitalcamerasifeachonecosts$159.

SOLUTION

Goal:Youarebeingaskedtofindthetotalcostof15digitalcameras.

Strategy:Usemultiplicationsinceyouareaskedtofindatotalandyouhave15camerascosting$159each.

Implementation:$159×15=$2,385.Hence,thetotalcostof15digitalcamerasis$2,385.

Evaluation:Youcancheckyouranswerbyestimation:160×15=$2,400.Since$2,400iscloseto$2,385,youranswerisprobablycorrect.

Usedivisionwhenyouaregiventhetotalnumberofitemsandanumberofgroupsandneedtofindhowmanyitemsin

eachgroup,orwhenyouaregiventhetotalnumberofitemsandthenumberofitemsineachgroupandneedtofindhowmanygroupsthereare.

EXAMPLE

Theshippingdepartmentofabusinessneedstoship192pairsofchildren’sshoes.Iftheyarepacked12pairsperbox,howmanyboxeswillbeneeded?

SOLUTION

Goal:Youarebeingaskedtofindhowmanyboxesareneeded.

Strategy:Hereyouaregiventhetotalnumberofpairsofshoes,192,andthecompanyneedstopack12pairsineachbox.Youareaskedtofindhowmanyboxes(groups)areneeded.Inthiscase,usedivision.

Implementation:192÷12=16boxes.Hence,youwillneed16boxes.

Evaluation:Check:16boxes×12pairsofshoesperbox=192pairsofshoes.

Nowyoucanseehowtodecidewhatoperationtousetosolvearithmeticorpre-aglebraproblems.

TRYTHESE

1.Ifsevenmountainbicyclescost$1,288,howmuchdoeseachonecost?

2.Ifyoucanburn12caloriesbyrunningatabriskpacefor1minute,howmanycaloriescanyouburnifyourunfor20minutes?

3.Asalespersontravelsthefollowingmilesduringafour-daytrip:

Findthetotalnumberofmilesthesalespersontraveledonthetrip.

4.Foraspecificyear,Facebookhad92,208,000visitors.TheMySpacewebsitehad27,966,000fewervisitsduringthatyear.HowmanyvisitorsdidMySpacehave?

5.Iftheaverageyearlyphonebillforaspecificyearis$588,whatisthemonthlyrateforthephoneservice?

6.Abookcompanyshipsitsbooksinboxesthathold24books.Howmanyboxesareneededtoship336books?

7.Billpurchaseseightvideogamesfor$18each.Findthetotalamounthespentforthegames.

8.Ifyouhad$357inyourcheckingaccount,andyouwrotechecksfor$81and$116,whatwouldyourbalancebe?

9.Adamdecidestosave$130eachmonthforayear.Howmuchmoneywillhehaveatyear’send?

10.Abusinesspersonmailedfivepackagescosting$8,$14,$18,$3,and$6.Findthetotalcostofthepostagebill.

SOLUTIONS

1.$1,288÷7=$184

2.12×20=240calories

3.852+347+521+276=1,996miles

4.92,208,000−27,966,000=64,242,000

5.$588÷12=$49

6.336÷24=14boxes

7.$18×8=$144

8.$357−$81−$116=$160

9.$130×12=$1,560

10.$8+$14+$18+$3+$6=$49

StillStrugglingIfyougetthewronganswer,therearetwoplacesyoucouldhavemadeamistake.First,youcouldhaveperformedthewrongoperation.thatis,maybeyoudividedwhenyoushouldhavemultiplied.second,youcouldhavemadeamistakeinperformingtheoperationorperhapspressingthewrongkeyifyouareusingacalculator.itisbesttodotheproblemoverratherthantryingtofindyourmistake.thismethodworksbestiftheproblemrequiresseveralstepsastheonesfoundinlaterchaptersinthebook.

RefresherI:Decimals

Toaddorsubtractdecimals,placethenumbersinaverticalcolumnandlineupthedecimalpoints.Addorsubtractasusualandplacethedecimalpointintheanswerdirectlybelowthedecimalpointsintheproblem.

EXAMPLE

Findthesum:98.145+6.8372+421.6

SOLUTION

EXAMPLE

Subtract351.2−45.18

SOLUTION

Tomultiplytwodecimals,multiplythenumbersasisusuallydone.Countthenumberofdigitstotherightofthedecimalpointsintheproblemandthenhavethesamenumberofdigitstotherightofthedecimalpointintheanswer.

EXAMPLE

Multiply53.61×4.8

SOLUTION

Todividetwodecimalswhenthereisnodecimalpointinthedivisor(thenumberoutsidethedivisionbox),placethedecimalpointintheanswerdirectlyabovethedecimalpointinthedividend(thenumberunderthedivisionbox).Divideas

usual.

EXAMPLE

Divide2511.2÷43

SOLUTION

Todividetwodecimalswhenthereisadecimalpointinthedivisor,movethedecimalpointtotheendofthenumberinthedivisor,andthenmovethedecimalpointthesamenumberofplacesinthedividend.Placethedecimalpointintheanswerdirectlyabovethedecimalpointinthedividend.Divideasusual.

EXAMPLE

Divide33.672÷7.32

SOLUTION

TRYTHESE

Performtheindicatedoperations

1.63.76+195.2+3.189

2.195.3−87.215

3.37.3×5.6

4.369.57÷97

5.327.6÷52

SOLUTIONS

StillStrugglingSometimesyouhavetoaddzerostodecimalnumbers.Zeroscanbeaddedafterthelastdigitontherightsideofthedecimalpoint.Forexample,0.63=0.630=0.6300=0.63000.

Thisrefresherreviewedhowtoadd,subtract,multiply,anddividedecimalnumbers.Whenperformingtheseoperations,itisnecessarytoputthedecimalpointinthecorrectplaceintheanswer.

SolvingWordProblemsUsingDecimals

NOTEIfyouneedtoreviewdecimals,completeRefresherI.Thissectionexplainshowtosolvewordproblemsusingdecimals.Manyreal-lifeproblemsinvolvedecimalnumbers.Forexample,problemsinvolvingmoneyusedecimals.

Inordertosolvewordproblemsinvolvingdecimals,usethesamestrategiesthatyouusedinthesectiononoperations.

EXAMPLE

Ifastockbrokerpurchases26sharesofstockatacostof$8.72pershare,whatisthetotalcostofthepurchase?

SOLUTION

Goal:Youarebeingaskedtofindthetotalcostofastockpurchase.

Strategy:Sinceyouneedtofindatotalandyouaregiventwodifferentitems(dollarsandshares),youmultiply.

Implementation:$8.72×26=$226.72

Evaluation:Youcancheckyouranswerusingestimation:$9×25=$225.Since$225iscloseto$226.72,theanswerseemsreasonable.

EXAMPLE

In1770,thepopulationofMainewas31.3thousandpeople,andthepopulationofNewHampshirewas62.4thousandpeople.HowmanymorepeoplelivedinNewHampshirethatyear?

SOLUTION

Goal:Youarebeingaskedtofindthedifferenceinthenumberofpeoplewholiveintwocolonies.

Strategy:Inordertofindthedifference,youneedtosubtractthetwopopulationvalues.

Implementation:62.4−31.3=31.1thousandpeople

Evaluation:Estimatetheanswerbyrounding62.4to60and31.3to30;thensubtract60−30=30.Since30iscloseto31.1,theanswerisprobablycorrect.

Sometimesawordproblemrequirestwoormoresteps.Inthissituation,youstillfollowthesuggestionsgivenatthebeginningofthischaptertodeterminetheoperations.

EXAMPLE

Findthetotalcostoffivepictureframesat$3.59eachandtwocandlesat$1.39each.

SOLUTION

Goal:Youarebeingaskedtofindthetotalcostoftwodifferentitems—fiveofoneitemandtwoofanotheritem.

Strategy:Usemultiplicationtofindthetotalcostofthepictureframesandthecandles,andthenaddtheanswers.

Implementation:Thecostofthepictureframesis5×$3.59=$17.95.Thecostofthecandlesis2×$1.39=$2.78.Addthetwoanswers:$17.95+$2.78=$20.73.Hence,thetotalcostoffivepictureframesandtwocandlesis$20.73.

Evaluation:Estimatetheanswer:Pictureframes:5×$3.50=$17.50;candles:2×$1.40=$2.80;totalcost:$17.50+$2.80=$20.30.Theestimatedcostof$20.30isclosetothecomputedactualcostof$20.73;therefore,theanswerisprobablycorrect.

TRYTHESE

1.TheDowJonesstockaveragesopenedat1,125.29pointsanddropped16.48points.Whatwastheclosingstockaverage?

2.Findthecostofeighthedgetrimmersifeachonecosts$35.75.

3.Findthetotalcostofanautomobiletripifthepersonpaid$156.73forgasoline,$362.58forlodging,$251.63formeals,and$154.26formiscellaneousexpenses.

4.Ifadriverdrives261.45milesin6.3hours,whatistheaveragespeedoftheautomobile?

5.Ifonekilogramweighsapproximately2.2046pounds,whatwouldbetheapproximatedweightinpoundsofanitemthatweighs12.7kilograms?

6.Findthecostoffouroutdoorchairsandtwosmalltablesifthechairscost$17.49eachandthetablescost$19.39each.

7.Harrietearns$10.75perhourandgets$16.73foreachhoursheworksover40hoursperweek.Ifsheworks46hoursoneweek,howmuchwouldsheearn?

8.Theweightofwateris62.5poundspercubicfoot.Findthetotalweightofatankfullofwaterifitholds20cubicfeetofwaterandthetankweighs36.8pounds.

9.Anairportlimousineservicecharges$15.50plus$5.65permiletotravelfromaperson’shometotheairport.Findthetotalcostofa12-miletrip.

10.Acellphonecompanychargesarateof$0.60forthefirsttwominutesand$0.15foreachminuteafterthat.Findthecostofa16-minutecall.

SOLUTIONS

1.1,125.29−16.48=1,108.81

2.$35.75×8=$286.00

3.$156.73+$362.58+$251.63+$154.26=$925.20

4.261.45÷6.3=41.5milesperhour

5.2.2046×12.7=27.99842pounds

6.$17.49×4=$69.96,$19.39×2=$38.78,$69.96+$38.78=$108.74

7.$10.75×40=$430.00,$16.73×6=$100.38,

$430.00+$100.38=$530.38

8.62.5×20=1,250pounds,1,250+36.8=1,286.8pounds

9.$5.65×12=$67.80,$67.80+$15.50=$83.30

10.$0.60×2=$1.20,$0.15×14=$2.10,$1.20+$2.10=$3.30

Thissectionexplainedhowtosolveproblemsusingdecimals.Manyreal-lifeproblemsinvolvemoney,soitisimportantforyoutoknowhowtofindthecorrectanswerswhendecimalnumbersareused.

RefresherII:Fractions

Inafraction,thetopnumberiscalledthenumeratorandthebottomnumberiscalledthedenominator.Toreduceafractiontolowestterms,dividethenumeratoranddenominatorbythelargestnumberthatdividesevenlyinto

bothnumbers.

EXAMPLE

SOLUTION

Tochangeafractiontohigherterms,dividethesmallerdenominatorintothelargerdenominator,andthenmultiplythesmallernumeratorbythatnumbertogetthenewnumerator.Thisprocedurewillbeusedinadditionandsubtractionoffractions.

EXAMPLE

SOLUTION

Divide32÷8=4andmultiply5×4=20.Hence, .

Animproperfractionisafractionwhosenumeratorisgreaterthanorequaltoitsdenominator.Forexample,20/3,6/5,

and3/3areimproperfractions.Amixednumberisawholenumberandafraction; , ,and aremixednumbers.

Tochangeanimproperfractiontoamixednumber,dividethenumeratorbythedenominatorandwritetheremainderasthenumeratorofafractionwhosedenominatoristhedivisor.Reducethefractionifpossible.

EXAMPLE

Change toamixednumber

SOLUTION

Tochangeamixednumbertoanimproperfraction,multiplythedenominatorofthefractionbythewholenumberandaddthenumerator.Thiswillbethenumeratoroftheimproperfraction.Usethesamenumberforthedenominatoroftheimproperfractionasthenumberinthedenominatorofthefractioninthemixednumber.

EXAMPLE

Change toanimproperfraction

SOLUTION

Inordertoaddorsubtractfractions,youneedtofindalowestcommondenominatorofthefractions.Thelowestcommondenominator(LCD)ofthefractionsisthesmallestnumberthatcanbedividedevenlybyallthedenominators.Forexample,theLCDof1/6,2/3,and7/9is18,since18canbedividedevenlyby3,6,and9.ThereareseveralmathematicalmethodsforfindingtheLCD;however,wewillusetheguessmethod.Thatis,justlookatthedenominatorsandfigureouttheLCD.Ifneeded,youcanlookatanarithmeticorpre-algebrabookforamathematicalmethodtofindtheLCD.

Toaddorsubtractfractions

1.FindtheLCD.

2.ChangethefractionstohighertermswiththeLCD.

3.Addorsubtractthenumerators.UsetheLCD.

4.Reduceorsimplifytheanswerifpossible.

EXAMPLE

SOLUTION

Use40astheLCD.

EXAMPLE

SOLUTION

Use36astheLCD.

Tomultiplytwoormorefractions,cancelifpossible,multiplynumerators,andthenmultiplydenominators.Cancelmeanstodivideoutthecommonfactors.

EXAMPLE

SOLUTION

Todividetwofractions,invertthefraction(turnthefractionupsidedown)afterthe÷signandmultiply.

EXAMPLE

SOLUTION

Toaddmixednumbers,addthefractions,addthewholenumbers,andsimplifytheanswerifnecessary.

EXAMPLE

SOLUTION

Tosubtractmixednumbers,subtractthefractions,borrowingifnecessary,andthensubtractthewholenumbers.

EXAMPLE

SOLUTION

(Noborrowingisnecessaryhere.)

Whenborrowingisnecessary,take1awayfromthewholenumberandaddittothefraction.Forexample

Anotherexample:

EXAMPLE

SOLUTION

Tomultiplyordividemixednumbers,changethemixednumberstoimproperfractions,andthenmultiplyordivideasshownpreviously.

EXAMPLE

SOLUTION

EXAMPLE

SOLUTION

Tochangeafractiontoadecimal,dividethenumeratorbythedenominator.

EXAMPLE

SOLUTION

Tochangeadecimaltoafraction,dropthedecimalpointandplacethenumberover10ifithasonedecimalplace,100ifithastwodecimalplaces,1,000ifithasthreedecimalplaces,etc.Reduceifpossible.

EXAMPLE

Change0.88toafraction.

SOLUTION

TRYTHESE

SOLUTIONS

StillStrugglingIfyouarehavingdifficultieswithfractions,youmayneedtofindanarithmeticorpre-alagebrabookandstudythesectiononfractions.

Thisrefresherreviewedthebasicoperationsofaddition,subtraction,multiplication,anddivisionoffractions.Also,itisimportanttoknowhowtochangefractionstodecimalsanddecimalstofractions.

SolvingWordProblemsUsingFractions

NOTEIfyouneedtoreviewfractions,completeRefresherII.Inordertosolvewordproblemsinvolvingfractions,usethesamestrategiesthatyouusedintheprevioussections.

EXAMPLE

Aplumberisinstallingwaterpipeinanewhouse.Heneedsfourpiecesmeasuring inches,

inches, inches,and inches.Howlongapipedoesheneedtocutallthepiecesfromit?

SOLUTION

Goal:Youareaskedtofindthelengthofapieceofpipenecessarytocutallthepiecesfromit.

Strategy:Sinceyouneedtofindatotalandallitemsareinthesameunits(inches),useaddition.

Implementation:

inches.

Evaluation:Youcanestimatetheanswersince in.isabout in., in.canbeusedasis;

in.isabout10in.;and isabout in.Hence, .Since in.iscloseto31in.,youranswerisprobablycorrect.

EXAMPLE

Abustravels milesin hours.Whatistheaveragespeedofthebus?

SOLUTION

Goal:Youareaskedtofindtheaveragespeedofthebus.

Strategy:Sinceyouaregivenatotaldistanceandthetimeittook,youdividethetotaldistancebythetimetogettheaveragespeed.

Implementation:

Hence,theaveragespeedis milesperhour.

Evaluation:Youcancheckbymultiplying .

EXAMPLE

IftheTigersare gamesbehindtheCougarsinthebaseballstandingsandtheWildcatsare5gamesbehindtheCougars,howmanygamesaretheWildcatsbehindtheTigers?

SOLUTION

Goal:YouareaskedtofindhowmanygamestheWildcatsarebehindtheTigers.

Strategy:SinceyouneedtofindhowmanygamesbehindtheWildcatsare,youusesubtraction.

Implementation: games.HencetheWildcatsare gamesbehindtheTigersinthestandings.

Evaluation:Youcancheckthesolutionbyadding

TRYTHESE

1.Atradesmancanassembleacablepulleysystemin hourswhilehisassistantcandothesamejobinhour.Howmuchfastercanthetradesmandothejob?

2.Onecubicfootofoilisabout gallons.Howmanycubicfeetofoilwoulda20-galloncontainerhold?

3.Joanneworked hoursonMonday, hoursonTuesday, hoursonWednesday,and3hoursonThursday.Findthetotalnumberofhourssheworkedthatweek.

4.AtraintravelsfromPittsburghtoChicagoin hourswhileanothertrainmadethesametripinhours.Howmuchfasterwasthesecondtrain?

5.Howmanyidentificationcardsthatare incheslongcanbecutfromapieceofcardstock incheslong?

6.Thescaleonamapstatesthat inchisequalto20miles.Findthedistanceinmilesbetweentwotownsifit

measures inchesonthemap.

7.Jeannecutthreepiecesofribbonthatmeasured inches, inches,and incheslong.Ifthetotal

lengthoftheribbonis incheslong,howmuchoftheribbonwasleft?

8.Eugenepurchasedalaptopcomputerfor$800.Hemadeadownpaymentof ofthepriceandpaidthebalanceineightmonthlyinstallments.Howmuchdidhepayeachmonth?

9.Ifa$264,000homeisassessedat ofitsvalue,findtheassessedvalueofthehouse.

10.Findthedistancearoundatriangularpieceofpropertyifthesidesmeasure feet, feet,and

feet.

SOLUTIONS

Inthissection,youlearnedhowtosolvewordproblemsusingfractions.

Summary

Chapter2explainedtheimportantwordsandconceptsthatwillenableyoutodeterminewhichoperations(addition,subtraction,multiplication,ordivision)tousewhensolvingwordproblemsusingwholenumbers,decimals,orfractions.

QUIZ

1.TheislandofPuertoRicocontains3,339squaremiles,whiletheislandofJamaicacontains4,244squaremiles.HowmuchlargeristheislandofJamaica?

A.7,583squaremiles

B.905squaremiles

C.6,354squaremiles

D.1,003squaremiles

2.Findthecostof8feetofribbonifitsellsfor$1.59perfoot.

A.$12.72

B.$14.52

C.$9.32

D.$16.82

3.ThelengthofLakeSuperioris350miles.ThelengthofLakeHuronis206miles,andthelengthofLakeErieis241miles.Findthetotallengthofallthreelakes.

A.834miles

B.973miles

C.797miles

D.743miles

4.Ifapersonearns$66,000ayear,whatistheperson’smonthlysalary?

A.$4,000

B.$4,200

C.$4,500

D.$5,500

5.Acarpentermadesixshelvesthatwere feetlongandthreeshelvesthatwere feetlong.Howmuchlumberdidheuse?

6.Aprofessorsaid ofhisstudentsarejuniors.Ifthereare72studentsinhisclasses,howmanyofthemarejuniors?

A.24

B.56

C.48

D.60

7.TochangeaFahrenheittemperaturetoaCelsiustemperature,subtract32°,andthentake oftheanswer.WhatistheCelsiustemperatureforaFahrenheitreadingof86°?

B.30°

D.44°

8.Apersonpurchasedadigitalcamerafor$25downandeightmonthlypaymentsof$16.65.Findthetotalcostofthecamera.

A.$41.55

B.$49.55

C.$158.20

D.$216.55

9.If4servingsofarecipecallfor cupsofflour,howmuchflourwillbeneededtomake12servings?

A.21cups

B.7cups

10.Apersonmadethefollowingpurchases:$18.77,$42.56,$51.75,and$14.36.Findthetotalamountspent.

A.$132.59

B.$127.44

C.$155.62

D.$142.73

chapter3SolvingPercentProblems

Thischapterreviewstheconceptofpercentandthethreetypesofpercentproblems.Finally,wordproblemsusingpercentsareexplained.

CHAPTEROBJECTIVES

Inthischapter,youwill

•Reviewhowtochangepercentstodecimals,changedecimalstopercents,changepercentstofractions,changefractionstopercents,andsolvethethreetypesofpercentproblems

•Learnhowtosolvewordproblemsusingpercents

RefresherIII:Percents

Percentmeanshundredthsorpartofahundred.Forexample,42%means0.42or42/100.Youcanthinkof42%asasquarebeingdividedinto100equalpartsand42%is42equalpartsoutof100equalparts.

Tochangeapercenttoadecimal,dropthe%signandmovethedecimalpointtwoplacestotheleft.Thedecimalpointin42%isbetweenthe2andthe%sign.Itisnotwritten.

EXAMPLE

Writeeachpercentasadecimal

a.63%

b.7%

c.346%

d.28.2%

SOLUTION

a.63%=0.63

b.7%=0.07

c.346%=3.46

d.28.2%=0.282

Tochangeadecimaltoapercent,movethedecimaltwoplacestotherightandaffixthepercentsign.

EXAMPLE

Changeeachdecimaltoapercent

a.0.64

b.0.02

c.6.71

d.0.159

SOLUTION

a.0.64=64%

b.0.02=2%

c.6.71=671%

d.0.159=15.9%

Tochangeapercenttoafraction,dropthepercentsignandplacethenumberinthenumeratorofafractionwhose

denominatoris100.Reduceorsimplifyifnecessary.

EXAMPLE

Changeeachpercenttoafraction

a.80%

b.55%

c.175%

d.5%

SOLUTION

Tochangeafractiontoapercent,changethefractiontoadecimalandthenchangethedecimaltoapercent.

EXAMPLE

Changeeachfractionormixednumbertoapercent

SOLUTION

Apercentwordproblemhasthreenumbers—thewhole,total,orbase(B);thepart(P);andtherateorpercent(R).Supposethatinaclassof25students,8areabsent.Nowthewholeortotalis25andthepartis8.Therateorpercentofstudentswhowereabsentis8/25=0.32=32%.Inapercentproblem,youwillbegiventwoofthethreenumbersandwillbeaskedtofindthethirdnumber.Percent

problemscanbesolvedbyusingapercentcircle.ThecircleisshowninFigure3-1.Inthetopportionofthecircle,writethewordpart(P).Inthelowerrightportionofthecircle,writethewordrate(R),and

inthelowerleftportion,writethewordbase(B).Putamultiplicationsignbetweenthetwolowerportionsandadivisionsignbetweenthetopandbottomportions.

FIGURE3-1

Ifyouareaskedtofindthepart(P),placetherate(R)inthelowerleftportionofthecircleandthebase(B)inthelowerrightportion.ThecircletellsyoutousetheformulaP=R×Bandmultiply.Ifyouareaskedtofindtherate(R),placethepart(P)inthetopportionofthecircleandthebase(B)inthelowerright

portion.ThecircletellsyoutousetheformulaR=P/Banddivide.Theanswerwillbeindecimalformandneedstobechangedtoapercent.Ifyouareaskedtofindthebase,placethepart(P)inthetopportionandtherate(R)inthebottomleftportion.Thecircle

tellsyoutousetheformulaB=P/Randdivide.SeeFigure3-2.

StillStrugglingBesuretochangethepercenttoadecimalorfractionbeforemultiplyingordividing.

FIGURE3-2

TypeI:FindingthePart

EXAMPLE

Find42%of36

SOLUTION

Since42%istherate,placeitinthelowerleftportionofthecircle,andsince36isthebase,placeitinthelowerrightportionofthecircleandthenmultiply.SeeFigure3-3.

FIGURE3-3

P=R×P=42%×36=0.42×36=15.12

StillStruggling

Thenumberaftertheword“of”isalwaysthebase.

TypeII:FindingtheRate

EXAMPLE

16iswhatpercentof20?

SOLUTION

Since16isthepart,placeitinthetopportionofthecircle,andsince20isthebase,placeitinthelowerrightportionofthecircle,andthendivide.SeeFigure3-4.

FIGURE3-4

TypeIII:FindingtheBase

EXAMPLE

48is60%ofwhatnumber?

SOLUTION

Since48isthepart,placeitinthetopportionofthecircle,andsince60%istherate,placeitinthelowerrightportionofthecircleandthendivide.SeeFigure3-5.

FIGURE3-5

TRYTHESE

1.Whatpercentof60is45?

2.Find13%of37.


4.150is25%ofwhatnumber?

5.Find84%of63.

6.72is24%ofwhatnumber?


8.16iswhatpercentof40?

9.15iswhatpercentof60?

10.Find15%of90.

SOLUTIONS

StillStruggling

Remember:thenumberaftertheword“of”isalwaysthebase,thenumberwiththepercentsign(%)isalwaystherate,andthenumberimmediatelyprecedingorfollowingtheword“is”isthepart.

Inthisrefresher,youhavereviewedhowtoconvertamongpercents,decimals,andfractions.Therearethreebasictypesofpercentproblems.Theyusethebase,therate,andthepart.Youwillbegiventwonumbersandbeaskedtofindthethirdnumber.

SolvingPercentWordProblems

NOTEIfyouneedtoreviewpercents,completeRefresherIII.Apercentproblemconsistsofthreevalues,thebase,therate,andthepart.Thebase(B)isthewholeortotal,andtherate(R)isapercent.Oneofthesethreewillbeunknown.Forexample,ifaboxcontains10calculators,thenthewholeis10.Iffourcalculatorsareplacedonastore’sshelf,then4isthepart.Finally,thepercentis4/10=0.40=40%.Thatis,40%ofthecalculatorswereplacedonthestore’sshelf.

Percentproblemscanbesolvedusingthecirclemethod.Figure3-6showshowtousethecirclemethodtosolvepercentproblems.Inthetopofthecircle,placethepart(P).Inthelowerleftportionofthecircle,placetherate(%),andinthelowerright

portion,placethebase(B).Nowifyouaregiventhebottomtwonumbers,multiply.Thatis,P=R×B.Ifyouaregiventhetopnumber,thepart,andoneofthebottomnumbers,dividetofindtheothernumber.Thatis,R=P/BorB=P/R.SeeFigure3-7.

FIGURE3-6

FIGURE3-7

Therearethreetypesofpercentwordproblems.Theyare

TypeI:Findingthepart

TypeII:Findingtherate

TypeIII:Findingthebase

Inordertosolvepercentproblems,readtheproblemandidentifythebase,rate,andpart.Oneofthethreewillbeunknown.Substitutethetwoknownquantitiesinthecircleandusethecorrectformulatofindtheunknownvalue.Besuretochangethepercenttoadecimalbeforemultiplyingordividing.

TypeI:FindingthePartInTypeIproblems,youaregiventhebaseandrateandyouareaskedtofindthepart.

EXAMPLE

Thereare40preownedautomobilesonalot.If30%ofthemarewhite,howmanyoftheautomobilesarewhite?

SOLUTION

Goal:Youarebeingaskedtofindthenumberofautomobilesthatarewhite.

Strategy:Drawthecircleandplace30%inthelowerleftportionofthecircleand40inthelowerrightportionofthecirclesinceitisthetotalnumberofautomobilesinthelot.Tofindthepart,useP=R×B.SeeFigure3-8.

FIGURE3-8

Implementation:SubstituteintheformulaandsolveforP.

P=R×BP=30%×40P=0.30×40P=12automobiles

Hence,12automobilesarewhite.

Evaluation:Since and of40=12,theansweriscorrect.

TypeII:FindingtheRate(%)InTypeIIproblems,youaregiventhepartandthewholeandyouareaskedtofindtherateasapercent.Theanswerobtainedfromtheformulawillbeindecimalform.Makesurethatyouchangeitintoapercent.

EXAMPLE

Apersonboughtatextbookfor$35andpaidasalestaxof$2.10.Findthetaxrate.

SOLUTION

Goal:Youarebeingaskedtofindtherate(%).

Strategy:Inthiscase,thebase(B)isthetotalcost,whichis$35,andthesalestax,$2.10,isthepart.Drawthecircleandput$35inthelowerrightportionand$2.10inthetopportion.Tofindtherate,use

.SeeFigure3-9.

FIGURE3-9

Implementation:

Thesalestaxrateis6%.

Evaluation:Tochecktheanswer,find6%of$35:6%×35=0.06×35=$2.10.Theansweriscorrect.

TypeIII:FindingtheBaseInTypeIIIproblems,youaregiventhepartandrateandareaskedtofindthebaseorwhole.

EXAMPLE

Asalespersonearnsa15%commissiononallsales.Ifthecommissionwas$2435.25,findtheamountofhissales.

SOLUTION

Goal:Youarebeingaskedtofindthetotalamountofsales.

Strategy:Inthistypeofproblem,youaregiventhepart(commission)andtherate.Place$2435.25inthe

topportionofthecircleandthe15%inthebottomleftportion.Use .SeeFigure3-10.

FIGURE3-10

Implementation:

Thetotalsaleswere$16,235.

Evaluation:Tochecktheanswer,find15%of$16,235:0.15×$16,235=$2435.25.Hence,theansweriscorrect.

Somepercentproblemsinvolvefindingapercentincreaseordecrease.Alwaysrememberthattheoriginalvalueisusedasthebaseandtheamountoftheincreaseordecreaseisusedasthepart.Forexample,supposeanalarmclocksoldfor$50lastweekandisonsalefor$40thisweek.Thedecreaseis$50−$40=$10.Thepercentofdecreaseis10/50=0.20or20%.

EXAMPLE

Willisincreasedthefibercontentofhisdietfrom12to15gramsaday.Findthepercentofincreaseinthedailyfiber.

SOLUTION

Goal:Youarebeingaskedtofindthepercentoftheincreaseintheamountoffiberheconsumed.

Strategy:Findtheincrease,andthenplacethatnumberinthetopportionofthecircle.Thebaseisthe

originalamount.Use .SeeFigure3-11.

FIGURE3-11

Implementation:Theincreaseis15−12=3grams.

Hence,therateofincreaseis25%.

Evaluation:Find25%of12:0.25×12=3.Thesolutioniscorrect.

TRYTHESE

1.Cindyearnsa17%commissiononallthesalesshemakes.Whatwashercommissionona$5,320sale?

2.Aquarterbackcompletes40%ofhispassesinagame.Ifhecompleted12passes,howmanydidhethrow?

3.If3gallonsofoilareremovedfromafull20-gallontank,whatpercentoftheoilremainsinthetank?

4.Atireisonsalefor$120.Ifitispurchasedwhenthesalepriceis20%offtheoriginalprice,whatwastheoriginalprice?

5.Findtherateofthesalestaxifthetaxonanitemcosting$32.60is$1.63.

6.Ona60-pointexam,Sammissed9questions.Whatwashispercentscore?

7.Ifthesalepriceofadeskwas$432,andthesalepricediscountwas20%offtheoriginalprice,findtheoriginalprice.

8.If30%ofInternetusersuseEnglish,inasurveyof1,600people,findthetotalnumberofpeoplewhousedEnglishontheInternet.

9.IfthemovieTitanicmade$600millionintheUnitedStatesand$1,848millionworldwide,whatpercentoftheincomewasmadeintheUnitedStates?

10.JeanBorotra,achampionshiptennisplayerfromFrance,wonfourgrandslamsinglestitles,ninegrandslamdoublestitles,andfivemixedgrandslamtitlesinhercareer.Whatpercentofherwinsweredoubles?

SOLUTIONS

Inthissection,youhavelearnedtosolvethebasictypesofpercentproblems.Therearethreeformulasthatareused.TheyareP=B×R;R=P/B;andB=P/R.

Summary

Inthischapter,thebasicconceptsofpercentwereexplained.Thebasicconversionsofpercentstodecimalsorfractionswereshown.Theconversionsofdecimalsorfractionstopercentswerealsoexplained.Finally,thesolutionstothethreebasictypesofpercentproblemswereshown.

QUIZ

1.Findthesalestaxonaloungechairthatcosts$39iftherateis6%.

A.$1.95

B.$0.24

C.$2.34

D.$1.95

2.Asalespersonreceivedacommissionof$80onasaleofanitem.Ifhiscommissionrateis16%,findtheamountofthesale.

A.$13

B.$12

C.$500

D.$12.80

3.Ifapersonearned$48,000ayearandreceiveda$1,200raise,whatwasthepercentincreaseinhersalary?

A.25%

B.0.25%

C.2.5%

D.250%

4.Ifafamilypurchasedahomefor$160,000andput18%down,howmuchwaslefttofinance?

A.$28,800

B.$128,000

C.$131,200

D.$32,000

5.Ifacalculatororiginallysoldfor$60andwasreduced25%forasale,whatwasthereducedprice?

A.$40

B.$45

C.$25

D.$15

6.Marytooka40-problemmathematicsquiz.Ifshereceivedagradeof85%,howmanyproblemsdidshemiss?

A.34

B.32

C.18

D.6

7.Adepartmentstorehas72employees.Onaverysnowyday,therewere18employeesabsent.Whatpercentoftheemployeeswereabsent?

A.25%

B.36%

C.75%

D.84%

8.Frankearned$1,800permonth.Ifhereceiveda6%salaryincrease,howmuchdoesheearnnow?

A.$108

B.$1908

C.$2118

D.$96

9.Acertainmixtureofpeanutsandcashewsconsistsof32%cashews.Ifthetotalweightofthemixtureis50pounds,howmanypoundsofthemixtureconsistsofpeanuts?

A.32pounds

B.16pounds

C.64pounds

D.34pounds

10.Apersonboughtanecklacefor$800.Ifshemadeadownpaymentof$250andpaidthebalancein11monthlyinstallments,howmuchdidshepayeachmonth?

A.$50.00

B.$95.45

C.$550.00

D.$65.00

chapter4SolvingProportionandFormulaProblems

Thischapterexplainshowtosolvewordproblemsusingproportionsandhowtoevaluateformulas.Manyreal-worldproblemscanbesolvedbyusingthesetwotechniques.

CHAPTEROBJECTIVES


•Solvewordproblemsusingproportions

•Solvewordproblemsusingformulas

Ratios

Inordertosolvewordproblemsusingproportions,itisnecessarytounderstandtheconceptofaratio.Aratioisacomparisonoftwonumbersbyusingdivision.Forexample,theratioof6to10is6/10,whichreducesto3/5.

Ratiosareusedtomakecomparisonsbetweenquantities.Ifyoudrive180milesin4hours,thentheratioofmilestohoursis180/4or45/1.Inotherwords,youaveraged45milesperhour.

Itisimportanttounderstandthatwhatevernumbercomesfirstinaratiostatementisplacedinthenumeratorofthefractionandwhatevernumbercomessecondintheratiostatementisplacedinthedenominatorofthefraction.Ingeneral,theratioofatobiswrittenasa/b.Ratioscanbewrittenwithacolon.Forexample,theratioof3to5canbewrittenas3:5.

Proportions

Aproportionisastatementofequalityoftworatios.Forexample,4/5=20/25isaproportion.Proportionscanalsobewrittenusingacolon.Forexample,theproportion4/5=20/25canbewrittenas4:5=20:25or4:5::20:25.

Aproportionconsistsoffourterms,anditisusuallynecessarytofindoneofthetermsoftheproportiongiventheotherthreeterms.Thiscanbedonebycross-multiplyingandthendividingbothsidesoftheequationbythenumericalcoefficientofthevariable.

EXAMPLE

SOLUTION

EXAMPLE

SOLUTION

Thestrategyusedtosolveproblemsinvolvingproportionsistoidentifyandwritetheratiostatementandthenwritetheproportion.Lettheunknowntermbex;thencross-multiplyandsolveforx.

EXAMPLE

Ifapersonburns110caloriesin8minutesofrunning,howmanycalorieswillthepersonburnifsherunsfor30minutes?

SOLUTION

Goal:Youarebeingaskedtodeterminehowmanycaloriescanbeburnedin30minutesofrunning.

Strategy:Writetheratio,andthensetuptheproportion.Theratiostatementis110caloriesto8minutes

or .Setuptheproportion.Itis .

Implementation:Solvetheproportion.

Therunnerwillburn412.5caloriesifsherunsfor30minutes.

Evaluation:

StillStrugglingNoticethatwhenyousetupaproportion,alwaysplacethesameunitsinthenumeratorsandthesameunitsinthe

denominators.inthepreviousproblem, .

EXAMPLE

Ifagrocerystoresellscannedpearsatapriceof4for$10,whatisthecostof10cans?

SOLUTION

Goal:Youarebeingaskedtofindthecostof10cansofpears.

Strategy:Writetheratio.Itis .Theproportionis .


Evaluation:

Theansweriscorrect.

EXAMPLE

Iffourgallonsofpaintcancover1,240squarefeet,howmanysquarefeetwillsevengallonsofpaintcover?

SOLUTION

Goal:Youarebeingaskedtofindhowmanysquarefeetsevengallonsofpaintwillcover.

Strategy:Writetheratio.Itis .Theproportionis .


Sevengallonswillcover2,170squarefeet.

Evaluation:

Theansweriscorrect.

EXAMPLE

Ifatreecastsashadowof10feetanda6-footpolecastsashadowof3.2feet,howtallisthetree?

SOLUTION

Goal:Youarebeingaskedtofindtheheightofthetree.

Strategy:Theratiostatementis .Theproportionis .

Implementation:

Thetreeis18.75feettall.

Evaluation:

Theansweriscorrect.

StillStrugglingAslongasyoukeepthesameunitsinthenumeratorsandthesameunitsinthedenominatorsofaproportion,itdoesnotmatterhowtheproportionissetup.Forexample,theproportionx/10=6/5willgivethesameanswerforxastheproportion10/x=5/6.

TRYTHESE

1.Onamap,thescaleis inch=150miles.Howfarapartaretwocitieswhosedistanceonamapisinches?

2.Ifarecipecallsfor2.4cupsofflouranditservessixpeople,howmanycupsofflourwillbeneededtoservetwopeople?

3.Ifamerchantcanorder12shirtsfor$280,howmuchwill15shirtscost?

4.Ifthreepoundsofgrassseedwillcover1,320squarefeet,howmanypoundswillbeneededtocover3,080squarefeet?

5.Ifapersondrivesanautomobile8,100milesin9months,abouthowmanymileswillthepersondrivetheautomobilein15months?

6.Samwantstowaterproofhispatiodeck.Iftwogallonsofsealantcancover500squarefeet,howmanygallonsshouldhebuyifhisdeckis1,460squarefeet?

7.Bettycanbicycle216milesin9days;howfarcanshetravelin14days?

8.Ifanauthorcanwritetwochaptersin11days,howmanydayswillittakehertocompletea16-chapterbook?

9.Sophiawantstosavemoneyforatrip.Ifshecalculatesthatshecansave$456inthreemonths,howmanymonthswillittakehertosaveforatripcosting$3,648?

10.IfAbigailcanpurchasesixconcertticketsfor$112.50,howmanyticketscanshepurchasefor$168.75?

SOLUTIONS

Thissectionexplainedhowtosolvewordproblemsusingproportions.Themostimportantpartissettinguptheproportion.Makesurethatyouhavethesameunitsinthenumeratorsofbothratiostatementsandthesameunitsinthedenominatorsoftheratiostatements.

Formulas

Inmathematicsandscience,manyproblemscanbesolvedbyusingaformula.Aformulaisamathematicalstatementoftherelationshipoftwoormorevariables.Forexample,thedistance(D)anautomobiletravelsisrelatedtotherate(R)ofspeedandthetime(T)ittravels.Insymbols,D=RT.Tosolveawordproblemusingaformula,simplyselectthecorrectformula,substitutethevaluesofthevariables,andevaluatetheformula.

Inordertoevaluateformulas,youusetheorderofoperations.Step1PerformalloperationsinparenthesesStep2RaiseeachnumbertoitspowerStep3PerformmultiplicationanddivisionfromlefttorightStep4Performadditionandsubtractionfromlefttoright

EXAMPLE

Findtheinterestonaloanwhoseprincipal(P)is$5,400atarateof6%foreightyears.UseI=PRT.

SOLUTION

Goal:Youarebeingaskedtofindtheinterest.

Strategy:UsetheformulaI=PRT.

Implementation:

Evaluation:Youcanestimatetheanswerbyrounding$5,400to$5,000andthenfinding6%of$5,000,whichis0.06×$5,000=$300.Theinterestforoneyearisabout$300.Theinterestforeightyearsthenis8×$300=$2,400.Sincethisiscloseto$2,592,theanswerisprobablycorrect.

EXAMPLE

FindtheFahrenheittemperature(F)whentheCelsiustemperature(C)is90°.

SOLUTION

Goal:YouarebeingaskedtofindaFahrenheittemperature.

Strategy:Usetheformula .

Implementation:

Evaluation:Youcanestimatetheanswerbymultiplying90°by1.5andadding30°.Thatis,901.5×+30=135+30=185°.Since185°iscloseto194°,youranswerisprobablycorrect.

StillStrugglingRemembertheorderofoperations.alwaysperformmultiplicationbeforeaddition.

EXAMPLE

Thedistance(d)anobjectfallsinfeetisd=32t2wheretisthetimeinseconds.Findthedistanceanobjectfallsinfiveseconds.

SOLUTION

Goal:Youarebeingaskedtofindthedistanceanobjectfallsinfiveseconds.

Strategy:Usetheformulad=32t2.

Implementation:

Evaluation:Estimatetheanswerbyrounding32to30andthenmultiplying30×25=750.Sincethisestimateiscloseto800,theanswerisprobablycorrect.

StillStrugglingRemembertheorderofoperations.Squarebeforemultiplying.

TRYTHESE

1.Findtheperimeter(P)ofasquarewhoseside(s)is24inches.UseP=4s.

2.Findthecurrent(I)inampereswhentheelectromotiveforce(E)is15voltsandtheresistance(R)is9ohms.

Use .

3.Findthevolume(V)ofacylinderincubicfeetwhentheheight(h)is15feetandtheradius(r)ofthebaseis4feet.UseV=3.14r2h.

4.FindtheCelsiustemperature(C)whentheFahrenheittemperature(F)is59°.Use .

5.Findtheforce(F)ofthewindagainstaflatsurfacewhosearea(A)is32squarefeetwhenthewindspeed(s)is30milesperhour.UseF=0.004As2.

6.FindtheFahrenheittemperature(F)whentheCelsiustemperature(C)is30°.Use .

7.Findthesurfacearea(A)ofacubeinsquarefeetwheneachside(s)measureseightinches.UseA=6s2.

8.Findtheamountofwork(W)donebyapplyingaforce(F)of80poundsmovingadistance(d)of12feet.UseW=Fd.

9.Findthedistance(D)anautomobiletravelsatarate(R)of42milesperhourin3.2hours(T).UseD=RT.

10.Findtheamountofinterest(I)earnedonaprincipal(P)of$8,220atarate(R)of9%foratime(T)offouryears.UseI=PRT.

SOLUTIONS

Inthissection,wordproblemsweresolvedbyusingformulas.Inordertouseformulascorrectly,youmustfollowtheorderofoperationstoevaluateformulas.

1.Performalloperationsinparentheses.

2.Raiseeachnumbertoapower.

3.Performmultiplicationanddivisionlefttoright.

4.Performadditionandsubtractionlefttoright.

Summary

Thischapterexplainedhowtosolvewordproblemsusingproportionsandformulas.Thesetypesofproblemsoccurinphysics,chemistry,andlifesciencescoursesaswellasinbusinessmathematicscoursesandotherareas.

QUIZ

1.Ifthreeouncesofacerealcontain210calories,howmanycalorieswouldbecontainedineightounces?

A.630

B.420

C.560

D.1,680

2.Ifapersoncanswim5lapsinapoolin3minutes,howmanylapscanthepersonswimin15minutes?

A.30

B.18

C.10

D.25

3.Iffourbottlesofwatercost$5.20,howmuchwill12bottlescost?

A.$15.60

B.$20.80

C.$10.40

D.$62.40

4.Ifthreeitemscost$25,howmanyitemscouldyoubuyfor$125?

A.9

B.15

C.12

D.18

5.Ifa6-footpolecastsashadowof3.5feet,howtallisatreethatcastsashadowof14feet?

A.18feet

B.12feet

C.20feet

D.24feet

6.Howfar(infeet)willanobjectfallinsixseconds?Used=32t2wheret=thetimeinseconds.

A.192feet

B.384feet

C.1,152feet

D.576feet

7.Ifapersontravelsadistanceof540milesat30milesperhour,findthetimeitwilltakethepersontogetthere.

Use whereD=distanceandR=rate.

A.27hours

B.18hours

C.14hours

D.9hours

8.TheformulaforfindingthevolumeofacylinderisV=πr2hwhereπ=3.14,r=thevalueoftheradius,andh=theheight.Whatisthevolumeofacylinderwhoseradiusisthreefeetandwhoseheightisfourfeet?

A.527.52feet

B.37.68feet

C.113.04feet

D.75.36feet

9.TheinterestonaloanisfoundbyusingtheformulaI=PRTwhereP=theprinciple,R=therate,andT=thetimeinyears.Howmuchinterestwouldapersonhavetopayona$3,250loanat5%forthreeyears?

A.$487.50

B.$1562.50

C.$975.00

D.$643.00

10.Findthedistancearoundacircularswimmingpoolifitsdiameterissevenfeet.UseC=πDwhereπ=3.14.

A.21feet

B.10.99feet

C.43.96feet

D.21.98feet

chapter5EquationsandAlgebraicRepresentation

Thischapterreviewshowtosolveequations.Manyofthewordproblemsinalgebracanbesolvedbysettingupanequationbasedonthewordsintheproblemandsolvingit.Inordertosetupanequation,youneedtotranslatetheinformationgivenintheproblemintosymbols.Thisiscalledalgebraicrepresentation.Thesetwoskillsareveryimportantwhensolvingwordproblems.

CHAPTEROBJECTIVES


•Solveequations

•Representwordstatementsusingletters(variables)andmathematicalsymbols

RefresherIV:Equations

Analgebraicexpressionconsistsofvariables(letters),numbers,operationsigns(+,−,×,÷),andgroupingsymbols.Hereareafewexamplesofalgebraicexpressions:

3x5(x−6)−8x29+2Anequationisastatementofequalityoftwoalgebraicexpressions.Herearesomeexamplesofequations:

5+4=93x−2=13x2+3x+2=0Anequationthatcontainsavariableiscalledaconditionalequation.Tosolveaconditionalequation,itisnecessaryto

findanumberthat,whensubstitutedforthevariable,makesatrueequation.Thisnumberiscalledasolutiontotheequation.Forexample,5isasolutiontotheequationx+3=8sincewhen5issubstitutedforx,itmakestheequationtrue;thatis,5+3=8.Theprocessoffindingasolutiontoanequationiscalledsolvingtheequation.Tocheckanequation,substitutethesolutionintotheoriginalequationandseeifit’satrueequation.Therearefourtypesofbasicequations.Inordertosolveeachtype,youperformtheoppositeoperationtobothsidesof

theequationastheoperationthatisbeingperformedonthevariable.Additionandsubtractionareoppositeoperations.Multiplicationanddivisionareoppositeoperations.Thenextfourexamplesshowhowtosolvebasicequations.

EXAMPLE

Solveforx:x−10=13

SOLUTION

EXAMPLE

Solveforx:x+16=34

SOLUTION

EXAMPLE

SOLUTION

EXAMPLE

Solveforx:8x=56

SOLUTION

StillStrugglingWhensolvingequations,thesamenumber(exceptzero)canbeaddedto,subtractedfrom,multipliedby,ordividedintobothsidesoftheequationwithoutchangingtheequalityoftheequation.

TRYTHESE

Solveeachequationforx

1.x+32=56

2.x−12=7

3.6x=72

4.x+4=16

SOLUTIONS

1.x+32=56x+32−32=56−32x=24

2.x−12=7x−12+12=7+12x=19

4.x+4=16x+4−4=16−4x=12

Morecomplexequationsrequireseveralstepstosolve.Thegoalistouseadditionand/orsubtractiontogetthevariablesononesideoftheequationandthenumbersontheothersideoftheequation.Thendividebothsidesbythenumberinfrontofthevariable.Thisnumberiscalledthenumericalcoefficientofthevariable.

EXAMPLE

Solveforx:3x+18=42

SOLUTION

Check:3x+18=423(8)+18=4224+18=4242=42

EXAMPLE

Solveforx:9x−15=4x+35

SOLUTION

StillStrugglingIfyougetthewronganswer,itisgenerallybettertostartoverandsolvetheequationagainratherthantryingtofindyourmistake.thisisespeciallytruewhenthereareseveralstepsinthesolution.

TRYTHESE

Solveeachequationforx:

1.12x+21=81

2.4x−10=14

3.15x−21=12x+18

4.7x+6=3x+48

5.11x−9=5x+33

SOLUTIONS

Manyequationscontainparentheses.Inordertoremoveparentheses,multiplyeachterminsidetheparenthesesbythenumberoutsidetheparentheses.Thisiscalledthedistributivepropertyofmultiplicationoveraddition.Forexample,

Whenyousolveanequation,removeparenthesesfirst,combineliketerms(i.e.,6x+8x),andthensolveasshowninthepreviousexamples.Liketermshavethesamevariablesthatareraisedtothesamepowers.

EXAMPLE

Solveforx:8(3x−4)=88

SOLUTION

EXAMPLE

Solveforx:6(2x+7)−10x=56

SOLUTION

TRYTHESE


1.4(x−10)+20=28

2.3(3x−7)=7x+33

3.9(4x−5)=99

4.8(2x−7)=7(2x+4)

5.2(4x+9)=5x−3

SOLUTIONS

Sometimeswhenyouaresolvingwordproblems,youwillneedtosolveanequationcontainingfractions.Itshouldbenotedthatfractiontermscanbewrittenintwoways.Seethenextexamples:

Tosolveanequationcontainingfractions,itisnecessarytofindthelowestcommondenominatorofallthefractions,andthenmultiplyeachtermintheequationbythelowestcommondenominator.Thisprocessiscalledclearingfractions.

EXAMPLE

SOLUTION

EXAMPLE

SOLUTION

TRYTHESE


SOLUTIONS

AlgebraicRepresentation

Whenyousolveanalgebrawordproblem,youmustfirstbeabletotranslatetheconditionsoftheproblemintoanequationinvolvingalgebraicexpressions.Recallthatanalgebraicexpressionwillconsistofvariables(letters),numbers,operationssigns(+,−,×,÷),andgroupingsymbolssuchasparentheses.Herearesomecommonphrasesthatareusedinalgebrawordproblems:AdditioncanbedenotedbysumaddedtoincreasedbylargerthanmorethanSubtractioncanbedenotedbylessthansubtractedfromdecreasedbyexceedsshorterthandifferencebetween

Multiplicationcanbedenotedbyproducttimesmultipliedbytwiceaslargethreetimesanumber1/2ofanumberDivisioncanbedenotedbydividedbyquotientofEqualscanbedenotedbyiswillbeisequalto

Herearesomeexamplesofhowthesephrasesaretranslatedintosymbols:

TRYTHESE

Writeeachinsymbols:

1.Anumberincreasedby10

2.Fourtimesanumberplus8

3.Sixlessthananumber

4.Fivelessthansixtimesanumber

5.Thesquareofanumberdecreasedby4

SOLUTIONS

1.x+10

2.4x+8

3.x−6

4.6x−5

5.x2−4

Inthepreviousexamples,onlyoneunknownwasused.Othertimes,itisnecessarytorepresenttworelatedunknownsbyusingonevariable.Considertheseexamples:“Thesumoftwonumbersis15.”Whenyouaregiventwonumberswhosesumis15andonenumberis,say,9,how

wouldyoufindtheothernumber?Youwouldfind15−9.Soifonenumberisx,theothernumberwouldbe15−x.“Onenumberis5morethananothernumber.”IfItoldyouonenumberis12,howwouldyoufindtheothernumber?You

wouldadd12+5.Soifonestatednumberisx,theothernumberwouldbex+5.“Onenumberistwotimesanothernumber.”IfItoldyouonenumberis4,howwouldyoufindtheothernumber?You

wouldmultiply4by2.Soifonenumberisx,theothernumberwouldbe2x.“Onenumberis7lessthananothernumber.”IfItoldyouonenumberwas22,howwouldyoufindtheothernumber?

Youwouldsubtract22−7.Soifonenumberisx,theothernumberisx−7.

StillStrugglingWhenyouarerepresentingtworelatednumbers,youhavechoicesonhowyoudoit.Forexample,iftheproblemsaysrepresenttwonumberssuchthatonenumberis7morethantheothernumber,youcouldrepresentthemasxandx+7orxandx−7.Eitherwayiscorrect.

TRYTHESE

Representeachusingsymbols:

1.Thesumoftwonumbersis24.

2.Onenumberis6lessthantheothernumber.

3.Thesecondnumberis5lessthanone-thirdofthefirstnumber.

4.Thesecondnumberis8morethantwicethefirstnumber.

5.Thesecondnumberisthreetimesthefirstnumber.

SOLUTIONS

1.Letx=thefirstnumberand24−x=thesecondnumber.

2.Thefirstnumberisx,andthesecondnumberisx−6.

3.Letx=thefirstnumberand =thesecondnumber.

4.Letx=thefirstnumberand2x+8=thesecondnumber.

5.Letx=thefirstnumberand3x=thesecondnumber.

Nowthatyouknowhowtotranslatewordphrasesintoalgebraicexpressions,thenextstepistotranslatewholesentencesintoalgebraicexpressionsusingtheequalsign.Considertheseexamples.“Fivetimesanumberincreasedby8isequalto38”translatesto

5x+8=38“Ninetimesanumberdecreasedby4isequalto32.”

9x−4=32“Thedifferencebetweenanumberandone-fourthitselfisequalto20.”

TRYTHESE

Translateeachintoanequation:

1.Threetimesanumberdecreasedby7is17.

2.If4isaddedtoanumber,youget16.

3.Thesumofanumberandthreetimesitselfisequalto32.

4.One-fourthanumberplus6is54.

5.If8isincreasedbytwiceanumber,thesumis26.

SOLUTIONS

1.3x−7=17

2.x+4=16

3.x+3x=32

5.8+2x=26

Finally,itisnecessarytobeabletowriteanequationfortworelatedunknownsusingonevariable.

EXAMPLE

Writeanequationforthisproblem:“Onenumberis8morethananothernumberandtheirsumis17.”

SOLUTION

Letx=thesmallernumberandx+8=thelargernumber.Theequationisx+x+8=17.

EXAMPLE

Writeanequationforthisproblem:“Onenumberisfourtimesaslargeasanothernumber.Iftwotimesthesmallernumberissubtractedfromthelargernumber,theansweris18.”

SOLUTION

Letx=thesmallernumberand4x=thelargernumber.Theequationis4x−2x=18.

TRYTHESE

Writeanequationforeach.Donotsolvetheequations.

1.Onenumberis6morethanthreetimesanothernumber.Findthenumbersiftheirsumis66.

2.Onenumberis ofanothernumber.Findthenumbersiftheirsumis36.

3.Whatnumberincreasedby ofitselfisequalto9?

4.Twotimesanumberis9morethan thenumber.Findthenumbers.

5.Acertainnumberexceedsanothernumberby10.Iftheirsumis63,findthenumbers.

SOLUTIONS

1.Letx=onenumberand3x+6=theothernumber.Theequationisx+3x+6=66.

2.Letx=onenumberand =theothernumber.Theequationis .

3.Letx=thenumberand =theothernumber.Theequationis .

4.Letx=thenumber.Theequationis .

5.Letx=onenumberandx+10=thelargernumber.Theequationisx+x+10=63.

Summary

Thefirstpartofthischapterexplainedhowtosolveequations.Manytypesofwordproblemsaresolvedusingequations.Itisaveryimportantalgebraictopic.

Thischapteralsoexplainsaveryimportantskillthatisusedtosolvewordproblemsinmathematics.Thatskillisbeingabletotranslatethewordsoftheproblemintomathematicalsymbolsandtosetupanequationusingthesesymbols.Oncetheequationisobtained,allthatisnecessarytogettheansweristoalgebraicallysolvetheequationforthevariable.

QUIZ

1.Thesolutiontotheequation11x+20=6x−25is

A.x=9

B.x=−11

C.x=−9

D.x=11

2.Thesolutiontotheequation6−x=−14−3xis

A.x=10

B.x=5

C.x=−3

D.x=−10

3.Thesolutiontotheequationx−9=15−3xis

A.x=6

B.x=3

C.x=8

D.x=2

4.Thesolutiontotheequation5(x−6)+3(2−x)=0is

A.x=14

B.x=12

C.x=−12

D.x=16

5.Thesolutiontotheequation9(7x−3)=31+8(7x−2)is

A.x=5

B.x=3

C.x=6

D.x=4

6.Thestatement3timesanumberxplus6canberepresentedas

A.6x+3

B.9x

C.3x+6

D.3x−6

7.Thestatement“12lessthanfivetimesanumberx”canberepresentedas

A.12x−5

B.5x−12

C.5·12x

D.(5+12)x

8.Ifthesumoftwonumbersis32,andonenumberisx,theothernumberis

A.32−x

B.x−32

C.32x

D.32÷x

9.Theequationrepresentedbythestatement“fivetimesanumberplus8isequalto45”is

A.5x−8=45

B.8·5x=45

C.5(x+8)=45

D.5x+8=45

10.Theequationrepresentedbythestatement“16minustwotimesanumberisequalto24”is

A.2−16x=24

B.16−2x=24

C.16x−2=24

D.2x−16=24

chapter6SolvingNumberandDigitProblems

Thischapterexplainshowtosolvenumberanddigitproblems.Thenumbersusedaremostoftenwholenumbersandareusuallypositive.Theproblemwillgiveyoutherelationshipbetweentwoormorenumbersinordertowriteanequation,andthensolvetheequation.Thenumbersweusetodayusethedigits0through9.Wehaveone-digitnumbers,two-digitnumbers,three-digitnumbers,

etc.Knowingwhateachdigitisinanumberwillenableyoutowriteanequationinordertosolvetheproblem.

CHAPTEROBJECTIVES


•solvewordproblemsaboutnumbers

•solvedigitproblems

NumberProblems

Thestrategyusedtosolvewordproblemsinalgebraisasfollows:

1.Representanunknownbyusingalettersuchasx.

2.Ifnecessary,representtheotherunknownsbyusingalgebraicexpressionsintermsofx.

3.Fromtheconditionsoftheproblem,writeanequationusingthealgebraicrepresentationoftheunknown(s).

4.Solvetheequationforx.

EXAMPLE

Onenumberis6lessthananothernumberandthesumofthetwonumbersis32.Findthenumbers.

SOLUTION

Goal:Youarebeingaskedtofindtwonumbers.

Strategy:Letx=onenumberandx−6=theothernumber.Sincetheproblemaskedforthesum,writetheequationasx+x−6=32.

Implementation:Solvetheequationforx:

Hence,onenumberis19andtheothernumberisx−6or19−6=13.

Evaluation:Checktheanswer:19+13=32

EXAMPLE

If5plusthreetimesanumberisequalto32,findthenumber.

SOLUTION

Goal:Youarebeingaskedtofindonenumber.

Strategy:Letx=thenumber.Fiveplusthreetimesanumberiswrittenas5+3x,andtheequationis5+3x=32.

Implementation:Solvetheequation.

Evaluation:Checktheanswer:5+3·9=32

EXAMPLE

Aprofessorhastwomathematicsclasseswithatotalof46studentsinbothclasses.Iftherearesixmorestudentsinoneclassthantheother,howmanystudentsareineachclass?

SOLUTION

Goal:Youarebeingaskedtofindthenumberofstudentsineachoftwoclasses.

Strategy:Letx=thenumberofstudentsinoneclassandx+6bethenumberofstudentsintheotherclass.Theequationisx+x+6=46.


Hence,thereare20studentsinoneclassand26studentsintheother.

Evaluation:Checktheanswer:20+26=46

EXAMPLE

Ifanumberisdecreasedby4andtwotimestheoriginalnumberisequaltosixtimestheothernumber,findthenumbers.

SOLUTION


Strategy:Letx=theoriginalnumberandx−4=theothernumber.

Nowtwotimestheoriginalnumberis2xandsixtimestheothernumberis6(x−4).Theequationis2x=6(x−4).


Hence,thefirstnumberis6andthesecondnumberis2.

Evaluation:Checktheanswer:6−4=2and2·6=6·2or12=12.

Somenumberproblemsuseconsecutiveintegers.Numberssuchas1,2,3,4,5,etc.,arecalledconsecutiveintegers.Theydifferby1.Consecutiveintegerscanberepresentedas:

Letx=thefirstinteger

x+1=thesecondinteger

x+2=thethirdinteger

etc.

Consecutiveoddintegersarenumberssuchas1,3,5,7,9,11,etc.Theydifferby2.Theycanberepresentedas:

Letx=thefirstoddinteger

x+2=thesecondconsecutiveoddinteger

x+4=thethirdconsecutiveoddinteger

etc.

Consecutiveevenintegersarenumberssuchas2,4,6,8,10,12,etc.Theyalsodifferby2.Theycanberepresentedas:

Letx=thefirsteveninteger

x+2=thesecondconsecutiveeveninteger

x+4=thethirdconsecutiveeveninteger

etc.

StillStrugglingYouneednotworrywhetheryouarelookingforconsecutiveevenoroddnumberssincetheproblemswillalwaysworkoutcorrectly.(thetextbookauthorshavemadethemupsothattheywill.)

EXAMPLE

Findthreeconsecutiveintegerswhosesumis96.

SOLUTION

Goal:Youarebeingaskedtofindthreeconsecutiveintegerswhosesumis96.

Strategy:Letx=thefirstinteger,x+1=thesecondinteger,andx+2=thethirdinteger.Theequationisx+(x+1)+(x+2)=96.

Implementation:Solvetheequation:

x+x+1+x+2=963x+3=96

3x+3−3=96−33x=93x=31

x+1=31+1=32x+2=31+2=33

Evaluation:Checktheanswer:31+32+33=96

EXAMPLE

Ifthesumoftwoconsecutiveoddintegersis36,findthenumbers.

SOLUTION

Goal:Youarebeingaskedtofindtwoconsecutiveoddintegerswhosesumis36.

Strategy:Letx=thefirstconsecutiveoddintegerandx+2=thesecondconsecutiveoddinteger.Sincethesumisequalto36,theequationisx+x+2=36.


Evaluation:17and19areconsecutiveoddintegers,andtheirsumis17+19=36.

TRYTHESE

1.Abaseballteamplayed27gamesandwonfivemoregamesthanitlost.Findthenumberofgamestheteamwon.

2.Ifthreetimesanumberplus10isequalto22,findthenumber.

3.Fourtimesanumberdecreasedby2isequalto26.Findthenumber.

4.If ofanumberis12lessthan ofthenumber,findthenumber.

5.Acarpenterwantstocuta42-inchpieceoflumberintothreepiecessothateachpieceissixincheslongerthantheprecedingone.Findthelengthofeachpiece.

6.Thedifferenceoftwonumbersis45,andonenumberissixtimestheothernumber,findthenumbers.

7.Findtwonumberswhosesumis25andwhosedifferenceis3.

8.Fiftynotebooksareplacedintotwoboxessothatoneboxhassixmorenotebooksthantheotherbox.Howmanynotebooksareineachbox?

9.Ifthesumofthreeconsecutiveintegersis81,findthenumbers.

10.Thesumoftwoconsecutiveevenintegersis62.Findthenumbers.

SOLUTIONS

1.Letx=thenumberofgamestheteamlostandx+5=thenumberofgamestheteamwon.

Theteamwon16gamesandlost11games.

2.Letx=thenumber,then3x+10=22.

3.Letx=thenumber,then4x−2=26

4.Letx=thenumber

5.Letx=thelengthofthefirstpieceoflumber,x+6=thelengthofthesecondpiece,andx+12=thelengthofthethirdpiece.Thenx+x+6+x+12=42inches.

6.Letx=onenumberand6x=theothernumber.Then6x−x=45.

7.Letx=onenumberandx−3=theothernumber.Thenx+x−3=25.

8.Letx=thenumberofnotebooksplacedinoneboxandx+6=thenumberofnotebooksplacedintheotherbox.Thenx+x+6=50.

9.Letx=thefirstinteger,x+1=thesecondinteger,andx+2=thethirdinteger.Thenx+x+1+x+2=81.

10.Letx=thefirstintegerandx+2=thesecondinteger.Thenx+x+2=62.

Inthissection,youlearnedhowtosolvenumberproblems.Eachproblemgivesyoutherelationshipbetweentwoormorenumbers.Fromthisinformation,youcansetupanequationandsolveforthenumbers.Consecutivenumbersincreaseby1.Consecutiveevennumbersandconsecutiveoddnumbersincreaseby2.

DigitProblems

Thesymbols0,1,2,3,4,5,6,7,8,and9arecalleddigits.Theyareusedtomakeournumbers.Anumbersuchas28iscalledatwo-digitnumber.Theeightistheonesdigitandthetwoisthetensdigit.Theonesdigitisalsocalledtheunitsdigit.Thenumber28meansthesumof2tensand8onesandcanbewrittenas or28.Thenumber537iscalledathree-digitnumber.Thesevenistheonesdigit,thethreeisthetensdigit,andthefiveisthehundredsdigit.Itcanbewrittenas or500+30+7=537.Adigitproblemwillsometimesaskyoutofindthesumofthedigits.Inordertodothis,justaddthedigits.Forexample,

thesumofthedigitsofthenumber537is5+3+7=15.

Sometimesdigitproblemswillaskyoutoreversethedigits.Ifthedigitsofthenumber arereversed,thenewnumberis63or .Usingthisinformationandthematerialintheprevioussection,youwillbeabletosolvedigitproblems.

EXAMPLE

Thesumofthedigitsofatwo-digitnumberis9.Ifthedigitsarereversed,thenewnumberis63morethantheoriginalnumber.Findtheoriginalnumber.

SOLUTION

Goal:Youarebeingaskedtofindacertaintwo-digitnumber.

Strategy:Letx=thetensdigitand9−x=theonesdigit.Theoriginalnumbercanbewrittenas10x+(9−x),andthenumberwiththedigitsreversedcanbewrittenas10(9−x)+x.Sincethenewnumberis63morethantheoriginalnumber,anequationcanbewrittenas

newnumber=originalnumber+63

10(9−x)+x=10x+(9−x)+63


Thetensdigitis1andtheonesdigitis9−1=8.Thenumber,then,is18.

Evaluation:Take18andreversethedigitstoget81.Subtract81−18=63.Hencethesumofthedigits1+8is9andthedifferenceofthetwonumbersis63.

EXAMPLE

Inatwo-digitnumber,thetensdigitis3lessthantheonesdigit.Ifthedigitsofthenumberarereversed,thesumoftheoriginalnumberandthenewnumberis77.Findtheoriginalnumber.

SOLUTION


Strategy:Letx=theonesdigitandx−3=thetensdigit.Thenumber,then,is10(x−3)+x.Whenthedigitsarereversed,thenewnumberis10x+(x−3).Sincetheirsumis77,theequationis10(x−3)+x+10x+x−3=77.


Theonesdigitis5andthetensdigitisx−3=5−3=2.

Thenumberis25.

Evaluation:Thetensdigitis3lessthantheonesdigit.Whenthedigitsof25arereversed,theansweris52;hence,25+52=77.

EXAMPLE

Inathree-digitnumber,theonesdigitisequaltothetensdigitandthehundredsdigitis5morethantheonesdigit.Iftheorderofthedigitsisreversed,twicethenewnumberis268lessthantheoriginalnumber.Findtheoriginalnumber.

SOLUTION

Goal:Youarebeingaskedtofindathree-digitnumber.

Strategy:Letx=theonesdigitandx=thetensdigit,sincetheyareequal.Thehundredsdigitisx+5sinceitis5morethantheonesdigit.Thenumber,then,is100(x+5)+10x+x.Whenthedigitsarereversed,thenewnumberis268lessthantheoriginalnumber.Theequationis100(x+5)+10x+x=2(100x+10x+x+5)+268.


Hence,theonesdigitis2,thetensdigitis2,andthehundredsdigitis2+5=7.Thenumberis722.

Evaluation:Thenumberis722,andreversingthedigits,youget227.Now722−2·227=722−454=268.

TRYTHESE

1.Thesumofthedigitsofatwo-digitnumberis10.If18isaddedtotheoriginalnumber,thenewnumberwillhavethesamedigits,buttheywillbereversed.Findtheoriginalnumber.

2.Thesumofthedigitsofatwo-digitnumberonaracecaris15.Ifthedigitsarereversed,thenewnumberis9morethantheoriginalnumber.Findtheoriginalnumber.

3.Ifthedigitsofatwo-digitnumberarereversed,thenewnumberis10morethantwicetheoriginalnumber.Thesumofthedigitsoftheoriginalnumberis8.Findtheoriginalnumber.

4.Thetensdigitofatwo-digitnumberis3morethantheonesdigit.Ifthenumberisonemorethaneighttimesthesumofthedigits,findthenumber.

5.Inatwo-digitnumber,theonesdigitis5morethanthetensdigit.Ifthenumberisthreetimesthesumofitsdigits,findthenumber.

6.Inatwo-digitnumber,thesumofthedigitsis7.Ifthedigitsarereversed,threetimesthenewnumberis13lessthantheoriginalnumber.Findtheoriginalnumber.

7.Thesumofthedigitsofatwo-digitnumberonafootballjerseyis11.Ifthetensdigitis3morethantheonesdigit,findthenumber.

8.Inatwo-digitnumber,theonesdigitis4lessthanthetensdigit.Thenumberis3lessthanseventimesthesumofthedigits.Findthenumber.

9.Thesumofthedigitsinatwo-digitnumberonabaseballjerseyis5.Iftheonesdigitis1morethanthetensdigit,findthenumber.

10.Inatwo-digitnumber,thesumofthedigitsis9.Ifthedigitsoftheoriginalnumberarereversed,thenewnumberis45morethantheoriginalnumber.Findtheoriginalnumber.

SOLUTIONS

1.Letxbethetensdigitand10−xbetheonesdigit.Thenumberis10x+10−x.Reversingthedigits,thenewnumberis10(10−x)+x.Theequationis10x+10−x+18=10(10−x)+x.

Thenumberis46.

2.Letx=theonesdigitand15−x=thetensdigit.Thenumberis10(15−x)+x.Ifthedigitsarereversed,thenewnumberis10x+15−x.Theequationis10(15−x)+x+9=10x+15−x.

Hence,thenumberis78.

3.Letx=theonesdigitand8−x=thetensdigit.Thenumberis10(8−x)+x.Whenthedigitsarereversed,thenewnumberis10x+8−x.Theequationis2[10(8−x)+x]+10=10x+8−x.

Thenumberis26.

4.Letx=theonesdigitsandx+3=thetensdigit.Thenumberis10(x+3)+x.Thesumofthedigitsisx+3+x.Theequationis10(x+3)+x=8(x+3+x)+1.

Thenumberis41.

5.Letx=thetensdigitandx+5=theonesdigit.Thenumberis10x+x+5.Thesumofthedigitsisx+x=5.Theequationis10x+x+5=3(x+x+5).

Thenumberis27.

6.Letx=theonesdigitand7−x=thetensdigit.Thenumberis10(7−x)+x.Whenthedigitsarereversed,thenewnumberis10x+7−x.Theequationis10(7−x)+x=3(10x+7−x)+13.

Thenumberis61.

7.Letx=theonesdigitandx+3=thetensdigit.Sincethesumofthedigitsis11,theequationisx+3+x=11.

Thenumberis74.

8.Letx=theonesdigitandx+4=thetensdigit.Thenumberis10(x+4)+x.Thesumofthedigitsisx+4+x.Theequationis10(x+4)+x+3=7(x+4+x).

Thenumberis95.

9.Letx=theonesdigitand5−x=thetensdigit.Theequationisx−1=5−x.

Thenumberis23.

10.Letx=theonesdigitand9−x=thetensdigits.Thenumberis10(9−x)+x.Whenthedigitsarereversed,thenewnumberis10x+9−x.Theequationis10(9−x)+x+45=10x+9−x.

Thenumberis27.

Thissectionexplainedhowtosolveproblemsinvolvingdigits.Ournumbersystemconsistsof10digits,andeachnumberconsistsofaonesdigit,atensdigit,ahundredsdigit,etc.

Summary

Thischapterexplainedtwotypesofproblems,namelynumberproblemsanddigitproblems.Althoughbothsectionsusenumbers,theequationstosolvetheproblemsaresomewhatdifferent.Itisnecessarytobeawareofthedifference.Innumberproblems,sometimesyouarelookingforonenumber,twonumbers,orthreenumbers.Sometimesthenumbers

areconsecutivenumbers,orconsecutiveoddandevennumbers.Indigitproblems,youareusuallylookingfortherelationshipbetweenthedigitsofasinglenumber.

QUIZ

1.Ifthesumofthreeconsecutivenumbersis39,thelargestofthethreenumbersis

A.12

B.14

C.118

D.13

2.Ifonenumberissixtimesanothernumberandthesumofthenumbersis147,thesmallernumberis

A.21

B.14

C.28

D.31

3.Thesumoftwonumbersis39andonenumberis5morethantheothernumber.Thesmallernumberis

A.22

B.12

C.17

D.14

4.Ifthesumoftwoconsecutiveevennumbersis166,thesmallernumberis

A.84

B.86

C.80

D.82

5.Ifthesumoftwoconsecutivenumbersis157,thesmallernumberis

A.84

B.63

C.78

D.61

6.Thetensdigitofatwodigitnumberis4morethantheonesdigitofatwo-digitnumber.If13isaddedtothenumber,theansweris75.Findthenumber.

A.51

B.62

C.15

D.46

7.Thesumofthedigitsofatwo-digitnumberis12.Ifthedigitsarereversed,thenewnumberis36morethantheoriginalnumber.Findthenumber.

A.48

B.57

C.66

D.39

8.Inatwo-digitnumber,theonesdigitis4morethanthetensdigit.Threetimesthenumberis74morethanthenumber.Findthenumber.

A.26

B.59

C.15

D.37

9.Inatwo-digitnumber,thetensdigitis5lessthantheonesdigit.Ifthesumofthedigitsis27lessthantheoriginalnumber,findtheoriginalnumber.

A.27

B.38

C.49

D.61

10.Thesumofthedigitsinatwo-digitnumberis9.Ifthedigitsarereversed,thenewnumberis18morethantwicetheoriginalnumber.Findtheoriginalnumber.

A.27

B.45

C.36

D.72

chapter7SolvingCoinandAgeProblems

Thischapterexplainshowtosolvecoinandageproblems.Coinproblemsconsistofproblemsaboutmetalcoinssuchaspennies,nickels,dimes,etc.,butcouldalsoincludepapermoneyorstamps.Anyproblemsinwhichamoneyvaluecanbeassignedtoobjectscanbesolvedusingthesametechniquesassolvingacoinproblem.Coinproblemscanbesolvedbyusingthevaluesofthecoins,thensettingupandsolvingtheequation.

Ageproblemsusuallyincludefindingtheagesoftwopeople,suchastheageofafatherandhisdaughter.Sometimesageproblemsincludetheperson’spresentageandhispastorfutureage.Theequationissetupusingthisinformation.

CHAPTEROBJECTIVES


•Solveproblemsinvolvingcoins

•Solveageproblems

CoinProblems

Supposeyouhavesomecoinsinyourpocketorwallet.Inordertodeterminetheamountofmoneyyouhave,youmultiplythevalueofeachtypeofcoinbythenumberofcoinsofthatdenominationandthenaddtheanswers.Forexample,ifyouhavesixnickels,fourdimes,andtwoquarters,thetotalamountofmoneyyouhaveinchangeis

6×5¢+4×10¢+2×25¢=30¢+40¢+50¢=120¢or$1.20.

Ingeneral,then,tofindtheamountofmoneyfor

Pennies—multiplythenumberofpenniesby1¢

Nickels—multiplythenumberofnickelsby5¢

Dimes—multiplythenumberofdimesby10¢

Quarters—multiplythenumberofquartersby25¢

Halfdollars—multiplythenumberofhalfdollarsby50¢

Tosolveproblemsinvolvingcoins:

1.Letx=thenumberofonetypeofcoin(i.e.,pennies,nickels,dimes,etc.).Writethenumbersoftheothertypeofcoinsintermsofx.

2.Setuptheequationbymultiplyingthenumberofeachtypeofcoinbythevalueofthecoin.

3.Solvetheequationforx,thenfindtheothernumbers.

4.Checktheanswers.

Ifyouwanttoavoiddecimals,youcanworkwithcentsratherthandollars.Youcanchangedollarstocentsbymultiplyingby100.Youcanchangetheanswerbacktodollarsbydividingby100.

EXAMPLE

Apersonhas16coinsconsistingofquartersandnickels.Ifthetotalamountofthischangeis$2.60,howmanyofeachkindofcoinarethere?

SOLUTION

Goal:Youarebeingaskedtofindthenumberofquartersandthenumberofnickelsthepersonhas.

Strategy:Letx=thenumberofquartersand(16−x)=thenumberofnickels;thenthevalueofthequartersis25xandthevalueofthenickelsis5(16−x).Thetotalamountofmoneyincentsis

.Theequationis25x+5(16−x)=260.


Thereare9quartersand7nickels.

Evaluation:Thevalueof9quartersand7nickelsis9×25¢+7×5¢=225+35=260¢=$2.60.

EXAMPLE

Apersonhasfivetimesasmanypenniesashehasdimesandeightmorenickelsthandimes.Ifthetotalamountofthesecoinsis$1,howmanyofeachkindofcoindoeshehave?

SOLUTION

Goal:Youarebeingaskedtofindthenumberofnickels,pennies,anddimes.

Strategy:Letx=thenumberofdimes,5x=thenumberofpennies,andx+8=thenumberofnickels.Thenthevalueofthedimesis10x,thevalueofthepenniesis ,andthevalueofthenickelsis5(x+8).Thetotalamountis$1×100or100¢.Theequationis .


Thereare3dimes,15pennies,and11nickels.

Evaluation:Thevalueof3dimes,15pennies,and11nickelsis3×10¢+15×1¢+11×5¢=30+15+55=$1.00.

Othertypesofproblemsinvolvingvaluescanbesolvedusingthesamestrategyasthecoinproblems.Considerthenextexample.

EXAMPLE

Apersonbought10candybarsconsistingofcarameltwistscosting$0.88eachandchocolatemarshmallowbarscosting$1.19each.Ifthetotalcostofthecandywas$10.97,findthenumberofeachkindofcandybarthepersonbought.

SOLUTION

Goal:Youarebeingaskedtofindhowmanycarameltwistsandhowmanychocolatemarshmallowcandybarsthepersonbought.

Strategy:Letx=thenumberofcarameltwistsand(10−x)=thenumberofchocolatemarshmallowbars.Sincethecarameltwistscost$0.88each,thevalueofthecarameltwistsis0.88x,andsincethechocolatemarshmallowbarscosts$1.19each,thevalueofthechocolatemarshmallowbarsis$1.19(10−x).Theequationis0.88x+1.19(10−x)=10.97.


Thepersonboughtthreecarameltwistsandsevenchocolatemarshmallowbars.

Evaluation:Threecarameltwistsandsevenchocolatemarshmallowbarscost3×$0.88+7×$1.19=2.64+8.33=$10.97.

StillStrugglingTheprecedingproblemwasworkedoutindollarsratherthanincents.Eitherwayiscorrect.Theequationincentswouldbe88x+119(10−x)=1,097.

TRYTHESE

1.Apersonhastwiceasmanydimesasshehasnickelsandthreemorenickelsthanpennies.Ifthetotalamountofthecoinsis$1.01,findthenumberofeachtypeofcointhepersonhas.

2.Apersonhasfivemorequartersthanpennies.Ifthetotalamountofthecoinsis$2.29,findthenumberofpenniesandquartersthepersonhas.

3.Apersonbought10stampsconsistingof44¢stampsand50¢stamps.Ifthecostofthestampsis$4.64,findthenumberofeachtypeofthestampspurchased.

4.Ifapersonhasfourtimesasmanynickelsasquartersandthetotalamountofmoneyis$1.35,findthenumberofquartersandnickels.

5.AdrugstoresellsabottleofvitaminCfor$3.75andabottleofvitaminEfor$6.29.Ifapersonpurchasedthreebottlesandpaid$13.79,howmanybottlesofeachvitamindidthepersonpurchase?

6.Adairystoresoldatotalof63snowconesandpopsicles.Ifthesnowconescost$1.25eachandthepopsiclescost$0.75eachandthestoremade$62.25,findthenumberofeachsold.

7.Inachild’ssavingsbank,therearefivetimesasmanyquartersashalfdollarsandeightmoredimesthanhalfdollars.Ifthetotalamountofthemoneyinthebankis$6.35,findthenumberofeachtypeofcoininthebank.

8.Aclerkisgiven$120inbillstoputinacashdraweratthestartofaworkday.Therearethreetimesasmany$1billsas$5billsandsixfewer$10billsthan$5bills.Howmanyofeachtypeofbillarethere?

9.Achild’sbankcontains29coinsconsistingofnickelsanddimes.Ifthetotalamountofmoneyis$1.90,findthenumberofnickelsanddimesinthebank.

10.Apileof24coinsconsistsofdimesandnickels.Ifthetotalamountofthecoinsis$1.50,findthenumberofdimesandnickels.

SOLUTIONS

1.Letx=thenumberofnickels,2x=thenumberofdimes,andx−3=thenumberofpennies.Thevalueofthedimesis10·2x=20x,thevalueofthenickelsis5x,andthevalueofthepenniesis1·(x−3).Theequationis20x+5x+(x−3)=101.

2.Letx=thenumberofpenniesandx+5=thenumberofquarters.Thevalueofthepenniesis1xandthevalueofthequartersis25(x+5).Theequationisx+25(x+5)=229.

Therearefourpenniesandninequarters.

3.Letx=thenumberof44¢stampsand10−x=thenumberof50¢stamps.Thevalueofthe44¢stampsis44xandthevalueofthe50¢stampsis50(10−x).Theequationis44x+50(10−x)=464.

Therearesix44¢stampsandfour50¢stamps.

4.Letx=thenumberofquartersand4x=thenumberofnickels.Thevalueofthequartersis25xandthevalueofthenickelsis5·4xor20x.Theequationis25x+20x=135.

Thereare3quartersand12nickels.

5.Letx=thenumberofbottlesofvitaminCand3−x=thenumberofbottlesofvitaminE.ThevitaminCcosts375x,andthevitaminEcosts629(3−x).Theequationis375x+629(3−x)=1,379

TherearetwobottlesofvitaminCandonebottleofvitaminE.

6.Letx=thenumberofsnowconessoldand63−x=thenumberofpopsiclessold.Thesnowconescost125xandthepopsiclescost75(63−x).Theequationis125x+75(63−x)=6,225.

Thestoresold30snowconesand33popsicles.

7.Letx=thenumberofhalfdollars,5x=thenumberofquarters,andx+8=thenumberofdimes.Thevalueofthehalfdollarsis50x.Thevalueofthequartersis25·5x=125x,andthevalueofthedimesis10(x+8).Theequationis50x+125x+10(x+8)=635.

Thereare3halfdollars,15quarters,and11dimes.

8.Letx=thenumberof$5bills.Let3x=thenumberof$1bills,andx−6=thenumberof$10bills.Thevalueofthe$5billsis5x.Thevalueofthe$1billsis ,andthevalueofthe$10billsis10(x−6).Theequationis5x+3x+10(x−6)=120.

Thereare10$5bills,30$1bills,and4$10bills.

9.Letx=thenumberofnickelsand29−x=thenumberofdimes.Thevalueofthenickelsis5x,andthevalueofthedimesis10(29−x).Theequationis5x+10(29−x)=190.

Thereare20nickelsand9dimes.

10.Letx=thenumberofdimesand24−x=thenumberofnickels.Thevalueofthedimesis10xandthevalueofthenickelsis5(24−x).Theequationis10x+5(24−x)=150.

Thereare6dimesand18nickels.

Inthissection,youlearnedhowtosolvecoinproblems.Thetechniqueistosetuptheequationbyrepresentingthenumberofcoinsusingxandmultiplyingeachnumberofcoinsbytheirnumericalvalues:1¢forpennies,5¢fornickels,10¢fordimes,25¢forquarters,and50¢forhalfdollars.

AgeProblems

Whenyouencounteranageproblem,youwilloftenseethattheproblemgivesyouinformationabouttheageofapersoninthefutureorinthepast.Forexample,ifamotheristhreetimesasoldasherdaughter,theirpresentagescanberepresentedas

Letx=thedaughter’sageand

3x=themother’sage

Now,iftheproblemgivesyouinformationabouttheiragessevenyearsfromnow,youcanrepresenttheirfutureagesas

Letx+7=thedaughter’sfutureageand

3x+7=themother’sfutureage

Likewise,iftheproblemgivesyousomeinformationabouttheirages,10yearsago,youcanrepresenttheirpastagesas

Letx−10=thedaughter’spastageand

3x−10=themother’spastage

Thebasicstrategyforsolvingageproblemsistorepresentthepresentagesofthepeople,representthepastorfutureagesofthepeople,andthensetuptheequationandsolveit.

EXAMPLE

Afatherissixtimesasoldashisson;in20years,hewillbetwiceasoldashisson.Findtheirpresentages.

SOLUTION

Goal:Youarebeingaskedtofindthepresentagesofthefatherandhisson.

Strategy:Letx=theson’spresentageand6x=thefather’spresentage.In20years,theson’sagewillbex+20andthefather’sagewillbe6x+20.Ifthefatherwillbetwiceasoldashissonin20years,theequationistwotimestheson’sagein20years=thefather’sagein20yearsor2(x+20)=6x+20.


Evaluation:In20years,theson’sagewillbe5+20=25andthefather’sagewillbe30+20=50.Since,thefatherwillbetwiceasoldastheson.

EXAMPLE

Eliisnineyearsolderthanhissister.Insixyears,Eliwillbetwiceasoldashissister.Findtheirpresentages.

SOLUTION

Goal:YouarebeingaskedtofindthepresentagesofEliandhissister.

Strategy:Letx=Eli’ssister’sageandx+9=Eli’sage.Insixyears,theirageswillbex+6=Eli’ssister’sageand(x+9)+6=Eli’sage.Insixyears,EliwillbetwiceasoldmeanstwotimesEli’ssister’sageinsixyears=Eli’sagein6yearsor2(x+6)=(x+9)+6.


Evaluation:Insixyears,Eli’ssisterwillbe3+6=9years,andEliwillbe12+6=18,whichistwicehissister’sage.

EXAMPLE

Sarahis11yearsolderthanBeth.Ifthesumoftheiragesis67,findeachone’sage.

SOLUTION

Goal:YouarebeingaskedtofindtheagesofSarahandBeth.

Strategy:Letx=Beth’sageandx+11=Sarah’sage.Thenthesumoftheiragesisx+x+11=67.


Evaluation:Sarahis11yearsolderthanBeth,andthesumoftheiragesis39+28=67.

EXAMPLE

Amotheris36yearsoldandherdaughteris14yearsold.Inhowmanyyearswillthemotherbetwiceasoldasherdaughter?

SOLUTION

Goal:Youarebeingaskedtofindthenumberofyearsitwillbeuntilthemotheristwiceasoldasherdaughter.

Strategy:Letx=thenumberofyears.Thenthemother’sageinxyearswillbe36+xyears,andthedaughter’sageinxyearswillbe14+xyears.Ifthemotheristwiceasoldasthedaughterinxyears,theequationis2(14+x)=36+x.


2(14+x)=36+x28+2x=36+x

28+2x−x=36+x−x28+x=36

28−28+x=36−28x=8

Hence,ineightyearsthemotherwillbetwiceasoldasherdaughter.

Evaluation:Ineightyears,themotherwillbe36+8=44yearsold,andthedaughterwillbe14+8=22years,inwhichcasethemotherwillbetwiceasoldasherdaughter.

TRYTHESE

1.Mikeis32andJoanis22.HowmanyyearsagowasMiketwiceasoldasJoan?

2.BethiseightyearsolderthanMegan.Elevenyearsago,BethwasthreetimesasoldasMegan.Findtheirpresentages.

3.ThesumofJudyandSam’sagesis66.JudywastwiceasoldasSam15yearsago.Findtheirpresentages.

4.Afatheristhreetimesasoldashisdaughter.Fifteenyearsago,hewasninetimesasoldashisdaughter.Howoldaretheynow?

5.Awomanisfivetimesasoldasherneighbor’sson.In24years,shewillbetwiceasoldastheson.Howoldaretheynow?

6.BartisthreeyearsolderthanBret.Insevenyears,BartwillbetwiceasoldasBretwasoneyearago.Findtheirpresentages.

7.Afatherisfourtimesasoldashistwinsons.Ifthesumoftheiragesinthreeyearswillbe75,howoldaretheynow?

8.CindyissixyearsolderthanMindy.Fouryearsfromnow,CindywillbethreetimesasoldasMindywastwoyearsago.Findtheirpresentages.

9.SidisfiveyearsolderthanhisbrotherTim.Ifthesumoftheiragesis37,howoldaretheynow?

10.Pamiseightyearsolderthanherbrother.Fouryearsfromnow,thesumoftheirageswillbe30.Findtheirpresentages.

SOLUTIONS

1.Letx=thenumberofyearsagothatMikewastwiceasoldasJoan.32−xwasMike’sageatthattime,and22−xwasJoan’sageatthattime.SinceMikewastwiceasoldasJoan,theequationis32−x=2(22−x).

32−x=2(22−x)

32−x=44−2x

32−x+2x=44−2x+2x

32+x=44

32−32+x=44−32

x=12

Hence,12yearsago,MikewastwiceasoldasJoan.Thatis,Mikewas32−12=20yearsoldandJoanwas22−12=10yearsold.

2.Letx=Megan’sageandx+8=Beth’sage.Elevenyearsago,Megan’sagewasx−11andBeth’sagewasx+8−11=x−3.Atthattime,BethwasthreetimesasoldasMegan,sotheequationisx−3=3(x−11).

3.Letx=Judy’sageand66−x=Sam’sage.Fifteenyearsago,Judy’sagewouldhavebeenx−15andSam’sagewouldhavebeen66−x−15or51−x.SinceJudywastwiceasoldasMike,theequationisx−15=2(51−x).

4.Letx=thedaughter’sageand3x=thefather’sage.Fifteenyearsago,thedaughter’sagewasx−15,andthefather’sagewas3x−15.Sincethefatherwasninetimesasoldasthedaughter,theequationis3x−15=9(x−15).

5.Letx=theson’sageand5x=thewoman’sage.In24years,thesonwillbex+24yearsold,andthewomanwillbe5x+24yearsold.Sinceshewillbetwiceasoldastheson,theequationis5x+24=2(x+24).

6.Letx=Bret’sageandx+3=Bart’sage.Insevenyears,Bart’sagewillbex+3+7orx+10.SinceBartwillbetwiceasoldasBretwasoneyearago,theequationisx+10=2(x−1).

7.Letx=eachtwin’sageand4x=thefather’sage;inthreeyears,eachtwinwillbex+3yearsoldandthefather’sagewillbe4x+3.Sincethesumoftheiragesis75,theequationisx+3+x+3+4x+3=75.

8.Letx=Mindy’sageandx+6=Cindy’sage.Infouryears,Cindywillbex+6+4=x+10yearsold,andMindy’sagetwoyearsagowasx−2.SinceCindywillbethreetimesasoldasMindywastwoyearsago,theequationisx+10=3(x−2).

9.Letx=Tim’sageandx+5=Sid’sage.Ifthesumoftheiragesis37,thentheequationisx+x+5=37.

10.LetPam’sage=x+8andherbrother’sage=x.Infouryears,Pamwillbex+8+4=x+12yearsold,andherbrotherwillbex+4yearsold.Theequationisx+4+x+12=30,sincethesumoftheirageswillbe30.

Summary

Inthissection,youhavelearnedhowtosolveageproblems.Thekeytothesolutionistoletx=oneperson’sage,thenrepresenttheotherperson’sageintermsofx.Setuptheequationusingbothagesandtheconditionorconditionsgivenintheproblemandsolve.Besuretocheckyouranswers.

Thischapterexplainedhowtosolvecoinandageproblems.

QUIZ

1.Apersonhasthreemorequartersthannickels.Ifthetotalamountofmoneyis$1.95,findthenumberofnickelsthepersonhas.

A.2

B.4

C.6

D.8

2.Apersonhas11coinsinhispocketconsistingofdimesandquarters.Ifhehasonemorequarterthandimesandatotalof$2,howmanydimesdoeshehave?

A.6

B.4

C.3

D.5

3.Amoneyboxcontainssixmorepenniesthannickelsandsevenmoredimesthannickels.Ifthetotalamountofmoneyinthebankis$1.72,findthenumberofdimesinthebank.

A.4

B.6

C.13

D.12

4.Apersonhas15billsconsistingof$1billsand$5bills.Ifthetotalamountofmoneythepersonhasis$43,findthenumberof$5billsthepersonhas.

A.5

B.7

C.10

D.15

5.Apersonhastwiceasmanypenniesashehasquarters,andhehasfivefewerdimesashehaspennies.Ifhehasatotalof$1.38,howmanypenniesdoeshehave?

A.8

B.3

C.4

D.6

6.ThesumofBillandLonny’sagesis52.Sixyearsago,BillwasthreetimesasoldasLonny.FindBill’spresentage.

A.32

B.36

C.4

D.6

7.Bob’sbrotheris10yearsolderthanBob.Ifthesumoftheiragesis26,findBob’sage.

A.10

B.15

C.8

D.6

8.Maryisfourtimesasoldasheryoungersister.In10years,shewillbetwiceasoldashersister.HowoldisMarytoday?

A.15

B.20

C.25

D.30

9.ThesumofBrooke’sageandherbestfriend’sageis51,andthedifferenceintheiragesis3.Brookeistheolder.HowoldisBrooke?

A.24

B.25

C.26

D.27

10.Carrieisthreeyearsyoungerthanherhusband.Insevenyears,thesumoftheirageswillbe101.HowoldisCarrie?

A.36

B.50

C.32

D.42

chapter8SolvingDistanceandMixtureProblems

Thischapterexplainshowtosolvedistanceproblemsandmixtureproblems.Distanceproblemsusuallyinvolvetwovehiclesmovingeitherinthesamedirectionoroppositedirectionsandatdifferentspeeds.Also,theseproblemscouldincludeboatstravelingupanddownastreamtakingintoaccountthespeedofthecurrent,orairplanesflyingwithoragainstthewind.

Mixtureproblemsinvolvemixingtwosolutionsorsolidstogetathirdmixtureconsistingofbothitems.Mixtureproblemscouldalsoincludedilutingsolutions—thatis,makingthemweaker.

CHAPTEROBJECTIVES


•solvedistanceproblems

•solvemixtureproblems

DistanceProblems

ThebasicformulaforsolvingdistanceproblemsisDistance=Rate×TimeorD=RT.Forexample,ifanautomobiletravelsat30milesperhourfor2hours,thenthedistanceisD=RT=30×2=60miles.

Distanceproblemsusuallyinvolvetwovehicles(i.e.,automobiles,trains,bicycles,etc.)eithertravelinginthesamedirectionorinoppositedirectionsoronevehiclemakingaroundtrip.Theprocedureforsolvingdistanceproblemsis

1.Drawadiagramofthesituation.

2.Setupatableasshown.

3.Fillintheinformationinthetable.

4.Writeanequationforthesituation,andsolveit.

EXAMPLE

Apersonrodehisbikeonabiketrailatarateof10milesperhour.Whileonthetrail,hehadaflattireandhadtowalkbacktohisautomobileatarateof2milesperhour.Ifthetotaltimehetraveledwas2.4hours,howfardidheride?

SOLUTION

Goal:Youarebeingaskedtofindthedistancethatthepersonrodeuntilhegotaflattire.

Strategy:Thedistancehewalkedandrodearethesame,butthedirectionsaredifferent.SeeFigure8-1.

FIGURE8-1

Placetherateforriding10milesperhourandrateforwalking2milesperhourintheboxesunderRate.Lettbethetimeherodeand2.4−tbethetimethathewalked.PlacetheseintheboxesunderTime.Togetthedistance,multiplytheratesbythetimesandplacetheseexpressionsintheboxesunderDistance,asshown.

Sincethedistancesareequal,theequationis10t=2(2.4−t).


ThedistanceherodeisD=RTorD=10×0.4=4miles.

Evaluation:Youcanchecktheanswerbydeterminingthedistancethepersonwalked.

D=RT

D=2(2.4−0.4)

=2(2)

=4miles

Inthepreviousexample,thesamepersonmadearoundtrip.Inthenextexample,wehavetwovehiclesgoinginthesamedirection.

EXAMPLE

AboatercantravelfromPortClintontoSmithtonin3hours.Ifhegoes5milesperhourfaster,hecantravelthesamedistancein45minutesless.HowfarisitfromPortClintontoSmithton?(Ignorethecurrent.)

SOLUTION

Goal:Youarebeingaskedthedistancebetweenthetwoports.

Strategy:Inthiscase,bothtripsareinthesamedirection.SeeFigure8-2.LetR=therateonthefirsttripandR+5betherateonthesecondtrip.PlacethesevaluesinthetableunderRate.

FIGURE8-2

Placethetimes,3hoursand hours,inthetimeboxes.(Note .)

Togetthedistance,multiplytheratebythetimeforeachcase.

Sincethedistancesarethesame,theequationis .


Tofindthedistance,usetheformulaD=RT.

D=15×3=45miles

HencethedistancebetweenPortClintonandSmithtonis45miles.

Evaluation:Checktoseeiftheotherdistanceisthesame.

Anothertypeofdistanceproblemisonewheretwovehiclesaregoingintheoppositedirection.

EXAMPLE

Twohikers12.5milesapartbeginbywalkingtowardeachother,andtheymeetin2.5hours.Ifonehikerwalksonemilefartherinanhourthantheother,howfastdoeseachhikerwalk?

SOLUTION

Goal:Youarebeingaskedtofindthespeedinmilesperhouratwhicheachhikerwalks.

Strategy:Drawadiagramshowingthateachpersonwaswalkingtowardtheotherorintheoppositedirections.SeeFigure8-3.Letxbetherateofthefirsthikerandx+1betherateofthesecondhiker.PlacetheseexpressionsintheboxesunderRate.Thetimebothhikerswalkis2.5hours.PlacethisvalueintheboxesunderTime.ThenunderDistance,write2.5xand2.5(x+1).

FIGURE8-3

Sincethesumofthedistancesis12.5miles,theequationis2.5x+2.5(x+1)=12.5.

Implementation:Solvetheequationforx:

Evaluation:Inordertochecktheanswer,findthedistanceseachhikerwalked,thenseeifthesumisequalto12.5miles.

TRYTHESE

1.Agirlrantoherfriend’shomeattherateof5milesperhourandwalkedhomeattherateof3milesperhour.Ifittook12minutesfortheroundtrip,howfarawayisherfriend’shouse?

2.Onatrip,amotoristtravelsanaverageof30milesperhourintownand60milesperhouronthefreeway.Ifatripof60milestookhimanhourandahalf,howmanymilesdidhedriveonthefreeway?

3.AfreighttrainandanAMTRAKtrainleavetownsthatare450milesapartandtraveltowardeachother.Theypasseachotherin5hours.TheAMTRAKtraintravels20milesperhourfasterthanthefreighttrain.Wheredotheymeet?

4.Onepersonridingamotorcycleleaves90minutesafteranotherpersonridinganothermotorcycleleavefromthesameplace,ridinginthesamedirection.Ifthepersonridingthefirstmotorcycletravels30milesperhourandthepersonridingthesecondmotorcycletravelsat40milesperhour,howlongwillittakethesecondpersontoovertakethefirstperson?

5.Apersonridingamotorcycleleavesacityatthesametimeasanotherpersondrivinganautomobile.Theytravelinoppositedirections.Ifthepersonridingthemotorcycleistraveling25milesperhourandthepersondrivingtheautomobileistraveling45milesperhour,howlongwillitbebeforetheyare280milesapart?

SOLUTIONS

1.Minutesmustbechangedtohourssincetheratesaregiveninmilesperhour.Twelveminutes= houror0.2hour.Lett=thetimeittookthegirltoruntoherfriend’shouseand0.2−t=thetimeittookhertowalkbackhome.

Thedistancesareequalsincesheismakingaroundtrip.

2.Lett=thetimethedriverdroveintownand1.5−t=thetimethedriverdroveonthefreeway(onehourandahalf=1.5hours).

Sincethetotaldistanceis60miles,theequationis30t+60(1.5−t)=60.

3.Letx=therate(speed)ofthefreighttrainandx+20=thespeedoftheAMTRAKtrain.

Sincetheymeetatsomepoint,thetotaldistancebothtravelis450miles.

4.Lett=thetimethefirstmotorcycletravelsandt−1.5bethetimethesecondmotorcycletravels.Ninetyminutes

hours.

Theytravelthesamedistance.

Thesecondmotorcyclewillovertakethefirstmotorcycle4.5hoursafterstarting.

5.Lett=thetimebothdriverstravel.

Thetotaldistancethattheytravelis280miles.

Thetotaldistancethattheytravelis100+180=280miles.

Themotorcycleandtheautomobileare280milesapartafter4hours.

Anothertypeofdistanceprobleminvolvesanairplaneflyingwithoragainstthewindoraboatmovingwithoragainstthecurrent.Ifanairplaneisflyingwithaheadwind,thespeedoftheairplaneissloweddownbytheforceofthewind.Ifanairplaneisflyingwithatailwind,thespeedoftheairplaneisincreasedbythewind.Forexample,ifanairplaneisflyingatanairspeedof150milesperhourandthereisa30mile-per-hourtailwind,thenthegroundspeedoftheairplaneisactually150+30=180milesperhour.Theairspeedisthespeedoftheplaneasshownonitsspeedometer,butifyouwerestandingontheground,youwouldclockthespeedat180milesperhour.Iftheplanehadanairspeedof150milesperhouranditwasflyingwithaheadwindof30milesperhour,thegroundspeedoftheairplanewouldbe150−30=120milesperhour.Inordertosolvetheseproblemsusingalgebra,thedirectionofthewindmustbeparalleltothedestinationoftheairplane.Whenthissituationisnottrue,trigonometrymustbeused.

Inasimilarsituation,ifaboatismovingdownstreamat25milesperhour(indicatedonitsspeedometer)andthecurrentis3milesperhour,thentheactualspeedoftheboatis25+3=28milesperhoursincethecurrentisactuallypushingtheboat.Iftheboatisgoingupstreamagainstthecurrent,thenthecurrentispushingagainsttheboatholdingitback.Inthiscase,thespeedoftheboatis25−3=22milesperhour.

EXAMPLE

AnairplanefliesfromPittsburghtoPhiladelphiain2hoursandreturnsin2.5hours.Ifthewindspeedis15milesperhourblowingfromthewest,findtheairspeedoftheplane.

SOLUTION

Goal:Youarebeingaskedtofindtheairspeedoftheplane.

Strategy:Letx=theairspeedoftheplane.SincePhiladelphiaiseastofPittsburghandthewindisblowingfromwesttoeast,thegroundspeedfromPittsburghtoPhiladelphiaisx+15.ThegroundspeedfromPhiladelphiatoPittsburghisx−15.Thetimesaregiven.

Sincethedistancesareequal,theequationis2(x+15)=2.5(x−15).


Evaluation:Checktoseeifthedistancesarethesame.UseD=RT.

EXAMPLE

Aboat’sspeedometerreads22milesperhourgoingdownstreamandreachesitsdestinationinanhour.Ifthereturntriptakesoneandahalfhoursatthespeedof25milesperhour,howfastisthecurrent?

SOLUTION

Goal:Youarebeingaskedtofindthespeed(rate)ofthecurrent.

Strategy:Letx=therateofthecurrent;thenthespeedoftheboatdownstreamis22+xandupstreamis25−x.Thetimesaregiven.


Evaluation:Checktoseeifthedistancegoingdownstreamisequaltothedistancegoingupstream,usingD=RT.

TRYTHESE

1.Aplaneflieswithaheadwindof27milesperhourfromLeMonttoPleasantvillein5hoursandreturnsin3.3

hourswithatailwindof27milesperhour.Findthedistancebetweentheairports.

2.AplanefliesfromNewEagletoSouthPinein3hoursandreturnsin5hours.Ifthespeedofthewindis25milesperhouranditisblowinginthedirectionofSouthPinefromNewEagle,findtheairspeedoftheplane.

3.Aboat’sspeedonitsspeedometerreads12milesperhourgoingdownstream,anditreachesitsdestinationin1.6hours.Thereturntriptakes3hoursat10milesperhouronthespeedometer.Findthespeedofthecurrent.

4.IfaplanefliesfromUnitytoSouthChesterin6hourswithaheadwindof24milesperhourandreturnsin4.2hourswithatailwindof18milesperhour,findtheairspeedoftheplane.

5.IfaboattravelsupstreamfromAllentowntoBolderCityin3hoursandreturnsdownstreamfromBolderCitytoAllentownin1.8hours,findthespeedoftheboat(onitsspeedometer)ifthecurrentis2milesperhour.

SOLUTIONS

1.Letx=theairspeedoftheplane.

Tofindthedistancebetweentheairports,find5(x−27)

5(x−27)=5(131.82−27)

=5(104.85)

=524.25

Theairportsare524.25milesapart.

2.Letx=theairspeedoftheplane.

3.Letx=thespeedofthecurrent.

4.Letx=theairspeedoftheairplane.

Thedistancesarethesame.

5.Letx=thespeedoftheboat.

Thedistancesarethesame.

Inthissection,youlearnedhowtosolvewordproblemsinvolvingdistance.Youusethebasicformula,distance=rate×time.

MixtureProblems

Manyreal-lifeproblemsinvolvemixtures.Therearethreebasictypesofmixtureproblems.Onetypeusespercents.Forexample,ametalworkermaywishtocombinetwoalloysofdifferentpercentagesofcoppertomakeathirdalloyconsistingofaspecificpercentageofcopper.Inthiscase,itisnecessarytorememberthatthepercentofthespecificsubstanceinthemixturetimestheamountofmixtureisequaltotheamountofthepuresubstanceinthemixture.Supposeyouhave64ouncesofamixtureconsistingofalcoholandwater,and30%ofitisalcohol.Then30%of64ouncesor19.2ouncesofthemixtureisalcohol.Anothertypeofprobleminvolvesdilutingsolutions.Finally,mixtureproblemscanalsoincludemixingnuts,candies,etc.Thesetypesofproblemsareexplainedinthissection.

Atablecanbeusedtosolvethepercentmixtureproblemsandanequationcanbewrittenusing

Mixture1+Mixture2=Mixture3

Note:Thewordmixtureappliestoalloy,solution,etc.

EXAMPLE

Ametallurgisthastwoalloysofcopper.Thefirstoneis40%copperandthesecondoneis70%copper.Howmanyouncesofeachmustbemixedtohave24ouncesofanalloythatis50%copper?

SOLUTION

Goal:Youarebeingaskedtofindhowmuchofeachalloyshouldbemixedtoget24ouncesofanalloythatis50%copper.

Strategy:Letx=theamountofthe40%copperalloyand24−x=theamountofthe70%copperalloy;then

Theequationis


40%(x)+70%(24−x)=50%(24)

Changethepercentstodecimalsbeforesolvingtheequation.

Hence,16ouncesofthe40%alloyshouldbemixedwith8ouncesofthe70%alloytoget24ouncesofanalloythatis50%copper.

Evaluation:Checkthesolution:

40%(x)+70%(24−x)=50%(24)0.40(16)+0.70(24−16)=0.50(24)

6.4+5.6=1212=12

EXAMPLE

Apharmacisthastwobottlesofalcohol;onebottlecontainsa60%solutionofalcoholandtheotherbottlecontainsa85%solutionofalcohol.Howmuchofeachshouldbemixedtoget30ouncesofasolutionthatis75%alcohol?

SOLUTION

Goal:Youarebeingaskedtofindtheamountsofeachsolutionthatneedtobemixedtoget30ouncesofa75%alcoholsolution.

Strategy:Letx=theamountofthe60%solutionand30−x=theamountofthe45%solution;thensetupatableasfollows:

Theequationis


60%x+85%(30−x)=75%(30)

Changethepercentstodecimalsbeforesolvingtheequation.

Hence,12ouncesofthe60%solutionshouldbemixedwith18ouncesofthe85%solutiontoget30ouncesofa75%solution.


60%x+85%(30−x)=75%(30)

60%(12)+85%(30−12)=75%(30)

0.60(12)+0.85(18)=0.75(30)

7.2+15.3=22.5

22.5=22.5

Thesecondtypeofmixtureprobleminvolvesdilutingasolutionoralloy.Inthesetypesofproblemsyouareaddingaweakersolutionoralloytobringdowntheconcentrationofthesubstance.Hereyouletxbetheamountoftheweakersolutionoralloythatisbeingaddedtotheoriginalsolution.Again,theequationis

Mixture1+Mixture2=Mixture3

EXAMPLE

Howmuchwaterneedstobeaddedto32ouncesofa30%alcoholsolutiontodiluteittoa20%alcoholsolution?

SOLUTION

Letx=theamountofwaterthatneedstobeadded.Sincethereisnoalcoholinpurewater,thepercentis0%.

Hence,16ouncesofwatermustbeaddedtothe30%solutiontogetasolutionthatis20%alcohol.


30%(32)+0%x=20%(32+x)

0.30(32)+0=0.20(32+16)

0.30(32)=0.20(48)

9.6=9.6

Thethirdtypeofmixtureproblemconsistsofmixingtwoitemssuchascoffees,teas,candy,etc.,withdifferentprices.Theseproblemsaresimilartothepreviousones.Youcanusethisbasicequation:(Amountofitem1)(Itsprice)+(Amountofitem2)(Itsprice)=(Mixtureamount)(Itsprice)

EXAMPLE

Amerchantmixessomecandycosting$6apoundwithsomecandycosting$2apound.Howmuchofeachmustbeusedinordertomake25poundsofmixturecosting$4perpound?

SOLUTION

Goal:Youarebeingaskedtofindhowmuchofeachcandymustbemixedtogethertoget25poundsofcandycosting$4.

Strategy:Letx=theamountofthe$6candyand25−x=theamountofthe$2candy;then

Theequationis6x+2(25−x)=4(25).


Hence,12.5poundsofcandycosting$6perpoundmustbemixedwith12.5poundsofcandycosting$2apoundtoget25poundsofcandycosting$4apound.


6x+2(25−x)=4(25)

6(12.5)+2(25−12.5)=4(25)

75+25=100

100=100

TRYTHESE

1.Astoreownerwantstomixsomefudgethatsellsfor$4.50apoundwithsomefudgethatsellsfor$6apound.Howmuchofeachkindoffudgemusthemixinordertogeta25-poundmixturethatsellsfor$5.50apound?

2.Howmuchofasolutionthatcontains40%alcoholmustbemixedwithasolutionthatcontains72%alcoholtoget600millilitersofasolutionthatis54%alcohol?

3.Achemisthas15%and25%solutionsofglycerolandalcohol.Howmuchofeachshouldbemixedtoget10ouncesofa22%solution?

4.Howmanyouncesofwatermustbeaddedto32ouncesofa60%alcoholsolutiontodiluteittoa40%solution?

5.Agrocerwantstosellsomenutsfor$3apound.Howmanypoundsofnutsthatsellfor$5apoundshouldbemixedwithnutsthatsellfor$2apoundtogetamixtureof24poundsofnutsthatsellfor$3apound?

6.Agoldsmithwantstomake50ouncesofagoldalloythatis48%goldbymixinganalloythatcontains60%goldwithonethatcontains25%gold.Howmanyouncesofeachtypeshouldbemixed?

7.Acandymakerwantstomake50one-poundboxesofmixedcandythatsellfor$2abox.Hehasonhand20poundsofcandythatsellsfor$1.50apound.Whatshouldbethepriceoftheothercandythathewilluse?

8.Abakerwantstomix10poundsofcookiescosting$2apoundwithsomecookiescosting$3.50apound.Howmanypoundsofthe$3.50cookiesshouldbemixedwiththe10poundsof$2cookiestogetamixtureofcookiescosting$2.75apound?

9.Amerchantwantstosellsometeacosting$4apound.Shehas15poundsofteacosting$2.50apound.Howmanypoundsofteacosting$5perpoundshouldshemixwith15poundsofthe$2teatogetamixturecosting$4apound?

10.Howmuchofanalloythatis60%zincshouldbeaddedto120poundsofanalloythatis40%zinctogetanalloythatis54%zinc?

SOLUTIONS

1.Letx=theamountoffudgethatsellsfor$4.50apoundand(25−x)=theamountoffudgethatsellsfor$6apound.

Hence,togetthepropermixture,thestoreownershouldmix poundsofthe$4.50fudgewithpoundsofthe$6fudge.

2.Letx=theamountofthe40%solutionand(600−x)=theamountofthe72%solution.

Hence,337.5millilitersofthe40%solutionmustbemixedwith262.5milliliterstoget600millilitersofa54%solution.

3.Letx=theamountofthe15%solutionand(10−x)=theamountofthe25%solution.

Hence,thechemistwouldhavetomix3ouncesofthe15%solutionand7ouncesofthe25%solutiontoget10ouncesofthe22%solution.

4.Letx=theamountofwatertobeaddedtothesolution.A60%solutionofalcoholis40%water(100%−60%).

Hence,if16ouncesofwaterisaddedtoasolutionthatis60%alcohol,itwilldiluteittoasolutionthatis40%alcohol.

5.Letx=theamountofnutsthatsellfor$5apoundand24−x=theamountofnutsthatsellfor$2apound.

Hence,thegrocershouldmix8poundsofthe$5mixand16poundsofthe$2mixtoget24poundsofmixednutsthatsellfor$3apound.

6.Letx=theamountofthealloythatis60%goldand50−x=theamountofthealloythatis25%gold.

Hence,thegoldsmithshouldmix ouncesofthe60%alloywith ouncesofthe25%alloytoget50ouncesofa48%goldalloy.

7.Letx=thepriceofthemixturethathewilluse.Sincehehas20pounds,hewillneed30poundsoftheothermixture(50−20=30).

Hence,hewillneed30poundsofamixturethatcosts$2.33apound.

8.Letx=theamountofthe$3.50cookies.

Hence,thebakershouldadd10poundsofcookiesthatcost$3.50.

9.Letx=theamountofteacosting$5apound.

Hence,shemustmix22.5poundsofteacosting$5.

10.Letx=theamountofthe60%zincalloythatistobeadded.

Hence,280poundsof60%alloyshouldbeadded.

Mixtureproblemscanbesolvedbyusingthebasicequation:Mixture1+Mixture2=Mixture3.Mixtureproblemscanalsoincludetypesofproblemswhereastrongmixturemustbedilutedtomakeaweakerone.

Summary

Thischapterexplainedhowtosolvedistanceandmixtureproblems.

QUIZ

1.Aboattravelsdownstreamtoaparkinthreehoursandreturnstoitsdockinfivehours.Ifthecurrentis6milesperhour,findthespeedoftheboatonitsspeedometer.

A.24milesperhour

B.18milesperhour

C.26milesperhour

D.20milesperhour

2.EvelynandJillleavetheirofficeatthesametimeandtravelinoppositedirections.IfJilldrives8milesperhourfasterthanEvelyn,theywillbe184milesapartaftertwohours.HowfastwasJilldriving?

A.40milesperhour

B.42milesperhour

C.48milesperhour

D.50milesperhour

3.Maryleavesforatripdrivingat52milesperhour.One-halfhourlater,Bethleavesonthesameinterstatehighwaytraveling62milesperhour.HowmanymileswillBethhavetodrivebeforesheovertakesMary?

A.138.6miles

B.161.2miles

C.147.4miles

D.153.8miles

4.MikeleaveshishouseforBentleyville,whichis200milesaway.After3hours,hestopsforlunchfor30minutes;thenhedrives10milesslowerfortherestofthetrip.Ifthetriptakes5hours,whatwashisbeginningspeed?

A.41.5milesperhour

B.48.5milesperhour

C.47.8milesperhour

D.63.5milesperhour

5.Bobbikesonatrailatanaveragespeedof12milesperhour.HisfriendRuthbikesatanaveragespeedof10milesperhour.Iftheystartfromoppositeendsofa33-miletrail,howfarfromBob’sstartingplacewilltheymeet?

A.20miles

B.18miles

C.22miles

D.15miles

6.Achemistwantstomakea30-ouncesolutionofalcoholandwaterthatis48%alcohol.Howmuchofa30%alcoholsolutionshouldbemixedwitha60%alcoholsolution?

A.12ounces

B.8ounces

C.15ounces

D.10ounces

7.Ahardwarestoreownerwantstomixsomenailscosting$4apoundwithsomenailscosting$2.50apoundtoget30poundsofnailscosting$3apound.Howmanypoundsof$4nailswillheuse?

A.20pounds

B.16pounds

C.10pounds

D.8pounds

8.Howmuchmilkthatcontains5%butterfatmustbemixedwithmilkcontaining15%butterfattoget100gallonsofmilkthatis9%butterfat?

A.40gallons

B.32gallons

C.54gallons

D.60gallons

9.Afloralshopmanagerwantstomake10bouquetsofrosesanddaisiestosellfor$18abouquet.Iftherosessellfor$25abouquetandthedaisiessellfor$15abouquet,howmanybouquetsofroseswillthemanagerneed?

A.12bouquets

B.15bouquets

C.3bouquets

D.8bouquets

10.Howmanyquartsofanicedteadrinkthatsellsfor$2aquartmustbemixedwithalemonadedrinkthatsellsfor$1.20aquarttoget12quartsoflemonade/icedteadrinkthatwillsellfor$1.50aquart?

A.5quarts

B.6quarts

C.4.5quarts

D.8.5quarts

chapter9SolvingFinance,Lever,andWorkProblems

Thischapterexplainshowtosolvefinanceproblems,leverproblems,andworkproblems.Financeproblemsinvolveinvestingmoneyatspecificinterestratesandreceivingtheinterestfromtheseinvestments.Leverproblemsinvolveplacingpeopleorweightsonaboardthatsitsonafulcruminordertobalancetheboard.Ifthe

weightsaredifferentfromeachother,theycanbeplacedatvariousdistancesfromthefulcruminordertobalancethelever.Acommonuseoftheleverisachild’sseesaw.

Workproblemsinvolvetwoormorepeopleperformingajob.Eachpersonworksatadifferentrate.Whenthepeopleworktogether,thejobwilltakelesstimethantheworkersdoingtheentirejobalone.Theseproblemscouldalsoincludetwopipesfillingordrainingatankatthesametime.

CHAPTEROBJECTIVES


•solvefinanceproblems

•solveleverproblems

•solveworkproblems

FinanceProblems

Financeproblemsusethebasicconceptsofinvestment.Therearethreetermsthatareused.Theinterest,alsocalledthereturn,istheamountofmoneythatismadeonaninvestment.Theprincipalistheamountofmoneyinvested,andtherateorinterestrateisapercentthatisusedtocomputetheinterest.Thebasicformulais

Interest=Principal×Rate×TimeorI=PRT.Intheseproblems,theinterestusediscalledsimpleinterest,anditistheinterestforoneyear.Theproblemscanbesetup

usingatablesimilartotheonesusedinthepreviouslessons.Theequationisderivedfromthefollowing:Interestfromfirstinvestment+Interestfromsecondinvestment=Totalinterest.Note:Interestratesvaryfromtimetotime;however,itdoesn’tmatterwhattheratesare,theproblemsaredoneinthe

sameway.Inordertomakethematerialunderstandable,ratesbetween2%and10%havebeenused.Itistheprocedurethatisimportant,notthenumbers.

EXAMPLE

Apersonhas$8,000toinvestanddecidestoinvestpartofitat6%andtherestofitat .Ifthetotalinterestfortheyearfromtheamountsinvestedis$435,howmuchdoesthepersonhaveinvestedateachrate?

SOLUTION

Goal:Youarebeingaskedtofindtheamountsofmoneyinvestedat6%and .

Strategy:Letx=theamountofmoneyinvestedat6%and($8,000−x)=theamountofmoneyinvestedat

.Thensetupatableasshown.

Theequationis

Interestonthefirstinvestment+Interestonsecondinvestment=Totalinterest


Evaluation:Findtheinterestonbothinvestmentsseparatelyandthenaddthemtoseeiftheyequal$435.UseI=PRTwhereT=1.

Firstinvestment:I=$5,000(6%)=$300

Secondinvestment:I=$3,000( )=$135$300+$135=$435

EXAMPLE

Apersonhasthreetimesasmuchmoneyinvestedat8%ashehasat3%.Ifthetotalannualinterestfromtheinvestmentsis$540,howmuchdoeshehaveinvestedateachrate?

SOLUTION

Goal:Youarebeingaskedtofindhowmuchmoneyisinvestedat8%and3%.

Strategy:Letx=theamountofmoneyinvestedat3%and3x=theamountofmoneyinvestedat8%;then

Theequationis8%(3x)+3%(x)=540.Implementation:Solvetheequation:

Hence,thepersonhas$2,000investedat3%and$6,000investedat8%.Evaluation:Findtheinterestearnedoneachinvestment,thenadd,andseeifthesumis$540.UseI=PRTwhereT=1.

Firstinvestment:I=3%($2,000)=$60Secondinvestment:I=8%(6,000)=$480

$60+$480=$540

EXAMPLE

Aninvestorhas$600moreinvestedinstockspaying9%thanshehasinvestedinbondspaying3%.Ifthetotalinterestis$162,findtheamountofmoneyinvestedineach.

SOLUTION

Goal:Youarebeingaskedtofindtheamountofeachinvestment.

Strategy:Letx=theamountinvestedinbondsandx+600=theamountinvestedinstocks.

Theequationis3%x+9%(x+600)=$162.Implementation:Solvetheequation:

Hence,thepersonhas$900investedinbondsand$1,500investedinstocks.Evaluation:Findtheinterestforbothinvestmentsandthenaddtoseeiftheansweris$162.UseI=PRTwhereT=1.

Bonds:I=3%(900)=$27Stocks:I=9%(1,500)=$135

$27+$135=$162

EXAMPLE

Aninvestorhastwiceasmuchmoneyinvestedat7%ashehasinvestedat3%and$400moreinvestedat2%thanhehasinvestedat3%.Ifthetotalinterestfromthethreeinvestmentsis$84,findtheamountshehasinvestedateachrate.

SOLUTION

Goal:Youarebeingaskedtofindtheamountsofthethreeinvestments.

Strategy:Letx=theamountinvestedat3%,2x=theamountofmoneyinvestedat7%,andx+400=theamountofmoneyinvestedat2%.

Theequationis3%x+7%(2x)+2%(x+400)=$84.Implementation:Solvetheequation:

Hence,theinvestorinvested$400at3%,$800at7%,and$800at2%.Evaluation:Findthethreeinterestamounts,andaddtoseeifyouget$84.UseI=PRTwhereT=1.

Firstinvestment:I=3%(400)=$12Secondinvestment:I=7%(800)=$56Thirdinvestment:I=2%(800)=$16

$12+$56+$16=$84

TRYTHESE

1.Anindividualinvested$7,000,partat6%andtherestat3.5%.Ifthetotalinterestheearnedafteroneyearwas$357.50,findtheamountofeachinvestment.

2.Anindividualinvestedacertainamountofmoneyinasavingsaccountpaying2%and$1,800morethanthatamountinaone-yearCDpaying1.5%.Ifthetotalinterestforthetwoinvestmentswas$51.50,findtheamountofmoneysheinvestedineach.

3.Apersoninvestedsixtimesasmuchmoneyat asshedidat .Ifthetotalinterestfromtheinvestmentsattheendoftheyearwas$193,howmuchdidsheinvestateachrate?

4.Aninvestormadetwoinvestments,onepaying9%andonepaying4%.Ifthetotalamountinvestedwas$15,000andthetotalinterestsheearnedafteroneyearwas$800,findtheamountofeachinvestment.

5.Aninvestorhas$1,500lessinvestedat6%thanhehasinvestedat8%.Ifthetotalyearlyinterestfromtheinvestmentsis$190,findtheamountshehasinvestedateachrate.

6.Anindividualinvestedtwiceasmuchinbondspaying2%ashedidinstockspaying6%.Iftheinterestattheendoftheyearwas$468,findtheamountofmoneyheinvestedineach.

7.Apersoninvestedacertainamountofmoneyinanaccountpaying5%.Heinvestsfivetimesthatamountinto

anotheraccountpaying ,andheinvests$700morethantheamountinthe5%accountintoathirdaccountpaying8%.Ifthetotalyearlyinterestfromallthreeaccountswas$5,851,findtheamountheinvestedineachaccount.

8.Apersonhas$5,000investedat5%.Howmuchshouldbeinvestedat3%tohaveanincome(yearly)interestof$1,222?

9.Ms.Smithinvestedsomemoneyat6%andsomemoneyat9%.Iftheyearlyinterestonbothinvestmentsisthesameandthetotalamountoftheinvestmentsis$15,000,findtheamountofeachinvestment.

10.Aninvestorhasthreeinvestments.Hehastwicetheamountofmoneyinvestedat ashehasinvestedat1%and$600moreinvestedat2%ashehasat1%.Iftheyearlyinterestis$852,findtheamountofeachinvestment.

SOLUTIONS

1.Letx=theamountofmoneyinvestedat6%and$7,000−x=theamountofmoneyinvestedat3.5%.

2.Letx=theamountofmoneyinvestedat2%andx+$1,800=theamountofmoneyinvestedat1.5%.

3.Letx=theamountofmoneyinvestedat and6x=theamountofmoneyinvestedat .

4.Letx=theamountofmoneyinvestedat9%and($15,000−x)=theamountofmoneyinvestedat4%.

5.Letx=theamountofmoneyinvestedat8%andx−$1,500=theamountinvestedat6%.

6.Letx=theamountinvestedinstocksand2x=theamountofmoneyinvestedinbonds.

7.Letx=theamountofmoneyinvestedat5%,5x=theamountofmoneyinvestedat ,andx+$700=theamountofmoneyinvestedat8%.

8.Letx=theamountofmoneythepersonshouldinvestat3%.

9.Letx=theamountofmoneyinvestedat6%and$15,000−x=theamountofmoneyinvestedat9%.Sincetheinterestearnedonbothinvestmentsisthesame,theequationis6%(x)=9%($15,000−x).

10.Letx=theamountofmoneyinvestedat1%,2x=theamountofmoneyinvestedat ,andx+$600=theamountofmoneyinvestedat2%.

Inthissection,youhavelearnedhowtosolvefinanceproblems.ThebasicformulathatisusedisInterest=Principal×Rate×TimeorI=PRT.Sincetheinterestisyearly,thetime=1year.YougetthebasicequationfortheproblembyusingInterestfromfirstinvestment+Interestfromsecondinvestment=Totalinterest.

Thereareseveraldifferenttypesofproblems,sotheequationcandiffersomewhatfromthebasicone.

LeverProblems

Oneoftheoldestmachinesknowntohumansisthelever.Theprinciplesoftheleverarestudiedinphysics.Mostpeoplearefamiliarwiththesimplestkindoflever,knownastheseesaworteeterboard,oftenseeninparks.

Theleverisaboardplacedonafulcrumorpointofsupport.Onaseesaw,thefulcrumisinthecenteroftheboard.Achildsitsateitherendoftheboard.Ifonechildisheavierthantheotherchild,heorshecansitclosertothecenterinordertobalancetheseesaw.Thisisthebasicprincipleofthelever.

Ingeneral,theweightsareplacedontheendsoftheboard,andthedistancetheweightisfromthefulcrumiscalledthelengthorarm.Thebasicprincipleoftheleveristhattheweighttimesthelengthofthearmontheleftsideoftheleverisequaltotheweighttimesthelengthofthearmontherightsideofthelever,orWL=wl.SeeFigure9-1.

Givenanyofthethreevariables,youcansetupanequationandsolveforthefourthone.Unlessotherwisespecified,assumethefulcrumisinthecenterofthelever.

FIGURE9-1

EXAMPLE

Samweighs150poundsandsitsonaseesaw2feetfromthefulcrum.WheremustSally,whoweighs120pounds,sittobalanceit?

SOLUTION

Goal:YouarebeingaskedtofindthedistancefromthefulcrumSallyneedstosittobalancetheseesaw.

Strategy:UsetheformulaWL=wlwhereW=150,L=2,w=120,andletx=l.

WL=wl150(2)=120x

SeeFigure9-2.

FIGURE9-2


Hence,shemustsit2.5feetfromthefulcrum.Evaluation:Checkthesolution:

WL=wl150(2)=120(2.5)

300=300Thefulcrumofaleverdoesnothavetobeatitscenter,asshowninthenextexample.

EXAMPLE

Thefulcrumofaleveris4feetfromtheendofa10-footlever.Ontheshortendrestsa96-poundweight.Howmuchweightmustbeplacedontheotherendtobalancethelever?

SOLUTION

Goal:Youarebeingaskedtofindhowmuchweightisneededtobalancethelever.

Strategy:Letx=theweightoftheobjectneeded.Thisweightmustbeplacedat10−4=6feetfromthefulcrumsinceitisattheendofthelongerside.

WL=wl96(4)=x(6)

SeeFigure9-3.

FIGURE9-3

Implementation:Solvetheequation:96(4)=x(6)

64poundsneedstobeplacedatthe6-footendtobalancethelever.Evaluation:Checkthesolution:

WL=wl96(4)=64(6)384=384

EXAMPLE

Whereshouldthefulcrumbeplacedona12-footleverwitha40-poundweightononeendanda60-poundweightontheotherend?

SOLUTION

Goal:Youarebeingaskedtofindtheplacementofthefulcrumsothattheleverisbalanced.

Strategy:Letx=thelengthoftheleverfromthefulcrumtothe40-poundweightand(12−x)=thelengthoftheleverfromthefulcrumtothe60-poundweight.SeeFigure9-4.

FIGURE9-4

TheequationisWL=wl

40x=60(12−x)Implementation:Solvetheequation:

Hence,thefulcrummustbeplaced7.2feetfromthe40-poundweight.Evaluation:Checkthesolution:

WL=wl40(7.2)=60(12−7.2)40(7.2)=60(4.8)288=288

Youcanplacethreeormoreweightsonaleveranditstillcanbebalanced.Iffourweightsareused,twooneachside,the

equationisW1L1+W2L2=w1l1+w2l2

EXAMPLE

Ona15-footseesaw,Mary,weighing95pounds,sitsononeend.NexttoMarysitsHelen,weighing85pounds.Helenisfivefeetfromthefulcrumwhichisinthecenteroftheseesaw.OntheothersideattheendsitsCarol,weighing105pounds.WhereshouldJulie,weighing80pounds,sitinordertobalancetheseesaw?

SOLUTION

Goal:YouarebeingaskedtofindthedistancefromthefulcrumwhereJulieshouldsitinordertobalancetheseesaw.

Strategy:Letx=thedistancefromthefulcrumwhereJulieneedstosit.SeeFigure9-5.

FIGURE9-5

TheequationisW1L1+W2L2=w1l1+w2l295(7.5)+85(5)=80x+105(7.5)


Julieneedstosit4.375feetfromthefulcrum.Evaluation:Checkthesolution:

W1L1+W2L2=w1l1+w2l295(7.5)+85(5)=80(4.375)+105(7.5)

712.5+425=350+787.51,137.5=1,137.5

TRYTHESE

1.Mattweighs110poundsandsitsfourfeetfromthefulcrumofaseesaw.IfJeanweighs80pounds,howfarshouldshesitfromthefulcrumtobalancetheseesaw?

2.Atoneendofaleverisa15-poundweightwhichis10inchesfromthefulcrum.Howmuchweightshouldbeplacedontheotherend12inchesfromthefulcrumtobalancethelever?

3.Apersonplacesaleverundera100-poundrockthatis2.5feetfromthefulcrum.Howmuchpressureinpounds

mustthepersonplaceontheotherendoftheleverifitis4feetfromthefulcrumtolifttherock?

4.Whereshouldthefulcrumbeplacedunderaeight-footleverifthereisa32-poundweightononeendanda40-poundweightontheotherendinordertobalancethelever?

5.Ona12-footseesaw,Kelly,weighing96pounds,sitsononeend.Peggy,weighing84pounds,sitsinfrontofher,fourfeetfromthefulcrumwhichisinthecenteroftheseesaw.OntheothersideattheendsitsFran,whoweighs72pounds.WhereshouldCarol,whoweighs100pounds,sitinordertobalancetheseesaw?

SOLUTIONS

1.Letx=thedistanceJeanshouldsitfromthefulcrum.

Jeanshouldsit5.5feetfromthefulcrum.

2.Letx=theweightplacedontheothersideofthelever.

Aweightof12.5poundsshouldbeplaced12inchesfromthefulcrumtobalancethelever.

3.Letx=thepressureinpoundsneededtoliftthe100-poundrock.

Itwilltake62.5poundsofpressuretolifttherock.

4.Letx=thedistancefromthefulcrumwherea32-poundweightsitsand8−x=thedistancefromthefulcrumthe40-poundweightsits.

Thefulcrumshouldbeplaced4.44feetfromtheendoftheleverthathasthe32-poundweight.

5.Letx=thedistancefromthefulcrumwhereCarolshouldsit.

Carolshouldsit4.8feetfromthefulcrum.

Inthissection,youlearnedhowtosolvewordproblemsinvolvinglevers.ThebasicformulaisWL=wl.Inotherwords,iftheleveristobebalanced,theweightsmustbeproperlyplacedatspecificlengthsfromthefulcruminordertoaccomplishthis.

WorkProblems

Workproblemsinvolvepeopledoingajob.Forexample,ifFrankcancutalawnintwohoursandhisyoungerbrothercancutthesamelawninthreehours,howlongwillittakethemtocutthegrassiftheybothworktogether?Inthiscase,wehavetwopeopledoingthesamejobatthesametimebutatdifferentrates.

Anothertypeofprobleminvolvespipesfillingordrainingbodiesofwatersuchastanks,reservoirs,orswimmingpoolsatdifferentrates.Forexample,ifonepipecanfillalargetankinfivehours,andasmallerpipecanfillthetankinthreehours,howlongwouldittaketofillthetankifbothpipesareturnedonatthesametime?Again,wehavetwopipesdoingthesamejobatdifferentrates.

Thebasicprincipleisthattheamountofworkdonebyoneperson,machine,orpipeplustheamountofworkdonebythesecondperson,machine,orpipeisequaltothetotalamountofworkdoneinagivenspecifictime.Alsotheamountofworkdonebyasingleperson,machine,orpipeisequaltotheratetimesthetime.Thatis,

Rate×Time=Amountofworkdone

EXAMPLE

Petecancompleteajobinfourhours,andMattcandothesamejobinsixhours.Howlongwillittakethemiftheybothworktogetheratthejob?

SOLUTION

Goal:Youarebeingaskedtofindthetimeinhoursitwilltakebothpeopletocompletethejobiftheyworktogether.

Strategy:Letx=thetimeittakesthemiftheyworktogether.Now,inonehour,Petecancomplete of

thejobandMattcancomplete ofthejob.

Petedoes xor amountofworkandMattdoes xor amountofwork.Thesearethefractionalpartsofworkdonebyeach.Thenthetotalamountofworkdoneis100%or1.Theequationis

Implementation:Solvetheequation:TheLCDof4and6is12,soclearfractions:

Hence,ifbothworktogether,theycancompletethejobin2.4hours.Evaluation:Checkthesolution:

EXAMPLE

Onepipecanfillalargetankin10hoursandanotherpipecanfillatankin6hours.Howlongwillittakebothpipestofillthetankiftheyareturnedonatthesametime?

SOLUTION

Goal:Youarebeingaskedtofindthetimeinhoursitwouldtaketofillthetankifbothpipesarefillingthetankatthesametime.

Strategy:Letx=thetimeittakestofillthetankwithbothpipes.Inonehour,thefirstpipedoes ofthe

workandthesecondpipedoes ofthework.

Again,thetotalamountofworkdoneis100%or1.Theequationis


TheLCDis30.

Hence,ifbothpipesareturnedonatthesametime,itwouldtake3.75hours.Evaluation:Checkthesolution:

Asyoucansee,bothtypesofproblemscanbedoneusingthesamestrategy.Thenextexamplesshowsomevariationsofworkproblems.

EXAMPLE

Apersoncanpaintameetingroomin8hoursandherassistantcanpaintthesameroomin12hours.Ifonacertainday,theassistantshowsuptwohourslateandstartstowork,howlongwillittakebothpeopletopainttheroom?

SOLUTION

Goal:Youarebeingaskedtofindthetimeittakesbothworkerstopainttheroom.

Strategy:Letx=thetimeittakestopainttherestoftheroomwhenbothpeopleareworking.

Sincetheassistantstartstwohourslater,thefirstpainterhasalreadydone2. or ofthework;hence,theequationis


TheLCDis24.

Sincethefirstpainterhasalreadyworkedtwohours,thetimeittakestopaintthewholeroomis2+3.6=5.6hours.Evaluation:Checkthesolution:

EXAMPLE

Alargewatertankcanbefilledin12hoursanddrainedin30hours.Howlongwillittaketofillthetankiftheownerhasforgottentoclosethedrainvalve?

SOLUTION

Goal:Youarebeingaskedhowlonginhoursitwilltaketofillthetankifthedrainisleftopen.

Strategy:Letx=thetimeinhoursittakestofillthetank.

Sincethedrainisemptyingthetank,theequationis


TheLCDis60.

Hence,itwilltake20hourstofillthetank.Evaluation:Checkthesolution:

EXAMPLE

Sarahcandoajobin40minutesand,workingwithMillie,bothcandothejobin15minutes.HowlongwillittakeMillietodothejobalone?

SOLUTION

Goal:YouarebeingaskedtofindthetimeinminutesittakesforMillietocompletethejobalone.

Strategy:Letx=thetimeittakesMillietocompletethejob.

Theequationis


TheLCD=40x.

Hence,itwilltakeMillie24minutestodothejobalone.Evaluation:Checkthesolution:

TRYTHESE

1.Onepipecanemptyapoolin90minutes,whileasecondpipecanemptyitin120minutes.Ifbothpipesareopenedatthesametime,howlongwillittaketodrainthepool?

2.Joecancompleteaprojectin45minutesandhisbrotherClemcancompleteitin60minutes.Iftheybothworkontheprojectatthesametime,howlongwillittakethemtocompletetheproject?

3.Melissacancleanabarnin4.5hoursandherfathercancleanitin3hours.Howlongwillittakeiftheybothworktogether?

4.Samcanplowafieldin6hoursandhisbrotherBillcanplowitin7.5hours.Howlongwillittakethemtoplowitiftheyusetwoplowsandworktogether?

5.Sidcancompleteajobin150minutes,andifSidandBretbothworkonthejob,theycancompleteitin90minutes.HowlongwillittakeBrettocompletethejobbyhimself?

6.PipeAcanfillatankin12minutes.PipeBcanfillitin16minutes,andpipeCcanfillitin18minutes.Ifallthreepipesareopenedatthesametime,howlongwillittaketofillthetank?

7.Apipecanfillatankin60minutes,whilethedraincandrainitin75minutes.Ifthedrainisleftopenandthefillpipeisturnedon,howlongwillittaketofillthetank?

8.Onefaucetcanfillalargetubin64minutes,whileanotherfaucetcanfillthetubin96minutes.Howlongwillittaketofillthetubifbothfaucetsareopenedatthesametime?

9.Carlcanseedalargefieldinfourhours.Hissoncandothejobinthreehours.Ifthesonstartsanhourafterhisfather,howlongwillittaketoseedthefield?

10.CarolcanmakeacostumetwiceasfastasBencan.Iftheybothworktogether,theycanmakeitinthreehours.HowlongwillittakeCaroltomakethecostumeifsheworksalone?

SOLUTIONS

1.Letx=thetimeittakestoemptythepoolifbothpipesareopen.

2.Letx=thetimeittakesbothtocompletetheproject.

3.Letx=thetimeittakesbothpeopletocompletetheprojectiftheyworkonittogether.

4.Letx=thetimeitwilltakeSamandBilltoplowthefieldiftheybothworkonittogether.

5.Letx=thetimeittakesBrettocompletethejob.

6.Letx=thetimeittakesallthreepipestofillthetank.

7.Letx=thetimeittakestofillthetank.

8.Letx=thetimeittakestofillthetubifbothfaucetsareon.

9.Letx=thetimeittakesbothpeopletoseedthefield.Sincehisfatheralreadyhadworkedonehourbeforehisson

started,hedid ofthework.

10.Letx=thetimeittakesCaroltomakethecostumeand2x=thetimeittakesBentomakethecostume.

Inthissection,youhavelearnedhowtosolveproblemsrelatedtosomekindofwork.ThebasicformulaisRate×Time=Amountofworkdone.

Summary

Thischapterexplainedhowtosolvefinance,lever,andworkproblems.

QUIZ

1.Apersonhas$15,000investedat6%andanothersuminvestedat4%.Ifthetotalinteresthereceivedonboth

investmentswas$1,700,findtheamountofmoneyhehasinvestedat4%.

A.$18,000

B.$20,000

C.$17,000

D.$14,000

2.Aninvestorhassomemoneyinvestedat7%andsomemoneyinvestedat3%.Thetotalinterestonbothinvestmentsis$518.Ifthetotalamountofmoneyhehasinvestedis$9,000,findtheamounthehasinvestedat7%.

A.$6,000

B.$6,200

C.$2,800

D.$3,000

3.Aninvestorinvested$40,000,someat5%andsomeat9%.Theannualinterestonthe9%investmentis$2,480morethantheinterestonthe5%investment.Howmuchmoneywasinvestedat9%?

A.$24,000

B.$28,000

C.$30,000

D.$32,000

4.Apersonhasthreetimestheamountofmoneyinvestedat4%thanshehasinvestedat2%.Ifthetotalinterestis$420,howmuchmoneyisinvestedat2%?

A.$2,000

B.$5,000

C.$3,000

D.$9,000

5.An85-poundweightisplacedonaboard2feetfromthefulcrum.Howfarfromthefulcrummustan80-poundweightbeplacedinordertobalancetheseesaw?

A.3.325feet

B.2.625feet

C.2.875feet

D.2.125feet

6.A120-poundweightisplacedonan8-footboardwiththefulcrumatthecenter.Howmuchweightshouldbeplaced3feetfromthefulcrumtobalancethelever?

A.160pounds

B.155pounds

C.170pounds

D.140pounds

7.Ifa120-poundweightisplacedattheendofa12-footleveranda150-poundweightisplacedontheotherend,howmanyfeetfromthe120-poundweightshouldthefulcrumbeplacedinordertobalancethelever?

8.Marycandetailanautomobilein3hours.Ifshegetshelpfromhersister,theycandetailthecarin1.8hours.Howlongwillittakehersistertodetailtheautomobileifsheworksbyherself?

A.4hours

B.4.5hours

C.3.5hours

D.3hours

9.Joycecancleanthewindowsofabuildingin8hours.Herpartnercancleanthesamewindowsin4.8hours.Howlongwillittakethemtocleanthewindowsofthebuildingiftheybothworktogether?

A.5.4hours

B.6hours

C.3hours

D.4.2hours

10.Asmallpipecandrainatankin40minutesandalargepipecandrainitin24minutes.Ifbothpipesareopenedatthesametime,howlongwillittaketodrainthetank?

A.20minutes

B.10minutes

C.12minutes

D.15minutes

chapter10SolvingWordProblemsUsingTwoEquations

Manywordproblemsinalgebracanbesolvedbyusingtwoequationswithtwounknowns(usuallyxandy).Whenyouusetwounknowns,youletx=oneoftheunknownsandy=theotherunkown.Thenyoucanwritetwoequationsandsolvethemasasystemofequations.Eachproblemwillhavetwosolutions,oneforthevalueofxandoneforthevalueofy.

CHAPTEROBJECTIVES


•Solveasystemoftwoequations

•Solvewordproblemsusingtwoequations

RefresherV:SystemsofEquations

Twoequationswithtwovariables,usuallyxandy,arecalledasystemofequations.Forexample,x−y=32x+y=12

iscalledasystemofequations.Thesolutiontoasystemofequationsconsistsofthevaluesforthetwovariableswhich,whensubstitutedintheequations,makebothequationstrueatthesametime.Inthiscase,thesolutionforthesystemshownisx=5andy=2.Thiscanbeshownasfollows:

Inotherwords,inordertosolveasystemofequations,itisnecessarytofindavalueforxandavalueforywhich,whensubstitutedintheequations,makesthembothtrue.Thereareseveralwaystosolveasystemofequations.Themethodusedhereiscalledthesubstitutionmethod.Youcan

usethesesteps:Step1Selectoneequationandsolveitforonevariableintermsoftheothervariable.Step2Substitutethisexpressionforthevariableintheotherequationandsolveitfortheremainingvariable.Step3Selectoneoftheequations,substitutethevalueforthevariablefoundinStep2,andsolvefortheothervariable.

EXAMPLE

Solvethesystem:

3x−y=5x+2y=18

SOLUTION

Step1:Selectthesecondequationandsolveitforxintermsofy.

x+2y=18x+2y−2y=18−2y

x=18−2y

Step2:Substitute18−2yforxinthefirstequationandsolvefory.

Step3:Select3x−y=5,substitutey=7andsolveforx.

Hence,thesolutiontothesystemisx=4andy=7.

Youcancheckthesolutionbysubstitutingx=4andy=7intheotherequationandseeifitistrue.

x+2y=184+2(7)=184+14=18

18=18

EXAMPLE

Solvethesystem:

x+4y=34x−3y=−26

SOLUTION

Step1:Solvethefirstequationforx.

x+4y=3x+4y−4y=3−4y

x=3−4y

Step2:Substitute3−4yforxinthesecondequationandsolvefory.

Step3:Substitute2foryinthefirstequationandfindthevalueforx.

x+4y=3x+4(2)=3x+8=3

x+8−8=3−8x=−5

Youcancheckthesolutionbyusing4x−3y=−26whenx=−5andy=2.

4x−3y=−264(−5)−3(2)=−26

−20−6=−26−26=−26

StillStrugglingWhenselectinganequationandavariabletosolveforinstep1,youshouldlookforanequationthathasavariablewhosenumericalcoefficientis1.sincethisisnotalwayspossible,youcanstillusethesubstitutionmethodtosolvetheequationasshowninthenextexample.

EXAMPLE

Solvethesystem:

3x−5y=−72x+3y=−11

SOLUTION

Step1:Selectthesecondequationandsolvefory.

Step2:Substituteinthefirstequation.

Clearfractions.

Step3:Findy.

Thesolutionisx=−4andy=−1.Youcanchecktheanswer.

StillStrugglingItdoesn’tmatterwhichequationyouuseorwhichvariableyousolveforfirst.

TRYTHESE

Solveeachsystem.

1.2x−y=−15

x+3y=3

2.3x−2y=−1

x+y=13

3.x−y=6

5x+y=6

4.8x=y

2x+y=10

5.3x−2y=−13

−2x+5y=−17

SOLUTIONS

1.2x−y=−15

x+3y=3

Solvethesecondequationforx.

x+3y=3

x+3y−3y=3−3y

x=3−3y

Substituteinthefirstequationandsolvefory.

Findx.

Thesolutionisx=−6andy=3.

2.3x−2y=−1

x+y=13

Solvethesecondequationforx.

x+y=13

x+y−y=13−y

x=13−y

Substituteinthefirstequationandsolvefory.

Findx.

x+y=13

x+8=13

x+8−8=13−8

x=5

3.x−y=6

5x+y=6

Solvethesecondequationfory.

5x+y=6

5x−5x+y=6−5x

y=6−5x

Substituteinthefirstequationandsolveforx.

Findy.

Thesolutionisx=2andy=−4.

4.8x=y

2x+y=10

Substituteforyinthesecondequationandfindxsince8x=y.

Findy.

8x=y

8(1)=y

8=y

Thesolutionisx=1andy=8.

5.3x−2y=−13

−2x+5y=−17

Solveforxinthefirstequation.

Substituteforxinthesecondequation.

Findx.

Thesolutionisx=−9andy=−7.

Inthisrefresher,youlearnedhowtosolveasystemoftwoequationswithtwounknowns.Themethodofsolutioniscalledsubstitution.Thereareothermethodsthatcanbeusedtosolvethesesystems.Youcanfindthesemethodsinalgebrabooks.

SolvingWordProblemsUsingTwoEquations

NOTEIfyouneedtoreviewsystemsofequations,completeRefresherV.Manyoftheprevioustypesofproblemscanbesolvedusingasystemoftwoequationswithtwounknowns.Thestrategyusedtosolveproblemsusingtwoequationsis:Step1Representoneoftheunknownsasxandtheotherunknownasy.Step2Translatetheinformationaboutthevariablesintotwoequationsusingthetwounknowns.Step3Solvethesystemofequationsforxandy.Inthissection,asampleofeachtypeofproblemissolvedbyusingasystemoftwoequationswithtwounknowns.You

willfindtheseproblemsaresimilartotheonesintheprevioussections.Thiswasdonesothatyoucancomparethetwomethods(i.e.,solvingaproblemusingoneequationversussolvingaproblemusingtwoequations).Forsometypesofproblems,suchasleverandworkproblems,itisbettertouseoneequation.

EXAMPLE

Onenumberis16morethananothernumberandthesumofthetwonumbersis28.Findthenumbers.

SOLUTION


Strategy:Letx=thesmallernumberandy=thelargernumber.

Sinceonenumberis16morethantheothernumber,thefirstequationis

y=x+16

Sincethesumofthetwonumbersis28,thesecondequationis

x+y=28

Implementation:Solvethesystem:

y=x+16x+y=28

Substitutethevalueforyinthesecondequationandsolveforxsincey=x+16.

Findtheothernumber.

y=x+16y=6+16y=22

Hence,thenumbersare6and22.

Evaluation:Checkthesecondequation.

x+y=286+22=28

28=28

EXAMPLE

Thesumofthedigitsofatwo-digitnumberis14.Ifthedigitsarereversed,thenewnumberis18morethantheoriginalnumber.Findthenumber.

SOLUTION


Strategy:Letx=thetensdigit

y=theonesdigit

Then

10x+y=originalnumber

10y+x=newnumberwithdigitsreversed

Sincethesumofthedigitsofthenumberis14,thefirstequationis

x+y=14

Sincereversingthedigitsgivesanewnumberthatis18morethantheoriginalnumber,thesecondequationis

(10x+y)+18=(10y+x)


x+y=1410x+y+18=10y+x

Solvethefirstequationfory.

x+y=14x−x+y=14−x

y=14−x

Substituteinthesecondequationandfindx.

Findy.

x+y=146+y=14

6−6+y=14−6y=8

Hence,thenumberis68.

Evaluation:Checktheinformationinthesecondequation.

Originalnumber=68Reversednumber=86

Since86is18morethan68,theansweriscorrect.

EXAMPLE

Apersonhas12coinsconsistingofquartersanddimes.Ifthetotalamountofthischangeis$2.25,howmanyofeachkindofcoinarethere?

SOLUTION

Goal:Youarebeingaskedtofindhowmanycoinsarequartersandhowmanycoinsaredimes.

Strategy:Letx=thenumberofquarters

y=thenumberofdimes

25x=thevalueofthequarters

10y=thevalueofthedimes

Sincethereare12coins,thefirstequationis

x+y=12

Sincethetotalvalueofthequartersplusthedimesis$2.25or225¢,thesecondequationis

25x+10y=225


x+y=1225x+10y=225

Solveforyinthefirstequation.

x+y=12x−x+y=12−x

y=12−x

Substitutethisexpressionforyinthesecondequationandsolveforx.

Findy.

x+y=127+y=12

7−7+y=12−7y=5

Hence,thereare7quartersand5dimes.

Evaluation:Findthevaluesofeachandseeiftheirsumis$2.25.

7quarters=7×$0.25=$1.755dimes=5×$0.10=$0.50

$1.75+$0.50=$2.25

EXAMPLE

Samis10yearsyoungerthanhisbrother.Intwoyears,hisbrotherwillbethreetimesasoldasSam.Findtheirpresentages.

SOLUTION

Goal:YouarebeingaskedtofindthepresentagesofSamandhisbrother.

Strategy:Letx=Sam’sage

y=hisbrother’sage

x+2=Sam’sageintwoyears

y+2=hisbrother’sageintwoyears

SinceSamis10yearsyoungerthanhisbrother,thefirstequationis

x+10=y

Intwoyears,Sam’sbrotherwillbethreetimesasoldasSam,sothesecondequationis

y+2=3(x+2)


x+10=yy+2=3(x+2)

Substitutethevalueofyinthesecondequationandsolveforxsincex+10=y.

Selectthefirstequation,letx=3,andsolvefory.

x+10=y3+10=y

13=y

Hence,Sam’sbrotheris13yearsoldandSamis3yearsold.

Evaluation:Sam’sageis3,whichis10yearsyoungerthanhisbrotherwhois13yearsold.Intwoyears,Samwillbe5andhisbrotherwillbe15.HencehisbrotherwillbethreetimesasoldasSam.

EXAMPLE

Apersondrovehiscarfromhometoarepairshopat30milesperhourandwalkedhomeat3milesperhour.Ifthetotaltriptook33minutes,howfaristherepairshopfromhishome?

SOLUTION

Goal:Youarebeingaskedtofindthedistancefromtheperson’shometotherepairshop.

Strategy:Letx=thetimethepersondroveandy=thetimethepersonwalked.

Sincethetotaltimeis33minutesor hour,thefirstequationis

x+y=0.55

SincethedistancesareequalandD=RT,thesecondequationis

30x=3y


x+y=0.5530x=3y

Solvethefirstequationforyandsubstitutethevalueinthesecondequation,andthensolveforx.

x+y=0.55x−x+y=0.55−x

y=0.55−x

Then:

FindthedistanceusingD=RT.

D=RTD=30(0.05)=1.5miles

Evaluation:Thetimehewalkedis0.55−0.05=0.5hours.ThedistanceisD=RT.

D=3(0.5)=1.5miles

StillStrugglingInthepreviousexample,theratesaregiveninmilesperhourandthetotaltimeisgiveninminutes,i.e.,33minutes.therefore,itisnecessarytoconverttheminutestohourssothattheunitsintheproblemarethesame.

EXAMPLE

Amerchantmixessomecashewscosting$6apoundwithsomepeanutscosting$2apound.Howmuchofeachmustbeusedinordertomake25poundsofmixturecosting$3.50apound?

SOLUTION

Goal:Youarebeingaskedtofindhowmuchofeachkindofnutsshouldbeused.

Strategy:Letx=theamountof$6cashewsusedandy=theamountof$2peanutsused.

Sincethetotalamountofthemixtureis25pounds,thefirstequationis

x+y=25

Sincethecostofthemixtureis$3.50,thesecondequationis

6x+2y=25(3.50)


x+y=256x+2y=25(3.50)

Solvethefirstequationforx.Substituteinthesecondequationandsolvefory.

x+y=25x+y−y=25−y

x=25−y

Substitute:

Solveforx.

x+y=25x+15.625=25

x+15.625−15.625=25−15.265x=9.375pounds

Hence,9.375poundsofthe$6cashewsareneededand15.625poundsofthe$2peanutsareneeded.

Evaluation:Checkthesecondequation.

6x+2y=25(3.50)6(9.375)+2(15.625)=87.5

56.25+31.25=87.587.5=87.5

EXAMPLE

Apersonhas$8,000toinvestanddecidestoinvestpartofitat3%andtherestofitat .Ifthetotalinterestfortheyearis$330,howmuchdoesthepersonhaveinvestedateachrate?

SOLUTION

Goal:Youarebeingaskedtofindtheamountsofmoneyinvestedateachrate.

Strategy:Letx=theamountofmoneyinvestedat3%andy=theamountofmoneyinvestedat .

Sincethetotalamountofmoneyis$8,000,thefirstequationis

x+y=$8,000

Sincethetotalinterestis$330,thesecondequationis

3%x+ (y)=$330


Solvethefirstequationforx.Substituteinthesecondequationandsolvefory.

x+y=8,000x+y−y=8,000−y

x=8,000−y

Then:

Findx.

x+y=8,000x+2,000=8,000

x+2,000−2,000=8,000−2,000x=6,000

Hence,thepersonhas$6,000investedat3%and$2,000investedat .

Evaluation:Checkthesecondequation

TRYTHESEUsetwoequationswithtwounknowns.

1.Thelargeroftwonumbersis12morethanthesmallernumber.Thesumofthenumbersis50.Findthenumbers.

2.Aninvestorhas$10,000toinvestat5%and2%.Findtheamountofeachinvestmentifthetotalinterestperyearis$410.

3.JaniceistwiceasoldasJane,andthesumoftheiragesnextyearwillbe41.Findtheirpresentages.

4.Ayoungpersonboughtsomeapplesat$1eachandsoldthemfor$1.25eachatafleamarket.Hisprofitwas$6.50.Ifhegavetwoapplestohisfriends,howmanyapplesdidhebuy?

5.Apersonhas24coinsindimesandquarters.Ifthetotalamountofmoneyshehasis$4.65,howmanyquartersanddimesdoesthepersonhave?

6.Findtwoconsecutiveoddnumberswhosesumis88.

7.Harrybought12stamps.Ifhepurchasedtwomore50-centstampsthan25-centstampsanditcosthim$4.75,howmanyofeachkindofstampsdidhepurchase?

8.ThesumofMarci’sageandherbrother’sageis21.IfMarciis11yearsolderthanherbrother,findMarci’sage.

9.Thesumofthedigitsofatwo-digitnumberis15.Ifthedigitsarereversed,thenewnumberis9lessthantheoriginalnumber.Findthenumber.

10.Mr.Leeinvestedpartof$9,500intoanaccountthatpays2%interestandtherestofitintoanaccountthatpays4.5%interest.Ifthetotalinterestperyearhereceivesis$346.25,findtheamountofmoneyhehasinvestedineachaccount.

SOLUTIONS

1.Letx=thelargernumberandy=thesmallernumber.

x=y+12

x+y=50

Substitutey+12forxinthesecondequationandsolveforx.

Thelargernumberis31andthesmallernumberis19.

2.Letx=theamountofmoneyinvestedat5%andy=theamountofmoneyinvestedat2%.

$7,000shouldbeinvestedat5%and$3,000shouldbeinvestedat2%.

3.Letx=Janice’sageandy=Jane’sage;thenx=2yandx+1+y+1=41.

Janiceis26yearsoldandJaneis13yearsold.

4.Letx=thenumberofapplesheboughtandy=thenumberofappleshesold.

Hebought36apples.

5.Letx=thenumberofquartersthepersonhasandy=thenumberofdimesthepersonhas.

Thepersonhas15quartersand9dimes.

6.Letx=thefirstconsecutiveoddnumberandy=thesecondconsecutiveoddnumber.

Theconsecutiveoddnumbersare43and45.

7.Letx=thenumberof50-centstampsandy=thenumberof25-centstamps.

Harryboughtseven50-centstampsandfive25-centstamps.

8.Letx=Marci’sageandy=herbrother’sage.

x+y=21

x=y+11

x+y=21

y+11+y=21

2y+11=21

2y+11−11=21−11

2y=10

y=5

x+y=21

x+5=21

x+5−5=21−5

x=16

Marciis16yearsoldandherbrotheris5yearsold.

9.Letx=theonesdigitandy=thetensdigit.

Thenumberis87.

10.Letx=theamountofmoneyinvestedat2%andy=theamountofmoneyinvestedat4.5%.

Summary

Inthischapter,youlearnedhowtosolvewordproblemsusingtwoequationswithtwounknowns.Theseequationsarecalledasystemofequations.Thismethodisanalternativetothemethodsthatuseoneequation.

QUIZ

(Usetwoequationstosolvetheseproblems.)

1.Ifthesumoftwonumbersis51andthedifferenceis13,findthelargernumber.

A.19

B.16

C.32

D.35

2.Fourcomputersandsevenprinterscost$1,960,whilesevencomputersandfourprinterscost$2,770.Findthecostofonecomputer.

A.$180

B.$350

C.$600

D.$80

3.Aninvestorhasatotalof$11,000,partofwhichheinvestedat2%interestandtherestheinvestedat4.5%.Iftheyearlyinterestfromtheinvestmentis$305,findtheamountofmoneyinvestedat4.5%.

A.$5,200

B.$3,400

C.$5,800

D.$7,600

4.Ifapersoncantravel10milesupstreamin5hoursandthesamedistancedownstreamin1.25hours,findtherateofthecurrent.

A.8milesperhour

B.5milesperhour

C.7milesperhour

D.3milesperhour

5.Mollyhassomecoinsinherpurse.Shehastwomorequartersthandimesandtwotimesasmanypenniesasdimes.Ifshehasatotalof$1.98,howmanydimesdoesshehave?

A.4

B.5

C.6

D.8

6.Awomanisfiveyearsolderthanhersister.Twentyyearsago,shewastwiceasoldashersister.Findherage.

A.24

B.28

C.30

D.32

7.Thesumofthedigitsofatwo-digitnumberis8.Ifthedigitsarereversed,thenewnumberis36lessthantheoriginalnumber.Findthenumber.

A.44

B.53

C.71

D.62

8.Agrocerwantstomixsomecookiescosting$3perdozenwithsomecookiescosting$1.75perdozen.Ifshewantsatotalof10dozenthatsellfor$2.25perdozen,howmanydozensof$3cookieswillsheneed?

A.4

B.3

C.2.25

D.2

9.Findthesmalleroftwoconsecutiveevennumbersiftheirsumis86.

A.40

B.42

C.44

D.46

10.ThesumofHarry’sageandLarry’sageis92.Fouryearsago,HarrywasthreetimesasoldasLarry.FindHarry’sagenow.

A.25

B.21

C.67

D.63

chapter11SolvingWordProblemsUsingQuadraticEquations

Thischapterexplainshowtosolvewordproblemsbyusingaquadraticequationorseconddegreeequation.Thisequationhasanx2term.Therefreshersectionshowshowtosolveaquadraticequationbyfactoring.

CHAPTEROBJECTIVES


•Solveaquadraticequationbyfactoring

•Solvealgebraproblemsusingquadraticequations

RefresherVI:SolvingQuadraticEquationsbyFactoring

Anequationsuchas2x2+3x−5=0iscalledaquadraticequationoraseconddegreeequation.Thereisonevariable(usuallyx)andasecond-degreeterm(usuallyx2).Thereareseveralwaystosolvequadraticequations.Themethodshownherewillusefactoring.Ifyoucannotfactortrinomials,youwillneedtoconsultanalgebrabooktolearnthisskill.

Aquadraticequationcanbewritteninstandardformwherethex2termisfirst,thextermissecond,andtheconstanttermisthethird.Also,zeroisontherightsideoftheequation.Forexample,thequadraticequation2x+x2=8canbewritteninstandardformasx2+2x−8=0.Inordertosolveaquadraticequationbyfactoring,youshouldfollowthesesteps:Step1Writetheequationinstandardform.Step2Factortheleftsideoftheequation.Step3Setbothfactorsequaltozero.Step4Solveeachequation.

EXAMPLE

Findthesolutionto5x+x2=24.

SOLUTION

Step1:Writetheequationinstandardform.

x2+5x−24=0

Step2:Factortheleftside.

(x+8)(x−3)=0

Step3:Seteachfactortozero.

x+8=0andx−3=0

Step4:Solveeachequation.

Noticethattherearetwosolutions.Youcancheckeachvalueintheoriginalequation.

x=−8:5x+x2=24

5(−8)+(−8)2=24

−40+64=24

24=24

x=3:5x+x2=24

5(3)+(3)2=24

15+9=24

24=24

EXAMPLE

Solve6x2−24=7x.

SOLUTION

Step1:Writeinstandardform.

6x2−7x−24=0


(3x−8)(2x+3)=0

Step3:Setbothfactorsequaltozero.

3x−8=02x+3=0


EXAMPLE

Solve3x2=27.

SOLUTION

Step1:Writeinstandardform.

3x2−27=0


3(x+3)(x−3)=0

Step3:Dividebothsidesby3andsetbothfactorsequaltozero.

x+3=0x−3=0


EXAMPLE

Solvex2=8x.

SOLUTION

Quadraticequationsgenerallyhavetwodifferentsolutions;however,somehaveonlyonesolutionsincethesolutionsareequal.

StillStrugglingItshouldbenotedthatnotallquadraticequationscanbesolvedbyfactoring.However,forthepurposesofthisbook,the

solutionstothewordproblemsinthischaptercanbesolvedusingfactoring.

TRYTHESE

1.x2−12=4x

2.10x=x2+21

3.4x2+19x=5

4.6x2−31x=−35

5.4x2=16x

6.x2=49

7.x2+4x=5

8.x2+1=2x+25

9.x2=12x

10.3x2−2=5x

SOLUTIONS

Inthisrefresher,youlearnedhowtosolveaquadraticequationbyfactoring.Therearetwoothermethodsthatareusedtosolveaquadraticequation.Onemethodiscompletingthesquare.Theothermethodisusingthequadraticformula.Thequadraticformulacanbeusedtosolveallquadraticequations.Itcanbefoundinmostbasicalgebratextbooks.When

usingit,youfollowSteps1and2givenhereandthenusetheformulaforSteps3and4togetthesolution.

SolvingWordProblemsUsingQuadraticEquations

Manyproblemsinmathematicscanbesolvedusingaquadraticequation.Thestrategyyoucanuseis:

Step1Representtheunknownusingxandtheotherunknownintermsofx.Step2Fromtheproblem,writeexpressionsthatarerelatedtotheunknown.Step3Writethequadraticequation.Step4Solvethequadraticequationforx.

Recallthataquadraticequationhastwosolutions.(Note:Sometimesthetwosolutionsareequaltoeachother.)Bothsolutionscanbeanswerstotheproblems;however,manytimesonlyonesolutionismeaningful.Inthatcase,disregardthesolutionthatdoesnotmakesense.

EXAMPLE

Ifthesumoftwonumbersis18andtheproductofthetwonumbersis72,findthenumbers.

SOLUTION

Goal:Youarebeingaskedtofindtwonumberswhosesumis18andwhoseproductis72.

Strategy:Letx=onenumberand(18−x)=theothernumber.

Iftheproductofthetwonumbersis72,theequationisx(18−x)=72.


Hence,thetwonumbersare6and12.Evaluation:Checkthefactsoftheproblem.Thesum6+12is18andtheproductis6·12=72.

EXAMPLE

Iftheproductoftwoconsecutivenumbersis156,findthenumbers.

SOLUTION

Goal:Youarebeingaskedtofindtwoconsecutivenumberswhoseproductis156.

Strategy:Letx=thefirstnumberandx+1=thenextnumber.

Theequationfortheproductisx(x+1)=156.


Hence,thenumbersare12and13or−12and−13.

Evaluation:Findeachproduct:12·13=156,and−12·(−13)=156

EXAMPLE

Thesumoftwonumbersis20.Ifthesumoftheirreciprocalsis ,findthenumbers.

SOLUTION

Goal:Youarebeingaskedtofindtwonumberswhosesumis20andwhosesumoftheirreciprocalsis .

Strategy:Letx=onenumberand20−x=theothernumber.

Thereciprocalsare and .

Thenthesumofthereciprocalsis .


Divideby5.

Evaluation:Thesumof12+8=20.Thesumofthereciprocalsis .

TRYTHESE

1.Onenumberis3morethananothernumber,andtheproductofthetwonumbersis54.Findthenumbers.

2.Iftheproductoftwoconsecutiveevennumbersis168,findthenumbers.

3.If8issubtractedfromthesquareofanumber,theansweris28.Findthenumbers.

4.Mikecanpaintaroomin16minuteslesstimethanIke.Iftheybothpainttheroomatthesametime,itwilltakethem15minutes.Howlongdoesittakeeachonetopainttheroomindividually?

5.Onenumberis4morethananothernumber.Ifthesquareofthesmallernumberis2lessthanthreetimesthelargernumber,findthenumbers.

6.BeverlyistwoyearsolderthanMary.Iftheproductoftheiragesis48,findeachone’sage.

7.Twosquareplotsoflandcontain74squarefeet.Ifthesideofoneplotis2feetlongerthanthesideoftheotherplot,findthedimensionsofbothplots.(TheformulafortheareaofasquareisA=s2.)

8.Thesumofanumberanditsreciprocalis .Findthenumber.

9.Twoworkerscanassembleatrailerinsixhours.Ifittakesthesecondworkerninehourslongerthanthefirstworkertoassemblethetrailer,howlongwillittakeeachworkertodothejobiftheyworkalone?

10.Ifthesumofthesquaresoftwoconsecutivenumbersis85,findthenumbers.

SOLUTIONS

1.Letx=onenumberandx+3=theothernumber.

Theanswersare−6and−9,and6and9.

2.Letx=onenumberandx+2=theothernumber.

Theanswersare−14and−12,and14and12.

3.Letx=thenumber.

Theanswersare6and−6.

4.Letx=thetimeittakesMiketopainttheroomandx+16=thetimeittakesIketopainttheroom.

IttakesMike24minutestopainttheroomandIke40minutestopainttheroom.

5.Letx=thesmallernumberandx+4=thelargernumber.

Theanswersare5and9and−2and2.

6.Letx=Mary’sageandx+2=Beverly’sage.

Hence,Maryis6yearsoldandBeverlyis8yearsold.

7.Letxbethelengthofthesideofoneplotandx2bethearea.Let(x+2)bethelengthofthesideoftheotherplotand(x+2)2bethearea.

Hence,thesideofoneplotis7feetandthesideoftheotherplotis5feet.

8.Letx=thenumberand =thereciprocalofthenumber.

9.Letx=thetimethefirstworkertakestodothejobandx+9=thetimethesecondworkertakestodothejob.

Hence,itwilltakeoneworker9hourstoassemblethetrailerandtheotherworker18hourstodothejob.

10.Letx=thefirstnumberandx+1=thenextnumber.

Hence,theanswersare−7and−6and7and6.

Summary

Inthischapter,youlearnedhowtosolvewordproblemsusingaquadraticequation.Manyoftheseequationscanbesolvedbyfactoring.Itisimportanttorealizethatmanyquadraticequationscannotbesolvedbyfactoringsothequadraticformulacanbeused.Thisformulacanbefoundinanalgebrabook.

QUIZ

1.Iftheproductoftwopositiveconsecutiveoddnumberis323,findthelargerone.

A.19

B.21

C.15

D.17

2.Apersonhastwosquarefoundationsfortwosheds.Thetotaloftheareasofbothfoundationsis73squarefeet.Ifthesideofonefoundationis5feetlongerthanthesideoftheotherone,findthelengthofthesmallerfoundation.(UseA=s2.)

A.6feet

B.2feet

C.5feet

D.3feet

3.Ifthelengthofarectangleis5incheslongerthanitswidthandtheareaoftherectangleis24squarefeet,findthelengthoftherectangle.(UseA=lw.)

A.8feet

B.6feet

C.4feet

D.3feet

4.Ifthesideofasquareisincreasedby3inches,theareaofthesquareis324squareinches.Ifthesideofthesamesquareisdecreasedby3inches,theareaofthesquareis144squareinches.Findthemeasureofthesideofthesquare.(UseA=s2.)

A.7inches

B.15inches

C.18inches

D.12inches

5.IfDaveisfouryearsolderthanJimandtheproductoftheiragesis117,howoldisDave?

A.9

B.11

C.13

D.15

6.Ifthesumofsquareoftwoconsecutivenumbersis61,findthesmallernumber.

A.6

B.8

C.7

D.5

7.Onesideofasquareisthreeincheslongerthanthesideofanothersquare.Ifthesumoftheirareasis185squareinches,findthelengthofthesideofthelonger-sidedsquare.(UseA=s2.)

A.11squareinches

B.9squareinches

C.12squareinches

D.7squareinches

8.Bretis4yearsolderthanSam.IfSam’sageissquared,theresultis26morethanBret’sage.FindSam’sage.

A.5

B.6

C.8

D.3

9.Ifthedifferencebetweenanumberanditsreciprocalis ,findthewholenumber.

A.7

B.9

C.8

D.6

10.Twoworkersworkingtogethercancleanasmallofficebuildingin4.8hours.Oneworkercandoitin4hourslesstimethantheother.Findthetimeitwouldtaketheslowerworkertocleanthebuildingifheworksbyhimself.

A.8hours

B.10hours

C.12hours

D.14hours

chapter12SolvingWordProblemsinGeometry,Probability,andStatistics

Thischapterexplainshowtosolvewordproblemsingeometry,probability,andstatistics.Theseproblemsareonlyasampleofthetypesofproblemsthatyouwillfindinthesecourses,sincethereareentirebookswrittenonthesesubjects.

CHAPTEROBJECTIVES


•Solvewordproblemsingeometry

•Solvewordproblemsinprobability

•Solvewordproblemsinstatistics

SolvingGeometryProblems

Althoughthewordproblemsingeometryareforthemostpartdifferentfromthoseinalgebra,manyproblemsingeometryrequirealgebratosolvethem.Sinceitisnotpossibletoshowallthedifferenttypesofproblemsthatyouwillfindingeometry,afewofthemwillbeexplainedheresothatyoucanreachabasicunderstandingofhowtousealgebratosolvesomeoftheproblemsfoundingeometry.

Eachproblemisbasedonageometricprincipleorrule.Theprincipleswillbegivenhereineachproblem.

EXAMPLE

Findthemeasureofeachangleofatriangleifthemeasureofthesecondangleistwiceaslargeasthemeasureofthefirstangleandthethirdangleisthreetimesthemeasureofthefirstangle.

Geometricprinciple:Thesumofthemeasuresoftheanglesofatriangleis180°.

SOLUTION

Goal:Youarebeingaskedtofindthemeasuresofthethreeanglesofatriangle.

Strategy:Letx=themeasureofthefirstangle

2x=themeasureofthesecondangle

3x=themeasureofthethirdangle

SeeFigure12-1.

FIGURE12-1

Sincethesumofthemeasuresoftheanglesofatriangleis180°,theequationis

x+2x+3x=180°


Hence,themeasuresoftheanglesare30°,60°,and90°.

Evaluation:Checkthatthesumoftheanglesis180°.

30°+60°+90°=180°

EXAMPLE

Ifthelengthofarectangleisthreetimesitswidthandtheperimeteroftherectangleis104inches,findthemeasuresofitslengthandwidth.

Geometricprinciple:TheperimeterofarectangleisP=2l+2w.

SOLUTION

Goal:Youarebeingaskedtofindthelengthandwidthofarectangle.

Strategy:Letx=thewidthoftherectangleand3x=thelengthoftherectangle.

SeeFigure12-2.

FIGURE12-2

SincetheformulafortheperimeterofarectangleisP=2l+2w,theequationis

2(3x)+2(x)=104.


Hence,thelengthis39inchesandthewidthis13inches.

Evaluation:Usetheformulaforperimeterandcheckthatitis104inches.

P=2l+2wP=2(39)+2(13)

=78+26=104inches

EXAMPLE

Thebaseofatriangleis8incheslongerthanitsheight.Iftheareaofthetriangleis10squareinches,findthebaseandheightofthetriangle.

Geometricprinciple:Theareaofatriangleis bh.

SOLUTION

Goal:Youarebeingaskedtofindthemeasuresofthebaseandtheheight.

Strategy:Letx=themeasureoftheheightandx+8=themeasureofthebase.

SeeFigure12-3.

FIGURE12-3

Since ,theequationis .


Inthiscase,weignorex=−10sinceaheightcannotbeanegativenumber.Thebaseisx+8=2+8=10inches.Hence,theheightis2inchesandthebaseis10inches.

Evaluation:Findtheareaandcheckthatitis10squareinches.

TRYTHESE

1.Ifanangleexceedsitscomplementby32°,finditsmeasure.Geometricprinciple:Complementaryanglesareadjacentangleswhosesumis90°.

2.Theareaofarectangleis80squareinches.Thelengthis16incheslongerthanthewidth.Findthedimensionsoftherectangle.Geometricprinciple:TheareaofarectangleisA=lw.

3.Theperimeterofarectangleis64inchesandthelengthisthreetimesthewidth.Finditsdimensions.Geometricprinciple:TheperimeterofarectangleisP=2l+2w.

4.Theperimeterofarectangleis76inches.Ifthelengthis14inchesmorethantwicethewidth,finditsdimensions.Geometricprinciple:TheperimeterofarectangleisP=2l+2w.

5.Ifthesideofalargesquareisfourtimesaslongasthesideofasmallersquareandtheareaofthelargesquareis375squareincheslargerthantheareaofthesmallersquare,findthelengthofthesideofthesmallersquare.Geometricprinciple:TheareaofasquareisA=s2.

6.Thesumofthemeasuresoftheanglesofatriangleis180°.Ifthemeasureofthesecondangleistwiceaslargeasthemeasureofthefirstangleandthemeasureofthethirdangleis20°morethanthemeasureofthesecondangle,findthemeasuresoftheangles.

7.Thebaseofatriangleis11feetlongerthanitsheight.Ifitsareais30squarefeet,findthemeasuresofthebase

andheight.Geometricprinciple:Theareaofatriangleis .

8.Iftwosidesofatriangleareequalinlengthandthethirdsideis10inchesshorterthanthelengthofoneoftheequalsides,findthelengthofthesidesiftheperimeteris29inches.Geometricprinciple:Theperimeterofatriangleisequaltothesumofthelengthsofitssides.

9.Iftheareaofacircleis314squareinches,findtheradius.Geometricprinciple:TheareaofacircleisA=3.14r2.

10.Ifoneangleofatriangleis42°morethantwiceanotherangle,andthethirdangleisequaltothesumofthefirsttwoangles,findthemeasureofeachangle.Geometricprinciple:Thesumofthemeasureoftheanglesofatriangleis180°.

SOLUTIONS

1.Letx=themeasureofoneangleandx+32°=themeasureofthelargerangle.

Themeasuresoftheanglesare29°and61°.

2.Letx=thewidthoftherectangleandx+16=thelengthoftherectangle.

Thelengthoftherectangleis20inchesandthewidthis4inches.

3.Letx=thewidthand3x=thelength.

Thelengthis24inchesandthewidthis8inches.

4.Letx=thewidthoftherectangleand2x+14=thelengthoftherectangle.

Thelengthis30inchesandthewidthis8inches.

5.Letx=thelengthofthesideofthesmallersquareand4x=thelengthofthelargersquare.

Thelengthofthesideofthesmallersquareis5inches.

6.Letx=themeasureofoneangle

2x=themeasureofthesecondangle

2x+20=themeasureofthethirdangle

Themeasuresofthethreeanglesare32°,64°,and84°.

7.Letx=themeasureoftheheightandx+11=themeasureofthebase.

Theheightis4feetandthebaseis15feet.

8.Letx=thelengthofoneofthetwoequalsidesandx−10=thelengthofthethirdside.

Thelengthsofthesidesare13inches,13inches,and3inches.

9.Letx=themeasureoftheradius.

Theradiusis10inches.

10.Letx=themeasureofoneangle

2x+42=themeasureofthesecondangle

x+2x+42=themeasureofthethirdangle

Themeasuresoftheanglesare16°,74°,and90°.

Inthissection,youlearnedhowtosolvesomekindsofwordproblemsingeometry.Manyoftheseproblemsusegeometricformulasandsomebasicalgebra.

SolvingProbabilityProblems

Probabilitydealswithchanceevents,suchascardgames,slotmachines,andlotteriesaswellasinsurance,investments,andweatherforecasting.Aprobabilityexperimentisachanceprocessthatleadstowell-definedoutcomes.Forexample,whenadie(singularfordice)isrolled,therearesixpossiblewell-definedoutcomes.

Theyare1,2,3,4,5,6

Whenacoinisflipped,therearetwopossiblewell-definedoutcomes.Theyareheads,tails

Thesetofallpossibleoutcomesofaprobabilityexperimentiscalledthesamplespace.Eachoutcomeinasamplespace,unlessotherwisenoted,isconsideredequallylikelythatis,ithasthesamechanceofoccurring.Aneventcanconsistofoutcomesinthesamplespace.Thebasicdefinitionoftheprobabilityofaneventis

Thestrategywhendeterminingtheprobabilityofaneventis

1.FindthenumberofoutcomesineventE.

2.Findthenumberofoutcomesinthesamplespace.

3.Dividethefirstnumberbythesecondnumbertogetadecimalorreducethefractionifafractionanswerisdesired.

EXAMPLE

Adieisrolled;findtheprobabilityofgettinganevennumber.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilityofgettinganevennumber.

Strategy:Whenadieisrolled,therearesixoutcomesinthesamplespace,andtherearethreeoutcomesintheevent—thatis,therearethreeevennumbers:2,4,and6.

Implementation: or0.5.

Evaluation:Since2,4,and6arehalfofthenumbersinthesamplespace,theprobabilityiscorrect.

Whentwocoinsaretossed,thesamplespaceisHH,HT,TH,TT

EXAMPLE

Twocoinsaretossed.Findtheprobabilityofgettingtwoheads.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilityofgettingtwoheads.

Strategy:Therearefouroutcomesinthesamplespace,andthereisonlyonewaytwoheadscanoccur.

Implementation:P=(twoheads)= .

Evaluation:Lookingatthesamplespace,itisobviousthattheprobabilityofonechoicefromfour

outcomesis .

Whentwodicearerolled,eachdiecanhaveoneofsixoutcomes.Therefore,thereare6×6=36outcomesinthesamplespace.Theoutcomescanbearrangedinorderedpairssuchthatthefirstnumberisthenumberofspotsonthefirstdie,andthesecondnumberinthepairisthenumberofspotsontheseconddie.Forexample,theorderedpair(2,4)meansa2cameuponthefirstdieanda4cameupontheseconddie.Also,thesumofthenumbersforthisoutcomeis2+4=6.Thesamplespacefortwodiceisshownnext:

(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)

(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

EXAMPLE

Twodicearerolled;findtheprobabilityofgettingasumof6.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilityofgettingasumof6.

Strategy:Thereare36outcomesinthesamplespaceandfivewaystogetasumofsix.Theyare(1,5),(2,4),(3,3),(4,2),and(5,1).

Implementation: .

Evaluation:Usethesamplespacetoverifyyouranswer.

EXAMPLE

Twodicearerolled;findtheprobabilityofgettingasumgreaterthan9.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilityofgettingasumgreaterthan9.

Strategy:Asumgreaterthan9meansasumof10,11,or12.Theyare(4,6),(5,5),(6,4),(5,6),(6,5)and(6,6).Hence,therearesixwaystogetasumgreaterthan9,andthereare36outcomesinthesamplespace.

Implementation: .

Evaluation:Usethesamplespacetoverifytheanswer.

Probabilityproblemsalsouseordinaryplayingcards.Inadeckofcards,thereare52cardsconsistingoffoursuits:heartsanddiamonds,whicharered,andspadesandclubs,whichareblack.Inaddition,thereare13cardsineachsuit,acethroughtenandajack,aqueen,andaking(calledfacecards).SeeFigure12-4.

FIGURE12-4

EXAMPLE

Acardisdrawnfromadeck.Findtheprobabilitythatitisaking.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilitythattheselectedcardisaking.

Strategy:Thereare52outcomesinthesamplespace,andfourofthemarekings.

Implementation: .

Evaluation:Usethesamplespacetoverifytheanswer.

EXAMPLE

Acardisselectedfromadeck;findtheprobabilitythatitisadiamond.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilityofselectingadiamond.

Strategy:Thereare13diamondsinadeckof52cards.

Implementation: .

Evaluation:Usethesamplespacetoverifyyouranswer.

Theexamplesshownpreviouslyareexamplesofwhatiscalledclassicalprobability.Thenextexamplesarefromanotherareaofprobabilitycalledempiricalprobability.Empiricalprobabilityusesfrequencydistributions.Supposethatabagofmixedcandycontainedsixcaramels,threepeppermints,sevenchocolates,andninecoconutcreams.Thesamplespacecanberepresentedusingafrequencydistributionasshown.

Thisdistributioncanbeusedtosolveprobabilityproblems.

EXAMPLE

Supposeapersonselectsapieceofcandyfromthebag;findtheprobabilitythatitisacaramel.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilitythatthepieceofcandyisacaramel.

Strategy:Thereare6caramelsandatotalof25piecesofcandy,sotheprobabilityformulacanbeused.

Implementation: .

Evaluation:Theanswercanbeverifiedbylookingatthefrequencydistribution.

EXAMPLE

Usingthesamebagofcandy,findtheprobabilitythatapersonselectsapeppermintorachocolate.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilitythatthepieceofcandyisapeppermintorachocolate.

Strategy:Thereare25piecesofcandyandthereare3peppermintsand7chocolates.

Implementation: .

Evaluation:Youcanverifytheanswerbylookingatthefrequencydistribution.

EXAMPLE

Inaclassroom,thereare20juniorsand8seniors.Ifastudentisselectedatrandomtoreadapassage,findtheprobabilitythatthestudentisasenior.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilitythatthestudentisasenior.

Strategy:Thereareatotalof28studentsintheclassand8areseniors.

Implementation: .

Evaluation:Theanswercanbeverifiedbylookingattheproblem.

Therearefourbasicrulesforprobabilityproblems:Rule1:Theprobabilityofanyeventisanumberfromzeroto1.Thismeansthatananswerinaprobabilityproblemcanneverbelessthanzero(i.e.,negative)orgreaterthan1.Rule2:Iftheprobabilityofaneventiszero,theeventcannotoccur.Forexample,ifyourollasingledie,findtheprobabilityofgettinga9.Sincea9cannotoccurwhenyourollasingledie(aregulardiehasonlysixsidesandsixnumbers),P(9)=0/6=0.Rule3:Iftheprobabilityofaneventis1,theeventiscertaintooccur.Forexample,ifyoutakeoutalloftheblackcardsfromadeckof52cards,youhave26redcardsleft.Nowifyouselectonecard,theprobabilitythatitwillberedwillbeP(redcard)=26/26=1.Inotherwords,aredcardiscertaintooccur.Rule4:Thesumoftheprobabilitiesofalltheeventsinthesamplespacewillbe1.Inotherwords,ifyoutakeeacheventinthesamplespace,finditsprobability,andaddallthevalues,youwillalwaysget1.Forexample,ifyourollasingledie,theprobabilityofgettingeachnumberis1/6,andsincetherearesixpossibleoutcomes,thesumoftheseprobabilitieswillbe1/6+1/6+1/6+1/6+1/6+1/6=6/6=1.

Anotherimportantaspectofprobabilityisthattheclosertheprobabilityofaneventisto1,themorelikelytheeventwilloccur.Ontheotherhand,theclosertheprobabilityofaneventistozero,thelesslikelytheeventwilloccur.Sometimesinprobabilityproblems,youwillbeaskedtofindtheprobabilitythatoneeventoranothereventwilloccur.

Theword“or”inthiscasemeanstoaddtheindividualprobabilities.Forexample,ifyoudrawonecardfromthedeck,theprobabilitythatitwillbekingoraqueenwillbe4/52+4/52=8/52=2/13sincetheindividualprobabilitiesare4/52and4/52.Therearefourkingsandfourqueens.Noticethatthesetwoeventscannotoccuratthesametime.Theyarecalledmutuallyexclusiveevents.Now,whatifyoudrawasinglecardfromadeckandyouareaskedtofindtheprobabilityofgettinga7oraclub?Inthiscase,therearefour7s,soP(7)=4/52,andthereare13clubs,soP(club)=13/52.Ifyouaddtheprobabilitiesofgettinga7oraclub,youwillget4/52+13/52=17/52.Thisisthewronganswer,sincethe7ofclubswascountedtwice.Inotherwords,thesetwoeventsarenotmutuallyexclusive.Whentwoeventsarenotmutuallyexclusive,youmustsubtracttheprobabilitythattheeventsoccuratthesametime.SoP(7ofclubs)=1/52.Hence,P(7orclub)=4/52+13/52−1/52=16/52=4/13.

Thetworulesaresummarizedasfollows:

Whentwooutcomesaremutuallyexclusive,P(AorB)=P(A)+P(B).

Whentwooutcomesarenotmutuallyexclusive,P(AorB)=P(A)+P(B)−P(AandB),whereP(AandB)istheprobabilitythattheoutcomesoccuratthesametime.

EXAMPLE

Drawacardfromadeck.Findtheprobabilitythatitisaredcardoranace.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilitythatthecardselectedisaredcardoranace.

Strategy:Thereare26redcardsandfouraces;however,twooftheacesarered.

Implementation:

.

Evaluation:Youcanlookatthesamplespaceandcounttheredcardsandthetwoacesthatarenotred.

Youget28differentcards.Hence,theansweris or .

Anothertypeofprobabilityproblemhappenswhenyouperformtheprobabilityexperimentmorethanonce.Forexample,supposeyourolladiethreetimesandyouareaskedtofindtheprobabilityforacertainoutcomesuchasgettingthree6s.Otherexamplesmightbedrawingtwocardsfromadeckorflippingfivecoins.Inthesetypesofproblems,youhavetodeterminewhetherornottheoutcomeofthefirsttimeyouperformtheexperiment

affectsorchangestheprobabilityoftheoutcomeofthesecondtimeyoudotheexperiment.Forexample,whenyouflipacointwiceorrolladiethreetimes,theoutcomeofthefirsttimedoesnotaffecttheoutcome

ofthesecondtimeyoudotheexperiment.Whenyouflipacoin,theprobabilityofgettingaheadeachtimeisalwaysone-half.Nomatterhowmanytimesyourolladie,theprobabilityofgettinga3willalwaysbe1/6.Inthesecases,theoutcomesaresaidtobeindependentofeachother.Whenyoudrawtwocardsfromadeckandreplacethefirstcardbeforeyouselectthesecondcard,theoutcomesare

independent,butifyoudonotreplacethecardbeforeselectingthesecondcard,theprobabilitychanges.Theseoutcomesaresaidtobedependent.Thesetworulescanbesummarizedasfollows:Whentwoeventsareindependent,P(AandB)=P(A)×P(B).Whentwoeventsaredependent,P(AandB)=P(A)×P(BgiventhatAhasoccurred).

EXAMPLE

Drawtwocardsfromadeckwithoutreplacement.Findtheprobabilityofgettingtwokings.

SOLUTION

Goal:Youarebeingaskedtofindtheprobabilityofgettingtwokingswhentwocardsaredrawnfromadeckwithoutreplacingthefirstcardafteritisdrawn.

Strategy:Therearefourkingsinadeckof52cardsso .Nowifakingoccursonthefirstdraw,

therearethreekingsleftand51cardsremaininginthedeck.So .

Implementation:Applytherule

.

Evaluation:Thesetypesofproblemsaredifficulttoevaluate,sousealittlecommonsenseorreasoningandcheckyourarithmetic.

Noticethatifthefirstcardisreplacedafterthefirstdraw,theoutcomesareindependentand

.

TRYTHESE

1.Asingledieisrolledonce;findtheprobabilityofgetting

a.a3

b.anumbergreaterthan2

c.anumberlessthan7

d.anumbergreaterthan6

2.Twodicearerolled;findtheprobabilityofgetting

a.asumof9

b.doubles

c.asumgreaterthan10

d.asumlessthan4

3.Acardisdrawnfromadeck;findtheprobabilityofgetting

a.the7ofspades

b.ajack

c.aclub

d.aheartoraclub

e.aredcard

4.Acouplehasthreechildren;findtheprobabilitythatthechildrenare

a.allgirls

b.allboysorallgirls

c.exactlytwogirlsandoneboy

5.Twodicearerolled;findtheprobabilityofgettingasumof8or10.

6.Inacoolerthereareninecansofcolaandsixcansofcherrysoda.Ifapersonselectsacanofsodawithoutlookingatit,findtheprobabilitythatitisacanofcola.

7.Twodicearerolled;findtheprobabilityofgettingasumgreaterthan8ordoubles.

8.Aboxcontainsthreeorangeballs,twoblueballs,andoneredball.Iftwoballsareselectedwithoutreplacement,findtheprobabilityofgettingtwoorangeballs.

9.Adieisrolledthreetimes;findtheprobabilityofgettinganevennumberallthreetimes.

10.Adieisrolledtwice.Findtheprobabilityofgettingthesamenumbertwice.

SOLUTIONS

1.

a.Therearesixoutcomesinthesamplespaceandoneoutcomeisa3;therefore, .

b.Therearesixoutcomesinthesamplespaceandtherearefouroutcomesthataregreaterthan2;thatis,3,4,5,

and6;hence, .

c.Therearesixoutcomesinthesamplespaceandsixnumberslessthan7;hence,

.

d.Therearesixoutcomesinthesamplespaceandnonumbersaregreaterthan6;hence,

.

2.

a.Thereare36outcomesinthesamplespaceandtherearefourwaystogetasumof9:(3,6),(4,5),(5,4),and

(6,3);hence, .

b.Thereare36outcomesinthesamplespaceandsixwaystogetdoubles:(1,1),(2,2),(3,3),(4,4),(5,5),and(6,

6);hence .

c.Thereare36outcomesinthesamplespaceandtwosumsgreaterthan10—thatis,asumof11,or12:(5,6),(6,

5),and(6,6).Hence, .

d.Thereare36outcomesinthesamplespaceandthreewaystogetasumof3or2:(1,2),(2,1),and(1,1).

Hence, .

3.

a.Thereare52outcomesinthesamplespaceandone7ofspades;hence, .

b.Thereare52outcomesinthesamplespaceandfourjacks;hence, .

c.Thereare52outcomesinthesamplespaceand13clubs;hence, .

d.Thereare52outcomesinthesamplespaceand13heartsand13clubs;hence,

.

e.Thereare52outcomesinthesamplespaceand26redcards(13diamondsand13hearts);hence,

.

4.Thesamplespaceforthreechildrenis

a.Thereareeightoutcomesinthesamplespaceandonewaytogetallgirls:GGG;hence, .

b.Thereareeightoutcomesinthesamplespaceandtwowaystogetallboysorallgirls:BBBandGGG;hence,

.

c.Thereareeightoutcomesinthesamplespaceandthreewaystogettwogirlsandoneboy:GGB,GBG,BGG;

hence, .

5.Thereare36outcomesinthesamplespaceandfivewaystogetan8andthreewaystogeta10;hence,

.

6.Thereare9+6=15cansinthecoolerand9ofthemarecola;hence .

7.Thereare36outcomesinthesamplespaceand10waystogetasumgreaterthan8.Thereare6waystogetdoubles,but(5,5)and(6,6)havebeencountedtwice,so

P(sumgreaterthan8or .

8.P(2orangeballs)=P(orange)×P(orange,giventhatanorangeballhasoccurred)= .

9.P(3evennumbers)= .Theeventsareindependent.

10.Inthiscase,anynumbercanoccurthefirsttime,butonthesecondroll,theoutcomehastomatchthenumber

thatoccurredthefirsttime.Thatis .Hence, .

Inthissection,youlearnedtosolvesimpleprobabilityproblems.Herethesolutionsareobtainedbydeterminingthe

numberofoutcomesinthesamplespace.Thisnumberisplacedinthedenominatorofthefraction.Thenumberofoutcomesdesiredisplacedinthenumeratorofthefraction.Thefractionisreducedifpossible.Severalprobabilityruleswerepresentedinthissectionandrulesfordeterminingtheprobabilityofeventswhenorisusedaregiven.Finally,whentheprobabilityexperimentisperformedmorethanonce,twoadditionalruleswereexplained.

SolvingStatisticsProblems

Statisticsisthescienceofconductingstudiestocollect,organize,analyze,summarize,anddrawconclusionsfromdata.Thedatacanbenumberssuchasweights,temperatures,testscores,etc.,orobservationssuchascolorsofautomobiles,politicalaffiliations,etc.Agroupofdatavaluescollectedforaparticularstudyiscalledadataset.Statisticsisusedinalmostallfieldsofhumanendeavor.

Instatistics,therearethreecommonlyusedmeasuresofaverage.Theyarethemean,median,andmode.Themeanisthesumofthedatavaluesdividedbythetotalnumberofdatavalues.

EXAMPLE

Findthemeanof9,23,15,20,and18.

SOLUTION

Goal:Youarebeingaskedtofindthemeanforthegivendataset.

Strategy:Addthevaluesanddividethesumby5(therearefivedatavalues).

Implementation:

9+23+15+20+18=8585÷5=17

Themeanis17.

Evaluation:Themeanwillfallbetweenthelowestandhighestvaluesand,mostofthetime,somewherenearthemiddleofthevalues.

Themedianisavaluethatfallsinthecenterofthedataset.Youmustfirstarrangethedatainorderfromthesmallestdatavaluetothelargestdatavalue.

EXAMPLE

Findthemedianfor17,24,22,16,and7.

SOLUTION

Goal:Youarebeingaskedtofindthemedianforthegivendataset.

Strategy:Arrangethedatavaluesinorderandfindthemiddlevalue.

Implementation:

7,16,17,22,24

Since17isthemiddlevalue,themedianis17.

Evaluation:Checktoseeifthedatavaluesarearrangedcorrectly;thenmakesureyouhavefoundthemiddlevalue.

Ifthenumberofdatavaluesisodd,asinthepreviousexample,themedianwillbeoneofthevalues;however,ifthenumberofdatavaluesiseven,themedianwillfallhalfwaybetweenthemiddletwovalues,asshowninthenextexample.

EXAMPLE

Findthemedianfor86,23,52,63,44,and91.

SOLUTION

Goal:Youarebeingaskedtofindthemedianforthegivendataset.

Strategy:Arrangethedatainorder;thenfindthemiddlepoint.

Implementation:

23,44,52,63,86,91

Themiddleofthedataishalfwaybetween52and63;hence,themedianis

Evaluation:Checkthesolution.

Thethirdmeasureofaverageiscalledthemode.Themodeisthedatavaluethatoccursmostoften.

EXAMPLE

Findthemodeof19,24,16,18,19,and27.

SOLUTION

Goal:Youarebeingaskedtofindthemodeforthegivendataset.

Strategy:Findthevaluethatoccursmostoften.

Implementation:

Itishelpful,althoughnotnecessary,toarrangethedatainorder:

16,18,19,19,24,27

Since19occurstwiceandthatismoreoftenthananyothernumber,19isthemode.

Evaluation:Theanswerisobvious.

EXAMPLE

Findthemodefor5,6,8,9,9,9,10,10,12,12,12,and16.

SOLUTION


Strategy:Analyzethedataandseewhatvalueoccursmostoften.

Implementation:Inthiscase,thevaluesof9and12occurthreetimes.Hence,thedatahastwomodes.Theyare9and12.

Evaluation:Theanswerisobvious.

EXAMPLE

Findthemodefor103,206,87,54,and153.

SOLUTION


Strategy:Findthedatavaluethatoccursmostoften.

Implementation:Inthiscase,eachdatavalueoccursonlyonce.Hence,wesaythatthereisnomode.

Twothingsshouldbenoted:

1.Themodeofadatasetcanbeasinglevalue,morethanonevalue,ornovalueatall.

2.Themean,median,andmodeforadataset,inmostcases,willnotbeequal.

Inadditiontothemeasuresofaverage,statisticiansalsousemeasuresofvariationtodescribeadataset.Thetwomostoftenusedmeasuresofvariationaretherangeandthestandarddeviation.Thesemeasuresdescribethespreadofthedataaboutthemean.Looselyspeaking,thelargertherangeorstandarddeviation,themorevariableorspreadoutthedataisintheset.Therangeisfoundbysubtractingthesmallestdatavaluefromthelargestdatavalue.

EXAMPLE

Findtherangefor17,32,19,16,and15.

SOLUTION

Goal:Youarebeingaskedtofindtherangeforthegivendataset.

Strategy:Subtractthesmallestdatavaluefromthelargestdatavalueintheset.

Implementation:Thesmallestdatavalueis15,andthelargestdatavalueis32,sotherangeis32−15=17.

Evaluation:Redotheproblem.

Therangeisaroughestimateofvariation,sostatisticiansalsousewhatiscalledthestandarddeviation.Thestandarddeviationcanbecomputedbyusingthefollowingprocedure:

1.Findthemeanforthedataset.

2.Subtractthemeanfromeachvalueinthedataset.

3.Squarethedifferences.

4.Findthesumofthesquares.

5.Dividethesumbyn−1,wherenisthenumberofdatavalues.

6.Takethesquarerootoftheanswer.(Youmayneedacalculatorforthisstep.)

EXAMPLE

Findthestandarddeviation:14,22,16,28,and20.

SOLUTION

Goal:Youarebeingaskedtofindthestandarddeviationforthegivendataset.

Strategy:Usetheproceduregivenpreviously.

Implementation:

1.Findthemean:

14+22+16+28+20=100

100÷5=20

2.Subtractthemeanfromeachdatavalue:

14−20=−6

22−20=2

16−20=−4

28−20=8

20−20=0

3.Squaretheanswers:

(−6)2=36

22=4

(−4)2=16

82=64

02=0

4.Findthesumofthesquares:

36+4+16+64+0=120

5.Dividethesumbyn−1,wheren=5andn−1=5−1=4:

120÷4=30

6.Findthesquarerootof30:

(rounded)

Thestandarddeviationis5.48.

Evaluation:Thestandarddeviationcanbeestimatedbydividingtherangeby4.Inthiscase,therangeis28−14=14.Thus,14÷4=3.5.Sincethisisonlyaroughestimate,weareintheballpark.

Roughlyspeaking,mostofthedatavalueswillusuallyfallbetweentwostandarddeviationsofthemean.

TRYTHESE

Forthedataset28,13,19,24,18,and24,findeach:

1.Themean

2.Themedian

3.Themode

4.Therange

5.Thestandarddeviation

SOLUTIONS

1.28+13+19+24+18+24=126

126÷6=21

Themean=21.

2.13,18,19,24,24,28

Themiddlevalueishalfwaybetween19and24;hence,themedianis(19+24)÷2=43÷2=21.5.

3.Thevaluethatoccursmostoftenis24,sothemodeis24.

4.Therangeis28−13=15.

5.Tofindthestandarddeviation,followthesesteps:

Findthemean.Itis21,asfoundinanswer1.

Subtractthemeanfromeachdatavalue:

28−21=7

13−21=−8

19−21=−2

24−21=3

18−21=−3

24−21=3

Squarethedifferences:

72=49

(−8)2=64

(−2)2=4

32=9

(−3)2=9

32=9

Findthesumofthedifferences:

49+64+4+9+9+9=144

Dividethesumby6−1+5

Findthesquarerootof28.8:

=5.37(rounded)

Hence,thestandarddeviationis5.37.

Inthissection,youlearnedhowtosolveproblemsusingstatistics.Therearethreemeasuresofaverage.Theyarethemean,median,andmode.Therearetwomeasuresofvariation.Theyaretherangeandstandarddeviation.Thesearethecommonstatisticalmeasuresthataremostoftenused.

Summary

Thischapterexplainedhowtosolvethreespecialtypesofproblems.Theyaregeometryproblems,probabilityproblems,andstatisticsproblems.Geometryproblemsusebasicgeometricprinciples.Probabilityandstatisticsproblemsuseformulas.

QUIZ

1.Asinglecardisselectedfromadeckofcards.Findtheprobabilitythatitisaclub.

2.Twodicearerolled;findtheprobabilityofgettingasumof11orasumoflessthan4.

3.Asingledieisrolled;findtheprobabilityofgettinga7.

A.0

C.1

D.Cannotbecomputed

4.Findthemeanof156,170,192,and146.

A.166

B.142

C.163

D.175

5.Findthemedianof12,5,10,and16.

A.10

B.7.5

C.10.5

D.11

6.Findthemedianof56,18,44,22,and65.

A.41.5

B.44

C.40

D.48.2

7.Findthemodeof19,37,15,14,and18.

A.19

B.18

C.20.6

D.nomode

8.Findthemodeof6,5,8,4,5,9,and12.

A.5

B.8.5

C.8

D.7

9.Findtherangeof8,14,10,8,and22.

A.10

B.14

C.8

D.22

10.Findthestandarddeviation(roundedtoonedecimalplace)of34,36,24,18,26.

A.3.6

B.5.2

C.13.7

D.7.4

FinalExam

1.ThesizeofCubais42,031squaremiles,andthesizeofGreatBritainis88,407squaremiles.HowmuchlargerisGreatBritain?

A.46,376squaremiles

B.54,327squaremiles

C.35,162squaremiles

D.130,438squaremiles

2.FindthetotaloftheareasoftheSeaofJapan,whichis391,100squaremiles,andtheHudsonBay,whichis281,900squaremiles.

A.109,200squaremiles

B.673,000squaremiles

C.432,100squaremiles

D.323,300squaremiles

3.Ifapersonpays$324amonthonaloan,howmuchwillthepersonpayinayear?

A.$27

B.$336

C.$3,888

D.$5,428

4.Howmanyboxesareneededtopackage448bottlesofshampooif14bottlescanfitinabox?

A.32

B.16

C.28

D.6,272

5.Apersontraveledfromherhometoabakery,adistanceof miles.Thenshewenttohersalon,adistanceof

milesfromthebakery.Howfardidshetravelinall?

6.Ageneratoruses gallonofgasolineperhour.Howmanygallonsofgasolineareusedifitisrun hours?

B.3gallons

D.5gallons

7.Ataxiservicecharges$8plus75centspermiletorentataxi.Howmuchdoesapersonpayfora16-miletrip?

A.$12

B.$8.75

C.$14.25

D.$20

8.Howmanypiecesofwood feetlongcanbecutfromaboardthatis10feetlong?

A.3

B.4

C.5

D.6

9.Aclerksold poundsofpeanuts, poundsofcashews,and poundsofalmonds.Howmanypoundsofnutsweresoldinall?

10.Mikeis feettallandCindyis feettall.HowmuchtallerisMike?

11.Apersontraveled374.4mileson16gallonsofgasoline.Howmanymilespergallondidthepersonget?

A.22.6milespergallon

B.23.4milespergallon

C.21.5milespergallon

D.24.7milespergallon

12.Julie’sbicyclespeedometerread534.2milesbeforeshestartedherride.Whenshefinished,herspeedometerread551.6miles.Howfardidshetravel?

A.18.4miles

B.14.4miles

C.17.4miles

D.16.4miles

13.Thevalueofahomehasincreased15%.Howmuchisthehomeworthnowifitsoriginalpricewas$71,875?

A.$10,781.25

B.$68,475

C.$62,500

D.$82,656.25

14.Apersonreceivesa4%raise.Findthenewsalaryifheearns$32,000now.

A.$1,280

B.$28,250

C.$33,280

D.$30,100

15.Apersondrove386.1milesandgot23.4milespergallon.Howmanygallonsdidthepersonuse?

A.16.5

B.18.3

C.23.4

D.20.5

16.Whatisthesellingpriceofacameraifthesalestaxis$14.52andtherateis6%?

A.$276.98

B.$242

C.$87.12

D.$321

17.Inordertogetalightbluepaint,2gallonsofwhitepaintaremixedwith5gallonsofbluepaint.Togetthesamecolor,howmanygallonsofwhitepaintareneededtobemixedwith22gallonsofbluepaint?

A.8.8gallons

B.6.4gallons

C.7gallons

D.6gallons

18.Mikebought14candybarsandpaid$25.50.Ifsomeofthebarscost$1.25andtherestcost$2.25,howmanyofthe$2.25candybarsdidhebuy?

A.6

B.4

C.5

D.8

19.Onamap,thescaleis inch=30miles.Findtheactualdistancebetweentwocitiesiftheyare3inchesapart.

A.60miles

B.54.8miles

C.72miles

D.120miles

20.Threeyearsago,Harrywastwiceasoldashisbrother.Ifthedifferenceintheiragesis8years,howoldisHarrytoday?

A.10

B.15

C.19

D.21

21.Onepipecanfillatankin16hoursandanotherpipecanfillthetankin20hours.Ifbothpipesareopened,howlongwillittaketofillthetank?(Roundtheanswertoonedecimalplace.)

A.12.3hours

B.10.6hours

C.9.3hours

D.8.9hours

22.Aleveris10feetlong.Whereshouldthefulcrumbeplacedinordertobalance40poundsatoneendand160poundsfromtheotherend?

A.6feetfromthe40pounds

B.8feetfromthe40pounds

C.3feetfromthe40pounds

D.5feetfromthe40pounds

23.Iftheproductoftwopositiveconsecutiveevennumbersis3,024,findthelargernumber.

A.46

B.54

C.56

D.48

24.Ifthelengthofarectangularplatformis6feetmorethanitswidthandtheareaoftheplatformis391squarefeet,findthelengthoftheplatform.

A.15feet

B.23feet

C.19feet

D.17feet

25.Lorihastwosavingsaccounts.Oneaccountpays4.6%interestandtheotherpays2.5%.Ifthetotalinvestmentis$19,000andthetotalinterestis$664,findtheamountofmoneyLorihasinvestedat4.6%.

A.$8,000

B.$9,000

C.$5,200

D.$6,500

26.Achild’sbankcontains42coinsconsistingofnickelsandquartersonly.Findthenumberofnickelsitcontainsifthetotalamountinthebankis$5.10.

A.15

B.19

C.27

D.31

27.Thesumofthedigitsofatwo-digitnumberis12.Ifthedigitsarereversed,thenewnumberis36morethantheoriginalnumber.Findthenumber.

A.48

B.39

C.93

D.84

28.Inatwo-digitnumber,thetensdigitis5morethantheonesdigit.Ifthedigitsarereversed,thenewnumberis45lessthantheoriginalnumber.Findtheoriginalnumber.

A.72

B.61

C.94

D.83

29.Anairplanetook10hourstoflyadistanceof750miles,flyingagainstthewind.Ifthereturntriptook6hoursflyingwiththewind,findthespeedofthewind.

A.20milesperhour

B.25milesperhour

C.15milesperhour

D.30milesperhour

30.Twopeopleleavetwotownsthatare200milesapartanddrivetowardeachother.Ifonepersondrives8milesperhourslowerthantheother,andtheymeetintwohours,howfastwastheslowerdrivergoing?

A.42milesperhour

B.46milesperhour

C.50milesperhour

D.54milesperhour

31.Howfarwillanautomobiletravelin hoursataspeedof32milesperhour?(UseD=RT.)

A.96miles

B.99miles

C.108miles

D.116miles

32.Findtheinterestonaloanof$9650at4%forsevenyears.(UseI=PRT.)

A.$386

B.$2,702

C.$3,160

D.$4,825

33.Findtheareaofatrianglewhosebaseis16feetandwhoseheightis9feet.

A.72squarefeet

B.144squarefeet

C.25squarefeet

D.50squarefeet

34.Findthedistanceanobjectfallsin12seconds.

A.4,608feet

B.192feet

C.96feet

D.2,304feet

35.Twoanglesofatriangleareequalinmeasure.Ifthethirdangleis15°greaterthantheotherangles,findthemeasureofthethirdangle.Thesumofthemeasuresoftheanglesofatriangleis180°.

A.70°

B.55°

C.50°

D.65°

36.Astoreownerhaseightmorescarvesthanshehasjackets.Findthenumberofjacketsshehasifshehasatotalof40items.

A.8

B.16

C.20

D.24

37.Whentwodicearerolled,theprobabilityofgettingasumof10is

38.Theprobabilityofgettingan8whenasingledieisrolledis

B.0

D.1

39.Whenacardisselectedfromadeck,theprobabilityofgettinganaceandaredcardis

40.Acommitteeconsistsoffivewomenandfourmen.Ifachairpersonisselected,findtheprobabilitythatitisawoman.

41.Whenthreecoinsaretossed,theprobabilityofgetting0,1,2,or3headsis

A.0

D.1

42.Thesumoftheprobabilitiesofalltheeventsinthesamplespacewillalwaysbe

A.0

B.1

D.Itvaries

43.Aprofessorhas10booksonashelfinhisoffice.Threearecalculusbooks,twoarealgebrabooks,andfivearestatisticsbooks.Ifheselectsabookatrandom,whatistheprobabilitythatheselectsanalgebrabookorastatisticsbook?

44.Findthemeanof19,27,14,19,16,20,and18.

A.18

B.18.5

C.19

D.Nomean

45.Whichofthefollowingstatementsistrue?

A.Themean,median,andmodeofadatasetwillalwaysbeequal.

B.Themean,median,andmodeofadatasetcanneverbeequal.

C.Ifthedatainthedatasetarewholenumbers,themeanwillalwaysbeawholenumber.

D.Noneoftheabovestatementsistrue.

46.Findthemedianof42,87,16,23,27,52,63,and20.

A.41.25

B.42

C.27

D.34.5

47.Findthemodeof20,7,19,11,17,and19.

A.17.5

B.15

C.19

D.16.5

48.Findthemodeof32,52,43,38,and41.

A.Nomode

B.43

C.42

D.38

49.Findtherangeof38,52,75,19,63,and37.

A.31

B.56

C.14

D.49

50.Findthestandarddeviationroundedtotwoplacesof12,18,20,23,and27.

A.20

B.5.61

C.31.5

D.19.5

AnswerstoQuizzesandFinalExam

Chapter1

1.B

2.D

3.D

4.D

5.C

6.A

7.B

8.C

9.D

10.C

Chapter2

1.B

2.A

3.C

4.D

5.A

6.D

7.B

8.C

9.D

10.B

Chapter3

1.C

2.C

3.C

4.C

5.B

6.D

7.A

8.B

9.D

10.A

Chapter4

1.C

2.D

3.A

4.B

5.D

6.C

7.B

8.C

9.A

10.D

Chapter5

1.C

2.D

3.A

4.B

5.C

6.C

7.B

8.A

9.D

10.B

Chapter6

1.B

2.A

3.C

4.D

5.C

6.B

7.A

8.D

9.B

10.A

Chapter7

1.B

2.D

3.C

4.B

5.A

6.B

7.C

8.B

9.D

10.D

Chapter8

1.A

2.D

3.B

4.C

5.B

6.A

7.C

8.D

9.C

10.C

Chapter9

1.B

2.B

3.D

4.C

5.D

6.A

7.C

8.B

9.C

10.D

Chapter10

1.C

2.B

3.B

4.D

5.A

6.C

7.D

8.A

9.B

10.C

Chapter11

1.A

2.D

3.A

4.B

5.C

6.D

7.A

8.B

9.C

10.C

Chapter12

1.A

2.C

3.A

4.A

5.D

6.B

7.D

8.A

9.B

10.D

FinalExam

1.A

2.B

3.C

4.A

5.B

6.B

7.D

8.D

9.C

10.A

11.B

12.C

13.D

14.C

15.A

16.B

17.A

18.D

19.D

20.C

21.D

22.B

23.C

24.B

25.B

26.C

27.A

28.D

29.B

30.B

31.C

32.B

33.A

34.D

35.A

36.B

37.C

38.B

39.A

40.D

41.D

42.B

43.A

44.C

45.D

46.D

47.C

48.A

49.B

50.B

SuggestionsforSuccessinMathematics

1.Besuretoattendeveryclass.Ifyouknowaheadoftimethatyouwillbeabsent,tellyourinstructorandgettheassignment.Ifitisanemergencyabsence,gettheassignmentfromanotherstudent.Trytodotheproblemsbeforethenextclass.Ifpossible,gettheclassnotesfromanotherstudent.

2.Readthematerialinthetextbookseveraltimes.Writedownorunderlinealldefinitions,rules,andsymbols.Trytodothesampleproblems.

3.Doallassignedhomeworkassoonaspossiblebeforethenextclass.Concentrateonmathematicsonly.Getallofyourmaterialsbeforeyoustartdoingyourhomework.Makesureyouwritetheassignmentonthetopofyourhomework.Readthedirections.Copyeachproblemonyourhomeworkpaper.Makesurethatyouhavecopieditcorrectly.Donotusescratchpaper.Workouteachproblemindetailanddonotskipsteps.Writeneatlyandlargeenough.Checktheanswerwiththeoneinthebackofthebookorreworktheproblemagain.Ifyoudidnotgetthecorrectanswer,trytofindyourmistakeorstartover.Don’tlookforshortcuts,becausetheydonotalwayswork.Writedownanyquestionsyouhaveandaskyourinstructororanotherstudentatthenextclassperiod.Ifyouarehavingdifficultywiththeproblem,consultyourtextbookandnotes.Don’tgiveuptooquickly.

4.Alwaysreviewbeforeeachexam.Youcanusuallyfindarevieworchaptertestattheendofeachchapterinthebook.Ifnot,youcanmakeupyourownreviewbyselectingseveralproblemsfromeachsectioninthebooktotry.Ifyoucan’tgetthecorrectanswer,asktheteacheroranotherstudenttohelpyoubeforetheexam.Ifyouhavemadestudycards,reviewthem.

5.Onthedayofthetest,arriveearly.Lookoveryournotesandstudycards.Bringallnecessarymaterialssuchaspencils,protractor,calculator,textbook,etc.,toclass.Whenyougetthetest,lookovertheentiretestbeforeyougetstarted.Readthedirections.Worktheproblemsthatyouknowhowtodofirst.Donotspendtoomuchtimeonanyoneproblem.Afteryouhavefinishedthetest,iftimeallows,checkeachproblem.Whenyougetthetestback,checkyourmistakesandstudythetypesofproblemsthatyouhavemissed,becausesimilarproblemsmaybeonthefinalexam.

6.Ifyouhavedifficultywithmathematics,arrangeforatutor.Someschoolshavelearningcenterswhereyoucanreceivefreetutoring.

7.Finally,makesurethatyouareinthecorrectclass.Youcannotskipmathclasses.Mathematicsissequentialinnature.Whatyoulearntoday,youwillusetomorrow.Whatyoulearninonecourse,youwilluseinthenextcourse.

GOODLUCK!

A

addition,20ageproblems,137–145algebraicexpression,84algebraicrepresentation,97–102angleofatriangle,264–265area:

circle,268square,268triangle,266–268

B

base,48

C

Celsiustemperature,77checkinganequation,84classicalprobability,276clearingfractions,93–94coinproblems,128–136conditionalequation,84consecutiveevenintegers,111consecutiveintegers,110consecutiveoddintegers,110crossmultiplication,66–67current,77

D

dataset,284decimals,24–27

addition,24division,25multiplication,25subtraction,24wordproblems,27–30

decimalstofractions,36denominator,30digitproblems,116–124digits,116dilutedsolution,166–167distance,74,150distanceproblems,150–163division,22downstream,158drawapicture,9–10

E

empiricalprobability,276equation,84estimation,2evaluation,2

F

factoring,246–248Fahrenheittemperature,75financeproblems,180–190findapattern,3–5,10–12formula,74formulaproblems,75–79fractionproblems,30–42fractions,30–42

addition,32–33division,33–34multiplication,33subtraction,33wordproblems,39–42

fractionstodecimals,36fractionstohigherterms,31frequencydistribution,276fulcrum,190

G

goal,2groupingsymbols,84,90guessandtest,7–8

H

headwindoperations,75HowtoSolveIt,2

I

implementation,2improperfractions,31improperfractionstomixednumbers,31interest,75,180

L

leverproblems,190–197lowestcommondenominator(LCD),32

M

makeanorganizedlist,6–7measuresofaverage:

mean,284median,285mode,286

measuresofvariation:range,287standarddeviation,287–290

mixednumbers,31,34–36addition,34division,35–36multiplication,35subtraction,34–35

mixednumberstoimproperfractions,32mixtureproblems,164–176multiplication,21

N

numberproblems,108–116numerator,30

O

operations,19–20,75

P

part,55–57percent,46–62

decimaltopercent,46–47fractiontopercent,47–48percenttodecimal,46percenttofraction,47wordproblems,55–62

percentdecrease,59–60percentincrease,59–60percentproblems,48–54perimeterofarectangle,265Polya,1principal,75,180probability,272–273probabilityexperiment,272–273

event,273outcome,273samplespace,273

probabilityrules,278proportion,66

Q

quadraticequations,246–251

R

rate,48–49,74–75,180ratio,66reducingfractions,30–31return,180reversingdigits,116

S

seconddegreeequation,246solution,84solveasimplerproblem,12–13solvinganequation,84–97solvingprobabilityproblems,272–280solvingproblemsingeometry,264–272solvingproblemsusingproportions,66–74solvingproblemsusingquadraticequations,252–260solvingproblemsusingtwoequations,223–241standarddeviation,287–288statistics:

average:mean,284median,285mode,286

range,287spread,287standarddeviation,287–288

strategy,2substitution,214subtraction,20–21systemsofequations,214

T

tailwind,158time,74–75

U

upstream,158

W

workbackwards,13workproblems,77,197–209

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