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MathWordProblemsDeMYSTiFieD®
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MathWordProblemsDeMYSTiFieD®
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.
MathWordProblemsDeMYSTiFieD®
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
2.Thenextnumberinthesequence12346710111516is
A.17
B.18
C.20
D.21
3.Thenextnumberinthesequence364861210is
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
Inthischapter,youwilllearnhowto
•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.
3.Whatpercentof64is48?
4.150is25%ofwhatnumber?
5.Find84%of63.
6.72is24%ofwhatnumber?
7.Whatpercentof35is21?
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
Inthischapter,youwilllearnhowto
•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 .
Implementation:Solvetheproportion.
Evaluation:
Theansweriscorrect.
EXAMPLE
Iffourgallonsofpaintcancover1,240squarefeet,howmanysquarefeetwillsevengallonsofpaintcover?
SOLUTION
Goal:Youarebeingaskedtofindhowmanysquarefeetsevengallonsofpaintwillcover.
Strategy:Writetheratio.Itis .Theproportionis .
Implementation:Solvetheproportion.
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
Inthischapter,youwilllearnhowto
•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
Solveeachequationforx:
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
Solveeachequationforx:
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
Inthischapter,youwilllearnhowto
•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.
Implementation:Solvetheequation.
Hence,thereare20studentsinoneclassand26studentsintheother.
Evaluation:Checktheanswer:20+26=46
EXAMPLE
Ifanumberisdecreasedby4andtwotimestheoriginalnumberisequaltosixtimestheothernumber,findthenumbers.
SOLUTION
Goal:Youarebeingaskedtofindtwonumbers.
Strategy:Letx=theoriginalnumberandx−4=theothernumber.
Nowtwotimestheoriginalnumberis2xandsixtimestheothernumberis6(x−4).Theequationis2x=6(x−4).
Implementation:Solvetheequation.
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.
Implementation:Solvetheequation:
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
Implementation:Solvetheequation:
Thetensdigitis1andtheonesdigitis9−1=8.Thenumber,then,is18.
Evaluation:Take18andreversethedigitstoget81.Subtract81−18=63.Hencethesumofthedigits1+8is9andthedifferenceofthetwonumbersis63.
EXAMPLE
Inatwo-digitnumber,thetensdigitis3lessthantheonesdigit.Ifthedigitsofthenumberarereversed,thesumoftheoriginalnumberandthenewnumberis77.Findtheoriginalnumber.
SOLUTION
Goal:Youarebeingaskedtofindatwo-digitnumber.
Strategy:Letx=theonesdigitandx−3=thetensdigit.Thenumber,then,is10(x−3)+x.Whenthedigitsarereversed,thenewnumberis10x+(x−3).Sincetheirsumis77,theequationis10(x−3)+x+10x+x−3=77.
Implementation:Solvetheequation:
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.
Implementation:Solvetheequation:
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
Inthischapter,youwilllearnhowto
•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.
Implementation:Solvetheequation:
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 .
Implementation:Solvetheequation:
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.
Implementation:Solvetheequation:
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.
Implementation:Solvetheequation:
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.
Implementation:Solvetheequation:
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.
Implementation:Solvetheequation:
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.
Implementation:Solvetheequation:
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
Inthischapter,youwilllearnhowto
•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).
Implementation:Solvetheequation:
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 .
Implementation:Solvetheequation:
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).
Implementation:Solvetheequation:
Evaluation:Checktoseeifthedistancesarethesame.UseD=RT.
EXAMPLE
Aboat’sspeedometerreads22milesperhourgoingdownstreamandreachesitsdestinationinanhour.Ifthereturntriptakesoneandahalfhoursatthespeedof25milesperhour,howfastisthecurrent?
SOLUTION
Goal:Youarebeingaskedtofindthespeed(rate)ofthecurrent.
Strategy:Letx=therateofthecurrent;thenthespeedoftheboatdownstreamis22+xandupstreamis25−x.Thetimesaregiven.
Implementation:Solvetheequation:
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
Implementation:Solvetheequation:
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
Implementation:Solvetheequation:
60%x+85%(30−x)=75%(30)
Changethepercentstodecimalsbeforesolvingtheequation.
Hence,12ouncesofthe60%solutionshouldbemixedwith18ouncesofthe85%solutiontoget30ouncesofa75%solution.
Evaluation:Checkthesolution:
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.
Evaluation:Checkthesolution:
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).
Implementation:Solvetheequation:
Hence,12.5poundsofcandycosting$6perpoundmustbemixedwith12.5poundsofcandycosting$2apoundtoget25poundsofcandycosting$4apound.
Evaluation:Checkthesolution:
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
Inthischapter,youwilllearnhowto
•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
Implementation:Solvetheequation:
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
Implementation:Solvetheequation:
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)
Implementation:Solvetheequation:
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
Implementation:Solvetheequation:
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
Implementation:Solvetheequation:
TheLCDis24.
Sincethefirstpainterhasalreadyworkedtwohours,thetimeittakestopaintthewholeroomis2+3.6=5.6hours.Evaluation:Checkthesolution:
EXAMPLE
Alargewatertankcanbefilledin12hoursanddrainedin30hours.Howlongwillittaketofillthetankiftheownerhasforgottentoclosethedrainvalve?
SOLUTION
Goal:Youarebeingaskedhowlonginhoursitwilltaketofillthetankifthedrainisleftopen.
Strategy:Letx=thetimeinhoursittakestofillthetank.
Sincethedrainisemptyingthetank,theequationis
Implementation:Solvetheequation:
TheLCDis60.
Hence,itwilltake20hourstofillthetank.Evaluation:Checkthesolution:
EXAMPLE
Sarahcandoajobin40minutesand,workingwithMillie,bothcandothejobin15minutes.HowlongwillittakeMillietodothejobalone?
SOLUTION
Goal:YouarebeingaskedtofindthetimeinminutesittakesforMillietocompletethejobalone.
Strategy:Letx=thetimeittakesMillietocompletethejob.
Theequationis
Implementation:Solvetheequation:
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
Inthischapter,youwilllearnhowto
•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
Goal:Youarebeingaskedtofindtwonumbers.
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
Goal:Youarebeingaskedtofindatwo-digitnumber.
Strategy:Letx=thetensdigit
y=theonesdigit
Then
10x+y=originalnumber
10y+x=newnumberwithdigitsreversed
Sincethesumofthedigitsofthenumberis14,thefirstequationis
x+y=14
Sincereversingthedigitsgivesanewnumberthatis18morethantheoriginalnumber,thesecondequationis
(10x+y)+18=(10y+x)
Implementation:Solvethesystem:
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
Implementation:Solvethesystem:
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)
Implementation:Solvethesystem:
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
Implementation:Solvethesystem:
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)
Implementation:Solvethesystem:
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
Implementation:Solvethesystem:
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
Inthischapter,youwilllearnhowto
•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
Step2:Factortheleftside.
(3x−8)(2x+3)=0
Step3:Setbothfactorsequaltozero.
3x−8=02x+3=0
Step4:Solveeachequation.
EXAMPLE
Solve3x2=27.
SOLUTION
Step1:Writeinstandardform.
3x2−27=0
Step2:Factortheleftside.
3(x+3)(x−3)=0
Step3:Dividebothsidesby3andsetbothfactorsequaltozero.
x+3=0x−3=0
Step4:Solveeachequation.
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.
Implementation:Solvetheequation:
Hence,thetwonumbersare6and12.Evaluation:Checkthefactsoftheproblem.Thesum6+12is18andtheproductis6·12=72.
EXAMPLE
Iftheproductoftwoconsecutivenumbersis156,findthenumbers.
SOLUTION
Goal:Youarebeingaskedtofindtwoconsecutivenumberswhoseproductis156.
Strategy:Letx=thefirstnumberandx+1=thenextnumber.
Theequationfortheproductisx(x+1)=156.
Implementation:Solvetheequation:
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 .
Implementation:Solvetheequation:
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
Inthischapter,youwilllearnhowto
•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°
Implementation:Solvetheequation:
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.
Implementation:Solvetheequation:
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 .
Implementation:Solvetheequation:
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
Goal:Youarebeingaskedtofindthemodeforthegivendataset.
Strategy:Analyzethedataandseewhatvalueoccursmostoften.
Implementation:Inthiscase,thevaluesof9and12occurthreetimes.Hence,thedatahastwomodes.Theyare9and12.
Evaluation:Theanswerisobvious.
EXAMPLE
Findthemodefor103,206,87,54,and153.
SOLUTION
Goal:Youarebeingaskedtofindthemodeforthegivendataset.
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!
Index
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