GRE Math Review 1 Arithmetic.docx

GRE Math Review 1 Arithmetic.docx

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GREMathReview1Arithmetic

GRADUATERECORDEXAMINATIONS®

MathReview

Chapter1:

Arithmetic

Copyright©2010byEducationalTestingService.Allrightsreserved.ETS,theETSlogo,GRADUATERECORDEXAMINATIONS,andGREareregisteredtrademarksofEducationalTestingService（ETS）intheUnitedStatesandothercountries.

TheGRE®MathReviewconsistsof4chapters:

Arithmetic,Algebra,Geometry,andDataAnalysis.Thisistheaccessibleelectronicformat（Word）editionoftheArithmeticChapteroftheMathReview.Downloadableversionsoflargeprint（PDF）andaccessibleelectronicformat（Word）ofeachofthe4chaptersoftheMathReview,aswellasaLargePrintFiguresupplementforeachchapterareavailablefromtheGRE®website.Otherdownloadablepracticeandtestfamiliarizationmaterialsinlargeprintandaccessibleelectronicformatsarealsoavailable.Tactilefiguresupplementsforthe4chaptersoftheMathReview,alongwithadditionalaccessiblepracticeandtestfamiliarizationmaterialsinotherformats,areavailablefromE T SDisabilityServicesMondaytoFriday8:

30amto5pmNewYorktime,at16 0 9-7 7 17 7 8 0,or18 6 6-3 8 78 6 0 2（tollfreefortesttakersintheUnitedStates,U STerritories,andCanada）,orviaemailatstassd@ets.org.

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TableofContents

OverviewoftheMathReview5

OverviewofthisChapter5

1.1Integers6

1.2Fractions11

1.3ExponentsandRoots16

1.4Decimals20

1.5RealNumbers24

1.6Ratio30

1.7Percent31

ArithmeticExercises39

AnswerstoArithmeticExercises44

OverviewoftheMathReview

TheMathReviewconsistsof4chapters:

Arithmetic,Algebra,Geometry,andDataAnalysis.

Eachofthe4chaptersintheMathReviewwillfamiliarizeyouwiththemathematicalskillsandconceptsthatareimportanttounderstandinordertosolveproblemsandreasonquantitativelyontheQuantitativeReasoningmeasureoftheGRE®revisedGeneralTest.

ThematerialintheMathReviewincludesmanydefinitions,properties,andexamples,aswellasasetofexerciseswithanswersattheendofeachchapter.Note,however,thatthisreviewisnotintendedtobeallinclusive.Theremaybesomeconceptsonthetestthatarenotexplicitlypresentedinthisreview.Ifanytopicsinthisreviewseemespeciallyunfamiliarorarecoveredtoobriefly,weencourageyoutoconsultappropriatemathematicstextsforamoredetailedtreatment.

OverviewofthisChapter

ThisistheArithmeticChapteroftheMathReview.

Thereviewofarithmeticbeginswithintegers,fractions,anddecimalsandprogressestorealnumbers.Thebasicarithmeticoperationsofaddition,subtraction,multiplication,anddivisionarediscussed,alongwithexponentsandroots.Thechapterendswiththeconceptsofratioandpercent.

1.1Integers

Theintegersarethenumbers1,2,3,andsoon,togetherwiththeirnegatives,

negative1,negative2,negative3,dotdotdot,and0.

Thus,thesetofintegersis

dotdotdot,negative3,negative2,negative1,0,1,2,3,dotdotdot.

Thepositiveintegersaregreaterthan0,thenegativeintegersarelessthan0,and0isneitherpositivenornegative.Whenintegersareadded,subtracted,ormultiplied,theresultisalwaysaninteger;divisionofintegersisaddressedbelow.Themanyelementarynumberfactsfortheseoperations,suchas

7 + 8 = 15,

78minus87=negative9,

7minusnegative18=25,and

7times8=56,

shouldbefamiliartoyou;theyarenotreviewedhere.Herearethreegeneralfactsregardingmultiplicationofintegers.

Fact1:

Theproductoftwopositiveintegersisapositiveinteger.

Fact2:

Theproductoftwonegativeintegersisapositiveinteger.

Fact3:

Theproductofapositiveintegerandanegativeintegerisanegativeinteger.

Whenintegersaremultiplied,eachofthemultipliedintegersiscalledafactorordivisoroftheresultingproduct.Forexample,

2times3times10=60,

so2,3,and10arefactorsof60.Theintegers4,15,5,and12arealsofactorsof60,since

4times15equals60and5times12=60.

Thepositivefactorsof60are1,2,3,4,5,6,10,12,15,20,30,and60.Thenegativesoftheseintegersarealsofactorsof60,since,forexample,

negative2timesnegative30=60.

Therearenootherfactorsof60.Wesaythat60isamultipleofeachofitsfactorsandthat60isdivisiblebyeachofitsdivisors.Herearefivemoreexamplesoffactorsandmultiples.

ExampleA:

Thepositivefactorsof100are1,2,4,5,10,20,25,50,and100.

ExampleB:

25isamultipleofonlysixintegers:

1,5,25,andtheirnegatives.

ExampleC:

Thelistofpositivemultiplesof25hasnoend:

0,25,50,75,100,125,150,etc.;likewise,everynonzerointegerhasinfinitelymanymultiples.

ExampleD:

1isafactorofeveryinteger;1isnotamultipleofanyintegerexcept

1andnegative1.

ExampleE:

0isamultipleofeveryinteger;0isnotafactorofanyintegerexcept0.

Theleastcommonmultipleoftwononzerointegersaandbistheleastpositiveintegerthatisamultipleofbothaandb.Forexample,theleastcommonmultipleof30and75is150.Thisisbecausethepositivemultiplesof30are30,60,90,120,150,180,210,240,270,300,etc.,andthepositivemultiplesof75are75,150,225,300,375,450,etc.Thus,thecommonpositivemultiplesof30and75are150,300,450,etc.,andtheleastoftheseis150.

Thegreatestcommondivisor（orgreatestcommonfactor）oftwononzerointegersaandbisthegreatestpositiveintegerthatisadivisorofbothaandb.Forexample,thegreatestcommondivisorof30and75is15.Thisisbecausethepositivedivisorsof30are1,2,3,5,6,10,15,and30,andthepositivedivisorsof75are1,3,5,15,25,and75.Thus,thecommonpositivedivisorsof30and75are1,3,5,and15,andthegreatestoftheseis15.

Whenanintegeraisdividedbyanintegerb,wherebisadivisorofa,theresultisalwaysadivisorofa.Forexample,when60isdividedby6（oneofitsdivisors）,theresultis10,whichisanotherdivisorof60.Ifbisnotadivisorofa,thentheresultcanbeviewedinthreedifferentways.Theresultcanbeviewedasafractionorasadecimal,bothofwhicharediscussedlater,ortheresultcanbeviewedasaquotientwitharemainder,wherebothareintegers.Eachviewisuseful,dependingonthecontext.Fractionsanddecimalsareusefulwhentheresultmustbeviewedasasinglenumber,whilequotientswithremaindersareusefulfordescribingtheresultintermsofintegersonly.

Regardingquotientswithremainders,considertwopositiveintegersaandbforwhichbisnotadivisorofa;forexample,theintegers19and7.When19isdividedby7,theresultisgreaterthan2,since

2times7islessthan19,butlessthan3,since

19islessthan3times7.Because19is5morethan

2times7,wesaythattheresultof19dividedby7isthequotient2withremainder5,orsimply2remainder5.Ingeneral,whenapositiveintegeraisdividedbyapositiveintegerb,youfirstfindthegreatestmultipleofbthatislessthanorequaltoa.Thatmultipleofbcanbeexpressedastheproductqb,whereqisthequotient.Thentheremainderisequaltoaminusthatmultipleofb,or

r = aminusqb,whereristheremainder.Theremainderisalwaysgreaterthanorequalto0andlessthanb.

Herearethreeexamplesthatillustrateafewdifferentcasesofdivisionresultinginaquotientandremainder.

ExampleA:

100dividedby45is2remainder10,sincethegreatestmultipleof45that’slessthanorequalto100is

2times45,or90,whichis10lessthan100.

ExampleB:

24dividedby4is6remainder0,sincethegreatestmultipleof4that’slessthanorequalto24is24itself,whichis0lessthan24.Ingeneral,theremainderis0ifandonlyifaisdivisiblebyb.

ExampleC:

6dividedby24is0remainder6