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ContentsHithere!

MeetMagoosh

Chapter1:MeettheGREChapter2:HowtoStudy

1-MonthGREStudySchedule

Chapter3:GREQuantitativeReasoningMeettheGREQuantitativeSectionQuantitativeQuestionTypesQuantitativeConcept#1:Fractions/Ratios/Percents

Fractions/Ratios/PercentsPracticeQuestions

Fractions/Ratios/PercentsAnswersandExplanations

QuantitativeConcept#2:IntegerPropertiesandNumberSenseIntegerPropertiesandNumberSensePracticeQuestions

IntegerPropertiesandNumberSenseAnswersandExplanations

QuantitativeConcept#3:AlgebraAlgebraPracticeQuestions

AlgebraAnswersandExplanations

QuantitativeConcept#4:ExponentsandRootsExponentsandRootsPracticeQuestions

ExponentsandRootsAnswersandExplanations

QuantitativeConcept#5:WordProblemsandStatisticsWordProblemsandStatisticsPracticeQuestions

WordProblemsandStatisticsAnswersandExplanations

QuantitativeConcept#6:GeometryGeometryPracticeQuestions

GeometryAnswersandExplanations

QuantitativeConcept#7:CountingandProbabilityCountingandProbabilityPracticeQuestions

CountingandProbabilityAnswersandExplanations

QuantitativeConcept#8:DataInterpretationDataInterpretationPracticeQuestions

DataInterpretationAnswersandExplanations

QuantitativeConcept#9:QuantitativeComparisonStrategies

QuantitativeComparisonPracticeQuestions

QuantitativeComparisonAnswersandExplanations

MathFactandFormulaCheatSheet

Chapter4:GREVerbalReasoningMeettheGREVerbalSectionVerbalQuestionTypesVocabularyontheGREFlashcardsVerbalQuestionTypes:TextCompletion

TextCompletionPracticeQuestions

TextCompletionAnswersandExplanations

VerbalQuestionTypes:SentenceEquivalenceSentenceEquivalencePracticeQuestions

SentenceEquivalenceAnswersandExplanations

VerbalQuestionTypes:ReadingComprehensionReadingComprehensionPracticeQuestions

ReadingComprehensionAnswersandExplanations

Chapter5:GREAnalyticalWritingAssessmentMeettheGREAWATheIssueTaskTheArgumentTask

PracticeWritingTasks

Chapter6:GREPracticeTestGREPracticeTestQuestions

GREAnalyticalWriting

GREQuantitativeSection1

GREVerbalSection1

GREQuantitativeSection2

GREVerbalSection2

AnswerKeyGREPracticeTestExplanations

GREAnalyticalWritingAssessment

GREQuantitativeSection1AnswersandExplanations

GREVerbalSection1AnswersandExplanations

GREQuantitativeSection2AnswersandExplanations

GREVerbalSection2AnswersandExplanations

OfftotheTest!Appendix:MagooshVocabularyWordLists

Hithere!First,wewanttotakethisopportunitytothankyouforcheckingoutthisbookbyMagoosh.You’llsoondiscoverthatthecontentofthesepages,combinedwithouronlineproduct,packsagiganticGRE-prepwallop.

Yousee,whatyou’reholdinginyourhandsisacompanionguidefortheMagooshGRE-prepprogramofferedonlineatmagoosh.com.You’renotgoingtofindthousandsofpracticequestionsinthesepages.Instead,you’regoingtofindtwohundredofthehighest-quality,best-explained,andmostrobustquestionsthathavehelpedthousandsofstudentssucceedontheactualGRE.Ontopofthethoroughexplanationsandstep-by-stepdirections,werankthedifficultylevelofeachquestionandtellyouwhatpercentageofouronlinestudentsansweredthequestion(oronelikeit)correctly.Thesestatsaretheretohelpyousizeupeachquestionbeforeyoudivein.Youcanusethemtofigureoutwhatactuallymakesaquestioneasy,medium,orhard.GettingafeelforthedifficultylevelofthequestionsontheGREwillhelpyouapproachthetestwithmoreconfidence.“Lookatthat,”you’llsaytoyourself.“Ijustsailedthroughahardquestionlikeitwasnobigdeal!”

WiththehelpofthisbookandalltheusefulfeaturesinMagooshonline,you’llrocktheGRE.

That’sbecausethepreparationyoudowithMagooshonlinemimicstherealexperienceofacomputer-basedtestliketheGRE.You’llhaveallthetoolsyouneedandknowwhattoexpectwhenyousitdowntotaketherealtest.Inadditiontothethousand-pluspracticequestionsonline,wealsoprovideatextexplanationandavideoexplanationforeveryquestion.Yes,everysingleone.Andwedon’tjustofferpracticequestions.Tohelpyoucoverthebasics,wealsoprovidemorethantwohundredvideolessonsthatcovereverytopiconthetest.

Wehopeyou’regettingthesensethatwereallycareaboutyoursuccess,becausewedo.That’swherethisbookcomesin.WestartedwonderinghowwecouldhelpyoustudyevenmoreefficientlyfortheGREandfiguredthatanactualbookmightcomeinhandy.Sometimesbookscangoplaceslaptopscan’t.YoucantossthisbookinyourbagandstudywheneveryoufindyourselflackinganInternetconnection,wantingtoreviewhandwrittennotes,orfeelinglikeyoumightlearnsomethingbetterfromaphysicalbook.ThisbookextendstheMagooshcontenttohelpkeepyoustudyingwheneverandwhereverit’sconvenient.

Magooshisn’tjustaboutpracticequestionsandtests,though.Wealso

http://www.magoosh.com

haveafreeblog(magoosh.com/gre)fullofgreatstudytipsandotherinfo.Infact,we’veusedsomeofourmostpopularpoststohelpwritethisbook.Inaddition,you’llfindfreemathandvocabularyflashcardsintheiOSandGooglePlayappstores.Youcanflipthroughtheseonlineresourceswheneveryouhaveafewfreeminutes.

Wehopeyoufindthisbookuseful,andwewouldlovetohearyourfeedback.Ifyouhaveanyquestions,comments,orwanttosendavirtualhighfive,getintouchwithusatbook@magoosh.com.Happystudying!

Allthebest,

TheMagooshTeam

http://www.magoosh.com/gre

MeetMagooshTheTeamWeatMagooshareabunchofeducationnerdswhogetsuperstokedabouthelpingstudentsachievetheiracademicdreams.

ThisisBhavinParikh,ourCEOandfounder.HehasaBS/BAineconomicsandcomputersciencefromDukeUniversityandanMBAfromtheHaasSchoolofBusinessatUniversityofCalifornia,Berkeley.Now,he’sonamissiontochangethewaypeoplelearnandhowtheythinkaboutlearning,whichiswhyhestartedMagooshin2009.Funfact:whenhe’snothardatworkwithourteam,you’llusuallyfindBhavinplayingultimateFrisbeeorSmashBros.

Andhere’stherestofthefantasticMagooshteam!

TheMagooshExpertsLearnmoreabouttheMagooshtest-prepexpertswhohelpedwritethisbook!Allthetips,tricks,andlessonsyou’reabouttoreadcamefromthemindsofthesetwoinstructors.Ifyouhaveanyquestionsforthem,sendanemailtobook@magoosh.com.We’realwayshappytospeakwithstudents!

ChrisLeleGREandSATCurriculumManageratMagoosh

ChrisistheGREandSATcurriculummanager(andvocabularywizard)atMagooshOnlineTestPrep.InhistimeatMagoosh,hehasinspiredcountlessstudentsacrosstheglobe,turningwhatisotherwiseadauntingexperienceintoanopportunityforlearning,growth,andfun.Someofhisstudentshaveevengoneontogetnearperfectscores.Chrisisalsoverypopularontheinternet.HisGREchannelonYouTubehasovereightmillionviews.

mailto:book@magoosh.com

MikeMcGarryGMATCurriculumManageratMagoosh

MikewrotetheGREQuantitativechapterofthisbook.AtMagoosh,hecreatesexpertlessonsandpracticequestionstoguideGREandGMATstudentstosuccess.HehasaBSinphysicsandanMAinreligion,bothfromHarvard,andovertwentyyearsofteachingexperience,specializinginmath,science,andstandardizedexams.Mikelikessmashingfoosballsintoorbit,anddespitehavingnoobviouscranialdeficiency,heinsistsonrootingfortheNYMets.

OurMissionWecreateproductsthatgivestudentseverywhereaccesstoenjoyable,affordable,andeffectivetestprep.

OurCoreValuesWewanttosharethesewithyousoyouknowwhatwe’reallabout.

Accessible>Exclusive

We’reopentoideasfromeveryone,insideandoutsideofMagoosh.

Challenge>Comfort

Wechallengeourselvestolearnnewskillsbytacklingtaskswe’veneverdonebefore.

Friendly>Formal

Wealwaysshowrespectandkindnesstoourteammates,customers,andpartners,whetheronlineoroffline.

Wow>Profit Wegoaboveandbeyondinourworkandneversay,“It’snotmyjob.”

Done>Perfect Wehaveabiastowardactionandwon’tdelayforperfectiontomorrowwhatcanbedonetoday.

Data>Intuition Werunexperimentstotestideasandgatherdata.

Passion>[Something]

Welovewhatwedo!Helpingstudentsistoomuchfuntobeconsideredwork.

Communication>Efficiency

Wesetclearexpectations,communicatewhenwe’vecompletedatask,andfollowupwhennecessary.

Change>StatusQuo

Weadapttodifficultsituationsandreevaluateourpriorities,sowe’llalwaysbeaworkinprogress.

OurProductsMagooshofferstestprepfortheGRE,GMAT,LSAT,SAT,ACT,TOEFL,IELTS,MCAT,andPraxis.Andwe’reexpandingtonewtestssoon!

MagooshGREOnlineOurpremiumonlineproductoffers:

MorethantwohundredlessonvideosontheGREQuantitativeReasoning,GREVerbalReasoning,andGREAnalyticalWritingAssessment(AWA)categories—that’sabouttwentyhoursofvideo!MorethanonethousandGREQuantitativeandGREVerbalpracticequestions,withvideoexplanationsaftereveryquestionMaterialcreatedbyexperttutorswhohavein-depthknowledgeoftheGREEmailsupportfromourexperttutorsCustomizablepracticesessionsandmocktestsPersonalizedstatisticsbasedonperformanceAccessanytime,anywherefromanInternet-connecteddevice

Whenyousignupforthepremiumproduct,youcanaccessallyourstatsatanytime.ThisiswhattheMagooshdashboardlookslike:

WhatStudentsSayaboutMagooshWe’reyourbiggestfanswhenitcomestoyourGREstudies.Andoursupportdoesn’tgounnoticed.Checkoutwhatsomeofourcurrentandformerstudentshavesaidaboutusbelowandonmagoosh.com/storiesandgre.magoosh.com/testimonials.

AdditionalResourcesfromMagooshInadditiontothisbookandourfullonlineGREprep,weoffermanyotherresources(yes,evenmore!)tohelpyougetthemostoutofyourGREprepjourney.

http://www.magoosh.com/storieshttp://gre.magoosh.com/testimonials

MagooshGREstudyplans:Whetheryouhaveone,three,orsixmonthstostudy,we’vecreatedthesedailyandweeklyguidestotellyouexactlywhatandwhenyoushouldstudysothatyou’llbefullypreparedfortestday.Findalloftheguidesbyvisitingthispageonline:magoosh.com/gre/gre-study-plans-and-guides.

MagooshGREappsforiOSandAndroid:MagooshPrep:PracticeGREQuantitative,GREVerbal,andGREAWAonthego.GREVocabularyFlashcardsfromMagoosh:Quizyourselfeverydaywiththeseflashcardstolearnthe1200+mostimportantwordsontheGRE.You’llmasterdifferenttiersofvocabdifficultyandunlocknewlevelsalongtheway.Playanopponentforanextrachallenge!

MagooshGREBlog:Visitmagoosh.com/grefortipsandadvicefromourtest-prepexpertsonhowtopreparefortestdayanddominatetheexam.

MagooshGREeBooks:TheUltimateGREGuide(magoosh.com/gre/ultimate-gre-guide)GREVocabularyeBook(magoosh.com/gre/2012/gre-vocabulary-ebook)GREVocabularyFlashcards(magoosh.com/gre/2013/gre-vocabulary-flashcards)GREMathFormulaeBook(magoosh.com/gre/2012/gre-math-formula-ebook)

MagooshGREYouTubechannel:LearnfromourexperttutorChrisLele,whose“VocabWednesday”videoswalkyouthroughsomeofthetrickiestvocabularyyou’llseeontheGRE.

MagooshGREforum(officialquestionsandexplanations):Onthispage,you’llfindlinkstoallofMagoosh’svideoexplanationsforofficialGREmaterial.Youcanleaveresponsestoquestionsorupvoteanswersthatyoufindparticularlyhelpful.(gre.magoosh.com/forum)

http://www.magoosh.com/gre/gre-study-plans-and-guideshttp://www.magoosh.com/grehttp://www.magoosh.com/gre/ultimate-gre-guidehttp://www.magoosh.com/gre/2012/gre-vocabulary-ebookhttp://www.magoosh.com/gre/2013/gre-vocabulary-flashcardshttp://www.magoosh.com/gre/2012/gre-math-formula-ebookhttp://gre.magoosh.com/forum

Whichofthefollowingequationsistrueforallpositivevaluesofxandy?

Chapter1

MeettheGRETheTestBreakdownThesectionsTheGREconsistsoftwoQuantitativeReasoningsections,twoVerbalReasoningsections,andoneexperimentalsection,whichcanbeeitherVerbalorQuantitative.Inaddition,therearetwotimedessay-writingassignmentsintheAnalyticalWritingAssessment(AWA)sectionoftheGRE.Theexperimentalsectionwon’tcounttowardyourscore,butyouwon’tknowwhichsectionistheexperimentalone.Youshouldtreatallsectionslikethey’retherealdeal.BoththeoverallGREVerbalandGREQuantitativescorescanrangefrom130to170.Theessaysarescoredfrom0to6inhalf-pointincrements.

NumberofquestionsandtimelimitTheGREQuantitativesectionscontaintwentyquestionseach.You’llbegiventhirty-fiveminutestocompleteeachsection.TheVerbalsectionsalsoconsistoftwentyquestionseach,butyou’llhavejustthirtyminutestocompletethosesections.

GREQuantitativeoverviewThetwosectionsinthiscategoryaremadeupofapproximatelysevenquantitativecomparisonquestionsandthirteennon–quantitativecomparisonquestions(don’tworry,we’llexplaincomparisonandnon-comparisonquestionssoon).Possiblequestiontypesincludethefollowing(thesefourexamplepracticequestionswillbeexplainedstartingonpage35;noneedtoworryaboutsolvingthemnow!):

Multiplechoice:astandardquestiontypeinwhichyoujusthavetoidentifytheonecorrectanswer.

2x2+6>40

Whichvaluesofxsatisfytheinequalityabove?

Indicateallsuchvalues.

–8

–6

–4

–2

2

4

6

8

Twotrainsstartingfromcities300milesapartheadinoppositedirectionsatratesof70mphand50mph,respectively.Howlongdoesittakethetrainstocrosspaths?

Multipleanswer:aquestiontypethatcanhaveuptotenanswerchoices;you’llhaveto“selectallthatapply,”whichmeansthatthenumberofcorrectanswersisn’tprovided.

Numericentry:anopen-endedquestiontypeinwhichyou’llhavetotypeinthecorrectvalue.

Quantitativecomparison:aquestiontypethatliststwoquantitiesand

ColumnA ColumnB

Thenumberofpositivemultiplesof49lessthan2000

Thenumberofpositivemultiplesof50lessthanorequalto2000

ThequantityinColumnAisgreater

ThequantityinColumnBisgreater

Thetwoquantitiesareequal

Therelationshipcannotbedeterminedfromthe

informationgiven

asksyoutocomparethemandselectoneofthefollowing:AisequaltoB,AisgreaterthanB,AislessthanB,ortherelationshipbetweenthetwoquantitiescannotbedeterminedfromtheinformationgiven.

Formoreinformationonquestiontypes,seetheinformationinchapter3.

Also,don’tforgetthatthere’sabasicon-screencalculatorthatyou’llhaveaccesstowhilecompletingtheGREQuantitativesections!Seepage33.

GREVerbaloverviewThetwosectionsinthiscategoryareeachmadeupofaboutsixtextcompletionquestions,foursentenceequivalencequestions,andtenreadingcomprehensionquestions.

Textcompletion:questionsthatcanhaveonetothreeblanksandrangefromshortsentencestofour-sentenceparagraphs.Fordouble-andtriple-blanktextcompletionquestions,youmustanswereachblankcorrectlytoreceivefullpoints—there’snopartialcredit!

Sentenceequivalence:questionsthathavesixpossibleanswerchoices.Foreverysentenceequivalencequestion,therewillbetwocorrectanswers.Toreceiveanycredit,youmustchoosebothcorrectanswers.

Readingcomprehension:passagesthatrangefromtwelvetosixtylines.Topicmatterisusuallyacademicinnatureandcoversareassuchasscience,literature,andthesocialsciences.Questiontypesinclude

standardmultiple-choicequestions,highlight-the-passagequestions,andmultiple-answerquestions,whichrequireyoutochooseanynumberofthreepossibleanswerchoices.

GREAWAoverviewAtthebeginningofyourGRE,you’llhavetowritetwoessays:theIssuetaskandtheArgumenttask.Theessaysaretimedatthirtyminuteseach.Neitherispartofyour130–170score.Eachessayreceivesascorerangingfrom0to6.Yourfinalessayscoreistheaverageofbothessayscores.

HowIstheGREScored?TheGREscalemayseemprettyarbitrary.Afterall,whohaseverseenatestgradedona130–170scale?Arangeof0–100,yes,but130–170?Weird!Well,accordingtoETS,whentheyrevisedtheGREscoringbackin2011,theywantedtosticktothreedigitssothatschoolswouldn’thavetooverhaulallthetextboxentriesthatcallforthatspecificnumberofdigits.Fairenough.Also,toavoidconfusion,ETSmadesurethecurrentscoringsystemdidn’toverlapnumberswiththeoldGREscoringsystem.

TheGREscoringsystemmakesupforthelimitedrange(just40points!)bygivingmoresignificancetotheextremeendsofthescale.Forexample,ontheGRE,thedifferencebetweenascoreof165andascoreof170willbethedifferencebetweenbeinginthe96thpercentileandbeinginthe99thpercentile.

Attheendoftheday,you’renotgoingtobetestedonthesestatisticalnuances.Theimportantthingtorememberisthatmanyadmissionsofficesbasetheirevaluationsonapercentilescore,whichyou’llalsoreceiveaspartofyourscorereport.

Questionsarestatic,buttheGREisadaptiveTheGREisadaptive,butnotinthewaysomeothertestsare.Manytests(suchastheGMAT,forinstance)willadapttoyourproficiencylevel.Themorequestionsyougetright,theharderthey’llget;ifyougetmorequestionswrong,they’llgeteasierandeasier.TheGREisn’tadaptiveinthisway.Questiondifficultywithinasectiondoesn’tchangedependingonwhetheryouanswerquestionscorrectly.However,yourperformanceonthefirstGREVerbalandQuantitativesectionswilldeterminewhetherthenextsectionsofthesametypeareeasierormoredifficultthanthefirst.

Questionshaverandomlevelsofdifficulty

Questionsmightbestatic,butthatdoesn’tmeanasectioncan’tbecomeprogressivelyharderoreasier,theoretically.ThereisnoorderofdifficultyontheGRE.Thefirstquestioncanbethehardestandthelastquestiontheeasiest.

EachquestionisweightedthesameEachquestionisbasicallyweightedthesame.Sothequestionthatseemslikeitwilltakefifteenminutestoanswerisworththesameasthequestionyoucananswerinfifteenseconds.

HowDifficultIstheGRE?Simplyput,theGREcanbeaverydifficulttest,especiallyifyou’vebeenoutofschoolforawhile.Typically,thistendstoholdtruerfortheGREQuantitativethanfortheGREVerbalorAWA.Solet’sbreakitdownbycategory.

HowhardisGREQuantitative?Thetruthis,assoonasyouleavecollege,thelikelihoodofusingmathdiminishesdrastically.Compoundingthe“rustymath-brainsyndrome”isthefactthatGREmathisdifferentfromthemathyouprobablydidincollege;it’smuchclosertothemathyoudidinyourjunioryearofhighschool.That’snottosayit’seasy.It’sgenerallymuchtrickierthananythingyoueversawinyouralgebraclass.Throwinthehigh-pressuretestingenvironment,andit’sunderstandablewhythemerementionofGREprepcanfillastudentwithutterdread.

HowhardisGREVerbal?Believeitornot,asyoureadmoreandareexposedtodifferentkindsoftext,thepartsofyourbrainthatarewiredforreadingandlanguageskillswillcontinuetoexpand.Ofcourse,knowingyourbrainisgettingbetterprobablydoesn’thelpyoufeelbetteraboutfacingafour-hundred-wordpassageonuser-experiencedesigntheoriesoratextcompletiontaskthatasksyoutodistinguishbetweenextenuatingandcorroborating.ThesimplefactisthattheGREVerbalisveryhard,evenforPhDcandidates.Thewritingisdenseandstylistic;thevocabularyisesotericanddaunting.

HowhardisGREAWA?TherealdifficultyoftheGREwritingsectionstemsfromrustywritingskills.

Formanystudents,ithasbeenyears—orevendecades—sincetheylastwroteafive-paragraphessay.Whetheryoufallintothatgroupornot,tryingtoscorea“6”ishard,evenforconfidentwriters.

Don’tworry!Withlotsofstudy,youcanstilldoverywellontheGRE!Remember,it’satestthatputsyouincompetitionwithothers.Youdon’thavetoanswereveryquestioncorrectlytoscorewell.Anotherwayoflookingatitisthis:justasyoustruggleonaverydifficultthree-blanktextcompletionoraprobabilityquestioninvolvingthecombinationsformula,sodo99percentoftheotherstudentstakingthetest.

Don’tforgettodothefollowing:

Giveyourselfplentyoftimetoprep.LearnhowtotaketheGRE(you’realreadydoingthatbyreadingthisbookandusingMagooshonline!).Believethatimprovementwillcomegradually.Attimesyoumayplateau.Sobepatient.

PerformanceStatisticsontheGRErevisedGeneralTest

VerbalReasoning

QuantitativeReasoning

AnalyticalWriting

NumberofTestTakers

1,585,305 1,587,610 1,579,373 51%female,43%male(6%didnotprovidetheirgender)

Mean 150 152 3.6

StandardDeviation

8 9 0.9

Datasource:ETS.org

IstheGREImportant?Thisisaveryinterestingquestion,andonethatevenweatMagooshdebateinternally.Somepeoplethinkit’snotimportantatallandthatit’sjustanarbitrarytestthatattemptstomeasureIQ.Ontheotherhand,somepeoplethinktheGREisavalidmeasureoftheintellectualskillsrequiredforsuccessingradschool.

Ultimately,sincemuchofthereadinginthetestitselfisliftedfromacademicjournals,beingabletounderstandsuchwritingwillmakeadifferenceingradschool.Themathsectionsmaynotbeasdefensible,especiallyforsomeonelookingtowritehisorherdissertationonmotifsoffifteenth-centuryfrescoes.Atthesametime,though,themathfoundonthetestisn’tthat“mathy”—it’smoreatestofyourlogicalreasoningwithnumbers,which,especiallyforthemajorityofyouwhowillhavetodabbleinthestatisticssideofthings,issomewhatrelevant.Thenthere’stheGREAWA,whichtestsyourabilitytowritecompetentlyaboutcomplexissues—maybenotsousefultoengineers,butnottotallyirrelevanttostudentswhohopetohavetheirworkpublishedinacademicjournals.

Fornow,theGREisimportantinthatit’sapieceofanapplication—thoughonlyarelativelysmallone—thatgivesgradschooladmissionsofficersasomewhataccuratesenseofanapplicant’sintellectualabilityinahighlyartificial—andstressful—environment.Thatsaid,relevantworkexperience,excellentlettersofrecommendation,andastrongundergraduateGPAcanstrengthenaweakscore,justasalackofanyrelatedexperienceoraterribleundergraduateGPAcanneutralizeaperfectscore.

Chapter2

HowtoStudyWhetheryou’retakingtheGREforthefirstorthefifteenthtime,you’llneedtocomeupwithaplanofactionwhenitcomestostudying.Youknowyourschedule,attentionspan,strengths,weaknesses,andstudyhabitsbetterthananyoneelse.Tocomplement,inthischapter,weofferallkindsofstudy-relatedtips,tricks,andthoughts,fromasamplestudyscheduletotime-savingtipstomorale-boostingreminders.Wewantyoutoknowthatyou’renotgoingthisalone.WeatMagooshwillhaveyourbackthewholeway.

FindtheTimetoStudyPossiblythemostchallengingaspectofpreparingfortheGREisexercisingtheself-disciplinetostudyforitinthefirstplace.Manypeopleconvincethemselvesthatitwillbeimpossibletocarveoutanytimetostudywhileworkingafull-timejobandmaintainingsometypeofwork-lifebalance,sotheydon’teventry.But,asitturnsout,they’rewrongabouttherebeingsolittletime.AccordingtoastudybytheBureauofLaborStatistics,workingadultsintheUShaveonaverageapproximatelyfourandahalfhoursofleisuretimeeachday.ThetricktoexcellingontheGREistoleveragethefreetimeyoudohaveandusetime-managementtechniquesthatworkforyou.Herearesomeideasonhowtodojustthat:

1. Takepublictransportation.Ifyouhavetheoptiontousepublictransportationtocommutetoandfromwork,takeit.Yes,itmaymakeyourcommutelongeranditmaybealittleinconvenient.However,itwillprovideyouwitharoutinestudytimewheretherearefewdistractions.Whenyou’resittingonthemetro,subway,orbus,there’snothingbettertodothanstudy,sotakeadvantageofthisalonetime.Anaddedbonus?Bytakingpublictransportation,youavoidthefrustrationofdrivingintraffic,savegas,andavoidwearandtearonyourcar.

2. Studyonthego—always.Downloadourflashcardappsorkeepasetofflashcardsinyourpurseorbackpacksoyoucanstudywheneveryouhappentohaveaspareminute.Ifyou’reanauditorylearner,youcanlistentorecordingsofvocabularywords.Anotheroptionistouseastudyapponyoursmartphone.Forexample,theMagooshGREappprovidesconvenientaccesstovideolessonsthat

canhelpyouprepare.Whetheryouprefertouseanapp,flashcards,oraudiorecordings,thekeyistotakeadvantageofanyopportunitytoincorporatemorestudyingintoyourday.

3. Spendanextrahouratwork.ConsiderarrivingatworkanhourearlierorstayingthereanhourlatersoyoucanstudyfortheGRE.Whyshouldyoudoitatwork?Somepeoplearetoodistractedathome,andifthat’syou,itmaybebettertoeitherstudyatyourdeskorinaconferenceroom—especiallyifthebuildingisniceandquietbeforeandafterbusinesshours.

4. Studyatyourpeakfocusinghours.Somepeoplestudybetterinthemorning,whileothersareabletofocusbetterintheeveningoratnight.Youknowyourselfandyourstrengthsandweaknessesbetterthananyoneelse.

5. Scheduletheexam.Onceyou’vedecidedtotaketheGRE,takeapracticetesttogaugehowmuchyouneedtoincreaseyourscore.Then,scheduletheexamandgiveyourselfapproximatelytwelveweeksorsotoprepare.Ifyouneedtoincreaseyourscoredramatically,youmaywanttoallowevenmoretimethanthat—twentyweeksoutmightbeabettertargetdate.Itmayseemprematuretoscheduletheexambeforeyou’veevenstartedstudyingforit,butthisisactuallyagreatmotivator.Onceyou’veregisteredandpaidfortheexam,you’remorelikelytotakeitseriously.Byregistering,you’vemovedyour“oneday”or“someday”toadefinitivedate.

PreparingfortheGREisatime-consumingtask.However,it’salsooneofthebestwaystoprepareforgraduateschoolandkick-startyourstudyhabits.Whetheryou’reconsideringafull-time,part-time,oronlinemaster’sprogram,carvingoutasetstudytimefortheGREwillalsohelpyoudevelopastudyregimenthatwillbenefityouingraduateschool.

Good,YouGotItWrongForsomestudents,gettinganswerswrongcanbeinterpretedtooeasilyassomekindofnegativepersonalmessage(e.g.,“I’mdumb,”“There’ssomethingwrongwithme,”etc.),anditbecomesanegativeandfrustratingexperience.

Buthere’sadifferentperspectiveaboutmistakes.Everymistake,everywronganswer,isanopportunityforgrowthandself-improvement.Thetrulyexcellentstudentlivesbythisveryhighstandard:absolutelynevermakethe

samemistaketwice.Thatrequiresincredibleperseverance,butevenfallingshortofthat,eachwronganswerisachancetoimproveandclarifysomenecessaryconceptaboutwhichyoupreviouslywereunclearorconfused.

Thinkofhowgratefulyouwouldbeif,beforesomeimportantevent,youhappenedtowalkbyamirrorandnoticeyouhadsomethingsmudgedacrossyourface.Nowyoucanwipethatsmudgeoffandimproveyourappearancebeforetheimportantevent.Metaphorically,everyquestionyougetwrongissuchamirror—achancetolookatyourself,removethesmudge,andimproveyourunderstanding.

Ourstudyplansaredesignedwiththisinmind.Wehaveyoujumpintomixedcontentquestionsrightaway,wellbeforeyouhaveachancetocompletealltheMagooshvideolessons.Onereasonis,ofcourse,thattheGREitselfwillthrownothingbutmixedcontentatyou,andwewantyoutogetcomfortablewiththis“gear-switching”asearlyinyourstudyprocessaspossible.Thatbeingsaid,weknowthismeansyou’llprobablygetmanyquestionswrongatthebeginning,andwebelievethisisagoodthing.Obviously,wearen’ttryingtopunishyouormakeyoufeelbadaboutyourself.Rather,weknowthatmakingmistakesandconsciouslyreflectingonthesemistakesisexactlywhatwillprimeyourmindforthecontentofthevideolessons.

Ittakesagooddealofconfidenceandemotionalsecuritytoadoptthisattitude—tolookbeyondthefrustrationofgettingquestionswrong,andtoembrace,withcourageandoptimism,theopportunityforself-improvementineachmistake.Inthefaceofanapparentlackofsuccess,it’sveryhardtomaintainanysortofcourageousoptimism.Ittakesaverystrongandsecureindividualwhocansay,“I’vegottentwohundredGREquestionswrongsofar,andthat’sgreat,becausefromthosemistakesIhavelearnedtwohundrednewconceptsthatIcanuse!”Butremember,ifyouwanttostandout,youmustbeastandout.Ifyouwantextraordinaryresults,youneedanextraordinaryperspective.WhereveryouareinyourGREpreparation,that’sexactlythekindofsuccesswewishforyou.

BeatingExamStressIfyou’realreadyoverwhelmedwithanxietyaboutthetest,inadditiontostudying,youshouldfocusonmanagingthatstress.Tocombatthisproblem,youmaywanttotrytomakefuturemocktestsasstressfulaspossible.Forinstance,whenyou’regivingyourselfapracticetest,don’tstandupexceptattheappointedbreaksoftheexam.Maybeeventurnonthetelevisioninthebackgroundjustsoyoucanlearntocopewithdistractions.

Anotherimportantwayofcopingwithstressistonoticeyourbreathing.Whenwebecomestressed,ourbreathbecomesshallowerandourbodiestenseup.Youcaneasestressbytakinglonger,deeperbreaths.Justafewdeepbreathsshouldcalmyournervesandhelprestoreyourbreathingand,mostimportantly,yourfocus.

TipstoStudyMoreEffectivelyWhenyou’retryingtofocusonstudying,andespeciallywhenyou’refollowingaprettyintensestudyplan,itcanfeellikethewholeworldistryingtodistractyou.Betweentextsfromfriends,socialmediamessagesfromfamily,andadorablepicturesofkittens,itcanbehardtostaytruetothestudyschedule.

Buttherearetried-and-truewaystocutthroughthedistractionsandfocusonthejobathand—studyingfortheGRE.Hereareafewtricks:

1. Changethewayyouthinkaboutstudying.Researchshowsthathowyoufeelaboutataskisjustasimportantashowyoudoit.Ifyoucanflipthatswitchinyourbrainthattakesyoufromthinking,“Ugh,studying!”tothinking,“Woohoo!LookatallthiscoolstuffIgettolearn!”you’llsetyourselfupforsuccess.

2. Lookforgoodstudyvibes.Whereyoustudyisimportant.Tryaquietroominyourhouseorastudycarrelatthelocallibrary.Trystudyinginafewdifferentspotsuntilsomethingfeelsright.Thencomebacktothatplaceoften.

3. Sticktoaschedulethatworks.Bedeliberateandworksignificantchunksofstudytimeintoyourdailyroutine.Everytimeyourepeatyourroutine,it’llgeteasier,untilitfeelstotallynatural.

4. Turnoffyourphone.Don’tputitonvibrateorairplanemode.Turnitoff.You’renotavailabletoothersduringstudytime.

5. Closethetabsonyourbrowsers.Yes,evenemail.Yes,evensocialmedia.Yes,evenfantasyfootball.Resisttheurgetosurfbyclosingalltabsatthestartofyourstudysession.

6. Addsomevarietytoyourstudyroutine.Trysomethingnewifyoukeepseeingspotswhilereadingthebook.Makeflashcards.Watchalesson.

7. Takebreaksandrewardyourself.Everyhourorso,makesureyougetupandgiveyourmindandbodyabreak.Takeawalk.Eatsomethingdelicious.Playwithapuppy.Breakshelpyourmindprocessallthatstuffyou’vebeenlearning.Youdeservebreaks!

Armyourselfwithstrategiestoavoiddistraction,andnothingcanstopyour

GREstudysessions—exceptyou,ofcourse,whenyoutakeabreak.Whichweabsolutelyrecommend!

HowtoRegisterandOtherCommonlyAskedQuestionsAswe’vementionedbefore,gettingyourtestdateandlocationallsetisagreatwaytobeginyourGREstudycycle.Herearesomefrequentlyaskedlogisticalquestionsaboutthetest.Knowingtheanswertotheseaheadoftimecanputyourmindateaseandletyoufocusontheimportantstuff—like,youknow,studying.

HowdoIregisterfortheGRE?YoucanregisterfortheGREbyvisitingets.org/greandcreatingaGREaccountwithETS.Thetestiscomputer-deliveredyear-round,withtheoptionoftakingituptothreetimesayear.

WhatdoIhavetobring?ThemostimportantthingyouneedtobringtothetestingsiteisavalidphotoIDthatincludesyourname,photo,andsignature.Ifyou’retestingoutsideoftheUnitedStates,werecommendbringingyourpassport.Ifyouarrivewithoutanyformofidentification,thetestingcenterwillturnyouaway.

Thenextmostimportantthingyouneedissomethingnutritioustoeat.TheGREisalongtest,andyou’llneedsomebrainfueltohelpyouavoidcrashingduringthesixty-five-linepassageontheuseofisotopedatinginglaciers.Werecommendbananas,dates,nuts,andothernaturalfoodswithahighcaloricdensity.However,youknowyourselfbest.IfyougotthroughcollegebykeepingyourselfawakeonSnickersbars,thenthat’swhatyourbodyisusedto.Ifthisisthecase,chompingonpistachiosforthefirsttimeinyourlife,rightbeforethetestbegins,maynotbethebestidea.

DoIneedtoknowwhichschoolsIwanttosendmyscorestothatday?HowmanyschoolscanIpick?TheGREhasafeaturecalledScoreSelect,whichallowsyoutosendyourscorestouptofourschoolsforfree.You’llgetthisoptionattheendofthetest.

Ifyouwanttosendyourscorestomorethanfourschools,you’llhaveto

paytwenty-fivedollarsperadditionalschool.Youdon’thavetoselectanyschoolswhenasked.However,thedownsidetonotselectingschoolsisthatsendingthescorereportsinthefuturewon’tbefree.

DoIneedtoknowthecodesoftheschoolsI’mapplyingto,orwillthatinformationbeprovided?Thetestwillprovideyouwiththerelevantcodesforeachschool.Sodon’tworry,youwon’thavetoburdenyourbrainwithevenmoreinformation.

WhenwillIreceivemyscores?Ifyoutakethecomputer-basedtest,you’llbeabletoseeyourunofficialGREVerbalandQuantitativescoresatthetestcenterthedayofthetest.YourofficialscoreswillbeavailableinyourETSaccountonlineabouttentofifteendaysafteryourtestdate,atwhichtimeyourscoreswillalsobesenttotheschoolsyouchoseontestday.Scoresofpaper-basedtestsareavailableapproximatelysixweeksafteryourtestdate.

What’sthetestingcenterlike?Usuallythelabsaredrabandsterile.Essentially,you’llbeaskedtoemptyyourpocketsandleaveeverythingbehindintheregistrationarea.Luckily,therewillbeatrustylockerinwhichyoucanputyourstuff.Yougettokeepthekeywhenyougointothetestingroom.You’reallowedtoruntothelockerduringbreaksintestingforanemergencysnack.

Theimportantthingistomentallyprepareforthetestingcenterexperience.Youdon’twantanysurprises:badtraffic,poorlymarkedbuildings,oratestingcenterstaffthatmovesatthepaceofglaciers.SomeGREtesttakersevenvisitthetestingcenteradaybeforetotracetheirexactsteps.

Howlongisthetest?Thetestwilltakeyouclosetofourhours.Thisincludescheckingintothetestingcenter,answeringafewbackgroundquestions,andthendivingintotwothirty-minuteessaysandfiveGREQuantitative/Verbalsections(includingthatoneexperimentalsection).

Thegoodnewsisthatyou’llgetonescheduledten-minutebreakafterthethirdsectionandone-minutebreaksbetweenalltheothersections.While

you’refreetotakeabreakatanyothertime,theclockwillkeeponrunningifyouchoosetodoso.Inotherwords,unlessyou’reabouttopassout,don’tgetoutofyourseat,exceptduringascheduledbreak.

StudyScheduleAtMagoosh,weknoweveryonehasdifferentneedsforstudying.Somepeopleliketodoalittleeachdayoveralongperiodoftime,whileothersprefertocraminalotofstudyingoverashortperiod.Weoffersuggestedstudyschedulesforalldifferenttypesoflearners.Fromourone-weekstudyscheduletooursix-monthschedule,we’vegotallofyoucovered.

Asitturnsout,ourone-monthscheduleisthemostpopularplanweoffer.Forthosewhoarefocusedandambitious,thisfour-weekstudyguideplanisgreat,aslongasyouhavethetimetoputin.Ifyoufeellikeyoumightnotbe100percentcommittedtoanintensestudyschedule,orifyouneedtoimproveyourscoredrastically,thenthisstudyplanfortheGREmaynotbeforyou.Ifthisisthecase,checkoutourtwo-tothree-monthstudyschedulesonmagoosh.com/gre.

http://www.magoosh.com/gre

1-MonthGREStudySchedule

WhatYou’llNeedThisbookAnonlineMagooshGREaccount(seethelastpageinthebookforyour20%offcoupon)MagooshGREMathFlashcards(availableatgre.magoosh.com/flashcards/math)TheOfficialGuidetotheGRErevisedGeneralTest(SecondEdition),fromETSAboutthreehourstostudyeachday(itsoundslikealot,butyougotthis!)Optional:theMagooshGREVocabularyFlashcardsapp,ortheMagooshGREVocabularyBuilderapp

Note:Forthoseofyouwithmorethanamonthtostudy,thisone-monthstudyschedulecanbestretchedoutovertwomonths.Magooshalsooffersavarietyofmorespecializedversions,includingdailyschedules,two-tothree-monthschedules,andsix-monthschedules.

http://www.gre.magoosh.com/flashcards/math

WeekOne

PrimaryGoalsBrushuponmathfundamentals.LearnbasicsforapproachingGREVerbalquestions.Readtwoarticlesfromrecommendedresources.Learnonehundredandfiftynewwords(eitherwithyourmagoosh.comaccountortheMagooshvocabularyapps).Remembertheimportanceofcontext.BeginwatchingMagooshlessonvideos(especiallythe“IntrototheGRE”module)andreadingthroughthisbook.

SecondaryGoalsGREQuantitative

Readchapters1and2.Alsoreadthebeginningofchapter3,uptothe“QuantitativeConcept#1”section.Logintoyouraccountatgre.magoosh.comandwatchallofthegeneralmathstrategieslessons,thearithmeticandfractionlessons,thepercentsandratiolessons,andtheintegerpropertieslessons.Note:Itwillbemorehelpfultostopandworkonpracticeproblemsasyougo.Pauseandtrysomeproblemsaftereverythreeorfourvideosthatyouwatch.You’llwanttomakesureyouapplythetechniquesyoulearnedabout.Watchingtoomanyvideosinarowcanresultinfalseconfidence.Practiceconceptsseeninlessonvideosbyusingthemagoosh.com“quiz”feature.Gothrougheveryeasyandmediumquestionrelatingtoarithmetic,percents,ratios,andpowersandrootsusingyouronlineMagooshaccount.Watchvideoswhenevernecessary.Returntoallquestionsyouinitiallymissandbeabletoanswerthemaccurately.

GREVerbalReadchapter4uptothe“VerbalQuestionTypes:TextCompletion”section.Gotomagoosh.comtowatchoverviewlessonsontextcompletionsallthewayupto“TextCompletionSentenceShifts.”Practicewitheasyquestionsintheonlineproduct.PracticethirtyonlinetextcompletionquestionsateasytomediumdifficultyWatchabouthalfoftheMagooshreadingcomprehensionlessonvideosonline.Practiceapproximatelythirtyreadingcomprehensionpracticequestionsonline,watchingvideoswhennecessary.Makeflashcardsofwordsyoudon’tknowandusereferencessuchasvocabulary.comtohelp.

Supplemental/OptionalReadtwoarticles,fourtofifteenpageseach,fromtheNewYorker/AtlanticMonthly/Economist.ReadingarticlesthataresimilarinstyletowhatisfoundontheGREVerbalwillhelpwithyourreadingcomprehension.Thesemagazineshavemanysucharticles,especiallyintheartsandcultureorscienceandtechnologysections.

http://gre.magoosh.comhttp://www.magoosh.com

Findfiftywordsyoudon’tknowinthearticles.Referencethosewordsusingvocabulary.comandmakeflashcardsonquizlet.com.Writeaquicksummary/reviewofoneofthetwoarticlesyouread.IncludeGREwordsyoustudiedthisweek.PracticethefirstfourdecksinMagoosh’sGREMathFlashcards(availableonlineandonmobile).

WeekTwo

PrimaryGoalsLearnonehundredandfiftynewvocabularywordsandreviewwordsfromlastweek.Makesureyou’recaughtuponyourassignedreadinginthisbook(chapters1,2,andpartsofchapters3and4).Makesureyou’reconfidentwiththematerialcoveredinlastweek’slessons.Revisitwhatevertopicsyoufeelneedreview.Manyconceptswe’llintroducethisweekwillbuildoffoflastweek’stopics.Workyourwaythroughthemagoosh.comlessonvideos,makingsureyoudothefollow-upquizzesaftereachlessongroup.Remember:practice,practice,practice!Don’tbecomeavideo-watchingzombie.Keepyourmindactivebypausinglessonvideosandtryingactualpracticeproblems!Learntheconceptscoveredinthisbook.

SecondaryGoalsFinishreadingthroughchapters3and4inthisbook.WorkthroughtheGREQuantitativeandGREVerbalpracticequestionsandcheckyouranswers.It’sokayifyou’renotgettingthehardandveryhardquestionscorrectyet.Thosewillseemeasierwithpractice.Onmagoosh.com,workthroughthealgebra,equations,andinequalitieslessons,thewordproblemlessons,andthepowersandrootslessons.Alsoonline,completeatleastseventy-fivepracticeGREQuantitativequestionsdealingwiththeaboveconcepts.FinishwatchingallGREVerballessonsonmagoosh.com.Completeatleastseventy-fivequestionsonmagoosh.comcoveringtextcompletions,sentenceequivalence,andreadingcomprehension.TakeandgradeapracticetestfromeitherthebackofTheOfficialGuideoritsCDsupplement.Watchthevideoexplanationsforthequestionsyoumissedorthatyouwerenot100percentconfidenton.Beforeyouwatcheachvideo,though,trytofigureoutforyourselfwhereyoumadeamistake.

Supplemental/OptionalReadtwoarticles.Makesurethecontentisdifferentfromthearticlesyoureadlastweek.Ifyoureadsomethingscientificlastweek,focusonbusinesstopics,history,orsocialcommentarythisweek.Writetwosummariesorreviews.RemembertoincludeGREvocabularywordsyouhavestudiedsofar.Takethefull-lengthpracticetestattheendofthisbook.Checkyouranswersandreviewyourmistakeswiththeanswerkeysection.Youcanalsoreviewyourmistakesbywatchinglessonvideosonmagoosh.com.PracticethenextfourdecksinourGREMathFlashcards.

http://www.magoosh.comhttp://www.magoosh.comhttp://www.magoosh.comhttp://www.magooshhttp://www.magoosh.com

WeekThree

PrimaryGoalsCompleteallmathmodulesexceptprobability.*Customizesessionstofocusonareaswhereyouneedthemostwork.Learntwohundrednewvocabularywords.Revisitlessonsthatcovertopicswhereyouneedmorereview.LearnabouttheAnalyticalWritingAssessment(AWA)portionoftheGRE.

*Whyskipprobability?Thetestwillhaveonlyoneortwoprobabilityquestions.BecauseprobabilityquestionsaresorareontheGRE,it’sbettertofocusonother“low-hangingfruit.”Ifyoufeelstrongatotherareasinmath,abasicoverviewofprobabilitycouldn’thurt.Otherwise,youmightwanttoskipstudyingit.

SecondaryGoalsReadchapter5ofthisbookandworkthroughboththepractice“AnalyzeanIssue”and“AnalyzeanArgument”writingtasks.Quizyourselfonvocabularyfromthefirsttwoweeks.Total:threehundredwords.Tryfiftywordsatrandom.Apassingrateis80%.CompleteonehundredandfiftyGREQuantitativeproblemsonmagoosh.combasedonthoseareasinwhichyouneedthemostpractice.Completeanotherseventy-five-plusGREVerbalquestionsonmagoosh.com.

Supplemental/OptionalReadthreearticles—themorechallenging,thebetter.Makesureyou’regettingyourvocabularyfromthesearticles,aswellasfrompracticequestions.Attempttousetwenty-fiveGREvocabularywordsinathree-pagereviewandsummaryofallthreearticles.TakeapracticetestfromTheOfficialGuide.ReviewyourmistakesbywatchingMagooshvideosonline.Forthebold,takeasecondpracticetestthisweek.FinishgoingthroughthedecksintheGREMathFlashcards.

http://www.magoosh.comhttp://www.magoosh.com

WeekFour

You’reinthehomestretchandhavedoneagoodjobgettingthisfar,butnowyouneedtoreallypushfullsteamahead.

PrimaryGoalsFeelconfidentinyourapproachtothedifferenttypesofquestions.Prepareyourself,asmuchaspossible,forthehigh-pressureenvironmentoftheactualtest.Onmagoosh.com,startdoingthehardquestions(andveryhard,ifyou’veansweredmorethan70percentofmagoosh.comquestionscorrectly).Ifyouweren’tabletocompletethe“hard”and“veryhard”questionsinthebookbefore,trythemagainnowandreviewyouranswers.

SecondaryGoalsTakeafull-lengthpracticetestonmagoosh.com.Evenifyou’veseenthequestionsinyourmocktestbefore,itdoesn’tmatter.Dothemagain.Learnonehundredandfiftynewwords.Alsomakesureyourevieweveryvocabularywordlistedintheappendixatthebackofthisbook.Takeafinalvocabularyquiz,testingyourselfonatleastsixhundredwordsyou’vestudied.KeepreviewingseveralMagooshGREMathFlashcardseachday.TakeanotherpracticetestinTheOfficialGuide.

Supplemental/OptionalTakeyetanothertestfromTheOfficialGuide.

http://www.magoosh.comhttp://www.magoosh.com

WhattoDoIfYouFallOfftheStudyWagonEventhebest-laidplansareimperfect.StudyingfortheGREisademandingprocessthattakestime,andevenafteralltheplanning,sometimesyou’regoingtofindyourselfveeringofftrack.Butyoudon’thavetogiveupcompletelyjustbecauseyoudidn’tmeetsomeofyourgoals.Instead,consideritagoodplacetostartover.

Youalreadyhaveaplanthatyouknowdidn’twork,soyou’llbeabletocraftabetterplanthistimearound.

Let’sexaminewhatmighthavethrownyouoffcourse,howtocorrecttheissue,andhowtomakeplanstobeginagain.

1. Bespecific.Partoftheproblemmighthavebeenthegoalsthatyousetforyourself.Goalsthataretoovagueortoobroadlendthemselvestofailure.Whenyourgoalsaren’tspecific,youwon’tknowwhattodonextorwhenyou’veachievedthem.

2. Berealistic.Trysettingsmaller,moreattainablegoals.Thispracticeallowsyoutoachievegoalswithgreaterease.Forexample,here’sahard-to-reachgoal:“IamgoingtogetaperfectscoreontheGREQuantitative.”Incontrast,here’samuchmoretangible,reasonablegoal:“Iamgoingtomemorizealltheprimenumbersfrom0to100andtheconversionsofcommonfractionstodecimals.”

3. Beflexible.Sometimeslifehappensandyouhavetoabandonyourstudiesforpersonalissues,familyemergencies,orprofessionaldeadlines.Thereareunexpectedeventsthatareimpossibletoplanforandcanthrowyourschedulecompletelyoutofwhack.There’sverylittlethatyoucandotoavoidthesehiccups.Onewaytomakethemlessofanissueistobuildaplanthathassomewiggleroominit.Plantomissdaysinyourstudyschedulebyscatteringfreedaysthroughout.Thesedayscanbeusedtoadjustyourschedulewhentheunexpectedhappens.

4. Beforward-thinking.Whateverpathyouhavechosen,anticipatetimeswhenyoumightfalloffthewagon.Didyoujustplanatrip?Areyougoingtoaweddingorsurprisingyourmomforherbirthday?Theseneedtobeaccountedforinadvance.Ifyouknowsomethingiscoming,makechanges—don’tjusttrotalongwiththesameplan.Byforeseeingbumpsintheroad,youcanaccountforthemandadjustnow.

Don’tbeatyourselfupoverasmallderailmentinplans.Goon,dustoffthoseGRE-prepmaterials,andgetbacktoit.Youhaveatesttodominate!

WhattoDoIfYouHitaStudyPlateauIfyou’vebeenstudyingfortheGREforagoodamountatime,it’slikelyyou’llstarttofeelfrustratedandunmotivated.Nomattertheskillyou’retryingtosharpen,therewillcomeamomentwhenyoufeelyou’vehitaplateau.Bynomeansshouldyougiveup.Yourbraincouldverywellbetellingyouthatyouneedtotakeabreakorthatyouatleasthavetomixitupalittle.

TheGREisnodifferent.Yourpracticetestscores,evenafteryou’vestudieddiligentlyforweeks,maynotbegoingup.It’seasytocometotheconclusionthatallyourworkmeansnothing.Butdon’tdespair:we’veputtogethersomeimportantpointerstoreviewwhenyoufeelyou’vehittheproverbialwall.

1. Gosomewherenew.It’spossiblethatGREprephasbecometooregimentedforyou.Structureanddisciplinearecriticaltosuccess,uptoapoint,butafterawhile,youmaystarttoloseinterestbecauseoftherepetition.Onegreatwaytobreakthemonotonyistostudyinadifferentplacethanusual.

2. Changeyourstudyroutine.YoumightstarteachstudysessionbydoingafewGREVerbalexercises.Youreviewyourmistakesandthenmoveontoasetofmathproblems.Afterfollowingthispatternforamonth,yourbrainstartstobecomebored.Surpriseit!Anyofthefollowingshoulddothetrick:

Doamini-testinwhichyouimmediatelyfollowupfivemathproblemswithalongreadingpassage.TryouttheMagooshGREPrepapp.SpendahalfhourtestingyourselfwithMagoosh’sVocabularyBuilderapporGREFlashcards.Reviewmaterialfromadayortwoago.Doyourememberwhatyoulearnedthatday?Revisitquestionsyoumissed.

3. Takeabreak.Ourbrains,likeourmuscles,needrest.Bytakingabreakfromsomethingyou’relearning,you’llhavetimetoprocessallthatinformation.Understandably,youdon’twanttotaketoomuchtimeoff.Butevenathree-tofour-daybreakfromstudyingvocabularywon’tcauseyoutoforgetallyou’velearned.Comingbackafterafewdays’rest,ontheotherhand,willgiveyoua

renewedperspective.Suddenly,thewordpolemical,whichyouwerehavingsomuchdifficultylearningbecauseyoucouldn’tgettheimageofapoleoutofyourhead,conjuresupnew—andmoreapplicable—images.

4. Focusondifferentpartsofthetest.Manystudentsbecometoofixatedondoingjustonequestiontypeorjustonesectiononthetest.Breaksfromstudyingthetestasawholearen’ttheonlywaytoprocessbetter:takingabreakfromaspecificsectioncanalsohelpcementwhatyou’velearned.

TheWeekBeforetheTestYou’veprobablyheardthisahundredtimesbefore:Alwaysgetagoodnight’ssleepthedaybeforesomethingimportant.PreppingfortheGREisnodifferent.TheGREisalong,taxingexperience.Andyoudon’twanttofindyourselfnoddingoffatanypointduringthefour-plushoursthatyou’llbesittinginthetestingcenter.Soagoodnight’srestiscrucialtoyourperformance.

Also,don’tdoanythingthatisn’tusuallypartofyourroutine.Forinstance,ifafriendasksyououtfordinner,reschedule.We’renottellingyoutobeantisocial,butpeopletendtostayuplaterwhenengagedinsocialactivities.Conversely,don’tturnoffyourphoneandhideunderthecovers,hopingforatwelve-hoursleepsession.Justtrytokeeptoyourweekdayroutinesasmuchaspossible.

Andifyou’renotanearlyriserbutwereforcedtomakeaneighta.m.testingappointment,besuretostartwakingupalittlebitearliereachdayduringtheweekbeforethetest.Thatway,you’llgetusedtowrestlingyourselfoutofbedatsixinthemorningbeforetestday.

Finally,don’tfeelobligatedtocram,anddefinitelydon’tstayuplatecramming.TheGREtestsknowledgebuiltupoveralifetime—oratleastafewmonthsofintensiveprepping.Crammingthenightbeforewon’tleadtoahigherscore,andbecauseitwillmostlikelyfrayyouralreadyfrazzlednerves,crammingmayactuallyhurtyourscore.Thatdoesn’tmeanyoushouldn’tdoafewpracticequestionsthedaybeforethetest.Butotherwise,trytorelax,asmuchasit’spossibletodoso.

HowtoStudyforaGRERetakeIfthissectionappliestoyou,thenit’sveryimportanttoaskyourselfwhatyoucandodifferentlyinpreparingfortheGREthistimearound.Ifyoufind

yourselfretakingthetest,thenmostlikelysomepartofyourstudyprocessthefirsttimearoundwasnotoptimal.Ofcourse,“something”isaterriblyvagueword.Sowe’lltrytohelpyoufigureoutwhatmighthavehappened.

Maybeyoudidn’ttakeenoughpracticetests.Nothingpreparesyoufortestdaybetterthanamockexam.Andnotjustanymockexam,mindyou.Magooshtendstobeasdifficultas,ifnotmoredifficultthan,theactualGRE.Ofcourse,nothingbeatstakinganofficialtest.TherearetwoETSmocktestsattheendoftheTheOfficialGuidetotheGREbookandandtwotestsonthePowerprepIICDthatcomeswiththebook.

Thistimearound,tryspacingoutpracticetestseveryfivedays.Theassumptionisthatyouhavealreadypreppedsufficientlyandthereforedon’tneedtolearnallthefundamentalsoveragain.Muchofyourtimeinbetweentestsshouldbefocusedondissectingyourperformance.Whatdidyoudowrong?Whatcouldyouhavedonebetter?Anyinsightsgleanedshouldbeusedinthefollowingtest;i.e.,you’reanticipatingmakingsimilarmistakesandwillthusbeonguard.

It’spossibleyouusedmediocreprepresources.Notallprepmaterialsareequal.IfyoufelttheGREwasverydifferentinthetypesofquestionaskedinyourpracticematerial,thenyoushouldconsiderusingnewmaterials.Tosolvethisproblem,youshoulddosomeresearchandusethematerialthatwillbesthelpyourscore.And,ifpossible,don’tjustsettleononeresource.

Itcanbehardtoidentifyyourweaknesses.Andifyoucan’tidentifythem,thenit’spossibleyoudidn’tfocusonthoseareaswhilestudying.

Keepinganerrorlogcanhelptremendously.Thislogcanbehighlysystematizedoritcanbeverybasic.Forinstance,youcansimplyfileadifficultquestioninyourmentalRolodex.Trytorememberhowthequestiontrickedyouandwhatyoushoulddothenexttimeyoucomeacrosssomethinglikeitinordertoavoidthesamemistake.Ofcourse,rememberingthingsisn’talwaysthebestwaytoimprove.Infact,werecommendthatyoufirstjotdownthequestionnumberandsourceofthequestioninanotebook.Thenanswerthefollowingquestions:

1. Whydidyoumissthequestion?2. Whywasyouranswerwrong?3. Whyisthecorrectanswercorrect?4. Whatwillyouavoiddoingthenexttimearound?

Comebacktothislogoften,especiallybeforeyoutakeapracticetest.Reviewyourerrors,sothatonthepracticetestyou’llbecarefulnottomakeasimilarmistake.Bythetimeyourtestrollsaround,you’llbeonguardagainstany

Eachmistakeisanopportunitytolearn.Duringyourworstmoments,while

preppingforanytest,andespeciallytheGRE,you’llfeelinclinedtohurlthebookor

laptopagainstthewall.Rememberthatevenifyoumissareallyeasyquestion,

doingsoisanopportunitytolearn.Thatis,youcantrytobetterunderstandwhatled

youtomissaneasy—orevendifficult—problem,sothatyoucanavoidthesame

mistakeinthefuture.

carelesserrors,andhopefullyyou’llneveragainhavetoaskyourselfhowtostudyforretakingtheGRE!

InSumWe’veprovidedawholebunchoftipsandtricksforstudyingfortheGRE.Nowit’stimetogettoit.TherestofthisbookisdedicatedtohelpingyouunderstandGREquestiontypesandpracticethethoughtprocessesandproceduresnecessarytoanswerthem.Remembertocomebacktothissectionwheneveryouneedaboostofconfidenceorapatontheback.WeknowthelifeofapersonstudyingfortheGREisbusyandcomplicated,butwealsoknowthatyoucantotallydothis!

Chapter3

GREQuantitativeReasoningBroughttoyoubyMikefromMagoosh

Didyouknow?TheancientGreekswerethefirsttodotruemathematics.

MeettheGREQuantitativeSectionWhatdointegralcalculus,trigonometry,andgeometryproofsallhaveincommon?Well,besidesbeingthreethingsyouprobablydon’tenjoydoingforfun,theyalsodon’tshowupontheGRE.That’sright:nohigher-levelmathwillshowuponthetest.Weknow.Phew,right?It’sokaytotakeaminutetosighwithreliefandjumpforjoyrightnow.

ThetypeofmaththatdoesshowupontheGREisthestuffyoumostlikelylearnedfrommiddleschooltojunioryearofhighschool.Togiveyoualittlerefresher,incaseyou’veforgottensomeofthisstuff,thecommontopicsincludethefollowing:basicpropertiesofshapes(circles,quadrilaterals,etc.),integerproperties,exponents,andwordproblems(includingratequestionsandprobability).Andreally,there’snotmuchbeyondthat.

WhatmakestheGREsotrickyisthecomplexityanddensityofmanyofthequestions.Onceyoufigureoutwhatthequestionisactuallyasking,though,themathinvolvedisn’tnearlyaslong-windedaswhatyouprobablydidinhighschool.Youknow,thosehomeworkquestionsthatyourteacherexactinglycheckedtomakesurethatyoufollowedeachofthethirteenstepstoarriveatthesolution.Infact,withtheGRE,youdon’tevenhavetoshowyourwork.Youjusthavetopicktherightanswer.

KeepreadingtofindouthowtoprepareforGREquantitativeproblems.We’veincludedtipsontopicssuchashowtopaceyourselfduringthetestandwhattodowhenfacingwhatseemslikeareallytoughquestion.

CommonlyTestedGREMathConceptsBasedonthepracticetestsinthesecondeditionofTheOfficialGuidetotheGRErevisedGeneralTest,wordproblemswillbeplentiful.Therearealsoafairnumberofalgebra-relatedquestions.Andevenafewcombinatoricsproblemsreartheirfearsomeheads.Ifyou’rewonderinghowmanyquestionsontheGREwilldealwithstatistics—atopicoftengivenscantattentioninmanyprepbooks—itcanpopupinspades.Mostinterestingly,coordinate

geometryonlyshowsuponce,andthereisn’tasinglerateproblem.

ThebreakdownofconceptsbasedonthefiftyquantitativequestionsinPracticeTest#2ofTheOfficialGuidedetailedbelowwilllikelybesimilartowhatyouseeontestday.Thatsaid,you’llprobablyseemorethanonecoordinategeometryquestionandmayalsoseearatequestion.Remember,justbecauseaconceptdidn’tshowuponthepublishedpracticetestdoesn’tmeanyouwon’tseethatconceptontestday.Interestproblems,bothsimpleandcompound,willprobablyshowup,too.

Whatisthetakeawayfromallthis?Ifyou’reweakinacertainarea—say,wordproblemsthatdealexclusivelywithaddingaseriesofnumbers—don’tsweatit.Youmayonlyseeoneproblemdealingwiththatconcept.Ontheotherhand,ifyou’reonlysomewhatcomfortablewithstatistics,it’sagoodideatostrengthenyourskillsinthatarea,asyou’relikelytoseeseveralquestionsonsuchabroadtopicontestday.

HowtoStudyGREMathSoyou’veboughtafewofthemajortestprepbooks,includingtheoneyou’reholdinginyourhandsrightnow,andyou’rereadytoripintotheGREQuantitative.You’llreadthrougheachbook,pagebypage,andbytheend,GREmathmasterywillbeyours.Ifonly!

StudyingfortheGREQuantitativeismuchmorecomplicatedthanjustreadingabook.Youcouldeasilygetthroughhundredsofpagesoftextandfeellikeyou’velearnedverylittle.TheGREQuantitativetakesknowledgeandpracticalapplication.

Twocyclists,MikeandDeborah,beginridingat11a.m.Mikeridesataconstantrateof40kilometersperhour(kmh),andDeborahridesataconstantrateof30kmh.AtnoonMikestopsforlunch.AtwhattimewillDeborahpassMike,giventhatshecontinuesataconstantrate?

Keepinmindthefollowingimportantpointsonhow(andhownot!)tostudyfortheGREQuantitative.

1. WatchfortheGREmathformulatrap.Youmayask,“Howcanformulasbebad?Aren’ttheythelifebloodofGREmath?”Actually,formulasaren’tallthathelpful.Andtheydefinitelyaren’tthecruxoftheGREQuantitative.TherealkeytobeingsuccessfulontheGREistohavesharpproblem-solvingskills.

Therealityisyoumustfirstdecipherwhatthequestionisasking.Alltoooften,studentslettheformulasdothethinking.Thatis,theyseeawordproblem—say,adistance/ratequestion—andinsteadofdeconstructingtheproblem,theyinstantlycomeupwithaformulaandstartplugginginnumbersfromthequestion.Inotherwords,theyexpectthequestiontofallneatlyintotheformula.

Toillustrate,takealookatthefollowingquestion:

Studentsaretemptedtoimmediatelyrelyonthed=rtformula.Theythink,DoIusetheformulaonceforMikeandonceforDeborah?Orforonlyoneperson?ButwhichpersondoIuseitfor?

Thisisanunfortunatequandary;thesolutiontothequestionactuallyreliesonfiguringouthowmanykilometersMikehasgoneinonehourandhowmanykilometersbehindhimDeborahis.There’snoformulaforthisconceptualstep.It’sonlyoncetheconceptualstephasbeencompletedthatwecanusethed=rtformula.Theanswer,bytheway,is12:20p.m.

Rememberthattheessenceofproblem-solvingisjustthat:solvingtheproblemlogically,soyoucanusetheformulawhenappropriate.

2. Usetrainingwheelstostart,thengetoutofyourcomfortzone!Manystudentslearnsomebasicconcepts,memorizeaformulaortwo,andfeelthattheyhavethehangoftheGREquantitativecontent.But,assoonastheyactuallysitdowntosolveafewproblems,theyfeelconfusedanduncertainofexactlywhat

problemtypethey’redealingwith.

Startingwithbasicproblemsisanexcellentstudystrategyearlyon.Youcanbuildoffthebasicconceptsinachapterandsolveproblemsofeasytomediumdifficulty.Thisphase,however,representsthe“trainingwheels.”

Actuallyridingabike,muchlikesuccessfullyansweringanassortmentofquestions,requiresyoutogetoutofyourcomfortzone.Inotherwords,youshouldtryafewquestionschosenatrandom.OpeningupTheOfficialGuideanddoingthefirstmathquestionsyouseeisagoodstart.Evenifyouhaven’tseentheconcept,justgoforit.You’llgetafeelforworkingthroughquestionswithlimitedinformation.

Sometimesstudentsdon’tlovethisadvice.Itmakesthemuncomfortabletotackleproblemstheyhaven’tstudiedforyet.Buttherealityisthatyoucanactuallysolvemanyproblemsbasedonwhatyoualreadyknow,especiallywhenyoutakeyourtimeandconsiderthequestioncarefully.

3. Avoidtunnelvision.Somestudentsbecomeobsessedwithacertainquestiontype,attheexpenseofignoringequallyimportantconcepts.Forinstance,somebegintofocusonlyonalgebra,forgettinggeometry,rates,counting,andmanyoftheotherimportantconcepts.

This“tunnelvision”isdangerous;muchasthe“trainingwheels”phaselullsyouintoafalsesenseofsecurity,onlydoingacertainproblemtypemakesitharderforyourbraintogetintothegroovewithothermathematicaltopics.

4. Don’tfocustoomuchonthereallyhigh-hangingfruit.Trynottospendalotoftimestudyingtopicsthatmaynotevenshowuponthetest.Thisisasubsetof“tunnelvision.”Forexample,somestudentsgettotallywrappedupinlearningpermutationsandcombinationsconcepts.Instead,theycouldbeusingthattimetostudyamoreimportantarea,suchasnumberpropertiesandgeometry,whichshowupmuchmorefrequentlyonthetest.

Wouldyouclimbtotheverytopofthetreetograbthemeagercombinations/permutationsfruit,whenrightwithinyourgrasparethelusciousnumberpropertiesfruit?

5. Bewaryoftest-prepmaterial.Manyofthesourcesoutthere

don’tofferpracticecontentthat’sasdifficultaswhatyou’llseeonthetest.Someoffertoofewsetswithamixtureofquestiontypes.Basically,theynevertakeyououtofthe“trainingwheels”phase.

Othercontenthasquestionsinwhichyoucaneasilyapplyaformulawithoutfirsthavingto“crack”theproblem.SuchmaterialwillleaveyoufeelingprettyunpreparedfortheactualGRE.

UnderstandWhereYouCanGoWrongGREmathcanbetricky,buttheteststillrewardsfundamentals.Ifyouhaveasolidgroundinginallofthefundamentals,youhaveastrongchanceofscoringabove160.Speedandconcentrationwillplayalargefactorinyourscore,soremembertowatchoutforthosecarelesserrors!

Thewronganswers—aswellasluckyguesses—canfallintoseveralcategorieslistedbelow:

1. Conceptualerrors.Gettingconfusedbyaconceptualquestionshouldbenoreasontogiveup,throwyourarmsintheair,andscream.Rather,youshouldhavetheexactoppositeattitude.Wow,IjustfoundsomethingthatIneedtoworkoninordertobetterunderstandtheconcepts.Doingsowillhelpmepreparefortestday.Sogobacktoyourresources,readupmoreonwhateverthatconceptmaybe,andfindpracticeproblemstohelpyouconquerthatconcept.

2. Fallingforthetrap.TheGREisatrickytest.ETShasdecadesofexperiencewritingquestionsmeanttotripuptesttakers.Didyouforgettoconsiderthatthevalueofxcouldbeafractionbetween0and1?Maybeyoumissedthewordisosceles,ormaybeyouonlyconsideredrectanglesandnotirregularquadrilaterals.

Whateverthecasemaybe,identifywhyyoufellforthetrapandtheassumptionsyoumadethatledyoutothewronganswer.Here’sahint:ifaquestionseemsalittletooeasy,thatmaybeasignthatthetestistryingtotrickyou.

3. Carelessmistakes.Carelesserrorsrangefromselectingthewronglettertomakingasmallcomputationalerror.Identifyingthetypeofcarelesserrorsyoumakecanhelpyouavoidrepeatingthosespecificmistakeswhenyou’reansweringquestions.

4. Misinterpretation.Sometimesyoumisreadtheproblemandendupsolvingadifferentproblem.ETScanactuallypredictwaysthat

testtakersmightmisinterpretaproblem,sotheyprovideawronganswerchoicethatmatchesthesecommonmisunderstandings.Tofightback,besuretoreadwordproblemscarefully,andpayattentiontosmallbutimportantwordssuchaspositive,integer,andnot.

HowtoApproachComplicatedMathProblemsEveryGREtestisgoingtohaveafewmathquestionsthatareverydifficult.Trynottogetflustered.You’llbeabletofigurethemoutwithpractice.

Belowarethreeimportantpointstokeepinmindwhenyou’redealingwithadifficultGREmathproblem.

1. Don’tgetrattled.It’sveryeasytobecomeanxiouswhenyoucomeacrossdifficultmathproblems,especiallywordproblems.Onereasonisthatwestartreadingandrereadingthesamequestion,hopingthatbythefourthreadwe’llfinallygetit.Atthispoint,rereadingisclearlyanexampleofdiminishing—andfrustrating—returns.Whattodo?

2. Readthequestioncarefully.Sometimesaproblemseemsmuchmorecomplicatedthanitactuallyis.Thereasonisthatwe’remisreadingawordorinsertingourownwordintothequestion.Wespendseveralminutestryingoutdifficultequationsonlytorealizethatnoneoftheanswerchoicesmatchesupwithouranswer.Toavoidthis,makesureyoudon’trushthroughthequestion.Instead,readcarefullyandknowwhatthequestionisaskingbeforeattemptingtoanswerit.

3. Letyourbraindecipherthequestion.Itcantakeaboutthirtyseconds—andacareful,calmrereading—tounderstandwhatthequestionisasking.Thenyou’llneedtofindthesolutionpath.Todoso,think—orevenwritedown—thenecessarystepstogettothesolutionstepbystep.

Onepieceofgoodnews:youcantakelongeroncomplicatedquestions.Afterall,therearen’ttoomanyofthem.Justmakesuretosolvetheeasyandmediumquestionsfirst.Remember,youcanalwayscomebacktoaquestion.Sometimesit’seasiertodecodethesecondtimearound.

TipsforQuicklySolvingGREMathProblemsStudentswhoarereasonablycomfortablewithmathitselfoftencomplain,“I

Inthefigure,ABCDisasquare,andallthedotsareevenlyspaced.Eachverticalorhorizontaldistancebetweentwoadjacentdotsis3units.Findtheareaoftheshadedregion.

60

72

81

96

120(Answerandexplanationbelow.)

couldsolvemostGREquantitativeproblemsifIhadenoughtime,butIalwaysseemtorunoutoftime,”or“IjustwishIcoulddoGREmathfaster.”Doesthissoundlikesomethingyou’vesaid?Thenthissectionisforyou.

Solvethisquestion.Allotyourselfastrictone-and-a-half-minutetimelimit.

ThebrainFirst,let’squicklyreviewafewthingsaboutthebrain.Beforewedo,though,pleaseknowthatwe’renotneurologists.Instead,ourgoalhereistointroduceyoutoanotherwayofthinkingabouttheproblemabove.Sobelowisjustalittlebackgroundtosetthescene.Thebrain,ofcourse,ismuchmorecomplexthantheamountofspacewehaveinthisbook!

Thecerebrum,the“intelligence”partofourbrain,isdividedintotwohalves—theleftandrighthemispheres.Eachhalfcontrolsallthesensationandmuscularmovementsontheoppositesideofthebody.Sowhenyoumoveyourleftleg,it’stherighthalfofyourbrainthatcontrolsit.Thetwohemispheresalsoprocessinformationverydifferently.

Thelefthemisphereisallaboutlogic,organization,precision,anddetail

management.Itspecializesindifferentiation,thatis,tellingtheexactdifferencebetweencloselyrelatedthings.It’sverygoodatfollowingclearrules,recipes,formulas,andproceduresinastep-by-step,logicalway.Theleftbraincontrolsthegrammarandsyntaxoflanguage.Ifsomeonesaid,“Iarehappy,”itwouldbeyourleftbrainthatrecognizedthatasincorrect.

Therighthemispherecontrolsintuitionandpattern-matching.Itspecializesinintegration,thatis,seeingtheunderlyingsimilarityorunitybehindthingsthatseemprettydifferent.It’softencalledthe“artistic”sideofthebrain,andit’sgoodatinterpretinginformationpresentedassymbolsorimages.It’salsogoodatfacialrecognitionandvoicerecognition.

MathandthebrainOkay.Sowhichhemisphereisbetterformath?Well,thedetail-management,organization,andprecisionoftheleftbrainareahugehelpinarithmeticandalgebra.Meanwhile,therightbrain’spattern-matchingskillscanplayaroleinsomebranchesofmath,suchasgeometry.Now,we’regoingtomakeanoversimplificationinordertoillustratetwowaysofapproachingGREmath.Imaginethatwe’reallsplitupinto“left-brainpeople”and“right-brainpeople,”eachwitha“dominant”hemispherethathasalargerinfluenceoverthewaywethink.Ingeneral,left-brainpeopleusuallyfeelreasonablycomfortablewithmathandcancertainlyfollowthemethodicalprocedureswithease.Typically,right-brainpeopletendtohaveamoredifficulttimekeepingallthedetailsstraight,althoughmanytimestheytuneintothe“bigpicture”ideasfaster.

Themagicofright-brainthinkingLeft-brainpeopleusuallyknowtherulesreasonablywell.They’reoftenthe“good-at-math”kindofstudents.FacedwithmostGREquantitativequestions,theycouldfigureouttherightanswer,givenenoughtime.Thecatch,ofcourse,isthattheydon’thaveunlimitedtimeontheGRE—justthirty-fiveminutesfortwentyquestions,or1:45perquestion.

AnoverwhelmingnumberofGREmathquestionsaredesignedspecificallytopunishsomeonewhoisoverlyleft-brained.Inotherwords,thequestionsarewrittentobelookedatindifferentways.Yes,themethodical,step-by-step,ploddingapproachescanwork,buttheytaketoomuchtime.Thebesttesttakerscanreframeaquestionorseeitinadifferentway—onethatcanbeansweredmorequickly.

Example:thepracticeproblem

Let’stakealookbackatthepracticeproblem.Yes,youcouldfigureoutseparatelytheareaofeachtriangle,eachsquare,eachtrapezoid,andthenaddallofthoseshapestogether.Thatwouldtakesometime.Solet’slookatitadifferentway.

Here’saright-brainsolutiontothisquestion:rearrangethepieces!

First,slidethelittletriangleupintothecorner.

Now,noticethattheleft-mosttrapezoidwouldfitnicelyintothatblankspaceontheupperright.

Now,flipthatremaininglongtrapezoidoverthediagonalBD:

Wouldyoulookatthat!Theshadedregionaccountsforexactlyhalftheareaofthebigsquare.Thebigsquareis12×12=144,sotheshadedregionis144÷2=72.Answer=(B).Onceyouseethetrick,thepattern,there’sonlythemostminimalofcalculationsneeded.

Anotherreasonablyquickright-brainapproachwouldbetosimplycountallofthelittletriangles.

Theoriginalshadedfigurecanbebrokeninto16equaltriangles.Thefullsquareis16littlesquaresor32triangles.Therefore,theareaoftheshadedfigureisexactlyhalftheareaofthesquare.

LearningtoseeSomeright-brainreadersmightcelebratesuchaprocess,whilesomeleft-brainpeoplemightbefrustratedorannoyedatthispoint.Theymaybethinking,Great!Nowthatit’spointedout,yes,that’sanefficientwaytosolve,buthowamIsupposedtoseethatonmyown?

Masteringthestrengthsandskillsofyournon-dominanthemisphereisneveraneasytask,butitcanbedone.

Thereareawildvarietyofthingsyoucandotoenhanceright-brainfunction.Readpoetry.Lookatart.Makeart.Readaboutpatternsincomparativemythology.Free-associate.Imagine.Followchainsofwordassociations.Slowdownandreallylookatthings.

SpecificTipsforDoingGREMathFaster1. Iftheproblemasksforthevalueofanexpressioninvolving

variables,chancesaregoodtherewillbesomewaytosolveforthevalueoftheexpressiondirectly,withoutsolvingfortheindividualvariables.

2. Fora“findtheareaoftheshadedregion”questioninwhichtheregionisparticularlycomplicated(asinthepracticequestion),lookforawaytorearrangeandsimplify.

3. Iftheproblemisageometryonestatedinwords,alwayssketcharoughdiagram,unlessyoucanvisualizethediagrameasily.

4. Iftheproblemispurelynumericaloralgebraic,considerwhethertherewouldbeawaytovisualizetheproblem(numberline,xy-plane,etc.)

Fromthispointforward,wheneveryoupracticeGREquantitativequestions,firstofall,alwayspracticeagainstastricttimelimit.Second,thecriterionisnolongerwhetheryougottheanswerrightornot.Evenifyougotthecorrectanswer,compareyoursolutiontotheofficialsolution.Ifyoursolutionwasaslowmethodicalapproachandtheofficialsolutionshowsashortcut,thenforyourpurposes,considerthisaquestionyougotwrong.Foreverysuchquestion,forceyourselftowritedowntheshortcutandconsiderwhatyoucouldhavedoneto“see”thatshortcut.Forceyourselftoputitintowordsandexplainit—apracticethatwillstrengthenyourinterhemisphericconnections.Asyoucollectmoreandmorewrite-ups,gobacktorereadthecollection.Keepdoingthisconsistently.Learnfromyourmistakes,andbeforeyouknowit,you’llstart“seeing”solutionstoGREquantitativeproblems!

PacingStrategiesEachGREQuantitativesectioncontainstwentyquestions.You’regiventhirty-fiveminutesforeachsection,whichworksoutto1:45perquestion.Belowaresomehelpfultipstohelpyouusethesethirty-fiveminuteswisely.

1. Goforthelow-hangingfruit.EachquestionwithinagivenGREQuantitativesectionisworththesamenumberofpoints.That’sanextremelyimportantpointtoremember.

That’sright,friends.Let’ssayETSdevisedaquestionsuchasthefollowing:

Evensomeonereallygoodatmathmighttakefiveminutestosolvethisquestion—iftheycansolveitall.Yet,thecorrectanswerwouldyieldtheexactsamenumberofpointsasthisquestion:

If2x=42,whatisthevalueofx?

Sowhat’sthetakeawayfromthis—otherthan“Factorialsscareme!”?

Well,whywastetimeonaverydifficultquestionwhenyoucansimplyskiptoaneasierquestion?Thinkofitthisway:inthirty-fiveminutesyouwanttoscoreasmanypointsasyoucan,andeachquestionisworththesame.

Ifyouwerepaidonethousanddollarsforeveryappleyoupickedfromatreeinthirty-fiveminutes,whatwouldyoudo?Youwouldgoforthelow-hangingfruit.Youwouldn’twasteyourtimeclimbingto

theverytopofthetreetopluckanapplethat’sworththesameamountofmoneyasanapplethatyoucansimplyreachoutandgrabwithbothyourfeetplantedontheground.

Ofcourse,afteracertainpoint—onceallthoseeasyquestionsarecomplete—youmustgrabthefruituphighandgoforthedifficultquestions.Butmakesureyou’veansweredtheeasyonesfirst.

2. Budgetyourtimewisely.Thereareeasyquestions,mediumquestions,anddifficultquestions.Easyquestionsshouldtakebetweenforty-fivesecondsandoneminute.Mediumquestionsshouldtakebetweenoneandtwominutes.Anddifficultquestionsshouldtakenolongerthanthreeminutes.Theratioofeasy,medium,anddifficultquestionsvariespersection,butingeneralyoucanexpecttoseeasmatteringofeach.Withinaneasysection,theratiowillskewtowardeasy;inadifficultsectionthatratiowillskewtowarddifficult.

3. Learntoletgoofaquestion.Ifyou’restaringataquestionandhavebeenunabletocomeupwithasolutionafteraminute,youshouldseriouslyconsidermovingontothenextquestion.Again,keepthelow-hangingfruitmetaphorinmind.

If,however,you’redealingwithadifficultmathquestion(andit’sclearthatit’sdifficultandthatyou’renotjustmissingsomethingobvious),thentakeacoupleofminutes,assomequestionswillclearlytakethatmuchtime.Don’tfreakoutoveraquestionthat’sclearlytortuousjustbecauseyou’vetakentwominutes.Aslongasyou’reheadedtowardthesolution,persevere.

4. Don’tbesloppy,butdon’tobsessovereasyquestions.Usingthetimebreakdownabove,wecanseethateasyquestionsshouldtakelessthanaminute.It’simportanttoanswerthesequestionsconfidentlyandmoveon.Ifyoudither,you’rewastingtimethatcouldbespentonamoredifficultquestion.That’snottosayyoushouldracethroughaneasyquestion,becausethatdefeatsthewholelow-hangingfruitlesson.Missingaquestionthatyoucouldeasilyhaveansweredcorrectlyhadyouspentthatextraseconddoesn’tmakesense,especiallyifyou’reracingtowarddifficultquestionsthatyoumaynotevenanswercorrectlyinthefirstplace.

5. Makesureyouguess.Aswe’veestablished,it’sokaytoskipsomequestions.Butmakesure,attheveryend,thatyouguesson

anyquestionsremaining,becausethere’snopenaltyforguessing.Soifyouskippedalotofquestions,giveyourselfenoughtimeattheendtoselectananswerforthequestionsyouleftblank.Alittlebitofluckcangoalongway!

CalculatorStrategiesCanIuseacalculatorontheGRE?Thisisaverypopularquestion,andtheansweroftenelicitsanaudiblesighofrelief.So,yes,youcanuseacalculator.

Butdon’tcelebratejustyet.

Firstoff,manyproblemsdon’trequireacalculator.Infact,usingacalculatormayverywellslowyoudownbecauseyoucaneitherdothearithmeticfasterinyourheadoronapieceofpaper.Then,there’salwaysthecaseofwhattocalculate.Whileacalculatorwon’tmakeacarelesserror,neitherwillittellyouhowtoansweraquestion.Basically,theGREmathisstilltestingyourabilitytologicallydeconstructaproblem.Inmanycases,thechallengeisn’tthemathbuttheapproachtoaproblem.

Whenisitadvantageoustouseacalculator?Therearetimeswhenthecalculationissimplytoodifficulttomultiplyonpaperandthequestionisn’taskingyoutoestimate.Problemssuchascompoundinterestfitthisdescription.Maybeyouhavetofindthehypotenuseofarighttrianglewithsidesof51and31.Figuringoutthesquarerootofalargenumbercouldbeverydifficultwithoutacalculator.

Ofcourse,iftheproblemasks,Whatistheunit’sdigitof31000?thenyouthenhavetocomeupwithacleverwaytoapproachtheproblem;acalculatordoesn’tholdthatmanydigits.

GettingafeelforthecalculatorThebestwaytodeterminewhetheryou’llbenefitfromacalculatoristotakeapracticetestusingMagoosh’sonlineGREprep.Bydoingso,youshouldgetafeelforthenumberandtypesofquestionsinwhichthecalculatorwillhelpyousavetimeandthoseinwhichusingitwillonlyeatuptime.

Totherightiswhattheonlinecalculatorlookslike.

You’veprobablyusedacalculatorbefore,sothedigits,decimalpoint,fouroperations(+,–,×,÷),andparenthesesbuttonsareallprobablyquiteintuitive.The±changesthesignoftheentrycurrentlyonthedisplay.The√buttontakesthesquarerootoftheentrycurrentlyonthedisplay.

ManystudentsareconfusedaboutCEvs.Cbuttons.TheCEbuttonis“clearentry”andCis“clearall.”What’sthedifference?Supposeforsomereason,youhavetocompute23×41×72—thatwouldbeanunlikelythingtocalculateontheGRE,butpretendyouhadtocalculatethatproduct.Let’ssayyoutype23×41×—butthen,byaccident,type27insteadof72.IfyouhitCE,thescreengoesto0,butthecalculatorremembersyourprevioussteps.AfterCE,youjusttype72and=andvoila!Thecorrectanswerof67,896appears.UsingtheCEbuttonisthewaytotellthecalculatortorememberthepreviousstepsofacalculation.Bycontrast,ifyouhitC,thecalculatorforgetseverythingandyou’restartingoverfreshwithabrandnewcalculation.UseCoften,wheneveryouwanttostartsomethingnew,anduseCEsparingly,onlywhenyouneedtochangethemostrecentinputinalongcalculation.

Onepersonwhodefinitelydidn’tneedacalculatorwasSrinivasaRamanujan(1887–1920),theIndiansupergeniuswhomademind-bogglingcontributionstomanydifferentbranchesofmathematics.

ThethreeMbuttonsarealsoconfusingandunderused.Thesememorybuttonsallowyoutostoreonenumberinmemoryandsaveitwhileyou’reperformingadifferencecalculation.TheM+addswhateverisonthedisplaytothememory.Thedefaultinmemoryis0.WhenyouuseM+toputsomenon-0valueintomemory,anMwillappearonthescreen,toletyouknowthatsomethingisinmemory.IfyoupressM+againwhenanothernumberisonthedisplay,thisactionaddsthenumbercurrentlyonthedisplaytothenumberinmemory.MRisthememoryrecallbutton,whichcallsthenumberfrommemorybacktothescreen.MCisthememoryclear:thisresetsthevalueinmemoryto0anditmakestheMonthescreengoawaybecausethere’snolongeranythingstoredinthememory.Thesebuttonscanbeveryusefulifyouhavetodoamulti-stepcalculation.

Forexample,supposeyouhadtheextremelyunlikelytaskofcomputingthevalueofthefollowing:

Youcancomputethisonthecalculatorasfollows:

1. Makesurememoryisempty(i.e.,noMonthescreen)2. 51×51=,thenM+.Atthispoint,Mwillappearonthescreen.Then

C.3. 68×68=,thenM+,thenC.4. 204×204=,thenM+,thenC.Atthispoint,thesumundertheradial

willbeinthememory.5. MRand√

Atthispoint,ifyoulike,youcanhitMCtoclearthememoryagain.Theadvantageofthememorybuttonsistobreakalongcalculationintobite-sizedpiecessoyoucanfocusonthepiecesoneatatime.ThedisadvantageisthatyoucanonlyusetheM+functionwhenyou’redoingaddition.

Finally,let’sdiscussthe“TransferDisplay”button.Onordinarymultiple-choicequestionsandmultiple-answerquestions,thishasnofunction,butonanumericentryquestion,thisbuttonwilltransferthenumberonthecalculator’sdisplaytotheanswerboxinthequestion.Thispreventsyoufrommakinganyawkwardcopyingmistakes.

Whichofthefollowingequationsistrueforallpositivevaluesofxandy?

QuantitativeQuestionTypesLet’stakealookatsomeofthetypesofquestionsyou’llseeintheGREQuantitative.Knowingbeforehandhowquestionsareformattedandwhatkindofanswersyou’llhavetoprovidecaneasesomestressandmakeyoufeelbetterprepared.

MultipleChoiceYou’vedefinitelyseenthistypeofquestionbefore.It’satypicalfive-choicemultiple-choicequestionwithonlyonerightanswer.

Here’sanexample—tryitoutforyourselfbeforecheckingtheexplanationbelow.

Thisquestionisreallytestingyourknowledgeofrootsandexponents.Ifanythingdoesn’tmakesensehere,youcanbrushuponalltherootsandexponentsrulesinthe“QuantitativeConcept#4:ExponentsandRoots”sectionlaterinthischapter.

2x2+6>40

Whichvaluesofxsatisfytheinequalityabove?

Indicateallsuchvalues.

MultipleAnswerTheGREcallsthese“multiple-choicequestions—selectoneormoreanswers.”Forbrevity—andclarity—atMagoosh,wecallthem“multiple-answerquestions.”Wedoinclude“multiple-choicequestions,”butweidentifythosetypeofquestionsthataskformorethanoneansweras“multiple-answerquestions.”

Imagineaquestionthathastenpossibleanswerchoices,anynumberofwhichcouldbecorrect—that’samultiple-answerquestion.So,unsurprisingly,theycanbealittlemoredifficultthanatypicalmultiple-choicequestion.

Thosewell-versedincombinationsandpermutationsknowthechanceofguessingcorrectlyonthisquestionis1in1,023,oddssoslimthequestionmightaswellhavebeenabigemptyfill-in-the-blank.Butthat’sokaybecauseyouwon’tbeguessing—you’llknowthematerial,soyou’llfeelconfidentandpreparedtotacklemultiple-answerquestions.

Luckily,mostmultiple-answerquestionswillhaveonlyfiveorsixpossibleanswerchoices,notten.Thebottomlineisthatifyouknowtheconceptbeingtestedandarecarefulandmethodical,thenyoushouldbeabletogetthiscumbersomequestiontypecorrect.

Hereisanexampleofamultiple-answerquestion:

–8

–6

–4

–2

2

4

6

8

Twotrainsstartingfromcities300milesapartheadtowardeachotheratratesof70mphand50mph,respectively.Howlongdoesittakethetrainstocrosspaths?

NotethatontheactualGRE,eachanswerchoicewillhaveasquarearoundit.Ifthere’sacirclearoundtheanswerchoice,thenit’smultiplechoice,businessasusual—oneansweronly.Butwhenyouseethesquare,youknowyou’redealingwithmultiple-answerquestions.Anothertelltalesignofthesequestionsisgiveawaytextsuchas“indicateallsuchvalues,”withthewordallunderlined.

Asforthisparticularmultiple-answerquestion,thefirsttrickistonoticethattheinequalityisn’tstatedinitssimplestform.Tosimplify,dividealltermsby2,thensubtract3frombothsides.Thisleadstoamuchmorestraightforwardinequality:x2>17.Ofcourse,[G]and[H]satisfythis,while[E]and[F]donot.Rememberthatwhenyousquareanegative,yougetapositive—forexample,(–6)2=+36.Thus,[A]and[B]alsowork.Thecompletecorrectanswersetforthismultiple-answerquestionis{A,B,G,H}.

Makesureyouwritesomethingdownwhentacklingmultiple-answerquestions.Tryingtojugglealltheinformationinyourheadwillsurelygetyouintroubleonmorechallengingquestions!

NumericEntry

Thisisaclassicproblemthatsendschillsupstudents’spines.Andnowwe’regoingtoaddanotherbone-rattlingelement:theemptybox.

That’sright—theGREwillhavefill-in-the-blank/empty-boxmathproblems,called“numericentry.”Therewon’tbetoomany,judgingfromofficialETSmaterials,butevenfacingafewquestionslikethisisenoughtounsettlemoststudents.

Let’sgobackandattacktheaboveproblem.Whenyouhaveanytwothings(trains,bicyclists,cars,etc.)headedtowardeachother,youmustaddtheirratestofindthecombinedrate.Thelogicbehindthecombinedrateisthatthetwotrains(asisthecasehere)arecomingfromoppositedirections,straightintoeachother.

Thisyields120mph,averyfastrate.(Incidentally,that’swhataccountsfortheseverityofhead-oncollisions.Don’tworry;thetrainsintheproblemwon’tcollide!)

Tofindthefinalanswer,wewanttoemployourniftyoldformulad=rt,wheredstandsfordistance,rstandsforrate,andtstandsfortime.

We’vealreadyfoundr,whichistheircombinedrateof120mph.They’re300milesapart,sothat’sd.Pluggingthosevaluesin,weget300=120t.Dividingbothsidesby120,wegett=2.5hrs.

Nowwecanconfidentlyfillthatboxinandletthetrainscontinueontheirrespectiveways.

QuantitativeComparisonQuantitativecomparisonisahugepartoftheGRE,makinguproughlyone-thirdofthequestionsintheGREQuantitative.Often,whenprepping,studentsforgetthisfactandspendmuchmoretimeonproblem-solving.Quantitativecomparisonisauniquebeast;whilethemathconceptsaretheexactsameasthosecoveredinproblem-solving,quantitativecomparisonquestionscanbeverytricky.Infact,theGREtestwritersworkveryhardtomakethesequestionsseemsimpleandstraightforward.Inreality,there’susuallyatraportwistwaitingtocatchtheunsuspectingtesttaker.

Theformatofquantitativecomparisonquestionswillalwaysbethesame:comparingtwoquantities(columnAvs.columnB),withthesamefouranswerchoicesthatevaluatetherelationshipbetweenthetwoquantities.However,thequantitiesforcolumnAandBcanbeanythingfromexpressionswithvariablestoreferencestoaquantityinageometricshape.

ColumnA ColumnB

Thenumberofpositivemultiplesof49lessthan2000

Thenumberofpositivemultiplesof50lessthanorequalto2000

ThequantityinColumnAisgreater

ThequantityinColumnBisgreater

Thetwoquantitiesareequal

Therelationshipcannotbedeterminedfromthe

informationgiven

Takealookatthisexampleonthefollowingpage:

Tostart,you’vegottoknowthatamultipleisanynumberthatresultswhenmultiplyinganinteger,x,by1,2,3,4….

Ifxisequalto5,thenthemultiplesof5wouldbe:

5×1=55×2=105×3=155×4=205×5=255×6=30⋯

Inthetableabove,youcanseethatanymultipleof5isdivisibleby5.Forinstance, .Therefore,1,000isamultipleof5.

Thequestionaboveasksyouhowmanymultiplesof49arelessthan2,000.Youcandivide2,000by49toseehowmanymultiplesof49arelessthan2,000.Doingsomaytakeawhile.Afasterwayistoseethat49isverycloseto50.Quickmathallowsyoutodeterminethat50×40is2,000.Therefore,49×40equals40lessthan2,000,or1,960.Ifyouweretomultiply49×41,youwouldbeadding1,960+49,whichequals2,009.Thisnumberisgreaterthan2,000.Therefore,youknowthatthereareonly40multiplesof49lessthan2,000.

WhataboutcolumnB?Well,you’vealreadyfiguredoutthat40×50

equals2,000.Buthereisthetrickypart.WhereascolumnAspecifiesthatthenumberhastobelessthan2,000,columnBsaysthenumberhastobelessthanORequalto2,000.Therefore,thereare40multiplesof50thatarelessthanorequalto2,000,etc.

Theansweris(C).

There’sagoodchancethatyourfirstinstinctwas(A).Clearly,49islowerthan50,soithastohavemoremultiples.Usually,whentheanswertoaquantitativecomparisonquestionappearsobviousatfirstglance,there’ssometwisttotheproblem.Inthiscase,thetwistwasthewordingincolumnB:“lessthanorequalto2000.”Sobewaryofanyquantitativecomparisonquestionsthatseemtooeasy.

QuantitativeConcept#1:Fractions/Ratios/Percents

Thefirstquasi-fractionswereusedbytheancientEgyptiansabout4000yearsago.Itwasthebrillianttwelfth-centuryMoroccanmathematicianAbuBakral-Hassarwhoinitiatedtheuseofthehorizontalbartoseparateanumeratorandadenominator.

We’reputtingthethreeconceptsoffractions,ratios,andpercentstogetherbecausethey’reintricatelylinked.Afterwebreakthesetopicsdownforyou,it’llbetimetopractice!

FractionsOfallthemathtopicsthatraisedread,fear,anxiety,andconfusion,fewdosoasconsistentlyasfractions.Whyisthatanyway?Well,atMagoosh,wehavethistheory.Thinkbacktowhenyoulearnedfractions—maybethethird,fourth,orfifthgrade.That’swhenfractionsareusuallytaught,buttherearetwoproblemswiththat.Firstofall,that’sbeforethetsunamiofpubertyhitandvirtuallyobliteratedallpreviouslyheldlogicalconnectionsinyourhead.Moreimportantly,fractions,likemanyothertopicsinmath,involvesophisticatedpatterns,butinthefourthgrade,thebrainisn’tcapableofabstraction,soinsteadyouhadtorelyonreproducingpatternsmechanically.Andwhenthosepatternsdon’tfitaparticularquestion,youbasicallydon’tknowwhattodo.

Thesolutiontogettingoverfractionitisistoapproachthosemechanicalprocedureswiththeunderstandingthatcomesfromhavinganadultbraincapableofseeingabstractpatterns.Whenyouunderstandwhyyoudoeachthing,then(a)youcanrememberitmuchbetterand(b)youcanfigureoutwhattodoinamomentofconfusion.

Soareyoureadytorekindleyourrelationshipwithfractions?Let’sstartwiththebasics.

Whatisafraction?Afractionisawayofshowingdivision.Thefraction meansthenumberyougetwhenyoudivide2by7.Thenumberatthetopofafractioniscalledthenumerator,andthenumberonthebottomofafractioniscalledthedenominator.

Thefraction alsomeanstwopartsofsomethingbrokenintosevenequal

parts.Imaginedividingsomethingwholeintosevenequalparts—oneofthosepartsis ofthewhole,so istwoofthoseparts,or2× .Thisdiagramrepresenting willprobablycallupdimmemoriesfromyourprepubescentmathematicalexperiences.

Noticethatifyouhavethefractions or ,theybothcanceldownto .Cancelingisdivision.Thus,whenyouhave ,andyoucancel(i.e.,divide)the4sinthenumeratorandthedenominator,theydon’tsimply“goaway,”butratherwhat’sleftinthenumeratoris4÷4=1,andwhat’sleftinthedenominatoris20÷4=5,soweget .

AddingandsubtractingfractionsWhenyouaddfractions,youcan’tsimplyaddacrossthenumeratoranddenominator.

Youmaydimlyrememberthatyoucanaddandsubtractfractionsonlywhenyouhaveacommondenominator.That’strue,butwhy?Believeitornot,thebasisofthisfactiswhat’scalledtheDistributiveLaw,a(b+c)=ab+ac.Forexample,ifyouadd3xand5x,youget3x+5x=8x.AccordingtotheDistributiveLaw,youcanaddtwotermsofthesamething,butifyouwanttoadd3x+5y,youcan’tsimplifythatanyfurtherandmustkeepitas3x+5y.Basically,youcan’taddapplesandoranges.

Whenthedenominatorsaren’tthesame,asintheexample ,thenyoucan’taddthemasis,butyoucantakeadvantageofasleekmathematicaltrick.Youknowthatanynumberoveritself,say ,equals1,andyoucanalwaysmultiplyby1andnotchangethevalueofsomething.Therefore,youcouldmultiplybysomefraction ,andthenmultiplybysomeotherfraction ,and

bothwouldretainthesamevalue.You’llwanttomultiplyeachtofindtheleastcommondenominator(LCD),whichhereis24.Thus,

Thesamethingworksforsubtraction:

MultiplyingfractionsThisistheeasiestofallfractionrules.Tomultiplyfractions,multiplyacrossthenumeratorsanddenominators.

What’salittletrickyaboutmultiplyingfractionsisknowingwhatyoucancancel.Ifyoumultiplytwofractions,ofcourseyoucancancelanynumeratorwithitsowndenominator,butyoucanalsocancelanynumeratorwithanydenominator.Here’sahorrendousmultiplicationproblemthatsimplifieselegantlywiththeliberaluseofcanceling.

DividingfractionsTogetstartedwithdivision,it’simportanttorememberthatmultiplyingbyisthesameasdividingby3.That’sjustthefundamentaldefinitionoffractionsasdivision.Thisalsomeansthatdividingby isthesameasmultiplyingby3.Thispatternsuggests,correctly,thatdividingbyafractionsimplymeansmultiplyingbyitsreciprocal:

Notice,asalways,thatyoucancelbeforeyoumultiply.Dividingafractionbyanumberfollowsthesamepattern:

Noticethesimilaritywiththepreviousrule:dividingby3meansthesamethingasmultiplyingby .Also,again,noticethatyoucancelbeforeyou

multiply.

RatiosSupposeabiggrouphastwokindsofmembers.Let’ssaythatallthebaseballplayersinacertainhighschoolstudentbodyaredividedintovarsityandjuniorvarsityplayers,twomutuallyexclusivegroups.Supposewe’retoldthattheratioofjuniorvarsitytovarsityplayersis3:5.Whatdoesthismean?Itwouldbewrongtoassumethatthereareonly3juniorvarsityplayersand5varsityplayers—it’shardtoplaybaseballwithsofewplayers!Instead,theratiotellsusaboutoverallmakeupofalargergroup.

Aratioof3:5couldmeanthatthenumbersofjuniorvarsityandvarsityplayersare6and9,or15and25,or30and50,or60and100.Inotherwords,itcouldmeananymultiplesof3and5.Whatitdoesmean,though,isthatifwemadeafractionwiththenumberofjuniorvarsityplayersinthenumeratorandthenumberofvarsityplayersinthedenominator,thatfractionwouldsimplifyto .

It’salsopossibletohavemorethantwogroupsrelatedinaratio.Let’ssaythatwecangroupalltheemployeesinacompanybyhowtheytraveltowork.GroupDaretheemployeesthatdrive,groupParetheoneswhotakepublictransportation,andgroupWaretheoneswhowalk.Let’ssaythat,atCompanyKX,theratioofDtoPtoWis2:5:1.Onceagain,thisdoesn’tnecessarilymeanthatthereareonly2+5+1=8employeesatCompanyKX.Instead,wecouldhaveanymultipleofthose:

8drivers,20onpublictransit,and4walkers14drivers,35onpublictransit,and7walkers26drivers,65onpublictransit,and13walkers

Also,thistripleratiodoesn’tgiveusjustonefractionbutseveral: ,,and ,forexample.There’salotofinformationinthatratio!

Mathematically,aratiobetweentwoquantitiesissimplyafraction:whetherwewrite3:5or ,we’reconveyingthesamebasicmathematicalinformation.Thegoodnewsisthatifyoulearnalltherulesfo