19TH EDmON sn.oon \Vt:t1tl Gt ...n. \!A-" Wt It WOW.f'h D fullr 10.,_ t.I.W Gil( a tfte.o n\ollltllf, 0\oant.Q'J\('- ..... ...,..... a Evctl)'lr>"C )Oilto ebou' IN -""'lei ,....,_tM_ IM'SbO" !)l)H 8Sc::rllttl U.. comMCI ..,IOOI.Iee 0t11t 01..0 newIOfel)d' or lhe ,_ 31(,') 1(,') ivNewGRE DiscreteQuantitative Questions Testing 'hctics PracticeExercises AnswerKeyAnswerExplanations Quantitative Comparison 169 170 181183 183 Questions187 Testing Tactics189 PracticeExercises206 AnswerKey 207 AnswerExplanations207 DataInterpretation Questions213 Testing Tactics PracticeExercisesAnswerKeyAnswer Explanations 214 218 223 223 Mathematics Review Arithmetic Algebra Geometry PART5:ModelTests Model Tests ModelTest1 AnswerKey AnswerExplanations ModelTest 2 AnswerKey AnswerExplanations 227 227 310 347 441 445 469 470 487 511 512 Acknowledgments - - - - - - - - -he authors gratefully acknowledge the following for permission toreprint read-ing passagesinthebook or on the CD. Book Passages Page34:FromClassic Authors oftheGilded Age by Darrel Abel.Copyright 1963 byBarron'sEducationalSeries. Page37:Excerptfrompp.140-1fromTheIndianinAmerica(NewAmerican NationSeries)byWilcombE.Washburn.Copyright1975byWilcombE. Washburn.Reprinted withpermission of HarperCollins Publishers. Page38:Fromthe NationalBiologicalInformation Infrastructure. Page107:From "SoMany FemaleRivals"by Christine Froula,The NewYorkTimes BookReview,February 7,1988. Page108:LadislasSegy,''AfricanSculpture Speaks,"Dover Publications,New York, 1958.Bypermissionof Helena Segy. Page450:JohnHopeFranklin,FromSlaverytoFreedom:AHistoryof Negro Americans,Alfred A.Knopf,Inc.,1947. Page454:Showalter,Elaine;ALiterature ofTheir Own.1977 PrincetonUniversity Press,2005renewed PUP,1999 exp.Paperback edition.Reprinted with permission of PrincetonUniversity Press. Page462:FromBlackLeadersof theTwentiethCentury.Copyright1982bythe BoardofTrusteesoftheUniversityofIllinois.Usedwithpermissionofthe University of IllinoisPress. Page492:From[VanGhent]EnglishNovel,Formand FunctionIE.Copyright 1953. Heinle/Arts &Sciences,a part of CengageLearning,Inc.Reproduced by per-mission.www. cengage. com/permissions Page495:FromVol.15of Americas,Copyright1963bytheOrganizationof AmericanStates. Page502:FromLiteraryWomen:TheGreatWritersbyEllenMoers.Copyright 1976,1977 by Ellen Moers.Used by permission of Doubleday, a division of Bantam Doubleday DellPublishing Group Inc. Page505:FromEyesonthe Prize:Civil RightsYears,editedby Clayborne Carsonet al.Copyright 1987 by PenguinBooks.Blackside,Inc.1991, withpermission. - -viNewGRE CD Passages From Black History and the Historical Profession,1915-1980. Copyright 1986,by the Boardof Trusteesof theUniversityof Illinois.Usedwithpermissionof theauthor and theUniversity of IllinoisPress. From India and Democracy by Sir George Schuster and Guy Wint, P10.Copyright 1941by Macmillan and Co. From"JamesLindandtheCureof Scurvy''byR.E.Hughes,courtesyof the National Center forBiotechnology Information. FromRodents:Their Lives and Habits by Peter WHanney,pp.103-4. Copyright 1975by Taplinger Publishing Company. FromThe Madwomaninthe Attic by Sandra M.Gilbert and SusanGubar,1979 by YaleUniversity Press,publisher. From"WhatAreGreenhouseGases?"bytheU.S.EnergyInformation Administration. Preface - - - - - - - - -saprospectivegraduatestudentconcernedwithprofessionaladvancement, you know theimportance of using good toolsand drawing on solid research. InthisNineteenth Edition of Barron'sGRE,weoffer youboth. Thisrevisioncontains the fruitsof our dose study of themajor changeseffective August1,2011,totheGREGeneralTestannouncedbytheEducationalTesting Service(ETS).WehavescrutinizedhundredsofactualGREquestions,traced dozensof GREreadingpassagestotheirsources,analyzedsubsetsof questionsby order of difficulty and question type. We have gone through allthe topics in the new analyticalwriting section,categorizing the actualissuesyou willencounter on your testandanalyzingtheargumentpassages,pinpointingtheirlogicalflaws.Inthe process,wehavecomeupwiththefollowingfeatures,whichshouldmakethis Nineteenth Edition particularly helpfulto you: TypicalGRE Questions Analyzed TheNineteenthEditiontakesyoustepbystepthroughmorethan1,000practice verbal and mathematical questions that simulate actual GRE questions, showing you how tosolvethem and how toavoid going wrong. Testing Tactics TheNineteenthEditionprovidesyouwithdozensof proven,highlightedtesting tacticsthat willhelp youattack the different typesof questionson the GRE. Comprehensive Mathematics Review The Nineteenth Edition presents you with extensivemathematicalreview of allthe topicsthatyouneedtoknow.Thisisespeciallyvaluableforcollegestudentsand adultswhohaven't takenmath since high school. GRE-Modeled Tests The NineteenthEditionoffersyouaDiagnostic TestgearedtothecurrentGRE,a testthat willenable youto pinpoint your areasof weaknessright away and concen-trate your review on subjects in which you need the most work, plus two Model Tests, allwithanswerscompletelyexplained,that informat,difficulty,andcontentecho today'sGRE. Twoadditional testsareon the accompanying CD-ROM (optional). - -viiiNewGRE Computer GREUpdate The NineteenthEditionintroducesyoutothelatest versionof thecomputer-based GRE-and, along with the accompanying CD-ROM (optional),explainseverything youneedtoknow about how totakethe computerized GRE. Analytical WritingUpdate The Nineteenth Edition alsoprovides you with an introduction tothe GRE analyt-icalwritingsection,familiarizingyouwiththerangeof topicscoveredandgiving youhelpfulhints on how to writeclear,cogent essaysin no time at all. This Nineteenth Edition once more upgrades what has long been a standard text. It reflectsthe contributions of numerous teachers, editors, and coaches, and the ded-ication of the staff at Barron's. It alsoreflects the forensic and rhetorical skillsof Lexy Green,Director of Debate at the College Preparatory School,to whom we owe spe-cialthanks. We,the authors,areindebted toalltheseindividualsfortheir ongoing effortstomakethisbook America'soutstanding GRE study guide. Timetable for a TypicalComputer-Based Graduate RecordExamination Total Time: 4hours /s. ect1onTime AllowedDescription 160minutesAnalytical Writing Essay1: Givingone'sperspective onanissue Essay 2:Analyzinganargument (30 minutes each) 1-minute break 230minutesVerbal Ability 6 text completionquestions 5 sentence equivalence questions 9reading comprehensionquestions 1-minute break 335minutesQuantitative Ability 8 quantitative comparisonquestions 9 discrete quantitative questions 3 data interpretationquestions 1 0-minute break 430minutesVerbal Ability 6 text completionquestions 5 sentence equivalencequestions 9readingcomprehension questions 1-minute break 535minutesQuantitative Ability 7 quantitative comparison questions 10 discrete quantitative questions 3 data interpretation questions 1-minute break 630 or 35minutesExperimental Section a third verbalor quantitative section NOTE:Sections2 through6 cancomeinany order-for example,Section 2 couldbe a Quantitative Ability sectionand theExperimentalSectioncouldbe any sectionexcept Section1. Althoughthe ExperimentalSection willnot count inyour score,it willlook identical to oneof the other sections-you won't know whichsectionit is,so youmust do your best onevery section of thetest. Prefaceix PART1 INTRODUCTION I DIAGNOSTICTEST you can find all NEW GRE books in pdf ETS revised GRE, Kaplan, Barron's, Princeton here:http://gre-download.blogspot.comWhatYouNeedtoKnow AbouttheGRE ANOVERVIEW OF THE COMPUTER-BASED GRE GENERAL TEST heGREGeneralTestisanexaminationdesignedbytheEducational TestingService(ETS)tomeasuretheverbal,quantitative,andanalytical writingskillsyouhavedevelopedinthecourseof youracademiccareer.High GRE scores strongly correlate with the probability of successin graduate school: thehigher you score,the morelikely you aretocomplete your graduate degree. Forthisreason,manygraduateandprofessionalschoolsrequireapplicantsto takethe GRE General Test,a test now givenonly on computer.(They may also requireyoutotakeaGRESubjectTestinyourparticularfield.Subject Tests currently areavailablein14fields.) Thecomputer-basedGREGeneralTestyoutakewillhavefiveor sixsections. There willalwaysbe one Analytical Writing section composed of two30-minute tasks(60 minutes) two 20-question Verbal Ability sections(30minutes each) two20-question Quantitative Ability sections(35minutes each) In addition,theremay be anunidentifiedExperimentalSection,whichwouldbeathirdverbalor quantitative section Occasionally,theremay be anidentifiedoptionalresearchsection(butnotif thereisanExperimental Section) The verbal section measures your ability to use words astoolsin reasoning; you are tested not only on the extent of your vocabulary but on your ability to discern the rela-tionshipsthatexistbothwithinwrittenpassagesandamongindividualgroupsof words. The quantitative section measures your ability to use and reason with numbers and mathematical concepts; you aretested not on advanced mathematical theory but ongeneralconceptsexpectedtobepartof everyone'sacademicbackground.The mathematics covered should be familiar to most students who took at leasttwoyears of math in a high school in the United States. The writing section measures your abil-ity to makerational assessmentsabout unfamiliar,fictitiousrelationships and tologi-cally present your perspective on an issue. CHAPTER1 4NewGRE COMMONLY ASKEDQUESTIONS ABOUT THE COMPUTER-BASED GRE How Does the GREDiffer fromOther Tests? Most tests college students take are straightforward achievement tests.They attempt to findout how much you havelearned,usually in a specific subject,and how well youcanapplythatinformation.Withoutemphasizingmemorizeddata,theGRE GeneralTestattemptstomeasureverbal,quantitative,andanalyticalwriting skills that you haveacquired overthe yearsboth in and out of school. Although the ETS claims that the GRE General Test measures skills that you have ?ver a long period, even a brief period of intensive study can make a great differencemyoureventualGREscores.Bythoroughlyfamiliarizingyourself with processof computer-based 'testing,the GRE testformat,andthe variousques-tiontypes,youcanenhanceyourchancesof doingwellonthetestandof being accepted by the graduate school of your choice. What IsIt Like to Take a Computer-Based GRE? If !oupurchasedtheversionof thisbookthatcomeswithaCD-ROM,thenby usmg_ that C_D,!ou can familiarize yourself with the icons that appear on the screen, practicearound the screen,and taketwoModel Tests,either inpractice or test-taking mode. Whether or not your versionof the book came with the optionalCD-ROM,youcangototheETS'sofficialGREwebsite- downloadtheirfreePOWERPRpII software,which mcludesa test preview tool and a practice test. When actually takethe GRE, you sit in a carrelin a computer labor testing center, a computer screen. Youmay be alone in the room,or other test-takers may be taking tests m nearby carrelw h1" . .s.Ityour mouse, you c Ick on an Icon to start hour test.section of the test isthe Analytical Writing section,and you will . ahved60h mmutesmwhichtocompletethetwowriting tasks.When you havefin-Isetet"ll bwn mgsection,youWI haveaone-minutebreaktotakeafewdeep

forthe next four or fivesections,each of which will consist of m.e-cOice verbal or quantitative questions. When the break isoverthe first Section _2 appears on the screen. Youanswer it, clicking on thenext NYAswerandthen,readytomoveon,youclickontheboxmarked ext.new question appears ondh screen,anyou gotroughthe processagainBe sure to answer every questionBeh.. theGREGal};.cause tere Isno penalty for an incorrect answer on uessb.est,when_ youdon'tknow ananswer,try tomakeaneducated a ydclearlyIncorrectchoices;if youcan'teliminateanychoices, WI guess,anmove on. At the end of the second. finishingtheth"d.section,you are giVenanother one-minute break. After Ifsection,youhaveatenm.bakTh. one-minute breaks-aftthch- mutereerewillbetwomore ere rourtand fifth sections. Why Do Some People Call the Computer-Based General Test aCAT? WhatYouNeedtoKnow About theGRE5 CATstandsforComputer-Adaptive Test.What doesthismean?It meansthatthe test adaptstoyour skilllevel:it iscustomized. What happens isthat after you complete the first quantitative or verbal section, the computerprogramassessesyourperformanceandadjuststhedifficultylevelof the questions you will have to answer in the second quantitative or verbal section. The morequestions you answer correctly in the firstsection,the harder willbethe ques-tions that you willbe given in the second section.However,the harder the questions are,themoretheyareworth.Soyourrawscoredependsonboththenumberof questions you answer correctly and the difficulty levelof those questions. Actually,theGREismuch lesscomputer-adaptivethanit usedtobe.It usedto adaptthe levelof questions youreceivedcontinuously;afterevery questionthe pro-gram would assessyour performance and determine the levelof difficulty of the next question.Now, it doesn't make that determination until you have completed an entire section. Can I TellHow WellI'm Doing on the Test fromthe Questions the Computer Assigns Me? Don't eventry;it neverpaystotry tosecond-guess the computer. There'snopoint in wasting time and energy wondering whether it'sfeeding you harder questionsor easierones.Letthe computer keeptrack of how wellyou'redoing- youconcen-trate on answering correctly asmany questions asyou can and on pacing yourself ShouldI Guess? Yes,youmust!Youarenot going toknow the correct answertoeveryquestionon theGRE. That's a given.But you shouldnever skip a question.Remember,there is no penalty foran incorrect answer.Soif a question has you stumped, eliminate any obviously incorrect answer choices,and then guessand don't worry whether you've guessedright or wrong.Your job isto get to the next question youcananswer.Just remember tousethe processof elimination to improve your guessing odds. How CanI Determine the Unidentified Experimental Section? Youcan't.Donot wasteevenone secondintheexamroomtryingtoidentifythe Experimental Section. Simply do your best on every section. Some people claim that mostoftenthe lastsectionistheExperimentalSection.Othersclaimthatthe sec-tionwithunusualquestionsisthe one that doesnot count.Ignore the claims:you haveno sure way to tell.If you encounter a seriesof questionsthat seem strange to you,do yourbest.Either theseareexperimentaland willnot count,in whichcase youhavenoreasontoworryaboutthem,orthey willcount,inwhichcasethey probably will seem just asstrange and troublesometo your fellow examinees. 6NewGRE TIPHow AreGREScores Calculated and WhenAre They Reported? After taking one of the Model Tests inthe back of thisbook and/or on the optional CD-ROM,it isimpossible to calculate exact scores,because there isno way to factor inthe difficulty levelof the ques-tions. To giveyourself aroughidea of how youdid,onboth the verbaland quantitative sections, assume that your raw score isequal to the number of cor-rect answers,and that your scaled score is equal to 130 plus your raw score.For example, if youansweredcorrectly 30 of the 40 quantitative questions, assume that your raw score wouldbe 30 and that your scaled score wouldbe160. On both the verbal and quantitative sections of the GRE, yourraw score isthe number of questions you answered correctly, adjusted for thedifficultylevelofthosequestions.Eachrawscoreisthen adjusted to a scaled score,which lies between 130 and 170. The writ-ten scorereportthat you willreceivein the mailwillincludeboth your scaledscoresand yourpercentilerank indicatingthepercent of examinees scoring below your scaled scoreson the General Test. Youranalyticalwritingscorewillbetheaverageof thescores assignedtoyouressaysbytwotrainedreaders.Thesescoresare rounded up to the nearest half-point. Your combined analytical writ-ing scorecan vary from0 to 6, with 6 the highest scorepossible. Assoon asyou have finishedtaking the test,the computer will calculate yourunofficial scaled scoresforthe verbaland quantita-tive sectionsand display them to you on the screen.Because your essaysaresent to for holistic scoring, you will not receive a score for the analytical writ-mg onthedayof thetest.Youshouldreceivein themailanofficial report contammg allthree scoresapproximately three weeksafter the test date. NOTE For allof the multiple-choice questions in the verbaland quantitative sections of the tests andpractice exercises in this book, the answer choices are labeled A,B, C,D,andE,and these letters are usedin the Answer Keysand the answer explanations. On an actual GREexam, these letters never appear on the screen. Rather, each choice ispreceded by ablank oval or square, and you willanswer a question by clicking withthe mouse on the ovalor square infront of your choice. GRE TEST FORMAT VerbalReasoning The. tWo verbal s.ections consist of a total of 40 questions. These questions mtobaste types:sentence completion questions and critical read-mg questions. Here ishow a 20-question verbal section generally breaks down: 10 sentence completion questions 10 critical reading questions(including logicalreasoning questions) Althoughtheamountof timespentoneachtypeof questionvaries from. persontoperson,ingeneral,sentencecompletionquestionstake lesstime toanswer than critical reading questions. SENTENCE COMPLETION QUESTIONS In sentence completionkd questions, you are aseto choose the best way to a sen.tence or short passage from which one, two,or three words avheomitted. Thesequestionstestacombinationof readingcom-preenston and vocabularykillY4 bb styleandtfthss.ou muste aleto recognize the logic, 'oneoesentencesothillbbl makesatyouweaetochoosetheanswerthat c:::vYo_u musftals?befi able to recognize differencesinusage.The anety otoptcsroma number of acad.fildsThd not, however, test specificcad.kn 1 effilc1e.eyo are familiar with the toi athemtcov:may feel more comfortable if you die any of the sentence! e sentenknceIsldidscussmg,but you should be able to han-H ..yourowege of the English 1 eretsatypicalsentencecom1..anguage. .p enon questionusinfth complenon formatsIn tht'sq.ke'gone oenewsentence Uestlon,you are asdto !'! _ _.una not one but two correct WhatYouNeedtoKnow About theGRE7 answers;both answersmust produce completed sentencesthat are like each other in meaning. This iswhat the test-makers calla sentence equivalent question. Select the two answer choices that, when used to complete the sentence, fit the meaning of the sentence as a whole and produce completed sentences that are alike in meaning.Although the two mismatched roommates are the proverbial odd couple-Felix ispedantic where Oscar is imprecise,where Oscar is slovenly,cuttivated where Oscar is uncouth-they nevertheless manage to share a smallapartmentdriving each other crazy. 0fastidiousOebullient Onice Ostoical Oegregious Click on your choices. Unlike Oscar,Felix isnot slovenly (messy and untidy); instead,he isa compulsive neatnik.Felixisfastidiousorniceinhishabits,excessivelysensitiveinmattersof taste.(Note the useof nice in a secondary sense.) Look at the same question,restructured into what the test-makers call a text com-pletionquestion.Inthistypeof question,youareaskedtofindonly onecorrect answer per blank.However, you must have a correct answer for each and every blank. For each blank select one entry fromthe corresponding column of choices. Fill all blanks in the waythat best completes the text.Although the two (i) roommates are the proverbial odd couple-Felix ispedantic where Oscar is imprecise, (ii) where Oscar is slovenly, cultivatedwhere Oscar is (iii) -they nevertheless manage to share a small apartmentwithout driving each other crazy.Blank(i)Blank(ii)Blank(iii) compatible curiousrefinedperipheral unkempt mismatched fastidious uncouth Click on your choices. 8NewGRE Seepage56forsentencecompletionquestiontacticsandpracticeexercisesthat willhelpyouhandle both of thenew sentence completionquestiontypes. CRITICAL READINGQUESTIONS Criticalreadingquestionstestyourabilitytounderstandandinterpretwhatyou read.Thisisprobablythemostimportantabilitythatyouwillneedingraduate schooland afterward. Althoughthepassagesmayencompassanysubjectmatter,youdonotneedto knowanythingaboutthesubjectdiscussedinthepassageinordertoanswerthe questionsonthat passage.The purpose of the questionistotest your reading abil-ity,not your knowledge of history,science,literature,or art. Both societies are territorial:they occupya particular home range,which they defend against intruders.Ukewise, both are cooperative:(40) members organize themselves intoworking groups that observe a clearly-defined dMsion of labor. Inaddftion, members of both groupscan conveyto each other a range of (45) basic emotions and personal information:animosfty, fright, hunger, rank within a particular caste, and abilftyto reproduce.Wilson readily concedes that, froma specialist's perspective, such a likeness(50) meyatfirstappearsuperficial, evenunscientifically glib.Nonetheless, in this eminent judgment, "His out of such deliberateoversimplification that the beginnings of a general theory are made.Which of the following statements best describes theorganization of the discussion of the importanceof the termtte/macaque comparison in the developmentof a unffied science of sociobiology?0He provides an example of a comparison and thenrejects fts implications.0He concedes that cunrent data are insufficient andmodffies his inttial assertion of their importance.0He acknowiedges hypothetical objections to thecomparison, but concludes by reaffirming ttssignificance.0He cttes critical appraisals ofthe comparison, butrefrains from making an appraisal of his own..0He notes an ambigufty in the comparison, but finallyconcedes fts validfty.Click on your choice.Theline.s arethepassage'sfinalsentences.Doesthe author acknowledge tothecomparison?Definitely.Doestheauthor concludeby reaffirmmg the szgnificance of thetermite/macaque comparison?Clearly hedoeshe .by quotinghecallsan eminent scholar),in doing so ngsup?ort to assertionthat suchoversimplified comparisonscan providethebasisforanImptalhTh h .or ant generteory.ecorrectansweristhethird cOice. The New GRE contains both 1 dc1c.1.ami Iar anunrami Iar question typesSome of the unramttarquestions11 al. c."cald"

veogtcreasonmgandargumentanalysis.Seepage82 cor cntireamg tacticsthat willhIhdl tionsIn add"It.IonChe p youane thenew logicalreasoning ques-.'seeapter 5 for add"t"ald. prepare you forthe remaindf th! :onta:ucs an. practice exercises that will er 0 e cnucal readmg portions of the test. WhatYouNeedtoKnow About theGRE9 Quantitative Ability The quantitative part of the GRE consistsof twomath sections,each with 20 ques-tions.Of the 40 questions,thereare 15quantitative comparison questions-? or 8 per section; 19 discrete quantitative questions, consisting of about 11multiple-choice ques-tions,4multiple-answerquestions,and4numericentryquestions,approxi-mately evenly split betweenthe two sections; 6 data interpretation questions-3 per section-all of which are discrete quan-titativequestions,mostly multiple-choice. In order toanswerthesequestions,youneedtoknow arithmetic,some very ele-mentary algebra,and a little geometry.Much of this material you learned in elemen-tary and middle school; the rest you learned during the first two years of high school. You do not need to know any advanced mathematics. The questions areintended to determine if you have a basic knowledge of elementary mathematics, and if you have theability toreasonclearly. If you haven'tdoneany mathematicsina while,gothroughthemathreviewin thisbook before attempting the Model Tests,and certainly before registering to take theGRE.If youfeelthatyourmathskillsarestillprettygood,youcantrythe Diagnostic Testfirst,and thenreadonly thosesectionsof themathreviewrelating tothosetopicsthat gaveyoutrouble. QUANTITATIVECOMPARISONQUESTIONS Of the 40 mathematics questions on the GRE,15are what isknown asquantitative comparisons.Unlessyouprepared fortheSATbefore2005,it isverypossiblethat youhaveneverevenseensucha question.Evenif youhavehad some contact with thistypeof question,youneedtoreviewthebasicideaand learntheessentialtac-ticsforanswering them. Therefore,readthese instructionsvery carefully. In thesequestionsthere are two quantities-Quantity A and Quantity B-and it isyour job to compare them.For these problems there areonly four possible answers: Quantity Aisgreater; Quantity B isgreater; The twoquantities areequal;and It isimpossible todetermine whichquantity isgreater. Inthisbook,thesefouranswerchoiceswillbereferredtoasA,B,C,andD, respectively.In some of the questions,information about the quantities being com-parediscenteredabovethem.Thisinformationmust betakenintoconsideration when comparing the twoquantities. 10NewGRE InChapter 9 you willlearn severalimportant strategies forhandling quantitative comparisons.For now,let's look at three examples to make sure that you understand the concepts involved. Quantity A (3+ 4)2 Quantity B 32+ 42 Evaluate each quantity:(3+ 4)2 =72 =49, whereas 32 + 42 =9+ 16=25. Since 49 > 25,Quantity A isgreater.The answer is A. Quantity A The average(arithmetic mean)of a andb a+b =16 Quantity B 8 Quantity A isthe averageof a andb:a ;b.Since wearetoldthat a + b =16, Q Aa+b16 uanttty1s -2- = 2 =8. So,Quantity A and Quantity B areequal. The answer isC. N?TE: We cannot determine the value of either a or b;all weknow isthat their sum 1s 1,6.Perhapsa= 10 andb =6, or a= 0 and b =16, or a=-4 and b = 20. It doesnt matter. The averageof 10 and 6 is8;the averageof 0 and168dh avf4d 20 1s, ante erage 0 - anIS 8.Since a+b is16, the average of a and b is8,all the time, nomatter what. The answer,therefore,isC. Quantity A a3 Quantity B If a =1, a3=1and a2- 1lh.th equal.'- n tts case,e quantities in the two columns are This means that theh' Th answer to tIsproblemcannot be A or BWh~ e answer can be A (or B)1'fQ.Y it isn't- not when a=1.on Y 1 uanttty A (or B)is greater all the time.But So,isthe answer C?Maybe.But forthe have to be equalall the ti. Arth;> answerto beC,the quantities would ,.mt.eey. WhatYouNeedtoKnow About theGRE11 No. If a =2, a3 =8, and a2 =4, and in this case the two quantities arenot equal The answer,therefore,isD. DISCRETE QUANTITATIVEQUESTIONS Of the40mathematicsquestionsontheGRE,19arewhattheETScallsdiscrete quantitative questions.More than half of those questions are standard multiple-choice questions, for which there arefiveanswer choices, exactly one of which iscorrect. The way to answer such a question isto do the necessary work, getthe solution, and then look atthefivechoicestofindyour answer.InChapter 8 wewilldiscussother tech-niquesforanswering these questions,but fornow let'slook at one example. EdisonHighSchoolhas 840 students,and theratio of thenumber of students takingSpanish to the number not takingSpanish is 4:3. How many of the students take Spanish? 280360480@560630 Tosolvethisproblemrequiresonly that youunderstand what a ratiois.Ignorethe factthat thisisa multiple-choice question.Don't evenlook at the choices. Let4xand3x bethenumberof studentstakingandnottakingSpanish, respectively. Then 4x + 3x =840::::} 7x =840::::}x =120. The number of students taking Spanish is4X120=480. Havingfoundtheanswertobe480,nowlookat the fivechoices.Theanswer isC. A secondtypeof discrete quantitative questionthat appearson the GREis what theETScallsa"multiple-choicequestion-morethanoneanswerpossible,"and whatforsimplicitywecallamultiple-answer question.Inthistypeof question there could beasmany as12choices,althoughusually there areno more than7 or 8.Any number of the answer choices,from just one to allof them, could be correct. Togetcredit forsucha question,youmustselectall of thecorrect answerchoices andnone of the incorrect ones.Here isa typical example. If x isnegative,whichof the followingstatementsmust be true? Indicate a// suchstatements. 0x2"isusedtoindicatethat one step mthe of aproblemfollowsimmediately fromtheprecedingone,andno explanationIS necessary.Youshould read as or or 3x=12=>X= 4 3x =12implies that x =4 3x = 12,which impliesthat x= 4 since 3x =12,then x =4. Hereisa sample solution tothefollowingproblemusing =>: What isthe valueof 2x2 - 5 when x= -4? x= -4 =>x2= (-4)2=16=>2x2 =2(16)=32=>2x2- 5 =32- 5 =27 When the reasonfora step isnotb----. d ....ovtous,___, Isnot use: rather,an explanation IS giVen,oftenmcludmgareferencetoaKEYFACTfiromCh11I 1apter.nmany tsho stepslareexplained,while others arelinked by the=>symbol,asin e roowmg examp e. IntroductiontoPart4145 Inthediagrambelow,if w=10, what isthe valueof z? ByKEYFACT J1,w+X+ y=180. Since isisosceles,x=y(KEY FACT ]5). Therefore,w+ 2y=180=>10+2y=180=>2y=170=> y= 85. Finally,since y+z = 180(KEYFACT 13),85+z = 180=>z=95. CALCULATORSON THE GRE You maynot bring yourowncalculatortousewhenyoutaketheGRE.However, startingin2011,forthe firsttime ever,you willhaveaccessto anonscreencalcula-tor.While you areworking on the math sections,one of the iconsat thetop of the screenwillbeacalculatoricon.During theverbaland writing sectionsof thetest, either that icon willbe greyed out (meaning that you can't dick on it)or it will sim-plynotbethereat all.During the math sections,however,you willbeabletoclick onthaticonatanytime;whenyoudo,acalculatorwillinstantlyappearonthe screen.Clicking the X inthe upper-right-hand corner of thecalculator willhideit. ...... Notethatwhenthecalculatorappearsonthescreen,itmay coverpartof thequestionortheanswerchoices.If thisoccurs, just click on the top of the calculator and drag it toa convenient location.If youusethe calculator to answer a question and then click NEXT to goto the next question,the calculator remains on the screen,exactly where it was, with the same numerical readout. Thisisactuallyadistraction.So,if youdousethecalculatorto answeraquestion,assoonasyouhaveansweredthatquestion, click onthe Xtoremovethe calculator fromthe screen.Later,it takesonly one click to get it back. Io.l

00080 G000J . Theonscreencalculatorisasimplefour-functioncalculator, a squarerootkey.Itisnot a graphing calculator;it isnot a Scientificcalculator.Theonlyoperationsyoucanperformwith onscreencalculatorareadding,subtracting,multiplying, dtvtding,andtaking squareroots.Fortunately,thesearethe only operationsyou willeverneed to answer any GRE question. 000GG (Transfer Display ) At thebottom of the onscreencalculator isa bar labeled TRANSFER DISPLAY. If youareusingthecalculator on anumeric entry question,andtheresultof your finalcalculationisthe answerthat you want toenter inthebox, onDISPLAY-the numbercurrentlydisplayedinthecalculatorsreadoutwtll Instantlyappearintheboxunderthequestion.Thissavesthefewsecondsthatit wouldotherwisetaketo enter your answer;moreimportant,it guaranteesthat you Won'tmak.. e an error typmg myour answer. Just because you have a calculator at your disposal does not mean that you should USe it very much.In fact,you shouldn't. The vastmajority of questionsthat appear onthe GRE do not require any calculations. Remember Use your calculator only when you need to. you can find all NEW GRE books in pdf ETS revised GRE, Kaplan, Barron's, Princeton here:http://gre-download.blogspot.comI l i! 'll I lj GeneralMath Strategies - - - - - - -n Chapters 8 and 9,you willlearn tactics that arespecifically applicable to discrete quantitative questions and quantitative comparison questions,respectively.In this chapter you willlearn severalimportant generalmath strategiesthat canbeusedon bothof thesetypesof questions. The directionsthat appear on the screen atthe beginning of the quantitative sec-tionsincludethe followingcautionary information: Figuresthataccompany questionsareintendedtoprovideinformationuseful in answeringthe questions. However,unlessanote statesthat afigureisdrawntoscale,youshouldsolve these problems NOT by estimating sizes by sight or measurement, but by using your knowledgeof mathematics. Despitethe factthat they aretelling you that you cannot totally rely ontheir dia-grams,if you learnhow to draw diagrams accurately, you cantrust the ones you draw. Knowingthebestwaysof handlingdiagramsontheGREiscriticallyimportant. Consequently,thefirstfivetacticsalldeal with diagrams. TACTIC 1. TACTIC 2. TACTIC 3. TACTIC4. TACTIC 5. Draw a diagram. Trust a diagram that has.been clrawn to scale. Exaggerate or change a diagram. Add a line to a diagram. Subtract to find shaded regions. . Toimplement these tactics,you needtobe ableto draw line segments and angles accurately,and youneedtobeabletolook atsegmentsandanglesandaccurately estimatetheir measures.Let'slook atthree variationsof the sameproblem. 1.If thediagonalof arectangleistwiceaslongastheshorterside,whatisthe degreemeasure of the angleit makes with the longer side? 2.Inthe rectangle below,what isthe valueof x? ~ } CHAPTER - -147 148NewGRE 3.Inthe rectanglebelow,what isthe valueof x?

LL_j Forthemoment,let'signorethecorrectmathematical way of solvingthisprob-lem.Inthe diagramin(3),the side labeled2 appearstobehalf aslong asthe diag-onal, which islabeled 4;consequently, you should assume that the diagram has been drawnto scale,and you should seethat xis about 30,certainly between25and 35. In(1)youaren't givena diagram,and in(2)the diagramisuselessbecause youcan see that it hasnot been drawn to scale(the side labeled 2 isnearly as long asthe diag-onal, which islabeled 4).However,if while taking the GRE, you see a question such as(1)or (2), you should be able to quickly draw on your scrap paper a diagram that looks just liketheone in(3),and then look at your diagramand seethatthemeas-ureof xis just about 30.If the answer choices forthese questions were @153045@6075 you would,of course,choose 30, B.If the choices were @202530@3540 youmight not bequite asconfident,but you should stillchoose30,hereC. When you take the GRE, eventhough you arenot allowedtohaverulersor pro-tractors,youshould be ableto draw your diagrams very accurately.For example, in ( 1)above,you shoulddraw a horizontal line,and then,either freehandor by trac-ing the corner of a pieceof scrappaper,draw a right angleon the line. The vertical line segment willbethe width of therectangle;label it 2. __,, Mark off that distancetwiceonapieceof scrappaper andusethat todraw the diagonal. scrap paper GeneralMathStrategies149 Youshould now havea diagramthatissimilar tothat in(3),and you should be abletoseethatxis about 30. Bytheway,xisexactly 30.Arighttrianglein whichonelegishalf thehypotenuse must be a 30-60-90 triangle, and that leg is opposite the 30 angle[see KEY FACT ]11]. Havingdrawnanaccuratediagram,areyoustillunsureastohowyoushould know that the value of xis 30 just by looking at the diagram? Youwill now learn not only how to look at any angle and know itsmeasure within 5 or 10 degrees,but how todraw any anglethat accurately. Youshould easily recognize a 90 angle and canprobably draw one freehand;but youcanalwaysjust tracethecorner of apiece of scrappaper.Todraw a 45angle, justbisecta90angle.Again,youcanprobably dothisfreehand.If not,ortobe evenmoreaccurate,drawarightangle,markoff thesamedistanceoneachside, drawa square,and then draw in the diagonal. / / / /45 L / / / / / / / Todraw other acuteangles,just dividethetwo45anglesinthe abovediagram withasmany linesasnecessary. 750 600II II 45 II30oII/ II// II//.15o II/ II/_....-..-----1 I//_..I,..... _...- 80 7060 I/150 45o IIII40 I1I III1//30 1l1/// IIII////20 1 I I;///_.......-1 II;//------1I _- - 10

:::::-Finally,to draw an obtuse angle,add anacute angleto a right angle. N.fle 1ust draw in some lines. ow,to estimate the measure oa gtvenang' estimate:65 I I \C\ estimate:110estimate:150 150NewGRE Totestyourself,findthemeasureof eachangleshown.Theanswersarefound below. (a)(b)(c)(d) q:JeJUO oOIU!lJl!MJlUO::> noAP!O'o09I(p)I(::>) oOl(q)o08(E)S.J;}MSUVTesting Tactics TACTIC DDraw a Diagram Onanyquestionforwhichafigureisnotprovided,drawone(asaccu-firatelydason your scrappaper- never attempt a geometry problemwithout rstrawmg a dtagram. isarea of a rectanglewhose lengthistwiceits widthand whose pen meter IS equal tothat of a squarewhoseareais1? 248 @16- @- -339 SOLUTION.Don't eventhinkfh' 0 answenngt1s questionuntil you havedrawn a and a rectangleand.labeled eachof them:each side of the suareis1'and if the wtdth of therectangleIsw,itslength(f)is2 w.q 2w P=4 P= 6w I 2w Now,writetherequired equation and solveit: =The area of the rectangle= ew= ( : )(= %, E. GeneralMathStrategies151 Bettydrove8miles west,6milesnorth,3miles east,and6 more milesnorth. Howmanymiles wasBetty fromher startingplace? c_________JI miles SOLUTION.Draw a diagram showing Betty'sroutefromA toB toCto D toE E= destination Now,extend line segment ED until it intersects AB at F.Then, AFE isa right trian-gle,whoselegsare5and12.The lengthof hypotenuseAE representsthedistance fromher starting point to her destination.Either recognizethat 6.AFE isa 5-12-13 righttriangleor usethe Pythagorean theorem: 52+122 = (AE)2(AE)2 = 25+144 = 169 =>AE = 13. Whatis thedifference inthedegree measures of theangles formedby the hour hand andtheminute hand of a clock at 12:35 and12:36? 1 5 5.5 @6 30SOLUTION. Draw a simple picture of a dock. The hour hand makes a complete rev-?lution, 360, once every 12 hours. So,in1 hour it goes through 360 +12 = 30o, and In oneminute it advancesthrough 30 +60 =0.5. The minute hand movesthrough 30aevery 5 minutes or 6per minute. So,inthe minute from12:35to12:36 (or any other minute),thedifference between the hands increasedby6o- 0.5o = 5.5o,C. 30 in1 hr.0.5 in1 min. 30 in 5 min. 6in1 min. 152New GRE NOTE:It wasnot necessary,and would havebeenmoretime-consuming,todeter-mine the anglebetweenthehandsat either12:35or12:36.(See TACTIC 6:Don't domorethan youhaveto.) Drawingsshouldnotbelimitedtogeometryquestions;therearemanyother questionson which drawingswillhelp. A jar contains10 redmarbles and30 greenones.How manyredmarbles must beadded to the jar so that 60%of the marbles willbered? SOLUTION.Letxrepresentthenumberof redmarblestobeaddedanddraw a diagram and labelit.' xRed 30Green 10 Red Fromthediagram_ it isclearthattherearenow 40+ xmarblesinthe jar,of which 10+xarered.Smcewewantthefractionof redmarblestobe60%,wehave 10+ X_ 0 603.. 40 + x- 60 Yo = 1 OO = S. Cross-multtplymg,weget: 5(10+ x)= 3(40 + x)=>50+ 5x=120+ 3x=> 2x= 70 =>x= 35. Of cou_rse,you could have set up the equation and solved it without the diagram, thedtagrammakesthe solution easierand youarelesslikely to make a careless mistake. TACTIC fl Trusta DiagramThatHasBeenDrawntoScale Whenever diagramshavebeen drawnto scale,they canbe trusted. This meansthat youcanlookatthediagramdf Id.anuseyoureyestoaccuratelyestimatethestzeso esan. lme segments.For example,inthefirstproblemdiscussedatthebegin-nmg of thtschapter,you could "see"that the measureof the angle wasabout 30. Totakeadvantage of this situation: If a diagramisgiventhat appearstobedrawnto scaletrust t"tIfd"' atagram IsgiVenthat hasnot been drawn to scale,try to draw it to scale on your scrappaper,and then trust it. Whennodiagramisprov"dddd d . 1 e,anyouraw one on yourscrappapertry to raw 1tto scale.' In5bela": wearetold that ABCD isa square and that dianalBD is 3.Inthe dtagram provtded, quadrilateral ABCD does indeed look like and General MathStrategies153 BD =3doesnot contradict any other information.Wecan,therefore,assumethat thediagramhasbeen drawnto scale. Inthefigureat theright,diagonalBD of square ABCD is3. Whatistheperimeter of the square? @4.5123J2@6J212J2

DC SOLUTION.Since this diagram hasbeen drawn to scale,you can trust it. The sides of thesquareappeartobeabout twothirdsaslong asthe diagonal,soassumethat eachsideisabout 2. Then the perimeter isabout 8. Which of the choicesisapprox-imately 8?Certainly not A or B.Since.J2""' 1.4,Choices C, D,and E areapprox-imately 4.2,8.4,and12.6,respectively.Clearly,the answermust be D. Direct mathematical solution.Lets bea side of the square. Then since !::.BCD is 33.fidh.fh. a 45-45-90righttriangle,s == .J2=-2-, antepenmeterotesquare1s4s" 4(3f) "6J2 .. Remember the goalof thisbook isto help you get credit forall the problems you know how todo,and,by using the TACTICS,toget credit formany that you don't know how to do.Example 5 istypical. Many students would miss this question.You, however,cannow answerit correctly,eventhough youmaynotrememberhowto solveit directly. In6.ABC,what is the value of x? @756045@3015 SOLUTION. If you don't see the correct mathematical solution, you should use TAC-!IC 2 and trust the diagram;but to dothat youbethat when youcopy lt onto your scrap paper you fix it.What's wrong wtth the way lt IS A drawnnow?AB == 8 andBC == 4,but in the figure,AB andBC are almost the same length. Redraw it sothat AB istwice aslong asBC Now,just look:xis about 60, B. Infact,xis exactly 60.If the hypotenuse of a rightis twicethe length of one of thelegs,thenit'sa 30-60-90 mangle, andtheangleformedbythe hypotenuse and that legis60(see 8 L.:,;...____._. Section11-J). 4 154NewGRE TACTIC2isequallyeffectiveonquantitativecomparisonquestionsthathave diagrams.Seepages9-11fordirectionsonhowtosolvequantitativecomparison questions. Quantity A AB A 1040 10Quantity B 10 Therearetwothingswrongwiththegivendiagram:LCis labeled 40'.but looksmuch more like60or 70,and AC andBC areeach labeled10,but BC IsdrawnmuchlongerWhenhd"b .youcopy tetagramontoyourscrappaper,e sure tocorrect these two mistakes:draw a triangle that hasa 40 angle and two sides of the same length. 10c400 10Now,it'sclear:AB CD Quantity A Quantity B 12.X y Quantity A Quantity B 13.The number of The number of odd positive even positive factorsof 30 factorsof 30 Questions14-15 refertothe following definition. {a,b}representsthe remainder when a isdividedby b. Quantity A 14.{103,3} Quantity B {lOS,5} 15. c and dare positive integers withc < d. Quantity AQuantity B {c,d} {dc}GeneralMathStrategies165 ANSWERKEY l.C6.E11.B 2.c7.c12.B 3.E8.2.513.c 4.1609.c14.A 5.D10.D15. A ANSWEREXPLANATIONS Twoasterisks(**)indicate an alternative method of solving. 1.(C)This isarelatively simple ratioproblem,but use TACTIC 7 and make sure you gettheunitsright. Todothis you need to know that there are100 centsin a dollar and12 inches in a foot. 3 dollars300 centsxcentspnce. weight8feet96 inches=16 inches Now cross-multiply and solve: 96x = 4800 =::}x = 50. 2.(C)Use TACTICS 2and 4.On your scrappaper,extend line segmentsOPandOR Q SquareOPQR,whose area is8,takesup most of quarter-circleOXY.Sothe area of the quarter-circle iscertainly between11and13. The areaof the whole circle is4timesasgreat:between 44 and 52.Check the fivechoices:they are approxi-mately 25,36,50,100,200. The answer isclearly C. **Another way touse TACTIC 4istodraw in line segmentOQ Sincethe area of the square is8,each sideis.J8, and diagonalOQ is .J8XJ2=.Jl6 = 4.ButOQ isalsoa radius,sothe area of thecircle isrc(4)2 =161t. 166New GRE 3.{E)Use TACTIC1:draw a picture representing a pile of books or abookshel 2xFrench 2000 XEnglish 1999 10English 7French Eng.Fr. Eng.Fr.I 10I 7 I X I 2xI 19992000 In the two yearsthe number of FrenchbooksDiana read was7+ 2x and thetotal bf 37 + 2x numer obooks was17 + 3x. Then 60% or- =To solvecross-multiply-517 +3x ' 5(7 + 2x)= 3(17 + 3x)~35+lOx= 51+ 9 x ~x=16. In2000,Diana read16 Englishbooksand 32 Frenchbooks,a total of 48books. 4.(D}.Use TACTIC 8.Systematically listthe numbersthat contain the digit1, wntmg ~many asyou need to seethe pattern.Between1 and 99the digit1 is used10times astheunits digit (1,11, 21,..., 91)and10times asthetensdigit (10,11,12,...,19)fora total of20 times.From 200 to 299,there are 20 more (the same 20 preceded by a 2).From100 to199 there are20 more plus 100 numbers where the digit1 isused in the hundreds place.Sothe total is 20 + 20 + 20+ 100=160. 5.~ 6 0Use_TACTIC2. Trust the diagram:AC, which isclearly longer thanOC, IS approximately aslong asradiusOE Therefore, AC must beabout10.Check the choices. They areapproximately 1.4, 3.1, 7,10,and14. The answer must be10. **The answeris10.Use TACTIC 4hd'd .. copy teIagramon your scrap paper an draw mdiagonalOB. Since the two diagonals of a rectangle areequaland diagonalOB .d's AC = OB = 10.'Isaram, General MathStrategies167 6.(E)Use TACTIC 5:subtract to findthe shaded area.The area of the square is4. The area of the equilateral triangle(seeSection11-J)is 22J3=4.J3=J3. 44 Sothearea of the shaded regionis4- J3 . 7.(C)Use TACTIC 6:don't do more than you haveto.In particular,don't solve forx. 5x +13= 31~5x =18~5x + 31= 18+ 31= 49 ~.Jsx+31=.J49= 7. 8.2.5Use TACTIC 6:don't do more than isnecessary.Wedon'tneedto know the valuesof a andb,only their average.Adding the twoequations, weget a+b5 6a + 6b = 30 ~a+ b = 5~-- =- = 2.5. 22 9.(C)Use TACTIC 5:to getDE,subtractOD fromradiusOE,which is4. Draw AO (TACTIC 4). B E Sincef:J.AOD isa30-60-90 right triangle,OD is2(one half of OA).So, DE= 4-2 =2. 10.(D)Use TACTIC 4and add some lines:connect the centersof thethree circlestoforman equilateral triangle whose sidesare2. Nowuse TACTIC 5 and findthe shaded areaby subtracting the area of the threesectorsfromthe area of the triangle. The area of the triangle is 22.J3 - = J3(seeSection11-J). 4 Eachsector isone sixth of a circle of radius1. Together they form one half of sucha circle,sotheir total area is-11t(1)2== ~ - Finally,subtract:the shaded area isJ3 1t 2 you can find all NEW GRE books in pdf ETS revised GRE, Kaplan, Barron's, Princeton here:http://gre-download.blogspot.com168NewGRE 11.(B)If youdon't seehow toanswerthis,use TACTIC 2:trust thediagram. Estimate themeasureof eachangle:forexample,a= 45,b = 70,c = 30, and d =120.Soc + d (150)isconsiderably greater thana+b (l15). ChooseB. **Infact,d by itself isequaltoa +b (anexterior angleof atriangleisequal to the sum of the opposite twointerior angles).Soc +d >a+b. 12.(B) the figure,it appearsthat x andy areequal,or nearly so.However, the gtveninformation statesthat BC >CD,but thisisnot clear fromthe dia- Use3:when you draw the figureon your scrap paper,exagger-atelt.Draw 1twuh BC much greater thanCD.Now it isclear that y isgreater. 13.(C) 8.Systematically list allthe factorsof 30,either individu-allyor mpatrS:1,30;2,15;3,10;5,6.Of the 8 factors,4areevenand 4 areodd. 14(Ah) Qua.ntity103 (1000)isdivided by 3,the quotient is333 and te remamder ts1Quantity B105d""blb ..tstvtstey 5,sothe remainder is0. Quanttty A tsgreater. 15.(A) A:c 18+w = 40 =>w = 22. DIRECT SOLUTION 2.Since5 is5 lessthan10,6 is4 lessthan10,and 7is3less than10,tocompensate,w must be 5 + 4+3=12morethan10. So,w = 10+ 12= 22. Judyisnow twice as oldas Adam,but 6 yearsago,she was 5 times asoldashe was.How oldis Judy now? 101620@24@32 SOLUTION. UseTACTIC1:backsolvestarting withC.If Judyisnow20,Adamis10,and6 ago,they would havebeen14 and 4.Since Judy would havebeenless 5 timesasold as Adam,eliminate c, D, and E,and try a smaller value.If Judy Isnow 16,Adamis8;6yearsago,they would havebeen10and 2. That'sit;10is5 times 2. The answerisB. (SeeSection11-H on word problems forthe correct algebraic solution.) Sometacticsallow youtoeliminatea fewchoicessoyoucanmakeanguess.On those problems where it can be used, TACTIC 1gets you the answer.The only reasonnot touseit on a particular problem IS that you caneastly solvetheproblem directly. TIPL_ __ Don't start withC if some of the other choices are much easier to work with. If youstart withB and it istoo small, youmay only get to eliminate two choices (AandB),instead of three,but it will save timeif plugging inChoice C would be messy. 172NewGRE If 3x = 2(5- 2x),thenx = @-100@1 77 SOLUTION. 10 7 Sincepluggingin0issomucheasierthanplugginginl, startwithB:thenthe 7 left-hand side of the equationis0and theright-hand sideis10. The left-hand side ismuch toosmall.Eliminate A and Band try something bigger- D,of course;it willbemuch easier todeal with1 than withlor!Q. Now the left-hand side is 3 77 andtheright-handsideis6.We'recloser,butnotthere.TheanswermustbeE.Noticethat wegotthe right answer without everplugging in one of thoseunpleas-ant fractions.Areyouuncomfortablechoosing Ewithout checkingit?Don'tbe.If youknowthattheanswerisgreaterthan1,andonly onechoiceisgreaterthan1,that choicehastoberight. Again,weemphasizethat,nomatter whatthechoicesare,youbacksolveonly if youcan'teasilydothealgebra.Moststudentswouldprobablydothisproblem directly: 3x =2(5- 2x)::::}3x =10- 4x::::}7x =10::::}x = 10 7 and savebacksolving fora harder problem.Youhavetodetermine whichmethod isbest foryou. TACTIC II ReplaceVariables withNumbers ofTACTIC 2 iscriticalforanyone developing goodtest-taking skills. This tacuccanbeusedwheneverthefivechoicesinvolvethevariablesinthequestion. There arethree steps: 1.Replaceeachletter with aneasy-to-usenumber. 2.Solvethe problem using thosenumbers. 3. eachof thefivechoiceswiththenumbersyoupickedtoseewhich chmce 1s equaltothe answer you obtained. Examples4and 5 illustrate the proper useofTACTIC 2. If aequal to the sum of b and c,whichof the followingisequal to the difference of b andc? a- b- ca- b + ca- c@a- 2ca- b- 2c DiscreteQuantitativeQuestions173 SOLUTION. Pickthreeeasy-to-usenumberswhichsatisfya=b +c:forexample,a=5, b = 3,c =2. Then,solvetheproblemwiththesenumbers:thedifferenceofbandc1s3-2= 1. Finally,check eachof thefivechoicestoseewhichoneisequalto1: @Doesa - b - c = 1? Doesa - b +c =1? Doesa - c =1? @Doesa - 2 c = 1? Doesa - b - 2 c = 1? TheanswerisD. NO.5- 3-2 =0 NO.5- 3+2=4 NO.5- 2=3 YES!5 - 2(2)= 5 - 4= 1 NO.5 - 3 - 2(2)= 2 - 4= -2 If thesumof fiveconsecutiveevenintegers ist, then,interms of t,whatis thegreatest of theseintegers? ""t - 20t - 10t""'t + 10"E't + 20 '01-5- -5- c5_g,-5- \5I5 SOLUTION. Pickfiveeasy-to-useconsecutive evenintegers:say,2,4,6,8,10. Thent,their sum,is30. Solvethe problem withthesenumbers:the greatest of theseintegersis10. .1020304050 Whent= 30,the fivechoteesare5' 5' 5' 5' 5 OnlySO,Choice E,isequal to10. 5 Of course,Examples 4and5 can be solved without using TACTIC 3ifyour alge-braskills are good.Here arethe solutions. SOLUTION 4.a = b +c::::}b = a_ c::::}b- c =(a- c) - c =a- 2c. SOLUTION 5Lt2n +4n +6andn +8 befiveconsecutive eveninte-.en,n +,,, gers,andlett betheir sum. Then, t=n +(n+2)+(n+4)+(n+6)+(n+8)=5n +20 2__..... n +8-- t- 20+8 =t- 20+40=t+ 20 So,n =t0---75555 Thf'dhalgebrayou can still use TACTIC e Important pomt 1s that 1you canto te'.f 2 andalways get the right answer.Of course, use 2 even 1you candothealgebraif youthink that byusing thistactic you w1llsolvetheproblem fasterorwillbe likelytomakeamistake.Thisisagoodexampleof whatwe mean when we say that with the proper use of these tactics, you can lllany questionsforwhich you may not know the correct mathematical soluuon.. 174NewGRE TIPL ----Replacetheletters withnumbers that are easy to use,not necessarily ones that make sense. It is perfectly OK to ignore reality.A school canhave5 students, apples can cost 1 0 dollars each,trains can go 5milesper hour or 1000 miles per hour- it doesn't matter. Examples6 and7 aresomewhat different.Youare askedtoreasonthrough word problemsinvolvingonlyvariables.Moststudentsfindproblemslikethesemind-boggling.Here,the useofTACTIC 2isessential.Without it,Example 6 isdifficult andExample7isnearlyimpossible.Thisisnotaneasytactictomaster,but with practice youwillcatchon. It a school cafeterianeedse cans of soup eachweek for eachstudent andif there ares studentsintheschool,for how many weeks will' x cans of souplast? esxxs e SOLUTION.

ex X @-es ex s Replacec, s, and x with three easy-to-usenumbers.If a school cafeteria needs 2 cansof soupeachweekforeachstudent,andif thereare5studentsinthe how many weekswill20cansof souplast? the cafeteria needs 2 X5 = 10 cans of soup per week, 20 cans will last 2 weeks. WhiChof the choicesequals2 whenc = 2,s = 5,and x= 20? - 200XS S1X csx- '- =50;- = -; - = 2;andex=8. cex8csThe answer is..::.,D. cs s NOTE: Youdonot need tohalf. h .get te exact vueoeachchoiCe.Assoon asyou seetatachoiCedoesnotequalthal1kih h .devueyouareoongfor,stop-ehmmate tat cOiceanmoveonForex1.h.. h amp e,mteprecedmgproblemitisclearthatcsx ISmucgreater than 2sor.. .d''. d .'e tmmate It tmmelately; you do not need to multiply It out toetermme that the valueis200. CAUTION Inthis typeof problemit t. ables by 1 s.

noa good Idea toreplaceany of the vari-.mcemultlplymg anddividing by1 give thesameresult, youwouldnot be able todistinguishbetweenexand..!!._ bothof ses' whichareequal to4 whene = 1,s = 5,andx = 20.It is also not a goodidea touse thesamenumber for different variablesexandxs areeachequal tox whene and 5 are equal.sc DiscreteQuantitativeQuestions175 A vendor sellshhot dogs ands sodas.If a hot dogcosts twiceasmuchasa soda, andifthe vendor takesina totalofd dollars,how many centsdoesa soda cost? 100ds+2hd(s+2h)@ 100d(s+ 2h)d s+2h100d100100(s+2h) SOLUTION. Replaceh,s, anddwiththreeeasy-to-usenumbers.Supposeasodacosts 50andahot dog $1.00. Then,if hesold2sodasand3hot dogs,hetook in 4 dollars. Whichof the choices equals50 whens = 2,h = 3,andd = 4? On!100d(A):100(4)=400=50. ys + 2h2 + 2(3)8 Now,practice TACTIC 3onthe followingproblems. Yannwillbex yearsoldy yearsfromnow.How oldwashez yearsago? x+y+zx+y-zx-y-z@y-x-zz-y-x SOLUTION. Assumethat Yannwillbe10in2years.How old washe3yearsago?If he willbe 10in2 years,heis8now and 3yearsagohewas5.Which of thechoicesequals5 whenx =10, y =2,andz=3?Only x- y- z,C. Standrove forh hours at a constant rateofr miles per hour.How many milesdidhego during thefinal20 minutes of hisdrive? 20rhr3rh@hr!_3203 SOLUTION. IfStandroveat60milesperhourfor2hours,howfardidhegointhelast 20 minutes?Since20minutesisi of anhour,hewent 20( i of 60)miles.Only 20whenr = 60andh = 2.Noticethathisirrelevant.Whether hehadbeen driving for 2 hours or 20 hours, the distance he covered in the last 20 minutes would be thesame. 176NewGRE TIPL_ __ In problems involving fractions,the best number to useisthe least common denominator of all the fractions.Inproblems involving percents,the easiest number to useis 1 00. (See Sections11-B and 11-C.) TACTIC II ChooseanAppropriateNumber TACTIC 3issimilar to TACTIC 2,in that wepick convenient numbers.However, hereno variableisgivenin the problem. TACTIC 3 isespeciallyusefulin problems involving fractions,ratios,and percents. AtHighSchooleachstudent studies exactly one foreignlanguage. Three-f1fthsof the students takeSpanish,and one-fourth of theremaining students takeGerman. If allof theothers takeFrench,what percent of thestudents take French? 101520@2530 SOLUTION. ThIdf31. eeastcommonenommatoroS and 4 Is20,soassumethatthereare20students at MadisonHigh.(Remember the numbers don't have to berealistic.) The numberof studentstakingSpanishis12( lof 20).Of theremaining8students, 15 2of them ( 4 of 8)takeGerman. The other 6takeFrench.Finally,6is30% of 20. The answerisE. 1994 to1995 the sales of a book decreased by 80%. If the sales 1n 1996 were the same as in1994, by what percent did th from1995 to 1996?ey Increase 80%100%120%@400%500% SOLUTION. thisprobleminvolvespercents,assumethat100copiesof thebook weresold 1994(and1996).Salesdroppedby80(80%of 100)to20in1995andthen Increasedby 80from20bk100 196 Th 'acto, m9.e percent increase was the actual increase80 the original amountX 100% =20X 100% =400%, D. DiscreteQuantitativeQuestions177 TACTICII EliminateAbsurdChoicesandGuess Whenyouhavenoideahowtosolveamultiple-choicequestion,youcanalways make an educated guess-simply eliminate all the absurd choices and then guess from amongtheremaining ones. During the course of a GRE, you will probably find at least a fewmultiple-choice questionsthat you don't know how tosolve.Since youarenot penalized forwrong answers,youaresurelygoingtoenteranswersforthem.Butbeforetakingawild guess,takea moment to look at the answer choices.Often twoor three of them are absurd.Eliminate thoseand then guessone of theothers.Occasionally,fourof the choicesareabsurd.When thisoccurs, your answer isnolonger a guess. Whatmakesachoiceabsurd?Lotsof things.Hereareafew.Evenif youdon't knowhow to solveaproblem youmay realizethat theanswermust bepositive,but some of thechoicesarenegative; theanswermust be even,but some of the choicesareodd; theanswermust belessthan100,but some choicesexceed100; a ratiomust belessthan1,but somechoicesaregreater than1. Let'slook at several examples.In a fewof them the information given isintention-ally insufficient to solve the problem; but you will still be able to determine that some of theanswersare absurd.In each casethe "solution" willindicate which choices you shouldhaveeliminated.Atthatpoint youwouldsimplyguess.Remember,onthe GRE when you guess,don't agonize. Just guessand moveon. A regioninside asemicircle of radiusr is shaded and youare asked for its area. ..!7tr2..!7tr2..!7tr2@1tT24323 SOLUTION. Youmayhavenoideahowtofindtheareaoftheshadedregion,butyou shouldknowthatsincetheareaof acircleis1tr2,theareaof asemicircleIS1h12r. 21tr2 Therefore,the area of the shaded region must beless tan 2nr, soe Immate C,D,andE.OnanactualGREproblem,youmaybeabletomakeaneducated guessbetween A and B.If so,terrific;if not,just chooseone or the other. 178NewGRE Theaverage (arithmeticmean)of 5,10,15,andzis20. Whatisz? @0@2025@4550 SOLUTION. If theaverageof fournumbersis20,andthreeof themarelessthan20,the other one must be greater than 20.Eliminate A and B and guess.If you further realize that since5 and10arealot lessthan 20,z willprobably be alot more than 20;eliminate C,aswell. If 25% of 260equals 6.5%of a,what isa? @10@65100@1301000 SOLUTION. Since 6.5% of a equals 25% of 260, which issurely greater than 6.5% of 260,a must begreater than 260.Eliminate A,B,C,and D. The answermust beE! Example14illustratesanimportant point.Evenifyouknow how tosolve a prob-lem,if you immediately seethat four of the fivechoices are absurd, just pick the fifth choiceand move on. A jackpot of $39,000istobedividedinsomeratio among threepeople. Whatisthe valueof thelargest share? @$23,400@$19,500$11,700@$7800$3900 SOLUTION. If theprizeweredividedequally,eachof thethreeshareswouldbeworth$13,000. If itisdividedunequally,thelargestshareissurelyworthmorethan$13,000. EliminateC,D,and E.In anactualquestion,you wouldbetold what theratiois,and that might enable you toeliminate Aor B.If not, you just guess. Ina certain club,theratioof thenumber of boys togirls is 5:3. What percent of themembers of the club are girls? @37.5%@50%60%@62.5%80% DiscreteQuantitativeQuestions179 SOLUTION. Sincethereare5boysforevery3girls,therearefewergirlsthanboys.Therefore, fewerthanhalf(50o/o)of themembersaregirls.EliminateB,C,D,andE.The answerisA. Inthefigurebelow,four semicircles are drawn,eachonecenteredat the midpoint of oneof thesides of square ABCD.Eachof the four shad_ed "petals"istheintersection of twoof thesemicircles.IfAB = 4,whatISthetotalarea of theshadedregion? AB 81t@32 - 81t16 - 81t@81t- 3281t- 16 SOLUTION. SinceAB =4,theareaof thesquareis16,andso,obviously,theareaof the shadedregionmust bemuch less.. Check eachchoice.Since 1tisslightly morethan 3(7t"" 3.14),81ttssomewhat greaterthan 24,approximately 25.. (A)81t""25.More than the areaof the wholesquare:waytoobtg. (B)32 - 81t""32 - 25=7. (C)16 - 81tisnegative. (D)81t- 32isalsonegative. (E)81t- 16"" 25- 16=9. NOTEThfhh brdA ismore than the area of the entire square :reeote cotces areasu. db1.d mmediatelyThe answer must be anC and Darenegative;they cane e tmmate 1 . BorE.If youthink the shaded area takesup lessthan half of the square,~ u e s sB;tf h..hhalffhareguessE(The answertsE). youtmk 1ttakesup more tanote squ' Nowuse TACTIC 4oneachof thefollowingproblems.Even_if youknowh1_ow .h.dhow many chotces you can e tm-tosolve them, don't.Practice thts teemque ansee !Uatewithout actually solving. Inthefigureat theright,diagonalEGof squareEFGH is~of diagonalAC of thesquareABCD. What is theratio 2 of thearea of the shadedregionto the area of ABCD? .J2 :13:4.J2 :2@1:21:2.J2 180New GRE SOLUTION. Obviously,the shaded regionissmaller than square ABCD, sothe ratiomust be lessthan1.EliminateA.Also,fromthediagram,itisclearthattheshadedregionismorethanhalf of squareABCD,sotheratioisgreaterthan0.5.EliminateD andE.Since3:4=.75andfi :2"" .71,BandCaretooclosetotellwhichiscorrect just by looking;soguess. The answer isB. Sharireceivesa commissionof 25 for every $20.00 worthof merchandise she sells. What percent isher commission? 1 _!_% 2_!_%5%@) 25%125% 42 SOLUTION. Clearly,acommissionof 25on$20isquite small.EliminateDandEand guessone of the smallpercents.If you realizethat1 o/o of $20 is20,then you know theanswerisalittlemorethan1o/o,and you shouldguess A(maybeB,but definitely not C). The answer is A. 1980 t?1990,Lior'sweight increased by 25%.If hisweight was k krlogramsrn1990, what was it in1980? 1.75k1.25k1.20k@.80k.75k SOLUTION. Since Lior's weight increased, his weight in 1980 waslessthan k.Eliminate A,B, and Cand guess.The answer isD. !he average of 10 numbers is -10.1f the sumof 6 of them is100, what rstheaverageof the other 4? -100-500@50100 SOLUTION. Since of all10numbers isnegative,soistheir sum.But the sum of the ? Ispositive,sothesum(andtheaverage)of theothersmustbenegative. Ehmmate C, D, and E.B iscorrect. DiscreteQuantitativeQuestions181 PracticeExercises DiscreteQuantitative Questions 1. Evanhas4times asmany books asDavid and 5 timesasmany asJason.If Jason hasmore than 40books,what isthe leastnumber of booksthat Evancould have? 200 205 210 @220 240 2.Judy plansto visit the National Gallery once eachmonth in 2012 except in July and August when she planstogothree times each. A singleadmissioncosts$3.50, a pass valid forunlimited visitsin any 3-month period canbepurchased for$18,and an annualpass costs$60.00. What isthe least amount,in dollars,that Judy can spend forher intended number of visits? dollars 3.Alisonisnow three timesasold asJeremy,but 5 yearsago,she was5timesasold ashe was. How old isAlisonnow? 10 12 24 @30 36 4.What isthe largest prime factor of 255? 5 15 17 @51 255 5.If cis the product of a andb,which of the followingisthe quotient of a andb? b2 -c 6.If w widgetscostc cents,how many widgets can you get ford dollars? lOOdw ---c dw IOOc 100cdw dw @) c cdw 7.If 120o/oof a isequalto80o/oof b,which of the followingisequaltoa+b? l.5a 2a 2.5a @3a 5a 182NewGRE 8.Inthefigurebelow,WXYZ isa square whose sidesare12.AB,CD,EF,andGH areeach8, andarethe diameters of the foursemicircles. What isthe area of the shadedregion? @144- 1287t 144- 647t 144- 327t @144- 167t 167t 9.If x andy areintegerssuch that x3= y2,which of the followingcould not bethe valueof y? Indicateall such values. ~ - 1 []]1 [98 [QJ12 [I] 16 [I] 27 10. What isa dividedby ao/oof a? @_!!_ 100 100 a 2 .!!.._ 100 100a 11. If an object ismoving at a speed of 36 kilo-m e t e r ~per hour,how many meters doesit travelmone second? .._____--.JImeters 12.On a certain French-American committee~ '3 of the membersaremen,andlof the men 8 are Americans.If lof the committee 5 membersareFrench,what fractionof the membersare American women? 13.For what valueof xis 82x-4=16x? 2 3 4 @6 8 14.If12a+3b= 1 and7b-2a=9,whatisthe average{arithmetic mean)of a andb? 0.1 0.5 1 @2.5 5 15.If xo/oof y is10, what is y? @10 X 100 X 1000 X X @-100 X -10 ANSWERKEY l.D 2.49.50 3.D 4.c 5.B 6.A 7.c 8.c 9.D,E 10.B 11.10 12.2 20 13.D ANSWEREXPLANATIONS 14.B 15.c Twoasterisks(**)indicate analternativemethod of solving. DiscreteQuantitativeQuestions183 1.(D)Test the answer choices starting with the smallest value.If Evanhad 200 books, Jason would have 40.But Jason hasmore than 40,so200 istoo small. Trying 205 and 210, weseethat neither isa multiple of 4,soDavid wouldn't havea whole num-ber of books.Finally,220 works.(Sodoes240,but weshouldn't eventestit since wewant the leastvalue.) **Since Jasonhasat least 41books,Evanhasat least 41X5 = 205.But Evan's total must be a multiple of 4and 5,hence of 20. The smallest multiple of 20 greater than205is220. 2.49.50Judy intends to gotothe Gallery16 timesduring the year.Buying a single admissioneachtime would cost16 X$3.50= $56, whichislessthanthe annual pass.If shebought a 3-month passfor June, July,and August,she would pay $18 plus$31.50 for9 singleadmissions(9X$3.50), fora total expense of $49.50, which isthe least expensive option. 3.(D)Use TACTIC1:backsolve starting with C.If Alisonisnow 24, Jeremy is8,and 5 yearsago,they would havebeen19 and 3, which ismore than5 timesasmuch. Eliminate A,B,and C, and try a bigger value.If Alison isnow 30, Jeremy is10,and 5 yearsago,they would havebeen 25and 5. That's it;25is5 times5. **If]eremy isnow x,Alisonis3x,and 5 yearsagothey werex- 5 and 3x- 5, respectively.Now,solve: 3x- 5= 5(x- 5)=>3x- 5= 5x- 25=> 2x = 20 =>x=10=>3x = 30. 4.(C)Test the choicesstarting with C:255is divisibleby17(255= 17X15),so thisisa possible answer.Does 255havea larger prime factor?Neither Choice D nor E isprime,sothe answer must be Choice C. 5.(B)Use TACTIC 2.Pick simple valuesfora,b,andc. Leta=3,b = 2,andc = 6. Thena +b = 3/2. Without these valuesof a,b,andc, only B isequalto 3/2. **c = ab =>a =~=>a +b =~+b =~-}=:2 6(A)Use TACTIC 2.If 2 widgetscost10 cents,then widgets cost5 centseach,and for3dollars,you can get 60. Which of the choicesequals60 whenw =2,c =10, andd = 3?Only A. ** widgetswx1OOdw =-=--=>x= centsc1OOdc 184NewGRE 7.(C)Since120% of80 = 80% of 120,leta= 80 andb = 120. Then a+b = 200, and 200+80= 2.5. 8.(C)If youdon't know howtosolvethis,you mustuse TACTIC 4and guessafter eliminating the absurdchoices.Which choicesareabsurd?Certainly,A and B, both of whicharenegative.Also,since Choice Disabout 94,whichismuch more thanhalf thearea of the square,it ismuch toobig.GuessbetweenChoice C (about 43)andChoiceE (about 50).If youremember that the way tofindshaded areasistosubtract,guessC. **The area of the square is122 =144. The area of eachsemicircleis81t,one-half the area of a circleof radius4.Sotogether the areasof the semicirclesis327t. 9.(D)(E)Test each choiceuntil you find allthe correct answers. (A)Could y = -1 ? Isthere aninteger x such that X> = (-1 )2 = 1?Yes,x = 1. (B)Similarly,if y = 1,x = 1. (C)Could y =8?Isthere aninteger x such that X>=(8)2 = 64? Yes,x = 4. (D)Could y = 12?Isthere aninteger such that X> = 122= 144?No,53= 125, which istoosmall,and 63 = 216, which istoobig. (E)Couldy=16?Isthere an integer xsuch that X>=162= 256?No, 63= 216, whichistoo small;and 73 = 343, which istoo big. (F)Could y = 27?Isthere aninteger x such that X>=272 =729? Yes,93= 729. The answer isD and E. 10.(B)a+(a%of a)= a+(_!!__xa)= a+(__)= a X100=100 100100a2a .**Use TACTICS2and 3:replacea by a number,and use100sincethe problem mvolvespercents.100 +(100% of 100)= 100 +100=1.Test each choice;which ones equal1 whena = 100.Both A and B: 1 OO = 1.Eliminate Choices CDand 100'' E,and test A and B with another value fora.50+(50% of 50)=50 +(25)= 2. ow,on y B works- = 2, whereas- =- . N1 ( 100501) 501002 11.10Setup a ratio: distance36 kilometers = 36,000 meters36,000 metersd == 10meters/secon 60 minutes3600 seconds time1 hour **Use TACTIC 1: Test choicesstarting with C: 100 meters/second=6000 meters/minute=360,000 meters/hour= 360 kilometers/hour. Not only isthat too big,it istoobig by a factor of 10. The answer is10. DiscreteQuantitativeQuestions185 12.;OUse TACTIC 3.The LCM of allthe denominatorsis120,soassumethatthe committee has120 members. Then there a r e ~X120= 80menand 40women. 3 Of the 80 men 30 (%X80) are American.Sincethereare72(%X120) French members,there are120 - 72= 48 Americans,of whom30 aremen,sothe other 18arewomen.Finally,the fractionof American womenis__!_!=_]__.12020 Thisisillustrated inthe Venndiagrambelow. Americans 13.(D)Usethe lawsof exponentsto simplifYthe equation,andthen solveit: 82x-4=l6x=>(23)2x-4= (24}X=>3(2x- 4)= 4x=> 6x- 12= 4x=> 2x =12=>x =6. 14.(B)Addthe two equations: lOa+lOb=10 =>a+ b = 1 =>a+ b = l. 22 Donot wastetime solving fora andb. 15.(C)Pick easy-to-usenumbers.Since100% of 10 is10, let x= 100 andy=10. Whenx = 100,Choices Cand E are each10.Eliminate Choices A,B,and D, andtry some other numbers:50% of 20is10.Of ChoicesCand E,only C= 20 whenx= 50. you can find all NEW GRE books in pdf ETS revised GRE, Kaplan, Barron's, Princeton here:http://gre-download.blogspot.comOuantitative ComparisonOuestions . -- - - - - - -bout15of the40questionsonthetwoquantitative sectionsof theG REare quantitativecomparisons.UnlessyoutooktheSATbefore2005,itisvery likely that you havenever seen questions of thistype and certainly never learned the correct strategiesforanswering them.Don't worry.In this chapter you willlearn all of thenecessarytactics.If youmasterthem,youwillquicklyrealizethatquantita-tivecomparisonsaretheeasiestmathematicsquestionsontheGREandwillwish thatthere weremorethan15of them. Beforethe firstquantitative comparison questionappearsonthescreen,you will seetheseinstructions. Directions: In the following question,there are two quantities, labeled Quantity A andQuantity B.Youaretocompare thosequantities,takingintoconsideration ~ yadditionalinformationgivenand decide whichof thefollowingstatements Is true: Quantity Aisgreater; Quantity B isgreater; The two quantities are equal;or It isimpossible to determine which quantity isgreater. Note: The given information, if any,is centered above the two quantities.If a sym-bol appearsmore than once,it representsthe same thing each time. .Beforelearning the different strategies forsolving thistypeof question,let'sclar-IfY theseinstructions.In quantitative comparison questions there aretwoquantities, anditisyour jobtocomparethem.The correctanswertoa quantitativecompari-sonquestionisone of the fourstatements listedinthedirectionsabove.Of course, onthecomputer screenthosechoiceswillnot belistedasA,B,C,andD.Rather, you will see an oval in front of each statement, and you will click on the ovalin front of thestatement you believeistrue. - -187 188NewGRE Rightnow,memorize the instructions for answering quantita-tivecomparison questions.When you take the GRE, dismiss the instructionsforthese questions immediately-do not spend even one second reading the directions (or looking at a sample problem). Youshouldclick on the ovalinfront ofif Quantity A isgreater.Quantity Aisgreater all thetime,no matter what. Quantity8isgreater.Quantity 8is greater all thetime,no matter what. The twoquantities areequal.The twoquantities areequalall thetime,no matter what.Itisimpossible todetermineTheanswer isnot oneof thefirst three choices. whichquantity isgreater. Thismeans,forexample,thatifyoucan find a singleinstance whenQuantity A isgreater than Quantity B,then you can immediately eliminate twochoices:the answer cannot be"Quantity B isgreater,"and the answer cannot be"The twoquantities areequal."In order forthe answertobe "Quantity B isgreater,"Quantity B would have tobegreaterall the time;but you know of one instance when it isn't.Similarly,since thequantitiesarenot equalall thetime,theanswercan'tbe'The twoquantities are equal." The correct answer,therefore,iseither "Quantity A isgreater"or "It isimpos-sibleto determine which quantity isgreater."If it turns out that Quantity Ais greater allthe time,then that isthe answer;if,however,you canfind a single instance where Quantity A isnot greater,the answer is"It isimpossible to determine which quantity isgreater." Byapplyingthetacticsthat youwilllearninthischapter,youwillprobably be abletodetermine whichof the choicesiscorrect;if,however,after eliminating two of thechoices,youstillcannotdeterminewhichansweriscorrect,quicklyguess betweenthetworemaining choicesand move on. Beforelearning the most important tacticsforhandling quantitative comparison questions,let'slook at twoexamplesto illustratethe preceding instructions. Quantity A x2 oQuantity Aisgreater. oQuantity8isgreater. oThe two quantities are equal. l0 andm :1= 1. Quantity A Letm = 222 = 4 Letm =.! (ir 1 -24 The answerisD. SOLUTION. Quantity A 13y Quantity B 23 = 8 (ir 1 -8 Compare B isgreater Aisgreater QuantityB 15y Eliminate Aand C B Use TACTIC1.There arenorestrictionson y,sousethe bestnumbers:1,0,-1. Quantity A Let y = 113(1) =13 Let Y=013(0) = 0 The answerisD. SOLUTION. Quantity A w+11 Quantity B 15(1) =15 15(0) = 0 Compare B is greater They're equal Quantity8 w-11 Eliminate Aand C B Use TACTIC1.There arenorestrictionsonw,sousethebestnumbers:1,0,-1. Quantity A Quantity8 CompareEliminate Letw- 1 1+11=121-11--10 Ais greater8and C Letw00+11-11 0-11--11 Ais greater Letw11+11-10-1- 11- 12Ais greater Guess.AWeletwbeapositivenumber,anegativenumber,and0.Eachrime, Awasgreater.That'snotproof,butitjustifiesaneducatedguess.[The answerzs A.Clearly11> -11d'fddh11] ', an1we aw to eacstde, we get:w +11>w- QuantitativeComparisonQuestions193 Quantity A The perimeter of a rectangle whoseareais18 SOLUTION. Quantity B The perimeter of a rectangle whoseareais28 What'sthisquestiondoing here?How canweuse TACTIC1?Wherearethevari-ablesthat we're supposed to replace? Well,each quantity isthe perimeter of a rectan-gle,andthe variablesarethe lengthsand widthsof theserectangles. QuantityAQuantity 8CompareEliminate ChoosearectangleChoose arectangle whoseareais18:whose areais 28: CJ4 I129 7 Theperimeter hereisThe perimeter hereQuantities A andAand8 9+2+9+2=22is7 + 4+ 7 + 4 = 22Bare equal KeepQuantity B,but takea different rectangleof area18whenevaluatingQuantity A: 0304 67 Perimeter = 3 + 6 + 3 + 6 = 18Perimeter = 22 TheanswerisD. SOLUTION. 2 a=-t Quantity A 3a 3 5 b = -t 6 8is greater c ='}_ b 5 Quantity 8 4c c Use 1.First,try the easiestnumber:lett,=0.Then a, a?d care each0, andmthiscase,thequantitiesareequal-.both0.EhmmateAandNow,try anothernumberfort.The obviouschotcets1,but thena,b,andc wtll allbefractions.Toavoidthis,lett=6.Then,a=t(6)= 4,b =%(6)= 5,and c:::(5)= 3.Thistime,3a = 3(4)=12 and 4b =4(3)= 12.Again,thetwoquan-tities areequaLChoose C. 194New GRE NOTE: Youshouldconsider answering thisquestion directly(i.e.,without plug-ginginnumbers),onlyifyouareverycomfortablewithboth fractionsand elementary algebra.Here'sthesolution: Therefore,2c =t, and 4c =2t.Since a=~t,3a =2t.So,4c =3a. The answer isC. 3 TACTIC II ChooseanAppropriateNumber This isjust like TACTIC1.We arereplacing a variable with a number, but the vari-ableisn'tmentioned inthe problem. Everyband member iseither15,16,or17 yearsold. One third of the band members are16,and twiceasmany band membersare16as15. Quantity A The number of 17-year-old band members QuantityB The totalnumber of 15- and16-year-old band members If thefirstsentenceof Example9hadbeen"Therearenstudentsintheschool band, allof whom are15,16, or 17 yearsold,"the problem would have been identi-caltothisone.Using TACTIC1,youcouldhavereplacednwithaneasy-to-use 1.. number,such as6,and solved:- (6)=2 are16 yearsold;1 is15, and the remammg 3 3are17. The answerisC. The point ofTACTIC 2isthat you can plug in numbers even if there are no van-abies.Asdiscussedin TACTIC3,Chapter 8,thisisespeciallyusefulonproblems involvingpercents,inwhichcase100isagoodnumber,andproblemsinvolving fractions,in which casethe LCD of the fractionsisa good choice.However,theuse ofTACTIC 2isnot limited tothese situations. Try using TACTIC 2 on the follow-ing threeproblems. The perimeter of a square and the circumference of a circle areequal. Quantity A The area of the circle Quantity B The area of the square QuantitativeComparisonQuestions195 SOLUTION. Firstuse TACTIC1,Chapter 7:draw a diagram. c = 21t(l) = 27tA = 1t(l )2 = 1t"' 3.14 Thenuse TACTIC 2:chooseaneasy-to-usenumber.Lettheradiusof the circle be1.Then itsarea is1t.Lets bethe side of the square: sA= s2 DP=4s 4s =27tz 6 =>s z 1.5=> areaof the square z (1.5)2 =2.25 The answer isA. Jen,Ken,and Lendivideda cashprize. Jentook50o/oof themoney and spent~of what shetook. Kentook 40o/oof themoney and spent-ff what hetook. Quantity A The amount that Jen spent Quantity B The amount that Kenspent SOLUTION.d UseTACTIC2.Assumetheprizewas$100.ThenJentook$50an 3.c ~ ( $ 5 0 )= $30.Kentook $40 and spent4 ($40)= $30. The answerts. ElianetypestwiceasfastasDelphine.. Delphine charges50o/omoreper pagethan Ehane. Quantity A Amount Eliane earnsin 9hours Quantity B Amount Delphine earnsin12hours spent 196NewGRE SOLUTION. UseTACTIC2.Chooseappropriatenumbers.AssumeDelphinecantype1 page per hour and Eliane can type 2. Assume Eliane charges$1.00 per page and Delphine charges $1.50. Then in9 hours,Eliane types18pages,earning $18.00. In12 hours, Delphine types12pages,earning12X$1.50= $18.00. The answerisC. TACTIC II Make theProblemEasier:DotheSameThingtoEachQuantity A quantitativecomparisonquestioncanbetreated asanequation or aninequality. Either: Quantity A< Quantity B,or Quantity A = Quantity B,or Quantity A >Quantity B In solving an equation or an inequality, you can always add the same thing toeach sideor subtractthesamething fromeachside.Similarly,insolvingaquantitative comparison,youcanalwaysaddthesamethingtoquantitiesAandB or subtract thesamething fromquantities A andB.Youcanalsomultiply or divideeachside of anequationor inequality bythesamenumber,but inthecaseofinequalities you candothisonlyifthenumber is positive.Sinceyoudon'tknow whether thequanti-ties are equal or unequal, you cannot multiply or divide by a variableunless you know that it is positive.If quantities A and B areboth positive you may square them or take their squareroots. Toillustratetheproperuseof TACTIC3,wewillgivealternativesolutionsto examples4,5,and 6,which wealready solvedusing TACTIC1. SOLUTION. Quantity A rri2Divide eachquantity byrri2(that'sOK- rri2 ispositive): m >0 andm ::f:. 1 Quantity B n? Quantity AQuantity B m2m3 z-=1-2=m mm Thisisamucheasiercomparison.Whichisgreater,m or1?Wedon'tknow.We knowm > 0 andm ::f:. 1,but it could be greater than1 or lessthan1.The answer is D. SOLUTION. Quantity A 13y QuantitativeComparisonQuestions197 QuantityB 15y Quantity AQuantity B Subtract13y fromeachquantity:13y- 13y = 015y- 13y = 2y Sincetherearenorestrictionson y,2y couldbegreaterthan,lessthan,orequal to0.The answerisD. SOLUTION. Quantity A W+ll Subtractw fromeachquantity: Quantity A QuantityB w- 11 Quantity B (w+11)-w=ll(w-11)-w=-11 Clearly11isgreaterthan -11. Quantity A isgreater. ,1h.htice TACTIC 3. Herearefivemore examp eson wICtoprac SOLUTION. Quantity A 111 -+-+-349 Subtract_!_ and_!_ fromeachquantity: 39 S.11 tnce- >- ,the answerisA. 45 Quantity A (43+ 59)(176) Quantity B 111 -+-+-935 Quantity AQuantity B Quantity B (43+59)(17+ 6) 198NewGRE Quantity AQuantity B SOLUTION. Divideeachquantity by(43+59): 6) + 6) Clearly,(17+ 6)>(17- 6).The answerisB. Quantity A (43- 59)(43- 49) Quantity B (43- 59)(43+ 49) SOLUTION. CAUTION (43- 59)isnegative,andyoumaynotdividethetwoquantitiesbya negativenumber. Theistonote that Quantity A,being the product of 2negative posmve,Quantity B,beingtheproduct of anegativenumber and a posmvenumber,tsnegative,and soQuantity Aisgreater. SOLUTION. Quantity A a2 Adda2 toeachquantity: a isa negativenumber Quantity A Sinceaisnegative,2a2 ispositive. The answer isA. Quantity A o 2 SOLUTION. Quantity A =5 Square each quantity: The answer isC. Quantity B -a2 Quantity B Quantity B 5 .[s Quantity B QuantitativeComparisonQuestions199 TACTICII Ask"CouldTheyBeEqual?"and"Must TheyBeEqual?" TACTIC 4 hasmany applications, but ismost useful when one of the quantities con-tains a variable and the other contains a number. In this situation ask yourself,"Could theybeequal?"If theansweris"yes,"eliminate A andB,andthenask,"Must they be equal?"If the second answer is"yes,"then Cis correct; if the second answer is"no," then choose D. When the answer to"Could they be equal?"is"no," weusually know rightaway whatthe correct answeris.In both questions,"Could theybeequal"and "Mustthey beequal,"the wordthey refers,of course,to quantities A andB. Let'slook at a fewexamples. The sidesof a triangleare3,4,andx SOLUTION. Quantity A X QuantityB 5 Couldtheybeequal?Couldx= 5?Of course.That'stheall-important3-4-5right triangle.Eliminate A and B.Must they beequal?Must x =5?If you'renot sure,try drawinganacuteoranobtusetriangle.TheanswerisNo.Actually,xcanbeany numbersatisfying:1< x< 7.(SeeKEYFACT J12,thetriangleinequality,and the figurebelow.)The answer isD. 44 Quantity A c 56< 5c < 64 4 QuantityB 12 SOLUTION. Couldthey beequal?Couldc = 12?If c = 12,then5c = 60,so,yes,they couldbe equal.Eliminate A and B.Must they be equal?Must c =12?Could c bemore or l.ess than12?BECAREFUL:5 X11= 55,whichistoosmall;and 5 X13= 65,whtch is toobig.Therefore,theonlyinteger thatc couldbeis12;butc doesn'thavetobe aninteger. The only restriction isthat 56 < 5 c < 64.If 5 c were58 or 61.6 or 63, then cwouldnot be12. The answerisD. 200NewGRE School A has100teachersand SchoolB has200 teachers. Eachschoolhasmore femaleteachersthanmaleteachers. Quantity AQuantityB The number of female teachersat School A The number of female teachersat SchoolB SOLUTION. Couldtheybeequal?Couldthenumberof femaleteachersbethesameinboth schools?No.Morethanhalf (i.e.,morethan100)of SchoolB's200teachersare female,but School A hasonly100teachersinall.The answerisB. SOLUTION. Quantity A m+ 2

(m+1)(m +2)(m +3)=720 Quantity B 10 Couldtheybeequal?Couldm +2= 10?No,if m+2= 10,thenm+1=9 and m + 3 =11, and 9 X10 X11=990, which istoo big. The answer isnot C, and since m +2clearly hastobesmaller than10,the answer isB. Quantity A The perimeter of a rectangle whose areais21 SOLUTION. QuantityB 20 theybee_qual?Couldarectanglewhoseareais21haveaperimeter of 20? Yes,If Itslength IS 7 and its width is37+3+7+320El"AdBMust h 1 ,=.Immatean. teybe If,roure sure that there isno other rectangle with anarea of21, rhen chooseC;If yourenot sure,guessbetweenCandD;if youknowthereareother rectanglesof area21,chooseD. There areother possibilities - lotsof them;here area 7x 3rectanleand a few other rectangles whose areasare21:g '-------:-:-----__J 1.51421QuantitativeComparisonQuestions201 TACTIC II Don'tCalculate:Compare Avoidunnecessarycalculations.Youdon'thavetodeterminetheexactvaluesof Quantity A andQuantity B;youjust havetocomparethem. TACTIC5isthespecialapplicationof TACTIC 7,ChapterI 0(Don'tdomore thanyouhaveto)toquantitativecomparisonquestions.Using TACTIC5allows youtosolvemanyquantitativecomparisonswithoutdoingtediouscalculations, thereby saving you valuable test time that you canuseon other questions.Before you startcalculating,stop,lookatthequantities,andaskyourself,"CanIeasilyand quicklydeterminewhichquantityisgreaterwithoutdoinganyarithmetic?" ConsiderExamples23and 24,whichlook verysimilar,butreallyaren't. Quantity A 37X43 Quantity A 37X 43 Quantity B 30X53 QuantityB 39X47 Example 23 is very easy. Just multiply:37X43=1591and 30X53=1590. The answerisA. Example24 iseveneasier.Don't multiply.In lesstime than ittakestodothemul-tiplications,even with the calculator, you can seethat 37 < 39and 43 < 47, soclearly 37X 43O The number ofThe number of Quantity AQuantity Bmultiplesof 6multiples of 9 2.lOx 10berween100berween100 X 9.and x +100andx+100 Quantity AQuantity B x+y=5 The timethat itThe timethat it y-x= -5 takestotype7takestotype6 Quantity A pagesat a rateofpagesat arate of Quantity B 3.6pagesper hour7pagesper hour 10.y0 cd9,11.001>9.896; since1 >0,1.234 > 0.8; and smce 3> -3, 3.01> -3.95.(Recall that if a decimal is written without a number totheleftof thedecimalpoint,youmayassumethata0isthere.So, 1.234 > 0.8.) 252NewGRE If the numbers to the left of the decimal point are equal (or if there are no numbers to the left of the decimal point), proceed asfollows: 1.If the numbers do not have the same number of digits to the right of the decimal point, add zerosto the end of the shorter one. 2.Now,compare the numbersignoring the decimal point. Forexample,tocompare1.83and1.823,adda0totheendof 1.83,forming 1.830.Now compare them,thinking ofthem aswhole numbers:since,1830 >1823, then1.830 >1.823. SOLUTION. Quantity A .2139 Quantity B .239 Donotthink that Quantity A isgreater because2139>239.Besuretoadd a 0 to theendof 0.239(forming0.2390)beforecomparing.Now,since2390>2139, Quantity B isgreater. KEYFACT83 There are twomethods of comparing positive fractions: 1.Convert them to decimals(by dividing),and use KEY FACT B2. 2.Cross-multiply. 13 For example,to compare3 and8, wehavetwo choices. ._!_3.31 1.Wnte 3 - .3333 ...and8 =.375.Smce.375>.333,then 8 >3. 2.Cross-multiply:_ ! ~~l. Since3X3>8x1thenl>_!3A8'83 KEYFACT84 When comparing positive fractions,there are three situations in which it is eas-ier just to look at the fractions,and not use either method in KEY FACT B3. 1.If the fractionshavethe samedenominator,the fractionwiththe larger numerator isgreater. Just as$9ismorethan$7,and 9booksaremore than 7books, 9fortiethsare more than 7fortieths_2__ >.]___.4040. 2.If the fractionshavethesamenumerator,thefractionwiththesmaller denominator is greater. MathematicsReview253 If youdivideacakeinto5equalpieces,eachpieceislargerthanthe pieces you would get if you had divided the cake into 10al. 11 d'II 33 equpteces:5 > 10, anstmt ar y5 > 10. 3.Sometimesthefractionsaresofamiliaroreasytoworkwith,youjust 31111(.101) know the answer.For example, 4 >5 and 20 >2smce 20 = 2 KEYFACTSB2,B3,and B4apply to positive decimalsand fractions. KEYFACT85 Clearly,any positive number isgreater than any negativenumber: 11 - > --25 and0.123> -2.56 For negative decimals and fractions,useKEY FACT A24, which states that if a>b,then -a< -b: 1111 - >- :=:::} -- < -- and0.83>0.829 :=:::} -0.83 - smce8X13> 20 X5. 08 213. -3 >- smce 20 X2>3X13. 20 254NewGRE !{ SOLUTION. Quantity A 11 Xy 0-y1, and soby KEYFACT A22,_!_ - _!_ ispositive. XJ Quantity A isgreater. Equivalent Fractions If Billand Al shareda pizza,andBillate_!_ thepizzaand Alate 4 of it,theyhad exactly the same amount.28 \Xli h''db. 1 4 e expresstIs1 eaY saymg that- and- areequivalent fractions:they have the exact same value.28 NOTE:If youmultiplyboththenumeratoranddenominatorof _!_ by4youget 2 4 8;and if youdivideboththenumeratoranddenominator of4by4youget!. This illustratesthenextKEYFACT. 82 KEYFACT86 Two fractions are equivalent if multiplying or divt"dibth thd d .ngoe numerator an enommator of the first one by the same numbthd er gtveseseconone. Consider the followingtwocases. 1.Whenthenumeratoranddenominatorof2h15h 8 areeacmu tipliedby1,te productsare3X1545d815345 =anX= 120.Therefore- andare '8120 equivalent fractions. MathematicsReview255 2.~and 428 arenot equivalent fractionsbecause2 mustbemultiplied by14to 35 get28,but 3must bemultipliedby15toget 45. KEYFACT87 Todetermineif twofractionsareequivalent,cross-multiply. The fractionsare equivalent if and only if the twoproducts are equal. Forexample,since120 X3=8 X45,then2 and 45 areequivalent. 8120 Since45X2of:. 3X28,then~and 28 arenot equivalentfractions. 345 A fractionisin lowest terms if no positive integer greater than1 isa factor of both thenumerator and denominator.For example,;Oisin lowest terms,since nointeger greaterthan1 isa factorof both 9 and 20;but_2_isnot in lowestterms,since3is 24 a factorof both9 and 24. KEYFACT88 Everyfractioncanbereduced to lowest termsby dividing thenumerator and thedenominator by their greatest common factor(GCF).If the GCF is1,the fractionisalready in lowest terms. Forany positiveintegern:n!,readn factorial,isthe product of allthe integers from 1 ton,inclusive. Whatisthevalueof 6! ? 8! _1 56 1 48 1 -8 1 @-4 3 -4 SOLUTION. Evenwith a calculator,you donot want tocalculate 6!023456 = 720)and 8! 720H,h 023-45678=40,320)and then takethe time toreduce 40320.eres te easy solution:' 6! 8! 1 --(jx5>< 4 X 3 X 2 X t= 8 X 7 = 56 1 256NewGRE '\ Arithmetic Operations with Decimals Arithmetic operations with decimalsshould bedone on your calculator,unlessthey aresoeasythat youcandothemin your head. Multiplying and dividing by powers of 10 isparticularly easy and doesnot require a calculator:they canbeaccomplished just bymovingthe decimalpoint. KEYFACT89 Tomultiply any decimal or whole number by apower of 10, move the decimal point asmany placestotheright asthere areOsin the power of 10, filling in with Os,if necessary. 1.35X 10 =13.51.35 X 100 =135vv I2 1.35 X 1000 =1350 23 X 10= 230 \j I '-..J 323 X 100 = 2300 v 2 23 X 1,000,000 = 23,000,000 6 KEYFACT810 Todivideany decimalor wholenumber by apower of 10,movethe decimal point as many places to the left as there are Osin the power of 10, filling in with Os,if necessary. SOLUTION. 67.8 +10 = 6.'l}67.8+100 = 0.678 v Quantity A 3.75 X104 I267.8+1000 = 0.0678 v 14 +10 = 1.4 u I 3 14 +100 = 0.14 '-..../214 +1,000,000 = 0.000014 6 Quantity B 37,500,000 + 103 Toevaluate A,movethe decimalpoint 4placestotheright:37,500. To evaluateQuantityB,movethedecimalpoint3placestotheleft:37,500.The answer isC. I I 1 MathematicsReview257 Arithmetic Operations with Fractions KEYFACT811 Tomultiplytwofractions,multiplytheirnumeratorsandmultiplytheir denominators: 343 X 412 5 x7 =5x7=35 KEYFACT812 31t3 X1t31t -x-=--=-525 X210 Tomultiplyafractionby any other number,writethat number asafraction whosedenominator is1: 33721 -X7=-X-=-5515 TACTIC ID Beforemultiplyingfractions,reduce.Youmayreducebydividinganynumerator andany denominator by a common factor. Expresstheproduct,xx 15, inlowest terms. 4916 SOLUTION. Youcoulduseyourcalculatortomultiply thenumeratorsand denominators: It isbetter,however,touse TACTIC Blandreduce first: 1rh1XlX5 4 x-Uf=4xlx2 .J- 2 1 5 8 360 576' 258NewGRE TACTIC m When aproblemrequiresyoutofinda fractionof anumber,multiply. 4 If -7 ofthe 350 sophomores atMonroeHighSchoolare girlsand!_ of '8 themplayonateam,howmany sophomore girlsdonotplay onateam? SOLUTION. 4 There are7 X350=200sophomore girls. 7 Of these,S X200=175playonateam.So,200- 175=25donotplayon ateam. The reciprocal of any nonzeronumber xis that number y such that xy =1.Since ( 1)1. xx= 1,thenxlsthereciprocalof x.Similarly,thereciprocalof thefraction a.h1: b.ab -b1sterractwn-, smce-. - =1. aba KEY FACT813 Tod i v i d ~any number by a fraction,multiply that num