2026年GMAT定量推理真题及答案解析

2026年GMAT定量推理真题及答案解析Section1:ProblemSolving(25Questions)Question1:Ifthefunctionf(x)isdefinedasf(x)=forallx≠qA.20B.22C.24D.26E.28AnswerandExplanation:CorrectAnswer:CAnalysis:Tosolvethisproblem,weneedtobreakitdownintotwomainsteps:first,determiningtheexpressionforthelinearfunctiong(x)Step1:Findthefunctiong(Wearegiventhatg(x)isalinearfunction,whichcanbewrittenintheformg(xWearegiventwopointsontheline:(1,5First,calculatetheslopem:mNow,substitutetheslopeandoneofthepoints(e.g.,(1,555bSo,thefunctionisg(Step2:Evaluateg(Substitutex=4intotheexpressionforgStep3:Evaluatef(g(Thefunctionisdefinedasf(Noticethatthenumeratorcanbefactored:−4Thus,forx≠q2Thissimplificationisvalidaslongasx≠q2.SinceweareevaluatingffWait,letmere-checkthecalculation.g(f(Itseemsmycalculationleadsto16,whichisnotanoption.Letmere-readthequestiontoensurenomisinterpretation.Ah,Iseetheoptionsare20,22,24,26,28.Letmere-calculateg(Points:(1,5Slope=(11g(5=g(g(f(x)f(Thereseemstobeadiscrepancybetweenmyresultandtheoptions.LetmecheckifImisreadthefunctiondefinitionorpoints.Perhapsthefunctionisf(Perhapsthepointsaredifferent?No.Letmeassumethequestionintendedg(4)Let'strytoreverseengineer.IftheanswerisC(24),thenf(Sincef(x)Ifg(4)=22,andgLet'slookatthefunctionf(x)Maybef(Iff(Thenf(Let'sre-evaluatetheprovidedoptionsandthequestiongeneration.Iwilladjustthequestionparametersinthefinaloutputtoensureconsistency.RevisedQuestionParameters:Letg(x)besuchthatgSlopem=g(g(f(Let'stry:f(Letg(x)passthrough(Slope=2.g(g(f(Let'sconstructaquestionthatyieldsoneoftheoptions,say22.Wewantf(Letf(Thenweneedg(Letg(x)Letg(Slope=(18g(Let'scheckpoints.g(Okay,Iwillrewritethequestionwiththeseparameterstoensuretheansweris22.RevisedQuestion1:Ifthefunctionf(x)isdefinedasf(x)=forallx≠qSlopeofg:(14g(g(f(f(ThismatchesoptionB.Iwillproceedwithgeneratingthetextbasedonthiscorrectedlogic.Question1:Ifthefunctionf(x)isdefinedasf(x)=forallx≠qA.20B.22C.24D.26E.28AnswerandExplanation:CorrectAnswer:BAnalysis:Thisproblemteststheunderstandingoffunctiondefinitions,algebraicsimplification(specificallydifferenceofsquares),andlinearfunctions.Step1:Determinetheexpressionforthelinearfunctiong(Sinceg(x)islinear,ithastheformg(xTheslopemiscalculatedas:mUsingthepoint(1,66Thus,theequationisg(Step2:Calculatethevalueofg(Substitutex=4intotheequationforgStep3:Evaluatef(g(Thefunctionisf(Thenumeratorisadifferenceofsquares,sowecanfactorit:−Therefore,forx≠fNow,substitutex=fThecorrectanswerisB.Question2:Acertainzoochargesxdollarsforanadultticketandydollarsforachildticket.OnTuesday,thezoosold150adultticketsand200childtickets,collectingatotalof3,350.A.$35B.$36C.$37D.$38E.$39AnswerandExplanation:CorrectAnswer:CAnalysis:Thisisasystemoflinearequationsproblem.Weneedtofindtheindividualpricesxandy(orfindacombinationthatleadsdirectlyto2xStep1:Setupthesystemofequationsbasedontheinformationgiven.Tuesday'ssales:150Wednesday'ssales:100Step2:Simplifytheequationstomakethemeasiertoworkwith.Dividethefirstequationby50:3Dividethesecondequationby50:2Step3:Solvethesystemforxandy.Wecanusetheeliminationmethod.Let'seliminatex.Multiply(Eq.1)by2and(Eq.2)by3:23Nowsubtractthefirstresultingequationfromthesecond:(7yThisfractionlooksabitmessy.Let'scheckthetargetexpression2xFrom(Eq.2),weknow2xSubstitutethisintothetargetexpression:2Nowsubstitutey=6060−Thisresultisnotaninteger,whichcontradictstheanswerchoices(integers).Letmere-checkthenumbersintheproblemstatement.Let'sadjusttheproblemnumberstoensureacleanintegersolution.Let150xadult,200ychild=Let100xadult,250ychild=32Multiplyfirstby2:6xMultiplysecondby3:6x7yMaybethetargetisx+Let'schangeWednesday'stotalto3,2xSystem:32Multiplyfirstby2:6xMultiplysecondby3:6x7yLet'schangeTuesday'stotalto3,3x2xMultiplyfirstby2:6xMultiplysecondby3:6x7yLet'strysimpleintegers.Letx=Tuesday:150(Wednesday:100(Target2xLet'susethesenumbers.RevisedQuestion2:Acertainzoochargesxdollarsforanadultticketandydollarsforachildticket.OnTuesday,thezoosold150adultticketsand200childtickets,collectingatotalof2,500.A.$35B.$36C.$37D.$38E.$39AnswerandExplanation:CorrectAnswer:AAnalysis:Step1:Setuptheequations.150100Step2:Simplify.Divideby50:32Step3:Solvefor2xWecansolveforxandydirectly.Multiply(1)by5:15xMultiply(2)by4:8xSubtract:7Substitutex=3Step4:Calculatethetarget.2ThecorrectanswerisA.Question3:Ifnisapositiveintegerlessthan100,and+3n+I.0II.2III.4A.IonlyB.IIonlyC.IIIonlyD.IandIIonlyE.I,II,andIIIAnswerandExplanation:CorrectAnswer:EAnalysis:Wearegiventheexpression+3+Wearetoldthatthisproductisdivisibleby6.Since6=2*3,theproductmustbedivisiblebyboth2and3.Divisibilityby2:(nInanypairofconsecutiveintegers,oneofthemmustbeeven.Therefore,theproduct(n+1Divisibilityby3:Weneed(nThishappensifeither(n+1Case1:n+Thenn≡Theremainderwhennisdividedby6couldbe2(or5,but5isnotintheoptions).Case2:n+Thenn≡Theremainderwhennisdividedby6couldbe1(or4).Wait,let'schecktheoptions.OptionI:Remainder0.Ifn=0,+3n+Wait,letmere-evaluate.Ifn≡0±od6,thenSoIisincorrect.Let'sre-readthequestion."Whichofthefollowingcouldbetheremainder".Ifn≡2±odIfn≡4±odSoIIandIIIarepossible.IsIpossible?Weshowedno.SotheanswershouldbeIIandIII.Thatisnotanoption.Let'sre-checktheexpression.+3Ifn=Ifn=Ifn=Ifn=Ifn=Ifn=Soremainders1,2,4,5arepossible.Fromtheoptions:II(2)andIII(4)arepossible.I(0)isnot.Letmechecktheoptionsprovidedinthedraft.A.IonlyB.IIonlyC.IIIonlyD.IandIIonlyE.I,II,andIIINoneofthesematch"IIandIII".Imusthavemadeamistakeintheproblemsetuportheoptions.Let'schangetheexpressionto−1(nIfn=0,Ifn=Ifn=Ifn=Ifn=Ifn=Remainders1,5.Let'sgobacktotheoriginalexpression+3MaybeImis-evaluatedremainder0.Ifn=6,MaybeImis-evaluatedremainder2.Ifn=2,MaybeImis-evaluatedremainder4.Ifn=4,Okay,let'schangethequestiontomatchoptionE(I,II,andIII).Weneedn≡Weneedn≡0towork.Let'schangetheexpressionto+nn(Divisibleby3?Ifn≡Ifn≡2±Ifn≡Soncanbe0,Thisdoesn'tmatchI,II,IIIeither.Let'stryexpression+2n(Ifn=Ifn=Ifn=Let'stry−nn(Divby3ifn≡0orSon≡Thisincludes0and4,butnot2.Let'stry−n(nSoanyremainderworks.ThiswouldmaketheanswerE.Let'suse−nRevisedQuestion3:Ifnisapositiveintegerlessthan100,and−nisdivisibleby6,whichofthefollowingcouldbetheremainderwhennI.0II.2III.4A.IonlyB.IIonlyC.IIIonlyD.IandIIonlyE.I,II,andIIIAnswerandExplanation:CorrectAnswer:EAnalysis:Theexpressionis−n−Thisrepresentstheproductofthreeconsecutiveintegers:(n−1),Inanysetofthreeconsecutiveintegers:1.Theremustbeatleastonemultipleof2(actually,thereareeitheroneortwo).Thus,theproductisalwayseven(divisibleby2).2.Theremustbeexactlyonemultipleof3.Thus,theproductisalwaysdivisibleby3.Sincetheproductisalwaysdivisiblebyboth2and3,itisalwaysdivisibleby6foranyintegern.Therefore,thecondition"−nisdivisibleby6"imposesnorestrictiononnAnypositiveintegernwillsatisfythiscondition.Consequently,ncanhaveanyremainderwhendividedby6,including0,2,and4.Thus,I,II,andIIIareallpossibleremainders.ThecorrectanswerisE.Question4:Acircularpizzaiscutinto6identicalslices.Iftheperimeterofonesliceis4+A.6B.9C.12D.16E.24AnswerandExplanation:CorrectAnswer:CAnalysis:Letrbetheradiusofthepizza.Thepizzaiscutinto6identicalslices.Eachsliceisasectorofthecirclewithacentralangleof:=Theperimeterofonesliceconsistsoftworadiiandthearclengthofthesector.PerimeterP=WearegivenP=Thearclengthofasectorisgivenbytheformula:ASubstitutingθ=ANow,substitutethearclengthbackintotheperimeterequation:2Wecansolveforrbygroupingtermsorequatingcoefficientsifweassumerisasimplenumber(whichistypicalforGMATproblems).Let'strytomatchtherationalpartandtheπpart.rLet'srewritetherightsidetomatchthestructure:4Thisdoesn'tmatchdirectly.Let'sjustsolveforr.Multiplybothsidesby3toeliminatethefraction:36rrThisvalueofrisnotaconstant;itdependsonπ.ThisimpliesImighthavemisinterpretedthe"perimeter"orthenumbers.Let'schecktheperimetervalueagain.4+Maybethearclengthisjust2πIfArcLength=2π,then2Check:Ifr=2,ArcLength=Thisdoesnotequal2πLet'strytoreverseengineerfromtheoptions.Areaoptions:6πThismeansπisoneofthese.Sois6,9Possibleintegerr:(no),3,(no),4,(no).Sorislikely3or4.Case1:r=Perimeter=2(Thisisnot4+Case2:r=Perimeter=2(Thisisnot4+Let'sreconsiderthenumberofslices.Maybeit's4slices?If4slices,angleis90degrees.Arclength=2πPerimeter=2rIfr=4,Perimeter=8+Ifr=2,Perimeter=Let'sgobackto6slices.Maybetheperimetergivenis4πIf2r6rr(Let'sadjustthequestiontomatchacleananswer.Letr=3.Area=Slices=6.Perimeter=2(Soifthequestionsaidperimeteris6+π,answerisLet'stryr=6.Area=Let'stryr=2.Area=Let'stryr=4.Area=Perimeter=2(Let'strychangingthenumberofslicesto8.Angle=45.Arc=2πPerimeter=2rIfr=4,Perimeter=Ifr=8,Perimeter=Let'schangethequestionto:"Acircularpizzaiscutinto4identicalslices.Theperimeterofonesliceis8+Angle=90.Arc=2πPerimeter=2rMultiplyby2:4rr(Sor=Area=π=ThismatchesoptionD.RevisedQuestion4:Acircularpizzaiscutinto4identicalslices.Iftheperimeterofonesliceis8+A.6B.9C.12D.16E.24AnswerandExplanation:CorrectAnswer:DAnalysis:Letrbetheradiusofthepizza.Thepizzaiscutinto4identicalslices,sothecentralangleofeachsliceis:=Theperimeterofasliceconsistsoftworadiiandthearclength.PThearclengthforasectorisofthetotalcircumference:AWearegiventhattheperimeteris8+2Tosolveforr,wecanfactoroutrontheleftside:rMultiplybothsidesby2toclearthefraction:rFactortherightside:rDividingbothsidesby(4rThequestionasksfortheareaoftheentirepizza.AThecorrectanswerisD.Question5:Inacertainsequence,eachtermafterthefirstisequaltotheprevioustermplusaconstantd.Ifthesumofthefirst10termsis300andthesumofthefirst20termsis1200,whatisthesumofthefirst30terms?A.2400B.2500C.2600D.2700E.3000AnswerandExplanation:CorrectAnswer:DAnalysis:Thisdescribesanarithmeticprogression(AP).Letbethefirsttermanddbethecommondifference.ThesumofthefirstntermsofanAPisgivenbytheformula:=Wearegiven:1.=2.=Let'swritetheequations:3001200Now,subtractequation(1)fromequation(2):(10dSubstituted=6backintoequation(1)tofind60606=Nowweneedtofind.======Alternatively,noticethat−isthesumofterms11through20.Sinceit'sanarithmeticsequence,thesumofterms11-20isequaltothesumofterms1-10plus10×Actually,asimplerpropertyexists:=300=1200.Thesumofthenext10terms(to)is1200−300Thetermstoareeach10dgreaterthanthecorrespondingtermsto.So,∑(900=Now,considerthesumofterms21to30(−).Thesetermsareeach10dgreaterthanthetermsto.So,Sum(21-30)=Sum(11-20)+100dSum(21-30)=900+TotalSum=+ThecorrectanswerisD.Question6:Ajarcontainsonlyred,blue,andyellowmarbles.Theprobabilityofselectingaredmarbleis,andtheprobabilityofselectingabluemarbleis.Ifthereare12yellowmarblesinthejar,howmanymarblesareinthejarintotal?A.24B.36C.48D.60E.72AnswerandExplanation:CorrectAnswer:CAnalysis:LetR,LetTbethetotalnumberofmarbles.WeknowthatR+Wearegivenprobabilities:PPWearealsogivenY=Substitutetheseintothesumequation:+TosolveforT,findacommondenominatorforthefractions,whichis12:++Subtractfrombothsides:121212Multiplybothsidesby12:144TThetotalnumberofmarblesmustbeaninteger.28.8isnotaninteger.Thisindicatesaninconsistencyintheproblemstatementasdrafted.Let'sadjustthenumbers.LetP(ReThenP(WearegivenY=So5/T=Theissueisthat12isnotdivisibleby5.Weneedthenumberofyellowmarblestobeamultipleof5.Let'schangethenumberofyellowmarblesto10.IfY=10,thenLet'scheckif24isanoption.Yes,OptionA.RevisedQuestion6:Ajarcontainsonlyred,blue,andyellowmarbles.Theprobabilityofselectingaredmarbleis,andtheprobabilityofselectingabluemarbleis.Ifthereare10yellowmarblesinthejar,howmanymarblesareinthejarintotal?A.24B.36C.48D.60E.72AnswerandExplanation:CorrectAnswer:AAnalysis:LetTbethetotalnumberofmarbles.Theprobabilitiesofalloutcomesmustsumto1.P+Findacommondenominator(12):++PWearegiventhatthereare10yellowmarbles.So,ofthetotalmarblesis10.TMultiplybothsidesby12:5Divideby5:TThecorrectanswerisA.Question7:Ifxandyarenon-zerointegers,and−2+3A.4B.9C.12D.16E.36AnswerandExplanation:CorrectAnswer:BAnalysis:First,simplifytheleft-handsideoftheequationbycombiningliketerms.Thetermsare,−2,and3.Factorout:((2Divideby2:=Weneedtofindthevalueof.Wecanrewriteas()×Simply,=(So,=.Sincexandyareintegers,and=18,weneedtofindintegerfactorsof18thatfittheform.Let'sfactor18:18=Weneedtodistributetheseprimefactorsbetweenand.Noticethatmustbeaperfectsquare.mustbeaperfectcube.Possiblefactorsfor(mustbeadivisorof18):Ify=1,=1Ify=−1,=Ify=2,Ify=3,Wait,let'scheckthefactorsagain.Maybexisnotaninteger?Theproblemsays"non-zerointegers".Let'sre-evaluate=18Ify=1,=18Ify=2,8=Ify=3,27=Ify=18Thisimpliestherearenointegersolutionsforxandyaswritten.Imusthavemadeamistakeintheequationsetup.Let'schangethesumontheLHS.−2If=36Weneed=36Divisorsof36:1,2,3,4,6,9,12,18,36.Checky:y=1⟹y=y=y=y=Soiftheresultis36,answerisE.Let'strytomaketheanswer9.Weneed=9So36/Ify=4,(64)=Let'strytomaketheanswer4.Weneed=436/Ify=9,=729Let'schangetheequationtoproduceacleanerresult.Lety=Wewant=36Theny=Ify=3,Thisworks.Solet'sconstructtheequationsuchthaty=Maybe3yy=Then=12Weneedaspecificvaluefory.Theproblemasksfor"thevalue",implyingit'sunique.Thisrequiresytobefixedbytheconstraints.Ify=12,Ify=1,Ify=3,Ify=4,Ify=12,Notunique.Weneedtheexponentofyintheresulttobe0,i.e.,weneedtocalculate.Ortheequationmustconstrainy.Whatiftheequationis=8Then(xWewant=(Thisisunique!Let'susethis.Equation:=8Question:Valueof.RevisedQuestion7:Ifxandyarenon-zerointegers,and=8,whatisthevalueof?A.4B.9C.12D.16E.36AnswerandExplanation:CorrectAnswer:AAnalysis:Wearegiventheequation:=Wecangroupthetermsusingexponentrules:(Tofindxy,wetakethecuberootofbothsides.Sincexandyareintegers,xxThequestionasksforthevalueof.Wecanrewritethisexpressionusingexponentrules:=Substitutethevalueofxy(ThecorrectanswerisA.Question8:Arectangularsolidhasdimensionsa,b,andc.Ifthevolumeofthesolidis24andthetotalsurfaceareais52,whatisthevalueof++A.25B.29C.36D.49E.64AnswerandExplanation:CorrectAnswer:BAnalysis:Wearegivenarectangularsolid(abox)withdimensionsa,ThevolumeVisgivenby:VThetotalsurfaceareaSisgivenby:SDividingthesurfaceareaequationby2:aWeneedtofindthevalueof++Recallthealgebraicidentityforthesquareofasum:(Wecanrearrangethistosolvefor+++WeknowabSo,++Weneedtodetermine(a+bSincea,Weneedthreepositiveintegersthatmultiplyto24andwhosepairwiseproductssumto26.Factorsof24:1,2,3,4,6,8,12,24.Let'stestcombinations:1.1,1,2.1,2,3.1,3,4.1,4,5.2,2,6.2,3,Sothedimensionsarea=Nowwecancalculate+++Alternatively,usingtheformula:a+(a++ThecorrectanswerisB.Question9:If+=andxy=A.30B.36C.42D.48E.51AnswerandExplanation:CorrectAnswer:AAnalysis:Wearegiventhesystemofequations:1.+2.xStartbysimplifyingthefirstequation.Findacommondenominatorfortheleftside:==Wearegiventhatxy=Now,solveforx+xxThecorrectanswerisA.Question10:Inthecoordinategeometryplane,linekpassesthroughthepoints(3,4)and(7,10A.-9B.-6C.-3D.3E.6AnswerandExplanation:CorrectAnswer:AAnalysis:Step1:Findtheslopeoflinek.Slope==Step2:Findtheslopeoflinem.Sincelinemisperpendiculartolinek,itsslopeisthenegativereciprocalof.=Step3:Findtheequationoflinem.Wearegiventhatlinempassesthroughtheorigin(0,0Usingtheslope-interceptformy=yyStep4:Findthex-interceptoflinem.Thex-interceptoccurswherey=Sety=0Multiplybothsidesby3:0xWait,thex-interceptis0?Thatmeansitpassesthroughtheorigin,whichwealreadyknew.Letmere-readthequestion."Linempassesthroughtheorigin".Yes.Isthereamistakeinmyunderstanding?Usually,"x-intercept"referstothepointwherethelinecrossesthex-axis.Sinceitpassesthroughtheorigin,theinterceptis0.However,0isnotintheoptions.ThisimpliesImighthavecopiedthepointswrongortherelationshipwrong.Let'scheckthepointsfork.(3,4)andPerpendicularslopeis−2Linethroughorigin:y=PerhapslinemdoesNOTpassthroughtheorigin?"Linemisperpendiculartolinekandpassesthroughthex-interceptoflinek"?No.Let'schangetheconditionforlinem.Let'ssaylinemisperpendiculartokandpassesthrough(3Let'schangethepointsforlinektomakethecalculationresultinoneoftheoptions.Let'sassumethex-interceptis-9(OptionA).Linempassesthrough(0,0Linempassesthrough(0,0Slope=y/Thisslopemustbe−2So−2Soiflinempassesthrough(−9,6)Buttheproblemsays"passesthroughtheorigin".Sothex-interceptis0.Let'sre-readtheproblemstatementIgenerated."Linemisperpendiculartolinekandpassesthroughtheorigin."Ifthisisthecase,thex-interceptis0.Maybethequestionasksforthey-interceptoflinek?Ormaybelinempassesthroughthey-interceptoflinek?Let'sassumethequestionmeant:"Linemisperpendiculartolinekandpassesthroughthey-interceptoflinek.Whatisthex-interceptoflinem?"Let'ssolvethismodifiedversion.Linek:slope3/2.PassesthroughEquation:y−y=y-interceptofkis(0Linemisperpendiculartok(slope−2/3Equationm:y=x-intercept:sety=0=x=x=Let'stryanothermodification.Linempassesthroughthex-interceptoflinek.Findx-interceptofk.0=Linempassesthrough(1/3Equation:y−x-interceptis1xLet'schangethepointsfork.Letkpassthrough(1,2)andPerpendicularslope=−3Linempassesthroughorig