2026年GRE《数学》Verbal真题及答案

2026年GRE《数学》Verbal真题

Section1

1.

QuantityA:

QuantityB:12

A.QuantityAisgreater.

B.QuantityBisgreater.

C.Thetwoquantitiesareequal.

D.Therelationshipcannotbedeterminedfromtheinformationgiven.

2.

Ifxisanintegergreaterthan1and=,whatisthevalueofx?

A.2

B.3

C.4

D.5

E.6

3.

Acertaincomputerstoresellsonlylaptopsandtablets.Theratiooflaptopstotabletsinthestoreis5:3.Ifthestorehas40morelaptopsthantablets,howmanytabletsdoesthestorehave?

A.40

B.50

C.60

D.80

E.100

4.

QuantityA:Thestandarddeviationoftheset10,10,10,10,10

QuantityB:Thestandarddeviationoftheset8,9,10,11,12

A.QuantityAisgreater.

B.QuantityBisgreater.

C.Thetwoquantitiesareequal.

D.Therelationshipcannotbedeterminedfromtheinformationgiven.

5.

Inthexy-plane,thelinewithequationy=3x−4intersectsthelinewithequationy=−x+batthepoint(2,k).Whatisthevalueofb?

A.2

B.4

C.6

D.8

E.10

6.

Acircleisinscribedinasquarethathasanareaof144.Whatistheareaofthecircle?

A.12π

B.24π

C.36π

D.48π

E.144π

7.

Ifaandbarepositiveintegerssuchthat−=45anda>b,whichofthefollowingcouldbethevalueofa?Indicateallsuchvalues.

A.7

B.9

C.12

D.23

E.24

8.

Arectangulartankisfilledwithwatertoadepthof4feet.Ifthebaseofthetankmeasures6feetby8feet,whatisthevolumeofthewaterinthetank,incubicfeet?

A.96

B.144

C.192

D.288

E.384

9.

QuantityA:(−3

QuantityB:−

A.QuantityAisgreater.

B.QuantityBisgreater.

C.Thetwoquantitiesareequal.

D.Therelationshipcannotbedeterminedfromtheinformationgiven.

10.

TheprobabilitythateventRwilloccuris0.45.TheprobabilitythateventSwilloccuris0.6.IfeventsRandSareindependent,whatistheprobabilitythatbothRandSwilloccur?

A.0.15

B.0.27

C.0.30

D.0.45

E.1.05

11.

Forallnumbersxandy,letx⋄ybedefinedas−2xy+.Whatisthevalueof(3⋄5)−(5⋄3)?

A.-16

B.-8

C.0

D.8

E.16

12.

Alistofnumbershasameanof20andastandarddeviationofd.Ifeachnumberinthelistisincreasedby5,whichofthefollowingstatementsistrue?

A.Themeanis20andthestandarddeviationisd.

B.Themeanis25andthestandarddeviationisd.

C.Themeanis25andthestandarddeviationisd+5.

D.Themeanis20andthestandarddeviationisd+5.

E.Themeanis25andthestandarddeviationcannotbedeterminedfromtheinformationgiven.

13.

IntriangleABC,themeasureofangleAisandthemeasureofangleBis.ThesideoppositeangleBhaslength10.WhatisthelengthofthesideoppositeangleC?

A.10

B.10

C.10

D.20

E.Itcannotbedeterminedfromtheinformationgiven.

14.

If=3andx+y=16,whatisthevalueofx−y?

A.4

B.6

C.8

D.10

E.12

15.

QuantityA:Thenumberofprimenumbersbetween50and60

QuantityB:Thenumberofprimenumbersbetween70and80

A.QuantityAisgreater.

B.QuantityBisgreater.

C.Thetwoquantitiesareequal.

D.Therelationshipcannotbedeterminedfromtheinformationgiven.

16.

Afactoryproduces500unitsofaproductperhour.Eachunitrequires2minutesofmachinetimeand5minutesoflabortime.Ifthefactoryoperatesfor8hours,howmanytotalminutesmachinetimeareconsumed?

A.1000

B.2000

C.4000

D.4800

E.8000

17.

Ifnisapositiveinteger,whatistheremainderwhenn(n+1)(n+2)isdividedby3?

A.0

B.1

C.2

D.Itcannotbedeterminedfromtheinformationgiven.

18.

Thegraphofthefunctionfisaparabolathatopensupward.Theminimumvalueoff(x)is-5,andthisminimumoccursatx=2.Whichofthefollowingcouldbetheequationforf?

A.f(x)=(x−2−5

B.f(x)=(x+2−5

C.f(x)=(x−2+5

D.f(x)=−(x−2−5

E.f(x)=−(x+2+5

19.

Ajarcontains4redmarbles,3bluemarbles,and2greenmarbles.Twomarblesaredrawnatrandomwithoutreplacement.Whatistheprobabilitythatbothmarblesarered?

A.

B.

C.

D.

E.

20.

QuantityA:15of30of400

QuantityB:30of15of400

A.QuantityAisgreater.

B.QuantityBisgreater.

C.Thetwoquantitiesareequal.

D.Therelationshipcannotbedeterminedfromtheinformationgiven.

Section2

Questions21to25refertothefollowinggraphs.

[GraphDescription1:Abarcharttitled"AnnualRevenueofCompanyX(2018-2023)".Thex-axisshowsyears2018through2023.They-axisshowsRevenueinmillionsofdollars.Thedatapointsare:2018:50,2019:65,2020:60,2021:80,2022:95,2023:110.]

[GraphDescription2:Apiecharttitled"RevenueDistributionbyDepartmentfor2023".Thechartisdividedintofoursectors:Electronics(40%),HomeAppliances(25%),Software(20%),Services(15%).]

21.

ForwhichyearwasthepercentincreaseinannualrevenueofCompanyXfromthepreviousyearthegreatest?

A.2019

B.2020

C.2021

D.2022

E.2023

22.

In2023,iftherevenuefromtheElectronicsdepartmentwas$44million,whatwasthetotalannualrevenueofCompanyX?

A.$100million

B.$110million

C.$120million

D.$130million

E.$140million

23.

Ifthetotalannualrevenuein2022was$95millionandtherevenuedistributionbydepartmentwasthesameasin2023,whatwastherevenue,inmillionsofdollars,fromtheSoftwaredepartmentin2022?

A.19.0

B.23.75

C.24.0

D.25.5

E.28.5

24.

Whatistheaverage(arithmeticmean)annualrevenueofCompanyXfortheyears2018through2023?

A.72.5

B.75.0

C.76.7

D.78.3

E.80.0

25.

In2019,therevenuefromtheHomeAppliancesdepartmentaccountedfor30%ofthetotalrevenue.IftherevenuefromtheHomeAppliancesdepartmentin2019was$19.5million,whatwasthetotalrevenuein2019?

A.$58.5million

B.$62.0million

C.$65.0million

D.$68.5million

E.$70.0million

26.

Ifthesumofthefirstntermsofasequenceisgivenby=+2n,whatisthe5thtermofthesequence?

A.11

B.13

C.15

D.17

E.35

27.

Arightcircularcylinderhasaheightof10andavolumeof250π.Whatisthecircumferenceofthebaseofthecylinder?

A.5π

B.10π

C.15π

D.25π

E.50π

28.

If|x−3|=5,whatarethepossiblevaluesofx?Indicateallsuchvalues.

A.-8

B.-2

C.2

D.8

E.10

29.

QuantityA:x

QuantityB:y

[DiagramDescription:AtrianglePQRisshown.AnglePQRisarightangle.SidePQisvertical,sideQRishorizontal.PointSliesonsidePR.AlineisdrawnfromQtoS.AnglePQSislabeledandangleSQRislabeled.]

A.QuantityAisgreater.

B.QuantityBisgreater.

C.Thetwoquantitiesareequal.

D.Therelationshipcannotbedeterminedfromtheinformationgiven.

30.

Aresearcherconductsasurveyof200people.70%ownacar,and40%ownabicycle.If25%ofthepeopleownbothacarandabicycle,howmanypeopleownneitheracarnorabicycle?

A.10

B.20

C.30

D.40

E.50

AnswerKeyandExplanations

1.Answer:A

Explanation:SimplifyQuantityA.

Findacommondenominatorforthenumerator:

+=+=

Nowdivideby:

QuantityAis12.QuantityBis12.Wait,letmerecheckthecalculation.

×27=4×3=12.

Let'sre-readthequestion.

QuantityA:

=1/3

=1/9

Sum=4/9

=1/27

(4/9)/(1/27)=(4/9)*27=4*3=12.

QuantityAis12.QuantityBis12.

SotheanswershouldbeC.

Self-Correction:Imadeamistakeintheinitialthoughtprocessorthequestiondesignintendedforatrick.Let'scheckthemathagain.

=1/3.=1/9.Sum=3/9+1/9=4/9.

=1/27.

(4/9)/(1/27)=(4/9)*27=4*3=12.

Thequantitiesareequal.

Correction:TheanswerkeysaysAinmydraftthought,butcalculationsaysC.LetmechangethequestiontomakeAgreaterortheanswertoC.

Let'sassumethequestioniscorrectaswrittenandtheanswerisC.

Wait,let'slookattheprompt.Igeneratedthequestion.Ishouldensurethequestionisrobust.

Let'schangeQuantityAto:.

Then(4/9)/(1/81)=(4/9)*81=4*9=36.

ThenA>B.

However,ImuststicktothequestionIwroteintheoutputblockifIwerejustanswering.ButIamgeneratingthetest.

Let'scheckthemathforthespecificquestionwrittenintheoutput:

Question1:.

Numerator:1/3+1/9=4/9.

Denominator:1/27.

Result:4/9*27=12.

QuantityBis12.

AnswerisC.

Wait,IwroteAnswer:Ainthedraftkey.IwillcorrecttheAnswerKeytoC.

Actually,let'smodifythequestionslightlyinthefinaloutputtoavoidtrivialequalityifIwant,orjustkeepit.EqualityisavalidGREquestion.IwillkeepthequestionasisandprovideAnswerC.

Re-evaluatingthedrafttext:Thedrafttexthas"Answer:A".Iwillcorrectitinthefinaloutputgenerationto"Answer:C".

2.Answer:B

Explanation:Rewritetheequationsobothsideshavethesamebase.

=(

=

Sincethebasesarethesame,settheexponentsequal:

x+3=3(x−1)

x+3=3x−3

6=2x

x=3

3.Answer:C

Explanation:LetLbethenumberoflaptopsandTbethenumberoftablets.

WearegiventheratioL:T=5:3,soL=T.

WearealsogivenL=T+40.

Substitutethefirstequationintothesecond:

T=T+40

Multiplyby3toclearthefraction:

5T=3T+120

2T=120

T=60

Thestorehas60tablets.

4.Answer:B

Explanation:Standarddeviationmeasuresthespreadofthedata.

QuantityA:Thesetis10,10,10,10,10.Allvaluesarethesame,sothereisnospread.Thestandarddeviationis0.

QuantityB:Thesetis8,9,10,11,12.Thevaluesarespreadoutaroundthemeanof10.Thestandarddeviationisgreaterthan0.

Therefore,QuantityBisgreater.

5.Answer:C

Explanation:Thepoint(2,k)liesontheliney=3x−4.Substitutex=2tofindk:

k=3(2)−4=6−4=2

Sothepointofintersectionis(2,2).

Thispointalsoliesontheliney=−x+b.Substitutex=2andy=2tofindb:

2=−2+b

b=4

6.Answer:C

Explanation:Theareaofthesquareis144,sothesidelengthofthesquareis=12.

Thecircleisinscribedinthesquare,sothediameterofthecircleisequaltothesidelengthofthesquare.

Diameterd=12,sotheradiusr==6.

Theareaofthecircleisπ=π()=36π.

7.Answer:B,D

Explanation:Factorthedifferenceofsquares:

−=(a−b)(a+b)=45

Weneedtofindpairsofpositiveintegerswhoseproductis45.Thefactorsof45are1,3,5,9,15,45.

Possiblepairsfor(a−b,a+b)are:

1.(1,45)

2.(3,15)

3.(5,9)

Sinceaandbarepositiveintegersanda>b,botha−banda+bmustbepositive,anda+b>a−b.

Wecansolveforaineachcaseusingthesystem:

a−b=facto

a+b=facto

Addingtheequations:2a=facto+facto⇒a=.

Case1:a===23.(Valid)

Case2:a===9.(Valid)

Case3:a===7.(Valid)

Sothepossiblevaluesforaare7,9,and23.

Lookingattheoptions:A(7),B(9),C(12),D(23),E(24).

ThecorrectchoicesareA,B,andD.

Wait,letmechecktheoptionsprovidedinthequestiontextIdrafted.

Draftoptions:A.7,B.9,C.12,D.23,E.24.

SotheanswerisA,B,D.

8.Answer:C

Explanation:ThevolumeofarectangularprismisV=length×width×height.

Here,thebaseis6×8andthedepthofwater(height)is4.

V=6×8×4=48×4=192

Thevolumeis192cubicfeet.

9.Answer:A

Explanation:Becarefulwithorderofoperations.

QuantityA:(−3.Here,-3israisedtothe4thpower.Sincetheexponentiseven,theresultispositive.(−3=(−3)×(−3)×(−3)×(−3)=81.

QuantityB:−.Withoutparentheses,theexponentappliesonlyto3.Thenthenegativesignisapplied.−=−()=−81.

81>−81,soQuantityAisgreater.

10.Answer:B

Explanation:Forindependentevents,theprobabilityofbothoccurringistheproductoftheirindividualprobabilities.

P(RandS)=P(R)×P(S)

P(RandS)=0.45×0.6=0.27

11.Answer:C

Explanation:First,simplifytheoperatordefinition.

x⋄y=−2xy+=(x−y

Nowcalculatethetwoquantities.

QuantityA:3⋄5=(3−5=(−2=4.

QuantityB:5⋄3=(5−3=(2=4.

Thequestionasksfor(3⋄5)−(5⋄3)=4−4=0.

12.Answer:B

Explanation:Addingaconstanttoeverynumberinasetshiftsthemeanbythatconstantbutdoesnotchangethespread(standarddeviation).

Originalmean=20.Newmean=20+5=25.

Originalstandarddeviation=d.Newstandarddeviation=d.

Thus,themeanis25andthestandarddeviationisd.

13.Answer:A

Explanation:Inanytriangle,thesideoppositethelargerangleislonger.

TheanglesareA=,B=.

Thesumofanglesis,soangleC=−(+)=.

Comparingtheangles:B()>A()>C().

Therefore,thesidesoppositethemsatisfyb>a>c.

Wearegivensideb=10(oppositeangleB).

Weareaskedforsidec(oppositeangleC).

Sinceb>c,sidecmustbelessthan10.

Wait,lookingattheoptions:A.10,B.10,C.10,D.20,E.Cannotbedetermined.

Basedonthelogic"sideoppositelargerangleislonger",weknowc<10.NoneofthespecificnumbersA,B,C,Darelessthan10(exceptifweconsidercalculationerrors).

Let'sre-readthequestion.

A=60,B=70.SideoppositeBis10.

C=50.

WeneedsideoppositeC.Let'scallitx.

ByLawofSines:=.

x=10.

Sincesin()<sin(),x<10.

Theoptionsprovidedare10,10,etc.Allare≥10.

Thissuggeststhereisaflawinthequestionoroptionsasdrafted.

Let'schangeAngleAtoandAngleBto.

ThenAngleCis.

SideoppositeBis10.WewantsideoppositeC.

AngleB>AngleC,sosideb>sidec.10>c.

Stillthesameissue.

Let'stry:AngleA=30,AngleB=60.ThenAngleC=90.

SideoppositeBis10.WewantsideoppositeC.

AngleC>AngleB,sosidec>sideb.c>10.

Options:10,10≈14.1,10≈17.3,20.

Let'scalculateexactly.30-60-90triangle.

Sideopposite30isa.Sideopposite60isa.Sideopposite90is2a.

Givensideopposite60is10.Soa=10⇒a=10/.

Sideopposite90is2a=20/.Notanoption.

Let'strystandard45-45-90.A=45,B=45.Sideb=10.Sidec=10.AnswerA.

Let'stry30-30-120.A=30,B=30,C=120.Sideb=10.Sidecisopposite120.

Lawofsines:10/sin(30)=c/sin(120).10/0.5=c/(/2).20=2c/.c=10.

ThismatchesoptionC.

RevisionforQuestion13:ChangeanglestoA=,B=.SideoppositeBis10.FindsideoppositeC.

ThenC=.UsingLawofSines,c=10.

Iwillusethisversionforthefinaloutput.

14.Answer:C

Explanation:Substitutex=3yintothesecondequation.

3y+y=16

4y=16

y=4

Nowfindx:

x=3(4)=12

Calculatex−y:

12−4=8

15.Answer:C

Explanation:QuantityA:Primenumbersbetween50and60.

51(divby3),52(even),53(Prime),54(even),55(divby5),56(even),57(divby3),58(even),59(Prime).

Thereare2primes:53,59.

QuantityB:Primenumbersbetween70and80.

71(Prime),72(even),73(Prime),74(even),75(divby5),76(even),77(divby7),78(even),79(Prime).

Thereare3primes:71,73,79.

Wait,QuantityBisgreater.

Letmerecheck.

50-60:53,59.Count=2.

70-80:71,73,79.Count=3.

SoB>A.

Letmecheck50-60again.53,59.Correct.

Letmecheck70-80again.71,73,79.Correct.

SoanswerisB.

Letmeadjustthequestiontomakethemequalorcorrecttheanswerkey.

Let'schangerangeBto80and90.

80-90:83,89.Count=2.

ThenA=B.

RevisionforQuestion15:ChangeQuantityBrangeto"between80and90".ThenAnswerisC.

16.Answer:D

Explanation:Thefactoryproduces500unitsperhourfor8hours.

Totalunits=500×8=4000units.

Eachunitrequires2minutesofmachinetime.

Totalmachinetime=4000×2=8000minutes.

Wait,lookingatoptions:A.1000,B.2000,C.4000,D.4800,E.8000.

Mycalculationis8000.ThatisoptionE.

Let'sre-read."Eachunitrequires2minutesofmachinetimeand5minutesoflabortime."

"howmanytotalminutesmachinetimeareconsumed?"

4000units×2min/unit=8000minutes.

Okay,answerisE.

Let'schangethequestiontomakeitmoreinterestingorcheckifImisread.

Maybe"Eachunitrequires2minutesofmachinetime...".

Okay,Iwillstickwith8000.TheanswerisE.

17.Answer:A

Explanation:Theproductn(n+1)(n+2)representsthreeconsecutiveintegers.

Inanysetofthreeconsecutiveintegers,theremustbeonemultipleof3.

Proof:Integersmodulo3are0,1,2.

Ifn≡0±od3,productisdivisibleby3.

Ifn≡1±od3,thenn+2≡0±od3.Productdivisibleby3.

Ifn≡2±od3,thenn+1≡0±od3.Productdivisibleby3.

Sincetheproductisalwaysdivisibleby3,theremainderisalways0.

18.Answer:A

Explanation:Thevertexformofaparabolaisf(x)=a(x−h+k,where(h,k)isthevertex.

Wearegiventheminimumvalueis-5atx=2.Sothevertexis(2,−5).

Sinceitopensupward,amustbepositive.

Substituteh=2,k=−1intothevertexform:f(x)=a(x−2−5.

Lookingattheoptions:

A.f(x)=(x−2−5.Herea=1(positive),vertex(2,−5).Matches.

B.f(x)=(x+2−5.Vertex(−2,−5).Incorrect.

C.f(x)=(x−2+5.Vertex(2,5).Incorrect.

D.f(x)=−(x−2−5.Opensdownward(a=−1).Incorrect.

E.f(x)=−(x+2+5.Opensdownward.Incorrect.

AnswerisA.

19.Answer:A

Explanation:Totalmarbles=4+3+2=9.

Probabilityoffirstmarblebeingred:P()=.

Afterdrawingonered,thereare3redmarblesleftand8totalmarbles.

Probabilityofsecondmarblebeingred:P()=.

Probabilityofboth:P(and)=×=.

Simplifythefraction:=.

20.Answer:C

Explanation:Multiplicationiscommutativeandassociative.

QuantityA:15.

QuantityB:30.

Thetwoexpressionsareidentical.

21.Answer:A

Explanation:Calculatethepercentincreaseforeachyearusing.

2019:==30.

2020:=−≈−7.7(Decrease).

2021:=≈33.3.

2022:==18.75.

2023:=≈15.8.

Comparingtheincreases:33.3%(2021)vs30%(2019).

Thegreatestincreaseisin2021.

Correction:Thecalculationshows2021isgreaterthan2019.

Let'scheckthemathagain.

2019:15/50=0.30.

2021:20/60=1/3=0.333...

So2021istheanswer.

Wait,lookingatmydraftanswerkeyIputA(2019).IwillcorrecttheanswerkeytoC(2021).

22.Answer:B

Explanation:ThepiechartshowsElectronicsis40%ofthetotalrevenuein2023.

LetTbethetotalrevenue.

0.40×T=44million.

T===44×2.5=110million.

23.Answer:A

Explanation:Totalrevenuein2022was$95million.

Theproblemstatestherevenuedistributionwasthesameasin2023.

Fromthepiechart,Softwareaccountsfor20%oftherevenue.

RevenuefromSoftware=0.20×95million.

95×0.2=19million.

24.Answer:C

Explanation:Sumtherevenuesanddivideby6.

Sum=50+65+60+80+95+110.

Sum=115+60+80+95+110=175+80+95+110=255+95+110=350+110=460.

Average===76.666...

Thisroundstoapproximately76.7.

25.Answer:C

Explanation:In2019,HomeApplianceswas30%oftotalrevenue.

0.30×Total=19.5million.

Total===19.5×=6.5×10=65million.

(Check:30of65is0.3×65=19.5.Correct.)

26.Answer:A

Explanation:Then-thtermofasequencecanbefoundby=−.

Wewantthe5thterm,.

=−.

Calculate:+2(5)=25+10=35.

Calculate:+2(4)=16+8=24.

=35−24=11.

27.Answer:B

Explanation:VolumeofacylinderisV=πh.

GivenV=250πandh=10.

250π=π(10).

Dividebyπ:250=10.

=25⇒r=5.

ThecircumferenceofthebaseisC=2πr.

C=2π(5)=10π.

28.Answer:B,D

Explanation:Theequation|x−3|=5meansx−3=5orx−3=−5.

Case1:x−3=5⇒x=8.

Case2:x−3=−5⇒x=−2.

Thepossiblevaluesare-2and8.

Options:B(-2)andD(8).

29.Answer:A

Explanation:IntrianglePQR,∠PQR=.

PointSisonthehypotenusePR.

ThesegmentQSisdrawntothehypotenuse.

Thereisageometricpropertythatthealtitudetothehypotenusecreatestwosmallertrianglesthataresimilartotheoriginaltriangleandtoeachother.

However,wecanuseasimplerapproachinvolvingcomparisonoflengthsorangles.

Actually,withoutspecificcoordinatesorsidelengths,wecannotdeterminetheexactvaluesofxandy.

Wait,isthereapropertyI'mmissing?

Inarighttriangle,thealtitudetothehypotenuseformstwoangles.

Let'sl