真题gre数学讲义1偶数或奇数

oddnumber奇 evennumber偶偶数＝偶数＋偶数或奇数＋奇数，奇数＝奇数＋偶数偶数＝偶数³偶数或奇数³偶数，奇数＝奇数³奇数偶数：2n（n为整数）n个连续的偶数相加等于零，则n为奇数相邻整数nn+1肯定是一奇一偶。整数n和整数n+2的奇偶性是一样的。Ifaandbarepositiveintegerssuchthata–banda/barebothevenintegers,whichofthefollowingmustbeanoddinteger?ab(a+b)(a+2)(b+2)（多选）Ifxandyareintegers2yxx^2y^2whichofthefollowingmustbeaneveninteger?xy2x+y^2+y^2+ （factor，divisor数就叫作N12=1*12=2*6=3*。、、、、、都是12的因数。质数（primenumber1N进行分解因数，如果除了1和它自身外，不能被其他自然数整除，那N就是质数。合数（compositenumber1的自然数N进行因factorinteger，primenumber等词，都要意识数值要可能需要进行分解质因数。个5。Ifyisthesmallestpositiveintegersuchthat3,150multipliedbyyisthesquareofaninteger,thenymustbe(A) (B) (C) (D) (E)Ifw,x,y,andzareintegerssuchthat1<w<x<y<zandwxyz=462,thenz(A) (B) (C) (D) (E)Ifxistheproductofthepositiveintegersfrom1to8,inclusive,andifi,k,m,andparepositiveintegerssuchthatx=2𝑖3𝑘5𝑚7𝑝,theni+k+m+p= (B) (C) (D) (E)12m2*3*5*7*11，所以12!9Ifp,s,andtarepositiveprimenumbers,p3st=728,whatisthevalueofp3s3t3思路：7282^3*7答案：2^3*7^3A. B. C. D. E.思路：606265，即n6265（正整数nn为完全平方数23是唯一的一对同为质数的相邻自然数。Apositiveintegerwithexactlytwodifferentdivisorsgreaterthan1mustaprimeanevenamultipleofthesquareofaprimethesquareofanoddIfxandyareprimenumbers,whichofthefollowingCANNOTbethesumofxandy? (B)13 (C)14 (D)16 (E)23Apositiveintegernissaidtobe“prime-saturated”iftheproductofallthedifferentpositiveprimefactorsofnislessthanthesquarerootofn.Whatisthegreatesttwo-digitprime-saturatedinteger?(A) (B) (C) (D) (E)比如：8=2*2*2，28的一个因数。若Howmanyfactorsdoes360 (D) (E)Whatisthegreatestprimefactorof2^100- 【解析】2^100-2^96=2^96*(2^4-1)=（2^96）*3*5,因此正确答案为CIftheintegernhasexactlythreepositivedivisors,including1andn,howmanypositivedivisorsdoesn^2have? 答案：63numberx^2factorsx>3Ifn=4p,wherepisaprimenumbergreaterthan2,howmanydifferentpositiveevendivisorsdoesnhave? 因子共有4个。 连续正整数consecutivepositiveIfnisanintegergreaterthan6,whichofthefollowingmustbedivisiblebyn(n+1)(n–n(n+2)(n–n(n+3)(n–n(n+4)(n–n(n+5)(n–n=m（m+4)(m+5)n3，4，6Ifnisapositiveinteger,2isnotafactorofn,3isnotafactorofn.ristheremainderwhen(n-1)(n+1)isdividedby24,whatisthevalueofr? 最大公约数greatestcommon例题：12与18例题：4和64646IfMistheleastcommonmultipleof90,196and300,whichofthefollowingisNOTafactorofM?n能被abn一定能够被a和bn812整除，n24整除； 用整数来表示余数：N/a=b„„3——N=a*b+3（余数为用小数来表示余数：N/a=b.14——N=a*b.14=a*b 0.14a）余数是非负整数（大于等于0的整数Whenpositiveintegerxisdividedbypositiveintegery,theremainderis9. Ifx/y=96.12,whatisthevalueofy? 0~59之间，120，12，24，36，48，60 因此概率P 考查点2：整除的表达 N/4=2a+ Whenpositiveintegerxisdividedby5,theremainderis3;andwhenxisdividedby7,theremainderis4.Whenpositiveintegeryisdividedby5,theremainderis3;andwhenyisdividedby7,theremainderis4.Ifx>y,whichofthefollowingmustbeafactorofx-y?(A) (B) (C) (D) (E)为b。则𝑚𝑛的个位数和𝑎𝑏的个位数相同。（333^777）/5的余数和（777^333）/53除以10余数是多少？求2345+1onesdigit？505393942542581258125A.B.C.D.E.可能的六位数中，能够被8整除的六位数的概率是多少Y=3时，3128=39，可以整除；Y=5时，5128=64，可以整除；Y=8时，812÷8不能整除。3-1001003答案：3*33=99，-3*33=-99，所以有66个，再加上0就是67个。(0可以整除任 答案67个。-45x997（98+42）÷7+1=x+y51，y+z52，x+z53,x+y+z5解析：x+y5a+1，y+z5b+2，x+z5c+3（a、b、c表示≥0的任意整数x+y+z=（x+y+y+z+x+z）/2==（5a+1+5b+2+5c+3）/2因为6是偶数，则5（a+b+c）是偶数，即a+b+c是偶数x+y+z（5a+5b+5c+6）/25/2所以（x+y+z）/5=5/2（ab）3=

*（a+b+c）+52

5215211~215+（2： decimal小 fraction分numerator分 denominator分考查点1：数位例题子：7654.3217：4：units(orones)3：tenths2：hundredths1：thousandthsdigit：s的是整数位（小数点前面的某位ths的是分位（小数点后面的某位。如：tenstenth是十Ifthetensdigitxandtheunitsdigityofapositiveintegernarereversed,theresultingintegeris9morethann.Whatisyintermsofx?10–9–x+x–x+Thefunctionfisdefinedforeachpositivethree-digitintegernbyf(n)=2x3y5z,wherex,yandzarethehundreds,tens,andunitsdigitsofn,respectively.Ifmandvarethree-digitpositiveintegerssuchthatf(m)=9f(v),thenm-v=89 例题：roundtotens保留至十位（即将个位化为0）roundupto rounddownto tothe Onarecenttrip,Cindydrovehercar290,roundedtothenearest10,andused12gallonsofgasoline,roundedtothenearestgallon.TheactualnumberofpergallonthatCindy’scargotonthistripusthavebeenbetween

Thesidesofasquareregion,measuredtothenearestcentimeters,are6centimeterslong.Theleastpossiblevalueoftheactualareaofthesquareregionis36.00sq35.00sq33.75sq30.25sq25.00sq =4.86*8,245,000=8.245* 1050–74iswrittenasanintegerinbasedecimalnotation,whatisthesumofthedigitsinthatinteger?Ifnisapositiveintegerandk=5.1*10n,k2=2.601*109,whatisthevalueof1：有限小数terminatingWhichofthefollowingfractionshasadecimalequivalentthatisaterminating

=（大-小）/小=（大-小）/大Attheendofthefirstquarter,thesharepriceofacertainmutualfundwas20percenthigherthanitwasatthebeginningoftheyear.Attheendofthesecondquarter,thesharepricewas50percenthigherthanitwasatthebeginningoftheyear.Whatwasthepercentincreaseinthesharepricefromtheendofthefirstquartertotheendofthesecondquarter?50%199𝑝∗550;40𝑝0 =(040110;=𝑝=(1+50%)10−(150%)10−Inacorporation,50percentofthemaleemployeesand40percentoftheemployeesareatleast35yearsold. If42percentofalltheemployeesareatleast35yearsold,whatfractionoftheemployeesinthecorporationarefemales?InTownX,64percentofthepopulationareemployed,and48percentofthepopulationareemployedmales.WhatpercentoftheemployedpeopleinTownXarefemales?2%sale设 price为xList=retail*（1+2%）=1.02xSale=retail*（1–5%）=𝑥𝑥 *100%=Thepositivenumbersw,x,y,andzaresuchthatxis20percentgreaterthany,y20percentgreaterthanz,andwis20percentlessthanx. Whatpercentgreaterthanzisw?W:X:Y=1：3：4Y:Z=3：53teaspoons=14tablespoons=1/4Accordingtotheinformationabove,howmanyteaspoonsareequalto3/8349Inacertaingame,theunitsofcurrencyofthreecountriesarethecrown,theshield,andthepound,respectively.If2crownsequal3shieldsand3shieldsequal4pounds,howmanycrownsequal18pounds?9OnJuly1oflastyear,thetotalnumberofemployeesatEwasdecreasedby10percent.Withoutanychangeinthesalariesoftheremainingemployees,theaverage(arithmeticmean)employeesalarywas10percentmoreafterthedecreaseinnumberofemployeesthanbeforethedecrease.ThetotalofthecombinedsalariesofalloftheemployeesatEafterJuly1lastyearwaswhatpercentofthatbeforeJuly1lastyear?设原来有剩下的：1/2*2/3*3/4=4 =（a+b（a-平方和：a^2+b^2=（a+b）^2–2ab或（a-b）^2+完全平方：a^2±2ab+（a+b（a^2b^2；立方差：a^3-b^3=（a-b（a^2+ab+b^2） 𝑎𝑚*𝑎𝑛=𝑎𝑚*𝑏𝑚=(𝑎∗(𝑎𝑚)𝑛=𝑎𝑚÷𝑎𝑛=

𝑎 ÷

=(= =指数𝑎𝑛a>0，则𝑎𝑛0Ofthefollowingvaluesofn,thevalueof(-1/5)-nwillbegreatestfor02336x=8100，求33x-36 3x 33x-3=33x=90=3=(3)=8100 3 （（5））=𝑘6kk𝑘6 Ifkisanintegerand(0.0025)(0.025)(0.00025)×10kisaninteger,whatistheleastpossiblevalueofk?

06 √𝑥2=|𝑥|（如果是未知数开根号，一定要注意加绝对值√𝑎*√𝑏=√𝑎÷√𝑏=√𝑎±√𝑏≠√𝑎±根号有可能估值，如：2<√5=3√8==3 √27==3 √64==3 √125==3 √216==1+1+（2

） 2x-10=√𝑥√10xx-10√𝑥)2-(√10)2，而(√𝑥)2-(√10)2=（√𝑥√10（√𝑥√10）=√𝑥所以√𝑥√101，√𝑥√102x=（√10+1）=2√10答案：2√10 求（1√2√3）（√2√3） （𝟏+√𝟐+√𝟑）−（√𝟐+ =1+2（√𝟐+√𝟑）+（√𝟐+√𝟑）−（√𝟐+√𝟑）=1+2√𝟐+1+2√𝟐 X是负数，求√（x−3）+√−𝑥∗|𝑥| √（x−3）+√−𝑥∗|𝑥|=|x−3|+x<03-x-x3-3- 若a<b，b<c，则a<c。如果a<b，那么a+c<b+c，a-c<b-c；如果aba+cb+c，a-cb-若a>b，c>0，则a*c>b*c；a*b>0，a>0，则b>0。Ifx<0and0<y<1,whichofthefollowinghasthegreatest (B)(

(E)If–4<x<7and–6<y<3,whichofthefollowingspecifiesallthepossiblevaluesof–24<xy<–28<xy<–42<xy<–24<xy<–42<xy<Ifkisanintegerand2<k<7,forhowmanydifferentvaluesofkisthereatrianglewithsidesoflengths2,7,andk？ 边，使另一边为0，再解。0x前的系数为正数，分解因式。x^2-x>2x^2-x-2>0，转化为(x-2)(x+1)>0包含x的因数的次数若为偶数，则不穿过，即奇过偶不过（1）𝑥 （2）−𝑥2+3x+10<（1）3𝑥1 （2）𝑥2𝑥7≥ 𝑥2Whichofthefollowingdescribesallvaluesofxforwhich1—𝑥2≥0？(A)x≤—1orx≥1(D)x≤—(E—1≤x≤13：𝑥2

>答案：-3<x<-2L1：𝑥2-x-6，L2：𝑥2–4。2

0x𝑥262

=𝑥)𝑥

>即（x-3）（x+2）（x+2）（x-2）即（x-3）（x-2）(𝑥+2)2>(𝑥2)20，则（x-3）（x-2）0x2x答案：x2或x（数值比较m>0andn>0,m<n,x>0A:𝑚xB:（数值比较p=r,A:B:5x+3y15，3x+5y15 若|x—a|>b，则x—a>b或x—a<—b。练习 性质2：| |=||+||，说明X和Y同号 Ifthesumoftwopositiveintegersis24andthedifferenceoftheirsquaresis48,whatistheproductofthetwointegers?(A) (B) (C) (D) (E)T8，D答案：-62ABC三个人，AB一起生产产品，43000个产品；BC一起生产产品，53500个产品，AC一起生产产品，62700个产品。求A一个人做1000个产品要多少小时？c4（a+b）=5（b+c）=6（a+c）=可解出：a250，b500，c2001000÷250=4小时4Twopositivenumbersdifferby12andtheirreciprocalsdifferby

4.Whatis52

5

一元二次方程：a𝑥2bxc0如果𝑏24𝑎𝑐≥0 x= 2

=

*𝑥2=a（x ：项数=𝑎𝑚;𝑎𝑛+Ifthesumof7consecutiveintegersis434,thenthegreatestofthe7integers(A) (B) (C) (D) (E)Markboughtasetof6flowerpotsofdifferentsizesatatotalcostof$8.25.Eachpotcost$0.25morethanthenextonebelowitinsize.Whatwasthecost,indollars,ofthelargestpot?V1思路：n=50，a1=1，a50=99。S=2

=V2思路：n=100，a1=1，a100=199，S=2

==50*50=例题Inthesequence1,2,4,8,16,32,...,eachtermafterthefirstistwicethepreviousterm.Whatisthesumofthe16th,17th,and18thtermsinthesequence?(A) (B) (C) (D) (E)Thesequencea1,a2,…,anissuchthatan=2an-1-Xforallpositiveintegersn≥2andcertainnumberX.Ifa5=99anda3=27,whatisthevalueofX? (E)45 ，求 + ） 𝑎1+𝑎2+𝑎3

=1

1+„„+

=

Forallrealnumbersv,theoperationv*isdefinedbytheequationv*=v-1/3v.If(v*)*=8,thenv= (B) (C) (D) (E)mmForallpositiveintegers =3mwhenmisodd =1/2mwhenmismm69Whichofthefollowingisequivalent 69定义：r*=1+1r*（1+1）* r*（1+1（𝑟 =（1+1）（1+1 =（1+1）（1+𝑟 =1 =2

1

答案：2+ 12341234ABD+∠BDE的和是多少？CBDB 180°，因为∠C35°，所以∠CBD∠CDB180°—35°=因为∠ABD180°—∠CBD，∠BDE180所以∠ABD+∠BDE180°—∠CBD+180°—∠CDB360°—=360°—145°=polygon多边形：triangle三角形quadrilal四边形pentagon五边形 AA obtuseangle钝角acuteangle锐角rightangle直角（3）直角三角形：a^2b^2=c^2锐角三角形：a^2b^2c^2钝角三角形:a^2+b^2<c^2 （4）sin30°= sin45°=√ sin60°=2（5）=2

**=2

=2=3=PQR中，QP3，PR4QR=2

*QP*PR

*QR*2QR=55Inthefigureabove,squareCDEFhasarea4.Whatistheareaof△ABF FFC 例题Ifeachsideof△ACDabovehaslength3andifABhaslength1,whatistheareaofregionBCDE? E例题Intherectangleabove,AB=6,AD=8.IfpointPisselectedfromsegmentBCatrandom,whatistheprobabilitythatthelengthofsegmentAPislessthan

例题Aladder25feetlongisleaningagainstawallthatisperpendiculartolevelground.Thebottomoftheladderis7feetfromthebaseofthewall.Ifthetopoftheladderslipsdown4feet,howmanyfeetwillthebottomoftheladderslip?(A) (B) (C) (D) (E)5所以，x= 53-9= 5 矩形perimeter=（+宽）*长area=*长对角线diagonal= 2+宽正方形=*2=（2）=上底*2+侧边（3）=*梯形（2）面积：（+下底）**2菱形（2）=*Thefigureshowsasquarepatiosurroundedbyawalkwayofwidthxmeters. ofthewalkwayis132squaremetersandthewidthofthepatiois5metersgreaterthanthewidthofthewalkway,whatistheareaofthepatio,insquaremeters?(A) (B) (C) (E)S直三角*4+S=S大设直角三角形长直角边=x，则短直角边=x-1，所以[x(x-1)/2]³4+1=25（或者根据勾股定理）解出x=4 答案：12 circle圆 center圆心 radius半径 考查点1：圆的周长和面积一个小白鼠在一个半径为2

feet240feet2

feetc2π*r2*2

*π=

Inthefigureshown,iftheareaoftheshadedregionis3timestheareaofthesmallercircularregion,thenthecircumferenceofthelargercircleishowmanytimesthecircumferenceofthesmallercircle?432有三个同心圆：C1、C2C3，C1是里面最小的圆。C1的面积=C2的面积—C1的面积=C3的面积—C2的面积，C2100πC1C3的半径（知道半径的比例，即使不知体数值，也可以求出来周长比150π。C1的半径是√505√2C3的半径是√150=5√2：5√6=2=2=π（2𝑟）–2*π𝑟2=2π𝑟=+2π*2r2*2πr8πr=r一个传送带上有A、B、C共3个，逆时针转动，可以知道A、B、C的C1.5A2³π³4=8π，6/8=3/4A2³π³6=12π，6/12=1/2弦chord：连接圆上任意两点的线段。 弧arc：圆上两点之间的圆周部分。性质1：直径对应的圆周角是90°性质 ：𝑛πr（n是圆心角 ：𝑛π𝑟2（n是圆心角O，OB、OC是半径。∠BOC130°，∠BAD50°。问∠ABD的大∠BDC=∠BOC*2

=∠BDA=180°-所以∠ABD180°115°50°SS长方形S半圆10*6–2

604.5πS圆S圆：S阴影 60

120答案：120O中，AC是直径，ABBC3√2根据直角三角形的性质，AC=6答案：9π正方形边长1，以正方形左上方和右下方的顶点为圆心作两个14 π-一个扇形面积41/242（2×π-1）=π-（

4 𝜋-2Inthecircleabove,PQisparalleltodiameterOR,andORhaslength18.WhatisthelengthofminorarcPQ?

O

RInthefigureabove,Aisthepointoftangencyfortwocirclesandalsothecenterofthethirdcircle.Iftheradiiofthreecirclesare1,whatistheexternalperimeteroftheA A apolygonisinscribedina acircleiscircumscribedabouta aInthefigureabove,acircleisinscribedinasquarewithsidebandasquarewithsideaisinscribedinthecircle.a=4,Whatistheareaofthesquarewithsideb?ab rectangularsolidabc2*（ab+ac+bc）正方体cube：体积=𝑎3，表面积=6𝑎2cylinder=π𝑟2*hAcylinderhas2congruentcircles等圆anda 球sphere：体积=4π𝑟3，表面积 3（x+2(y+2)(z+2)体积分别是：(3/π)^2*π*1090/π150/π634答案：600÷（1-1）=7200，7200*1= S16个圆柱体小包装盒在大包装盒中所占的空间比例为C。问SC哪个大，大多少？V长 2∗16

S=

2 CgreaterthanS=4𝜋66CS 已知点A（𝑥1,𝑦1），点B（𝑥2,𝑦2） |=√（𝑥1−𝑥2）+（𝑦1−（根据平面直角坐标系求两点距离的，可以求出对角线的长度 √（−1−3）+（3+1）=√32=x轴、y轴、x6、y4围成了一个矩形，在该矩形中，xy4xy=4xy44∗4

3直线方程:y=kx+bkslope斜率by-intercept纵截距。斜率的计算：k=𝑦1;𝑥1;

=直角三角形的对边ylyl2-0Thegraphofwhichofthefollowingequationsisastraightlinethatisparalleltolinelinthefigureabove?3y–2x=3y+2x=3y—2x=2y–3x=2y+3x=-3由于C也在该直线上，带入y=-5即可求得a=-2x04x+3y=48y16即直线：4x+3y=48y轴的交点是（0,16）y04x+3y=48x12即直线：4x+3y=48y轴的交点是（12,0）所以三角形面积=16*12÷2=96例题有一条直线L1，它经过直线L2：y0.5x1的x轴截距，并且与L1垂直。求另答案：y=2x–4例题例题Inthefigureabove,pointsPandQlieonthecirclewithcenterO.Whatisthevalueofs?1抛物线的方程y=a𝑥2bxca决定开口方向=−说明在y轴左侧）抛物线是否与x轴相交：Δ=𝑏24acΔ=0:抛物线与x轴有一个交点；Δ<0:抛物线与x轴没有交点；Δ>0:抛物线与x轴有两个交点。对称的表示：x-axisisperpendicularbisectorofsegmentAB；BisthereflectionofAaboutthex-axis；AandBaresymmetricaboutthex-点A（a,b）关于原点的对称点：（-a,-b）。例题Intherectangularcoordinatesystemabove,theliney=xisperpendicularbisectorsegmentAB(notshown),andthex-axisistheperpendicularbisectorofsegmentBC(notshown).IfthecoordinatesofpointAare(2,3),whatarethecoordinatesofpointC?(–3,(–3, (2,(3,(2,x 单利(SimpleInterests)=+本金*利率*例题Maryinvested$14,000for3yearsinaofdepositpaying9.25%simpleannualinterest.HowmanymoreinterestwouldMaryhavereceivediftheinterestrateonthishadbeen9.75%simpleannualinterest?例题A2-yearofdepositispurchasedforkdollars.Iftheearnsinterestatanannualrateof6percentcompoundedquarterly,whichofthefollowingrepresentsthevalue,indollars,oftheattheendofthe2(A) (B) (C) (D) （二）Work例题Asmallwaterpumpwouldtake3hourstofillanempty.Alargepumpwouldtake2hourstofillthesame.Howmanyhourswouldittakebothpumps,workingattheirrespectiveconstantrates,tofilltheemptyiftheybeganpumatthesametime?(A) (B) (C) (D) (E)例题Sixmachines,eachworkingatthesameconstantrate,togethercancompleteacertainjobin12days.Howmanyadditionalmachines,eachworkingatthesameconstantrate,willbeneededtocompletethejobin8days?(A) (B) (C) (D) (E)Whileworkingaloneattheirrespectiveconstantrates,computerXprocesses240filesin4hoursandcomputerYprocesses240filesin8hours.Ifallfilesprocessedbythesecomputersarethesamesize,howlongwouldittakethetwocomputers,workingatthesametimeandattheirrespectiveconstantrates,toprocessatotalof240files?22hr202hr4066hr20 R机器一小时的效率是完成1的任务量，S机器一小时的效率是完成1 两台机器共同运转的效率是每小时：1+1= 2小时候，完成了10=5的任务量，还剩1 答案：60A，B，A23个，B59个，A、B个，AB多花/A每一个花2/3分 B每一个花5/9分A450个需要时间：450*2/3300B450个需要时间：450*5/9250分钟所以AB多花了50分钟AB50=时间*IttakesCarlos9minutestodrivefromhometoworkatanaveragerateof22perhour.HowmanyminuteswillittakeCarlostocyclefromhometoworkalongthesamerouteatanaveragerateof6perhour?CirculargearsPandQstartrotatingatthesametimeatconstantspeeds.GearPmakes10revolutionsperminute,andgearQmakes40revolutionsperminute.HowmanysecondsafterthegearsstartrotatingwillgearQhavemadeexactly6morerevolutionsthangearP?68P11060/10=6秒/Q14060/40=1.5秒/次T秒之后，QP6圈(T/6)=(T/1.5)-6Susandroveatanaveragespeedof30perhourforthefirst30ofaandthenatanaveragespeedof60perhourfortheremaining30ofthetrip.Ifshemadenostopsduringthetrip,whatwasSusan'saveragespeed,inperhour,fortheentiretrip?Aboattraveledupstreamadistanceof90 atanaveragespeedof(v-3)perhourandthentraveledthesamedistancedownstreamatanaveragespeedof(v+3) perhour.Ifthetripupstreamtookhalfanhourlongerthanthetripdownstream,howmanyhoursdidittaketheboattotraveldownstream?

一个人的家距10 他前10min以15 /hour的速度开车，接下来的10min以昨天的速度开，问还要开多远才能到？ ÷20min= 15/h=0.2510-（10³0.25+10³0.5）=2.5一辆car用30分钟开了20 ，其中有x 用的速度是60 路用的速度是30 /h，问走x 130min2

10=1=10 答案：10Afterdrivingtoariverfrontparkinglot,Bobnstorunsouthalongtheriver,turnaround,andreturntotheparkinglot,runningnorthalongthesamepath.Afterrunning3.25south,hedecidestorunforonly50minutesmore.IfBobrunsataconstantrateof8minutesper,howmanyfarthersouthcanherunandstillbeabletoreturntotheparkinglotin50minutes?8x+（3.25+x）*8=例题1：AB地，乙车出现了一些故障，所以甲车先开，甲车的速答案：4hours

出发地到喷泉的路程：10*1= 3

÷6==5-1（10分钟）=2= Threegradesofmilkare1percent,2percent,and3percentfatbyvolume.Ifxgallonsofthe1percentgrade,ygallonsofthe2percentgrade,andzgallonsofthe3percentgradearemixedtogivex+y+zgallonsofa1.5percentgrade,whatisxintermsofyandz?y+2y+3y+r:0.4x+0.25y，b=0.6x，f=

0.3y

=一辆卡车的燃油箱能装15加仑的天然气，在城市里行驶平均每 车在城市里行驶60，在高速公 的天然气，平均走多少？ 加仑用了60 加仑用了

52一共用了（12+

1= BgallonB耗油—A405/27405/361511.25例题1：Anauctionhousechargesacommissionof15percentonthefirst$50,000ofthesalepriceofanitem,plus10percentontheamountofthesalepriceinexcessof$50,000.Whatwasthesalepriceofapaintingforwhichtheauctionhousechargedatotalcommissionof$24,000?theamountofthesalepriceAlice'stake-homepaylastyearwasthesameeachmonth,andshesavedthesamefractionofhertake-homepayeachmonth.Thetotalamountofmoneythatshehadsavedattheendoftheyearwas3timestheamountofthatportionofhermonthlytake-homepaythatshedidnotsave.Ifallthemoneythatshesavedlastyearwasfromhertake-homepay,whatfractionofhertake-homepaydidshesaveeachmonth?12*a*x=3*(1–a)*xThetollforcrossingacertainbridgeis$0.75eachcrossing.Driverswhofrequentlyusethebridgemayinsteadpurchaseastickereachmonthfor$13.00andthenpayonly$0.30eachcrossingduringthatmonth.Ifaparticulardriverwillcrossthebridgetwiceoneachofxdaysnextmonthandwillnotcrossthebridgeonanyotherday,whatistheleastvalueofxforwhichthisdrivercansavemoneybyusingtheM315岁。5yearsfromnowhisageis3timesasoldashisdaughter。Howmanyyearsfromnowon,hisagewillbetwiceashisdaughter?即今年M34岁，他女儿8岁。设从今年开始n年，他 ，34+n=2³（8+n，n=18所以，18年后，M是他女儿 的2倍A、 B、 C、 E、答案：B 知识点1：两个圆圈IAB（A∩B）+非A非Accordingtoasurvey,93percentofteenagershaveusedacomputertoygames,89percenthaveusedacomputertowritereports,and5percenthavenotusedacomputerforeitherofthesepurposes.Whatpercentoftheteenagersinthesurveyhaveusedacomputerbothtoygamesandto writereports?4040人当中：有数学的，也有英语学位的。有数学人占了50%15%，两个都没有的5%，求单独有英语的有多少人？答案：1835+-20人会说英语，24人会说法语，20+24-30=14 P 答案：全集I=A+B+C-（A∩B）-（A∩C）-（B∩C）+（A∩B∩C）+非非B非全集I=++三个特点都符合 A∩B=+Inamarketingsurveyforproductssomepeoplewereaskedwhichoftheproducts,ifany,theyuse.Ofthepeoplesurveyed,atotalof400useA,atotalof400useB,atotalof450useC,atotalof200useAandBsimultaneously,atotalof175useBandCsimultaneously,atotalof200useCandAsimultaneously,atotalof75useA,B,andCsimultaneously,andatotalof200usenoneoftheproducts.Howmanypeopleweresurveyed?选择S课程的有18人。核对发现每个学生都至少选择了一门课程，同时选（E∩M--5033s24z29bs两种分–（5+3）–（6+3）+3=Ofthe300subjectswhoparticipatedinanexperimentusingvirtual-realitytherapytoreducetheirfearofheights,40percentexperiencedsweatypa,30percentexperiencedvomiting,and75percentexperienceddizziness.Ifallofthesubjectsexperiencedatleastoneoftheseeffectsand35percentofthesubjectsexperiencedexactlytwooftheseeffects,howmanyofthesubjectsexperiencedonlyoneofthese例题展示1：Ashipmentofbannerscontainsbannersoftwodifferentshapes,triangularandsquare,andtwodifferentcolors,redandgreen.Inaparticularshipment26%ofthebannersaresquareand35%ofthebannersarered.If60%oftheredbannersintheshipmentaresquare,whatistheratioofredtriangularbannerstogreentriangularbanners?7

One-fifthofthelightswitchesproducedbyacertainfactoryaredefective.Four-fifthsofthedefectiveswitchesarerejectedand1/20ofthenondefectiveswitchesarerejectedbymistake.Ifalltheswitchesnotrejectedaresold,whatpercentoftheswitchessoldbythefactoryaredefective?20%*80%=80%*5%=20%-16%=100%-20%=4%÷80%=1/20=< 组合：m个元素（总体数量）n个元素（目标数量）组成一组。用C（combination）表示。𝐂𝐧 !排成一排。用P（permutation）表示。 !从甲地经乙、丙两地到，共有多少种不同的走法？加法原理：n类办法，在第一类办法中有m1m2种不同的方法，„„，在第n类办法中有mn种不同的方法，那么完成这件事共有N=m1+m2+m3+„+mn种不同方法。m2种不同的方法，„„，做³m2³m3³„³mn种不同的方法。n个步骤，步与步之间是连续的，只入。Thereare10childrenina 'sday-carecenter,andapairofchildrenistobeselectedtoyagame.Atmost,howmanydifferentpairsarepossible?(A) (B) (C) (D) (E)Jill,wholivesinCityC,nstovisit3differentcities,M,L,andS.ShenstovisiteachcityexactlyonceandreturntoCityCafterthe3visits.Shecanvisitthecitiesinanyorder.Inhowmanydifferentorderscanshevisitthe3(A) (B) (C) (D) (E)33Ifeachofthe12teamsparticipatinginacertaintournamentysexactlyonegamewitheachoftheotherteams,howmanygameswillbeyed?(A) (B) (C) (D) (E)解析：𝐶2=66Johnhas5friendswhowanttorideinhisnewcarthatcan modateonly3passengersatatime.Howmanydifferentcombinationsof3passengerscanbeformedfromthe5friends?385解析：𝐶3=105Acertainclubhas20members.Whatistheratioofthememberof5-membercommitteesthatcanbeformedfromthemembersoftheclubtothenumberof4-membercommitteesthatcanbeformedfromthemembersoftheclub?(A)16to (B)15to (C)16to (D)15to (E)5to解析：𝐶5：𝐶4= hasassignedadistinct3-digitcodenumbertoeachofits330employees.Eachcodenumberwasformedfromthedigits：2,3,4,5,6,7,8,9andnodigitappearsmorethanonceinanyonecodenumber.Howmanyunassignedcodenumbersarethere?(A) (B) (C) (D) (E)88Ifacodewordisdefinedtobeasequenceofdifferentletterschosenfromthe10lettersA,B,C,D,E,F,G,H,I,andJ,whatistheratioofthenumberof5-lettercodewordstothenumberof4-lettercodewords?(A)5to (B)3to (C)2to (D)5to (E)6to解析 = Eachparticipantinacertainstudywasassignedasequenceof3differentlettersfromtheset{A,B,C,D,E,F,G,H}.Ifnosequencewasassignedtomorethanoneparticipantandif36ofthepossiblesequenceswerenotassigned,whatwasthenumberofparticipantsinthestudy?(Note,forexample,thatthesequenceA,B,CisdifferentfromthesequenceC,B,A.)(A) (B) (C) (D) (E)88有0.1，0.001，0.0001，0.000001，0. 五个数值，Iftwoormorenumbersaddedtogether,howmanydifferentnumberswecanget？5两个数相加的可能性：𝐶2=1055三个数相加的可能性：𝐶3=55四个数相加的可能性：𝐶4=55五个数相加的可能性：𝐶5=5总共的可能性：10+10+5+1Acertainemploys6seniorofficersand4juniorofficers.Ifacommitteeistobecreatedthatismadeupof3seniorofficersand1juniorofficer,howmanydifferentcommitteesarepossible? (B) (C) (D) (E)解析：𝐶3*𝐶1=20*4 Themembershipofacommitteeconsistsof3Englishteachers,4Mathematicsteachers,and2SocialStudiesteachers.If2committeemembersaretobeselectedatrandomtowritethecommittee’sreport,whatistheprobabilitythatthetwomembersselectedwillbothbeEnglishteachers?(A)2/3(B)1/3(C)2/9(D)1/12(E)9解析：总的可能性：𝐶2=93符合条件的可能性：𝐶2=3概率是：3/36Acertainuniversitywillselect1of7candidateseligibletofillapositioninthemathematicsdepartmentand2of10candidateseligibletofill2identicalpositionsinthecomputersciencedepartment.Ifnoneofthecandidatesiseligibleforapositioninbothdepartments,howmanydifferentsetsof3candidatesaretheretofillthe3(A) (B) (C) (D) (E)解析：𝐶1* =7*45= Acertainrestaurantoffers6kindsofcheeseand2kindsoffruitforitsdesserttter.Ifeachdessertttercontainsanequalnumberofkindsofcheeseandkindsoffruit,howmanydifferentdesserttterscouldtherestaurantoffer?(A) (B) (C) (D) (E)（1）1cheese1𝐶1（6cheese1种）*𝐶1（2cheese1种）= （2）2cheese2𝐶2（6cheese2种）*𝐶2（2cheese2种）= 121527Fromagroupof3boysand3girls,4childrenaretoberandomlyselected.Whatistheprobabilitythatequalnumbersofboysandgirlswillbeselected?1

9

2

5

3解析：64个人的可能性：𝐶4=22女的可能性：𝐶2*6𝐶2=3*3 概率：9/15Thereare11womenand9meninacertainclub.Iftheclubistoselectacommitteeof2womenand2men,howmanydifferentsuchcommitteesare(A) (B)720(C)1,060(D) (E)解析：𝐶2*𝐶2=55*36= Thebuyerofanewhomemustchoose2of4typesofflooringand2of6paintcolors.Howmanydifferentcombinationsoftypeofflooringandpaintcolorareavailabletothehomebuyer?4+乘法原理解析：𝐶2*𝐶2=6*15=90 Aboxcontainsexactly24balls,ofwhich12areredand12areblue.Iftwoballsaretobepickedfromthisboxatrandomandwithoutrecement,whatistheprobabilitythatbothballswillbered?

4

Anystwill mendacombinationof3industrialstocks,2transportationstocks,and2utilitystocks.Iftheystcanchoosefrom5industrialstocks,4transportationstocks,and3utilitystocks,howmanydifferentcombinationsof7stocksarepossible?+乘法原理解析：𝐶3*𝐶2*𝐶2 Inaboxof12pens,atotalof3aredefective. Ifacustomerbuys2pensselectedatrandomfromthebox,whatistheprobabilitythatneitherpenwillbedefective?解析：12支钢笔：3个坏的，9从12支钢笔中抽2支钢笔总的可能性 =92支钢笔都不是坏的的可能性：𝐶29概率：36/66𝐶1

==10=2

2powersource44根电线选2根的可能性：𝐶2=6422powersource的可能性：𝐶2266begin这个单词中选字母重新排列，以元音开头的情况有多少种？4解析：2*4*3*2*1= 2*𝑃4=4第三、四个是0-9中任意一个数字，皆可重复，求所有可能的车牌号的数量？应该是26³10³10Acertainstockexchangedesignateseachstockwithaone-,two-,orthree-lettercode,whereeachletterisselectedfromthe26lettersofthealphabet.Ifthelettersmayberepeatedandifthesamelettersusedinadifferentorderconstituteadifferentcode,howmanydifferentstocksisitpossibletouniquelydesignatewiththesecodes? nstoassignidentificationnumberstoitsemployees.Eachnumberistoconsistoffourdifferentdigitsfrom0to9,inclusive,exceptthatthefirstdigitcannotbe0.Howmanydifferentidentificationnumbersarepossible?(A) (B) (C) (D) (E)解析：9*9*8*7Acertainrollercoasterhas3cars,andapassengerisequallylikelytoridein