真題gre數(shù)學(xué)講義1偶數(shù)或奇數(shù)

oddnumber奇 evennumber偶偶數(shù)＝偶數(shù)＋偶數(shù)或奇數(shù)＋奇數(shù)，奇數(shù)＝奇數(shù)＋偶數(shù)偶數(shù)＝偶數(shù)3偶數(shù)或奇數(shù)3偶數(shù)，奇數(shù)＝奇數(shù)3奇數(shù)偶數(shù)：2n（n為整數(shù)）n個(gè)連續(xù)的偶數(shù)相加等于零，則n為奇數(shù)相鄰整數(shù)nn+1肯定是一奇一偶。整數(shù)n和整數(shù)n+2的奇偶性是一樣的。Ifaandbarepositiveintegerssuchthata–banda/barebothevenintegers,whichofthefollowingmustbeanoddinteger?ab(a+b)(a+2)(b+2)（多選）Ifxandyareintegers2yxx^2y^2whichofthefollowingmustbeaneveninteger?xy2x+y^2+y^2+ （factor，divisor數(shù)就叫作N12=1*12=2*6=3*。、、、、、都是12的因數(shù)。質(zhì)數(shù)（primenumber1N進(jìn)行分解因數(shù)，如果除了1和它自身外，不能被其他自然數(shù)整除，那N就是質(zhì)數(shù)。合數(shù)（compositenumber1的自然數(shù)N進(jìn)行因factorinteger，primenumber等詞，都要意識(shí)數(shù)值要可能需要進(jìn)行分解質(zhì)因數(shù)。個(gè)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=2??3??5??7??,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（正整數(shù)nn為完全平方數(shù)23是唯一的一對(duì)同為質(zhì)數(shù)的相鄰自然數(shù)。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的一個(gè)因數(shù)。若Howmanyfactorsdoes360 (D) (E)Whatisthegreatestprimefactorof2^100- 【解析】2^100-2^96=2^96*(2^4-1)=（2^96）*3*5,因此正確答案為CIftheintegernhasexactlythreepositivedivisors,including1andn,howmanypositivedivisorsdoesn^2have? 答案：63numberx^2factorsx>3Ifn=4p,wherepisaprimenumbergreaterthan2,howmanydifferentpositiveevendivisorsdoesnhave? 因子共有4個(gè)。 連續(xù)正整數(shù)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? 最大公約數(shù)greatestcommon例題：12與18例題：4和64646IfMistheleastcommonmultipleof90,196and300,whichofthefollowingisNOTafactorofM?n能被abn一定能夠被a和bn812整除，n24整除； 用整數(shù)來表示余數(shù)：N/a=b??3——N=a*b+3（余數(shù)為用小數(shù)來表示余數(shù)：N/a=b.14——N=a*b.14=a*b 0.14a）余數(shù)是非負(fù)整數(shù)（大于等于0的整數(shù)Whenpositiveintegerxisdividedbypositiveintegery,theremainderis9. Ifx/y=96.12,whatisthevalueofy? 0~59之間，120，12，24，36，48，60 因此概率P 考查點(diǎn)2：整除的表達(dá) N/4=2a+ Whenpositiveintegerxisdividedby5,theremainderis3;andwhenxisdividedby7,theremainderis4.Whenpositiveintegeryisdividedby5,theremainderis3;andwhenyisdividedby7,theremainderis4.Ifx>y,whichofthefollowingmustbeafactorofx-y?(A) (B) (C) (D) (E)為b。則????的個(gè)位數(shù)和????的個(gè)位數(shù)相同。（333^777）/5的余數(shù)和（777^333）/53除以10余數(shù)是多少？求2345+1onesdigit？505393942542581258125A.B.C.D.E.可能的六位數(shù)中，能夠被8整除的六位數(shù)的概率是多少Y=3時(shí)，3128=39，可以整除；Y=5時(shí)，5128=64，可以整除；Y=8時(shí)，812÷8不能整除。3-1001003答案：3*33=99，-3*33=-99，所以有66個(gè)，再加上0就是67個(gè)。(0可以整除任 答案67個(gè)。-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的任意整數(shù)x+y+z=（x+y+y+z+x+z）/2==（5a+1+5b+2+5c+3）/2因?yàn)?是偶數(shù)，則5（a+b+c）是偶數(shù)，即a+b+c是偶數(shù)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分考查點(diǎn)1：數(shù)位例題子：7654.3217：4：units(orones)3：tenths2：hundredths1：thousandthsdigit：s的是整數(shù)位（小數(shù)點(diǎn)前面的某位ths的是分位（小數(shù)點(diǎn)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 例題：roundtotens保留至十位（即將個(gè)位化為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：有限小數(shù)terminatingWhichofthefollowingfractionshasadecimalequivalentthatisaterminating

=（大-?。?小=（大-小）/大Attheendofthefirstquarter,thesharepriceofacertainmutualfundwas20percenthigherthanitwasatthebeginningoftheyear.Attheendofthesecondquarter,thesharepricewas50percenthigherthanitwasatthebeginningoftheyear.Whatwasthepercentincreaseinthesharepricefromtheendofthefirstquartertotheendofthesecondquarter?50%199???550;40??0 =(040110;=??=(1+50%)10?(150%)10?Inacorporation,50percentofthemaleemployeesand40percentoftheemployeesareatleast35yearsold. If42percentofalltheemployeesareatleast35yearsold,whatfractionoftheemployeesinthecorporationarefemales?InTownX,64percentofthepopulationareemployed,and48percentofthepopulationareemployedmales.WhatpercentoftheemployedpeopleinTownXarefemales?2%sale設(shè) price為xList=retail*（1+2%）=1.02xSale=retail*（1–5%）=???? *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?設(shè)原來有剩下的：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） ????*????=????*????=(???(????)??=????÷????=

?? ÷

=(= =指數(shù)????a>0，則????0Ofthefollowingvaluesofn,thevalueof(-1/5)-nwillbegreatestfor02336x=8100，求33x-36 3x 33x-3=33x=90=3=(3)=8100 3 （（5））=??6kk??6 Ifkisanintegerand(0.0025)(0.025)(0.00025)×10kisaninteger,whatistheleastpossiblevalueofk?

06 √??2=|??|（如果是未知數(shù)開根號(hào)，一定要注意加絕對(duì)值√??*√??=√??÷√??=√??±√??≠√??±根號(hào)有可能估值，如：2<√5=3√8==3 √27==3 √64==3 √125==3 √216==1+1+（2

） 2x-10=√??√10xx-10√??)2-(√10)2，而(√??)2-(√10)2=（√??√10（√??√10）=√??所以√??√101，√??√102x=（√10+1）=2√10答案：2√10 求（1√2√3）（√2√3） （??+√??+√??）?（√??+ =1+2（√??+√??）+（√??+√??）?（√??+√??）=1+2√??+1+2√?? X是負(fù)數(shù)，求√（x?3）+√????|??| √（x?3）+√????|??|=|x?3|+x<03-x-x3-3- 若a<b，b<c，則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，則a*c>b*c；a*b>0，a>0，則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？ 邊，使另一邊為0，再解。0x前的系數(shù)為正數(shù)，分解因式。x^2-x>2x^2-x-2>0，轉(zhuǎn)化為(x-2)(x+1)>0包含x的因數(shù)的次數(shù)若為偶數(shù)，則不穿過，即奇過偶不過（1）?? （2）???2+3x+10<（1）3??1 （2）??2??7≥ ??2Whichofthefollowingdescribesallvaluesofxforwhich1—??2≥0？(A)x≤—1orx≥1(D)x≤—(E—1≤x≤13：??2

>答案：-3<x<-2L1：??2-x-6，L2：??2–4。2

0x??262

=??)??

>即（x-3）（x+2）（x+2）（x-2）即（x-3）（x-2）(??+2)2>(??2)20，則（x-3）（x-2）0x2x答案：x2或x（數(shù)值比較m>0andn>0,m<n,x>0A:??xB:（數(shù)值比較p=r,A:B:5x+3y15，3x+5y15 若|x—a|>b，則x—a>b或x—a<—b。練習(xí) 性質(zhì)2：| |=||+||，說明X和Y同號(hào) Ifthesumoftwopositiveintegersis24andthedifferenceoftheirsquaresis48,whatistheproductofthetwointegers?(A) (B) (C) (D) (E)T8，D答案：-62ABC三個(gè)人，AB一起生產(chǎn)產(chǎn)品，43000個(gè)產(chǎn)品；BC一起生產(chǎn)產(chǎn)品，53500個(gè)產(chǎn)品，AC一起生產(chǎn)產(chǎn)品，62700個(gè)產(chǎn)品。求A一個(gè)人做1000個(gè)產(chǎn)品要多少小時(shí)？c4（a+b）=5（b+c）=6（a+c）=可解出：a250，b500，c2001000÷250=4小時(shí)4Twopositivenumbersdifferby12andtheirreciprocalsdifferby

4.Whatis52

5

一元二次方程：a??2bxc0如果??24????≥0 x= 2

=

*??2=a（x ：項(xiàng)數(shù)=????;????+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=例題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 + ） ??1+??2+??3

=1

1+??+

=

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

1

答案：2+ 12341234ABD+∠BDE的和是多少？CBDB 180°，因?yàn)椤螩35°，所以∠CBD∠CDB180°—35°=因?yàn)椤螦BD180°—∠CBD，∠BDE180所以∠ABD+∠BDE180°—∠CBD+180°—∠CDB360°—=360°—145°=polygon多邊形：triangle三角形quadrilal四邊形pentagon五邊形 AA obtuseangle鈍角acuteangle銳角rightangle直角（3）直角三角形：a^2b^2=c^2銳角三角形：a^2b^2c^2鈍角三角形: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 例題Ifeachsideof△ACDabovehaslength3andifABhaslength1,whatistheareaofregionBCDE? E例題Intherectangleabove,AB=6,AD=8.IfpointPisselectedfromsegmentBCatrandom,whatistheprobabilitythatthelengthofsegmentAPislessthan

例題Aladder25feetlongisleaningagainstawallthatisperpendiculartolevelground.Thebottomoftheladderis7feetfromthebaseofthewall.Ifthetopoftheladderslipsdown4feet,howmanyfeetwillthebottomoftheladderslip?(A) (B) (C) (D) (E)5所以，x= 53-9= 5 矩形perimeter=（+寬）*長area=*長對(duì)角線diagonal= 2+寬正方形=*2=（2）=上底*2+側(cè)邊（3）=*梯形（2）面積：（+下底）**2菱形（2）=*Thefigureshowsasquarepatiosurroundedbyawalkwayofwidthxmeters. ofthewalkwayis132squaremetersandthewidthofthepatiois5metersgreaterthanthewidthofthewalkway,whatistheareaofthepatio,insquaremeters?(A) (B) (C) (E)S直三角*4+S=S大設(shè)直角三角形長直角邊=x，則短直角邊=x-1，所以[x(x-1)/2]34+1=25（或者根據(jù)勾股定理）解出x=4 答案：12 circle圓 center圓心 radius半徑 考查點(diǎn)1：圓的周長和面積一個(gè)小白鼠在一個(gè)半徑為2

feet240feet2

feetc2π*r2*2

*π=

Inthefigureshown,iftheareaoftheshadedregionis3timestheareaofthesmallercircularregion,thenthecircumferenceofthelargercircleishowmanytimesthecircumferenceofthesmallercircle?432有三個(gè)同心圓：C1、C2C3，C1是里面最小的圓。C1的面積=C2的面積—C1的面積=C3的面積—C2的面積，C2100πC1C3的半徑（知道半徑的比例，即使不知體數(shù)值，也可以求出來周長比150π。C1的半徑是√505√2C3的半徑是√150=5√2：5√6=2=2=π（2??）–2*π??2=2π??=+2π*2r2*2πr8πr=r一個(gè)傳送帶上有A、B、C共3個(gè)，逆時(shí)針轉(zhuǎn)動(dòng)，可以知道A、B、C的C1.5A23π34=8π，6/8=3/4A23π36=12π，6/12=1/2弦chord：連接圓上任意兩點(diǎn)的線段。 弧arc：圓上兩點(diǎn)之間的圓周部分。性質(zhì)1：直徑對(duì)應(yīng)的圓周角是90°性質(zhì) ：??πr（n是圓心角 ：??π??2（n是圓心角O，OB、OC是半徑?！螧OC130°，∠BAD50°。問∠ABD的大∠BDC=∠BOC*2

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

604.5πS圓S圓：S陰影 60

120答案：120O中，AC是直徑，ABBC3√2根據(jù)直角三角形的性質(zhì)，AC=6答案：9π正方形邊長1，以正方形左上方和右下方的頂點(diǎn)為圓心作兩個(gè)14 π-一個(gè)扇形面積41/242（2×π-1）=π-（

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

O

RInthefigureabove,Aisthepointoftangencyfortwocirclesandalsothecenterofthethirdcircle.Iftheradiiofthreecirclesare1,whatistheexternalperimeteroftheA A apolygonisinscribedina acircleiscircumscribedabouta aInthefigureabove,acircleisinscribedinasquarewithsidebandasquarewithsideaisinscribedinthecircle.a=4,Whatistheareaofthesquarewithsideb?ab rectangularsolidabc2*（ab+ac+bc）正方體cube：體積=??3，表面積=6??2cylinder=π??2*hAcylinderhas2congruentcircles等圓anda 球sphere：體積=4π??3，表面積 3（x+2(y+2)(z+2)體積分別是：(3/π)^2*π*1090/π150/π634答案：600÷（1-1）=7200，7200*1= S16個(gè)圓柱體小包裝盒在大包裝盒中所占的空間比例為C。問SC哪個(gè)大，大多少？V長 2?16

S=

2 CgreaterthanS=4??66CS 已知點(diǎn)A（??1,??1），點(diǎn)B（??2,??2） |=√（??1???2）+（??1?（根據(jù)平面直角坐標(biāo)系求兩點(diǎn)距離的，可以求出對(duì)角線的長度 √（?1?3）+（3+1）=√32=x軸、y軸、x6、y4圍成了一個(gè)矩形，在該矩形中，xy4xy=4xy44?4

3直線方程:y=kx+bkslope斜率by-intercept縱截距。斜率的計(jì)算：k=??1;??1;

=直角三角形的對(duì)邊ylyl2-0Thegraphofwhichofthefollowingequationsisastraightlinethatisparalleltolinelinthefigureabove?3y–2x=3y+2x=3y—2x=2y–3x=2y+3x=-3由于C也在該直線上，帶入y=-5即可求得a=-2x04x+3y=48y16即直線：4x+3y=48y軸的交點(diǎn)是（0,16）y04x+3y=48x12即直線：4x+3y=48y軸的交點(diǎn)是（12,0）所以三角形面積=16*12÷2=96例題有一條直線L1，它經(jīng)過直線L2：y0.5x1的x軸截距，并且與L1垂直。求另答案：y=2x–4例題例題Inthefigureabove,pointsPandQlieonthecirclewithcenterO.Whatisthevalueofs?1拋物線的方程y=a??2bxca決定開口方向=?說明在y軸左側(cè)）拋物線是否與x軸相交：Δ=??24acΔ=0:拋物線與x軸有一個(gè)交點(diǎn)；Δ<0:拋物線與x軸沒有交點(diǎn)；Δ>0:拋物線與x軸有兩個(gè)交點(diǎn)。對(duì)稱的表示：x-axisisperpendicularbisectorofsegmentAB；BisthereflectionofAaboutthex-axis；AandBaresymmetricaboutthex-點(diǎn)A（a,b）關(guān)于原點(diǎn)的對(duì)稱點(diǎn)：（-a,-b）。例題Intherectangularcoordinatesystemabove,theliney=xisperpendicularbisectorsegmentAB(notshown),andthex-axisistheperpendicularbisectorofsegmentBC(notshown).IfthecoordinatesofpointAare(2,3),whatarethecoordinatesofpointC?(–3,(–3, (2,(3,(2,x 單利(SimpleInterests)=+本金*利率*例題Maryinvested$14,000for3yearsinaofdepositpaying9.25%simpleannualinterest.HowmanymoreinterestwouldMaryhavereceivediftheinterestrateonthishadbeen9.75%simpleannualinterest?例題A2-yearofdepositispurchasedforkdollars.Iftheearnsinterestatanannualrateof6percentcompoundedquarterly,whichofthefollowingrepresentsthevalue,indollars,oftheattheendofthe2(A) (B) (C) (D) （二）Work例題Asmallwaterpumpwouldtake3hourstofillanempty.Alargepumpwouldtake2hourstofillthesame.Howmanyhourswouldittakebothpumps,workingattheirrespectiveconstantrates,tofilltheemptyiftheybeganpumatthesametime?(A) (B) (C) (D) (E)例題Sixmachines,eachworkingatthesameconstantrate,togethercancompleteacertainjobin12days.Howmanyadditionalmachines,eachworkingatthesameconstantrate,willbeneededtocompletethejobin8days?(A) (B) (C) (D) (E)Whileworkingaloneattheirrespectiveconstantrates,computerXprocesses240filesin4hoursandcomputerYprocesses240filesin8hours.Ifallfilesprocessedbythesecomputersarethesamesize,howlongwouldittakethetwocomputers,workingatthesametimeandattheirrespectiveconstantrates,toprocessatotalof240files?22hr202hr4066hr20 R機(jī)器一小時(shí)的效率是完成1的任務(wù)量，S機(jī)器一小時(shí)的效率是完成1 兩臺(tái)機(jī)器共同運(yùn)轉(zhuǎn)的效率是每小時(shí)：1+1= 2小時(shí)候，完成了10=5的任務(wù)量，還剩1 答案：60A，B，A23個(gè)，B59個(gè)，A、B個(gè)，AB多花/A每一個(gè)花2/3分 B每一個(gè)花5/9分A450個(gè)需要時(shí)間：450*2/3300B450個(gè)需要時(shí)間：450*5/9250分鐘所以AB多花了50分鐘AB50=時(shí)間*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?

一個(gè)人的家距10 他前10min以15 /hour的速度開車，接下來的10min以昨天的速度開，問還要開多遠(yuǎn)才能到？ ÷20min= 15/h=0.2510-（1030.25+1030.5）=2.5一輛car用30分鐘開了20 ，其中有x 用的速度是60 路用的速度是30 /h，問走x 130min2

10=1=10 答案：10Afterdrivingtoariverfrontparkinglot,Bobnstorunsouthalongtheriver,turnaround,andreturntotheparkinglot,runningnorthalongthesamepath.Afterrunning3.25south,hedecidestorunforonly50minutesmore.IfBobrunsataconstantrateof8minutesper,howmanyfarthersouthcanherunandstillbeabletoreturntotheparkinglotin50minutes?8x+（3.25+x）*8=例題1：AB地，乙車出現(xiàn)了一些故障，所以甲車先開，甲車的速答案：4hours

出發(fā)地到噴泉的路程：10*1= 3

÷6==5-1（10分鐘）=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

=一輛卡車的燃油箱能裝15加侖的天然氣，在城市里行駛平均每 車在城市里行駛60，在高速公 的天然氣，平均走多少？ 加侖用了60 加侖用了

52一共用了（12+

1= BgallonB耗油—A405/27405/361511.25例題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歲。5yearsfromnowhisageis3timesasoldashisdaughter。Howmanyyearsfromnowon,hisagewillbetwiceashisdaughter?即今年M34歲，他女兒8歲。設(shè)從今年開始n年，他 ，34+n=23（8+n，n=18所以，18年后，M是他女兒 的2倍A、 B、 C、 E、答案：B 知識(shí)點(diǎn)1：兩個(gè)圓圈IAB（A∩B）+非A非Accordingtoasurvey,93percentofteenagershaveusedacomputertoygames,89percenthaveusedacomputertowritereports,and5percenthavenotusedacomputerforeitherofthesepurposes.Whatpercentoftheteenagersinthesurveyhaveusedacomputerbothtoygamesandto writereports?4040人當(dāng)中：有數(shù)學(xué)的，也有英語學(xué)位的。有數(shù)學(xué)人占了50%15%，兩個(gè)都沒有的5%，求單獨(dú)有英語的有多少人？答案：1835+-20人會(huì)說英語，24人會(huì)說法語，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=++三個(gè)特點(diǎn)都符合 A∩B=+Inamarketingsurveyforproductssomepeoplewereaskedwhichoftheproducts,ifany,theyuse.Ofthepeoplesurveyed,atotalof400useA,atotalof400useB,atotalof450useC,atotalof200useAandBsimultaneously,atotalof175useBandCsimultaneously,atotalof200useCandAsimultaneously,atotalof75useA,B,andCsimultaneously,andatotalof200usenoneoftheproducts.Howmanypeopleweresurveyed?選擇S課程的有18人。核對(duì)發(fā)現(xiàn)每個(gè)學(xué)生都至少選擇了一門課程，同時(shí)選（E∩M--5033s24z29bs兩種分–（5+3）–（6+3）+3=Ofthe300subjectswhoparticipatedinanexperimentusingvirtual-realitytherapytoreducetheirfearofheights,40percentexperiencedsweatypa,30percentexperiencedvomiting,and75percentexperienceddizziness.Ifallofthesubjectsexperiencedatleastoneoftheseeffectsand35percentofthesubjectsexperiencedexactlytwooftheseeffects,howmanyofthesubjectsexperiencedonlyoneofthese例題展示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=< 組合：m個(gè)元素（總體數(shù)量）n個(gè)元素（目標(biāo)數(shù)量）組成一組。用C（combination）表示。???? !排成一排。用P（permutation）表示。 !從甲地經(jīng)乙、丙兩地到，共有多少種不同的走法？加法原理：n類辦法，在第一類辦法中有m1m2種不同的方法，??，在第n類辦法中有mn種不同的方法，那么完成這件事共有N=m1+m2+m3+?+mn種不同方法。m2種不同的方法，??，做3m23m33?3mn種不同的方法。n個(gè)步驟，步與步之間是連續(xù)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：??2=66Johnhas5friendswhowanttorideinhisnewcarthatcan modateonly3passengersatatime.Howmanydifferentcombinationsof3passengerscanbeformedfromthe5friends?385解析：??3=105Acertainclubhas20members.Whatistheratioofthememberof5-membercommitteesthatcanbeformedfromthemembersoftheclubtothenumberof4-membercommitteesthatcanbeformedfromthemembersoftheclub?(A)16to (B)15to (C)16to (D)15to (E)5to解析：??5：??4= 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. 五個(gè)數(shù)值，Iftwoormorenumbersaddedtogether,howmanydifferentnumberswecanget？5兩個(gè)數(shù)相加的可能性：??2=1055三個(gè)數(shù)相加的可能性：??3=55四個(gè)數(shù)相加的可能性：??4=55五個(gè)數(shù)相加的可能性：??5=5總共的可能性：10+10+5+1Acertainemploys6seniorofficersand4juniorofficers.Ifacommitteeistobecreatedthatismadeupof3seniorofficersand1juniorofficer,howmanydifferentcommitteesarepossible? (B) (C) (D) (E)解析：??3*??1=20*4 Themembershipofacommitteeconsistsof3Englishteachers,4Mathematicsteachers,and2SocialStudiesteachers.If2committeemembersaretobeselectedatrandomtowritethecommittee’sreport,whatistheprobabilitythatthetwomembersselectedwillbothbeEnglishteachers?(A)2/3(B)1/3(C)2/9(D)1/12(E)9解析：總的可能性：??2=93符合條件的可能性：??2=3概率是：3/36Acertainuniversitywillselect1of7candidateseligibletofillapositioninthemathematicsdepartmentand2of10candidateseligibletofill2identicalpositionsinthecomputersciencedepartment.Ifnoneofthecandidatesiseligibleforapositioninbothdepartments,howmanydifferentsetsof3candidatesaretheretofillthe3(A) (B) (C) (D) (E)解析：??1* =7*45= Acertainrestaurantoffers6kindsofcheeseand2kindsoffruitforitsdesserttter.Ifeachdessertttercontainsanequalnumberofkindsofcheeseandkindsoffruit,howmanydifferentdesserttterscouldtherestaurantoffer?(A) (B) (C) (D) (E)（1）1cheese1??1（6cheese1種）*??1（2cheese1種）= （2）2cheese2??2（6cheese2種）*??2（2cheese2種）= 121527Fromagroupof3boysand3girls,4childrenaretoberandomlyselected.Whatistheprobabilitythatequalnumbersofboysandgirlswillbeselected?1

9

2

5

3解析：64個(gè)人的可能性：??4=22女的可能性：??2*6??2=3*3 概率：9/15Thereare11womenand9meninacertainclub.Iftheclubistoselectacommitteeof2womenand2men,howmanydifferentsuchcommitteesare(A) (B)720(C)1,060(D) (E)解析：??2*??2=55*36= Thebuyerofanewhomemustchoose2of4typesofflooringand2of6paintcolors.Howmanydifferentcombinationsoftypeofflooringandpaintcolorareavailabletothehomebuyer?4+乘法原理解析：??2*??2=6*15=90 Aboxcontainsexactly24balls,ofwhich12areredand12areblue.Iftwoballsaretobepickedfromthisboxatrandomandwithoutrecement,whatistheprobabilitythatbothballswillbered?

4

Anystwill mendacombinationof3industrialstocks,2transportationstocks,and2utilitystocks.Iftheystcanchoosefrom5industrialstocks,4transportationstocks,and3utilitystocks,howmanydifferentcombinationsof7stocksarepossible?+乘法原理解析：??3*??2*??2 Inaboxof12pens,atotalof3aredefective. Ifacustomerbuys2pensselectedatrandomfromthebox,whatistheprobabilitythatneitherpenwillbedefective?解析：12支鋼筆：3個(gè)壞的，9從12支鋼筆中抽2支鋼筆總的可能性 =92支鋼筆都不是壞的的可能性：??29概率：36/66??1

==10=2

2powersource44根電線選2根的可能性：??2=6422powersource的可能性：??2266begin這個(gè)單詞中選字母重新排列，以元音開頭的情況有多少種？4解析：2*4*3*2*1= 2*??4=4第三、四個(gè)是0-9中任意一個(gè)數(shù)字，皆可重復(fù)，求所有可能的車牌號(hào)的數(shù)量？應(yīng)該是26310310Acertainstockexchangedesignateseachstockwithaone-,two-,orthree-lettercode,whereeachletterisselectedfromthe26lettersofthealphabet.Ifthelettersmayberepeatedandifthesamelettersusedinadifferentorderconstituteadifferentcode,howmanydifferentstocksisitpossibletouniquelydesignatewiththesecodes? nstoassignidentificationnumberstoitsemployees.Eachnumberistoconsistoffourdifferentdigitsfrom0to9,inclusive,exceptthatthefirstdigitcannotbe0.Howmanydifferentidentificationnumbersarepossible?(A) (B) (C) (D) (E)解析：9*9*8*7Acertainrollercoasterhas3cars,andapassengerisequallylikelytoridein