小站教育GMAT数学笔记

1、GMAT 数学笔记GMAT 数学备考关键词 一、知识点：准确掌握 二、词汇、表达法：读懂题目 三、熟练：平均两分钟一道题考试相关问题 一、时间与题量 二、题型 三、机经与换题库 四、其它If a and b are positive integers such that a b anda/b are both even integers, which of the following must be an odd integer?（A） a/2（B） b/2（C）(a+b)/2（D）(a+2)/2 （E） (b+2)/2If M is the least common multiple of

2、90, 196, and 300, which of the following is NOT a factor of M?(A) 600(B)700(C) 900(D) 2,100(E) 4,900复习注意事项*战略上重视*初等数学的思维*解法力求稳妥清晰*把握好 DS 题型*熟练重于技巧 推荐复习步骤*知识点查缺补漏*背熟词汇*复习课上所学*OG，及其它相关资料*机经：25 / 25文档可自由编辑打印第一章 算术1.integer (whole number): 整数* positive integer：正整数，从 1 开始，不包括 0。2.odd & even number 奇数

3、与偶数* 凡整数均具有奇偶性，如1 是奇数，0 是偶数。* 奇+奇=偶，奇+偶=奇若干个整数相乘，除非都是奇数，其乘积才会是奇数例： If a and b are positive integers such that a b and whicha are both even integers,bof the following must be an odd integer?(A） a(B） b(C)a + b(D)a + 2(E)b + 2222223. prime number & composite number 质数与合数* A prime number is a positi

4、ve integer that has exactly two different positive divisors,1 and itself.* A composite number is a positive integer greater than 1 that has more than two divisors.* The numbers 1 is neither prime nor composite, 2 is the only even primenumber.4. factor(divisor) & prime factor 因子和质因子* 一个数能被哪些数整除，这

5、些数就叫它的因子（因数、约数）。* 因子里的质数叫质因子（数）。例 1： If n=4p, where p is a prime number greater than 2, how many different positive even divisors does n have,including n?(A) 2(B) 3(C) 4(D) 6(E) 8例 2：If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n2 have?(A)

6、4(B) 5(C) 6(D) 8(E) 9例 3：1225 有几个因子？例：What is the greatest prime factor of 2100 - 296? (A) 2(B) 3(C) 5(D) 7(E) 11例：A positive integer n is said to be “prime-saturated” if the product of all the different positive prime factors of n is less than the square root of n. What is the greatest two-digit pr

7、ime-saturated integer?(A) 99 (B) 98 (C) 97 (D) 96 (E) 955. the greatest common divisor (GCD)& the leastcommon multiple(LCM) 最大公约数和最小公倍数例：If M is the least common multiple of 90, 196, and 300, which of the following is NOT a factor of M?(A) 600 (B)700 (C) 900 (D) 2,100 (E) 4,900例：What is the lowe

8、st positive integer that is divisible by each of the integers 1 through 7,inclusive?(A) 420 (B) 840 (C) 1,260 (D) 2,520 (E) 5,0406. decimals & fractions 小数和分数*相关词汇：reaccuring decimal ; terminating decimal ; numerator ;denominator ; improper fracion ; mixed number*整数位与分位： 后面加 s 的是整数位（小数点前面的某位），加

9、th 或 ths 的是 分位（小数点后面的某位），如 tens 是十位数，而 tenth 是十分位*What is the fractional part of .这样的表达法意为“谁的几分之几”*小数和分数的互相转换：例 1： 0373737=? (将其转换成一个分数)例 2：Which of the following fractions has a decimal equivalent that is a terminating decimal?(A) 10/189(B) 15/196(C) 16/225(D) 25/144(E) 39/1287. consecutive numbers

10、 连续数例 1：In an increasing sequence of 10 consecutive integers, the sum of the first 5 integers is 560. What is the sum of the last 5 integers in the sequence?(A) 585 (B) 580 (C )575 (D)570 (E) 565例 2：If n is an integer greater than 6, which of the following must be divisible by 3?(A) n(n+1)(n-4)(B) n

11、(n+2)(n-1)(C) n(n+3)(n-5) (D) n(n+4)(n-2)(E) n(n+5)(n-6)8. divisibility & remainder 整除及余数问题* 一个数是否能够被 5 整除，只要看它的最后一位（是 0 或 5）。* 一个数是否能够被 4 整除，只要看它的后两位（是否是 4 的倍数）。* 一个数是否能够被 8 整除，只要看它的后三位（是否是 8 的倍数）。* 一个数能否被 3 整除，取决于各位之和能否被 3 整除。* 一个数能否被 9 整除，也取决于各位之和能否被 9 整除。*0 能被所有数整除。* 余数包括 0，如 24 除以 6，商为 4 余数

12、为 0。 例：1912 257 的个位数字是几？例：If s and t are positive integers such that could be theremainder when s is divided by t?(A) 2(B) 4(C) 8(D) 20(E) 459.数字问题s = 64.12 ,which of the followingt例：1001 位数字组成的数，任意相邻的两位数字组成的数能被 17 或 23 整除， 这个 1001 位的数字以 6 开头，则它的最后六位是（）10.算术部分的几种常用方法*参数法 例：两个两位数个位与十位恰好颠倒，问下面哪个不能是两数之

13、和？ A.181，B.121，C.77，D.132，E.154解法：设两数分别为 ab 和 ba，则(ab)+(ba)=(10a+b)+(10b+a)=11(a+b)，即和 必为 11 的倍数，答案为 A。*代数法*试错法 例：× The product of the two-digit numbers above is the three-digit number ,where ,and are three different nonzero digits. If ×10, what is the two-digit number ?(A) 11(B) 12(C) 13(

14、D) 21(E) 31第二章 代数1Quadratic equations: 一元二次方程ax2+bx+c=0x = - b ±b2 - 4ac1,22a但一般更常用的是因式分解法：x2-2x-3=0(x-3)(x+1)=0x1=3, x2=-12.Simultaneous linear equations: 多元一次方程组* 基本方法：消元法。 例 1：3x+y=5(1)2x+y=4(2)(1)(2), 消去 y, 得 x=1,y=2* 注意：并不是任何二元一次方程组都有唯一解。 例 2: 3x+y=5(1)6x+2y=10(2) 上述方程有无穷多组解。 因此，方程的数量须等于未知

15、数的数量，此时多元一次方程有唯一的一组解。3. Simultaneous quadratic equations: 二元二次方程组 一般只考如下形式：a1x+b1y=c1(1)a2x2+b2x+a3y2+b3y=c2(2)即其中一个方程为一次。这种形式等价于一元二次方程，把（1）代入（2）即可。4. Inequalities: 不等式*不等式部分不会像中国高考那样考推导、证明，注意两边乘以负数变号等最基 本原则即可。5. Arithmetic sequence: 等差数列an=a1+(n-1)d sn=(a1+an)n/2 n=(an-a1)/d +16Geometric sequence:

16、等比数列an=a1qn-1s = a× 1 - qnn 1 1 - q当q<1 时， s = a1¥ 1 - q例： 1 + 12 221 1+ 3 + L + ¥ = ？2 2例： 0373737=? (将其转换成一个分数)7Sets: 集合例 1：全班 50 个人，选音乐课的有 20 人，选体育课的有 18 人，两课都选的有5 人，问两课都没选的几人？例 2： A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B

17、soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B sop. How many of the 200 households surveyed used both brands of soap?(A) 15(B)20(C)30(D)40(D)45例 3：五个人排队，甲不能在首位，乙不能在末位，有几种不同的排法？第三章 几何1 Lines & planes 直线与平面* 两直线平行并为第三条直线所截后，相应角的关系。* 直线与平

18、面的关系。例：If n distinct planes intersect in a line, and another line L intersects one of these planes in a single point, what is the least number of these n planes that L could intersect?(A) n (B) n1 (C) n2 (D) n/2 (E)(n1)/22. Triangles 三角形* 勾股定理：a2+b2=c2* 构成三角形的条件：两边之和大于第三边。* 三角形内部边和角的关系：大边对大角。3. Quad

19、rilaterals 四边形* parallelogram(平行四边形) : 面积=a×h; 周长=2（a+b）* rectangle(矩形) : 面积=a×h; 周长=2（a+b）* square(正方形) : 面积=a2 ; 周长=4a* trapezoid(梯形) : 面积=（a+b）×h/24.Circles 圆* 面积=R2* 周长=2R5.Polygons 多边形* 多边形内角和：（n-2）180º6.Rectangular Solids 长方体* 体积=a×b×c* 表面积=2（a×b+b×c+c&#

20、215;a）7.Cubes 正方体* 体积=a3* 表面积=6a28.Cylinders 圆柱* 体积=R2h* 表面积=2R2+2R×h 例：一个圆锥内接于一个半球，圆锥的底面与半球的底面重合，则圆锥的高与半 球的半径的比是多少？9.Coordinate Geometry 解析几何* 直线的标准方程：y=kx+b ；即斜截式，其中 k 为斜率 slope，b 为 y 轴截距y-intercept* 斜率的计算：K= (Y2-Y1)/( X2-X1)* 两点或一点加斜率确定一条直线。* 两直线垂直，其斜率的乘积为1。第四章 统计1. arithmetic mean (average)

21、 算术平均值1 nE=å ain i =12. median 中位数* The median is the middle value of a list when the numbers are in order.* 先排序，后取中。3. mode 众数* The mode of a list of numbers is the nmuber that occurs most frequently in the list.* A list of numbers may have more than one mode.4. expectation 期望* 期望就是算术平均值。5. de

22、viation 偏差di=ai-E6. variance 方差1 n 2D = å (ai - E )n i =17. standard deviation 标准差s = D例：.72,73,74,75,76.74,74,74,74,74.62,74,74,74,89The data sets , ,and above are ordered from greatest standard deviation to least standard deviation in which of the following ?(A) , (B) , ,(C) , , (D) , ,(E) ,

23、8. range 范围* 最大数减去最小数所得的差就是该组数据的范围。 例 1：150, 200, 250, nWhich of the following could be the median of the 4 integers listed above?. 175. 215. 235(A) only (B) only (C) and only (D) and only (E),and 例 2：The least and greatest numbers in a list of 7 real numbers are 2 and20,respectively. The median of

24、the list is 6,and the number 3 occurs most often in the list. Which of the following could be the average of the numbers in the list?. 0.5(A)only (B) andonly(C) and only(D) and only (E) ,and 第五章 数据充分性题*每道 DS 题的选项都是固定的：A Statement (1) ALONE is sufficient, but statement (2) alone is not suffici

25、ent. B Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C BOTH statements TOGETHER are sufficient, but NEITHER statementALONE is sufficient.D Each statement ALONE is sufficient.EStatement (1) and (2) TOGETHER are not sufficient.(additional data are needed).* DS 题的本质是一种判断

26、型的选择题，并非判断正误，而是判断根据条件给的信 息能否回答主题干里提出的问题。*要注意的几大问题：<1> 唯一性 例：x=?（1）x=2（2）x2=4<2> 否定性 例：x>0?（1）x2>0（2）x3<0<3> 不矛盾性例：A，B 两车在长直道路上相对行驶，现距离为 500 英里，问多长时间后相遇？(1) 其中一辆速度为 200 英里每小时。(2) 其中一辆速度为 300 英里每小时。<4> 独立性 例：x>0?（1）x=5（2）x3<0<1> If n is an integer, is n+1 o

27、dd? (1) n+2 is an even integer.(2) n-1 is an odd integer.<2> In PQR,if PQ=x, QR=x+2, PR=y, which of the three angles of PQRhas the greatest degree measure?(1) y=x+3 (2) x=2<3> Tom, Jane, and Sue each purchased a new house . The average (arithmetic mean )price of the three houses was $120

28、,000.What was the median price of the three houses?(1) The price of Toms house was $110,000. (2) The price of Janes house was $120,000.<4> 3.26, =?(1) 3.26 四舍五入到十分位后是 3.2。(2) 3.26 四舍五入到百分位后是 3.24。<5>If °represents one of the operations +, -,and ×,is k°(l +m)=(k°l)+(k&

29、#176;m)for all numbers k, l , and m?(1) k°1 is not equal to 1°k for some numbers k. (2) °represents subtraction.<6>On Jane's credit card account, the average daily balance for a 30-day billing cycle is the average of the daily balances at the end of each of 30 days. At the b

30、eginning of a certain 30-day billing cycle, Jane's credit card account had a balance of $600. Jane made a payment of $300 on the account during the billing cycle. If no other amounts were added to or subtracted from the account during the billing cycle, what was the average daily balance on Jane

31、s account for the billing cycle?(1) Janes payment was credited on the 21st day of the billing cycle.(2) The average daily balance through the 25th day of the billing cycle was$540.第六章 排列组合与概率1.Permutation & combination: 排列与组合m* P n = m!/(m - n)!Pmn 从 m 个元素中挑出 n 个并进行排列（需要考虑 n 个元素的内部顺序）的所有情况的数量。*

32、C n = m!/(m - n)!n!= Pn / n!m mCmn 从 m 个元素中挑出哪 n 个元素（不考虑 n 个元素的内部顺序）的所有情况的数量。*n m-nCm = Cm2Probability: 概率* 概率的古典定义：P(A)=A 所包含的基本事件数/基本事件总数。 例：掷一个骰子，掷出的是个奇数的概率是多少？练习题：<1> 一只袋中装有五只乒乓球，其中三只白色，两只红色。现从袋中取球两次， 每次一只，取出后不再放回。求：两只球都是白色的概率；两只球颜色不同的概率；至少有一只白球的 概率。<2> 从 5 位男同学和 4 位女同学中选出 4 位参加一个座谈会

33、，要求与会成员中 既有男同学又有女同学，有几种不同的选法？<3> 电话号码由四个数字组成，每个数字可以是 0,1,2,9 中的任一个数，求电 话号码是由完全不同的数字组成的概率。<4> 晚会上有 5 个不同的唱歌节目和 3 个不同的舞蹈节目，问：分别按以下要求各可排出几种不同的节目单？*3 个舞蹈节目排在一起；*3 个舞蹈节目彼此隔开；*3 个舞蹈节目先后顺序一定。<5> 6 张同排连号的电影票，分给 3 名男生和 3 名女生，如欲男女相间而坐，则 不同的分法数为多少？<6> 用 0,2,4,6,9 这五个数字可以组成数字不重复的五位偶数共有多少

34、个？<7> 从 6 双不同的手套中任取 4 只，求其中恰有一双配对的概率。<8> 3 封不同的信，有 4 个信箱可供投递，共有多少种投信的方法？<9> 3 个打字员为 4 家公司服务，现在每家公司各有 1 份文件需要录入，问每个打 字员都收到文件的概率？<10> A certain roller coaster has 3 cars, and a passenger is equally likely to ride in any 1 of the 3 cars each time that passenger rides the roller

35、coaster. If a certain passenger is to ride the roller coaster 3 times, what is the probability that the passenger will ride in each of the 3 cars?（A）0 （B）1/9（C）2/9 （D）1/3（E）1<11>A gardener is going to plant 2 red rosebushes and 2 white rosebushes. If the gardener is to select each of the bushe

36、s at random, one at a time, and plant them in a row, what is the probability that the 2 rosebushes in the middle of the row will be the red rosebushes?（A）1/12 （B）1/6 （C）1/5 （D）1/3 （E）1/2<12> If a committee of 3 people is to be selected from among 5 married couples so that the committee does no

37、t include two people who are married to each other,how many such committees are possible?（A）20 （B）40 （C）50 （D）80 （E）120<13> There are 5 cars to be displayed in 5 parking spaces with all the cars facing the same direction. Of the 5 cars, 3 are red, 1 is blue, and 1 is yellow. If the cars are id

38、entical except for color, how many different display arrangements of the 5 cars are possible?(A)20(B) 25(C) 40(D) 60(E) 125<14> How many different 6-letter sequences are there that consist of 1 A, 2 Bs, and 3Cs ?(A)6 (B) 60 (C) 120 (D) 360 (E) 720<15> A photographer will arrange 6 people

39、 of 6 different heights for photograph by placing them in two rows of three so that each person in the first row is standing in front of someone in the second row. The heights of the people within each row must increase from left to right, and each person in the second row must be taller than the pe

40、rson standing in front of him or her. How many such arrangements of the 6 people are possible?(A)5 (B) 6 (C) 9 (D) 24 (E)36<16>Pat will walk from Intersection X to Intersection Y along a route that is confined to the square grid of four streets and three avenues shown in the map above.How many

41、 routes from X to Y can Pat take that have the minimum possible length?(A) 6 (B) 8 (C) 10 (D) 14 (E) 16<17>In the integer 3589 the digits are all different and increase from left to right. How many integers between 4000 and 5000 have digits that are all different and that increase from left to

42、 right?数学词汇1数学符号等于：=equal to, the same as, is不等：>more than<less thanno less thanno more than加：+add, plus, more than; sum减：-minus, subtract; less than; difference乘：×multiply, times; product 除：÷divide; quotient 绝对值： absolute value平方：X2square立方：X3cube开平方：square root开立方： 3cube root平行：par

43、allel to垂直：perpendicular to2数字前缀1: uni,mono, 2: bi,di 3: tri,ter4: tetra,quad5: penta,quint 6: hex,sex7:sept,hapta8: oct9: enn 10:dec,deka3. 方程和函数equation方程solution, root, zero解 variable变量constant常量（数） term 项 coefficient 系数4. 数列和集合arithmetic progression等差数列 geometric progression等比数列 set集合subset 子集 s

44、equence 序列 term 序列中的项inclusive包含序列的首末项exclusive不包含序列的首末项5. 排列组合与概率permutation排列 combination 组合 probability，possibility概率6. 数论common division公约数common factor公因子composite number合数(质数和 1 以外的自然数)consecutive integer连续整数digit数字 divide除以 divisor因子,除数（evenly）divisible by可整除的even number偶数 factor因子 integer整数

45、irrational无理数least common multiple最小公倍数 multiple倍数,公倍数 natural number自然数 negative number负数 nonzero非零odd number奇数 positive number正数 prime factor质因子 prime number质数 quotient商 rational有理数 real number实数 remainder余数 whole number整数units'digit 个位数 tens'digit 十位数 hundreds'digit 百位数two-digit numbe

46、r两位数7. 单利复利和价格compound interest复利 cost成本 discount折扣down payment预付款,现付款interest rate利率 list price标价 margin利润 mark up涨价 mark down降价 markup毛利profit利润simple interest单利8. 其它代数addition加arithmetic mean算术平均数average平均数 base底数 closest approximation近似 decimal小数base 10 notation; decimal notation 十进制 decimal poi

47、nt 小数点 decreased 下降后的 decrease to从下降到 decrease by 下降了 define定义 denominator分母denote表示,代表 distinct不同的 expression表达式 fraction分数improper fraction假分数 increased增加后的 increase to 从增加到 increase by增加了in terms of用表达 least possible最小值 maximum最大值 minimum最小值 multiply乘multiplier 乘数 numerator 分子 per capita 人均 power

48、 指数proportional to正比于 proper faction真分数 ratio比率 reciprocal倒数 reduced降低后的rounded to the nearest tenth四舍五入到十分位 Successive;consecutive;in a row连续的 tenth十分位tenths'digit十分位tie平局times几倍two digits两个数字twice as many A as BA 是 B 的两倍3/2 as many A as BA 是 B 的 3/2 倍A is 20% more than BA 比 B 多 20%,(A-B)/B=20%

49、9. 几何abscissa横坐标 acute angle锐角 altitude高arc弧area面积angle bisector角平分线 bisect平分 center中心 chord弦circle圆 circumference圆周长 circumscribe外接,外切 clockwise顺时针 concentric circle同心圆 cone圆锥 congruent全等的 coordinate坐标counterclockwise逆时针 cube正方体 cylinder圆柱 decagon十边形 degree角度 diameter直径 diagonal对角线dimension大小,维度dis

50、tance距离due north正北方 equilateral triangle等边三角形 face面height高 hexagon六边形 hypotenuse斜边 isosceles triangle 等腰三角形 inscribe内接,内切 intersect相交length长度median of a triangle三角形的中线mid point中点 number lines数轴 obtuse angle钝角 octagon八边形 ordinate纵坐标 overlap交叠parallelogram平行四边形 pentagon五边形 perimeter周长 parallel lines 平

51、行线 perpendicular lines垂直线 plane 平面 polygon 多边形 quadrant象限whichof the following must be an odd integer?quadrilateral radiusradian四边形半径 弧度(弧长/半径)regularpolygon正多边形rectangular solidrectangle长方体长方形right angle right triangle squaresphere sidesurface area straight angle segment tangent trianglevertex（vert

52、ices）angle直角直角三角形 正方形 球边表面积 平角线段切线 三角形顶角作业<1>If a and b are positiveintegers such that a b anda are both even integers,b(A)a(B） b(C)a + b(D)a + 2(E)b + 222222<2>If y3, and 3x/y is a prime integer greater than 2, which of the followingmust be true?. x=y.y=1.x and y are prime integers(A) N