GMAT QA
P (not E) =
P (not E) = 1 - P(E)
Sum of the interior angles of a polygon with n sides
(n-2)*180 độ
|x + y| <= |x| + |y|
E.g. x = 10, y = -2 |10 - 2| <= |10| + |-2| 8 <= 12
# of multiples of x in the range
# of multiples of x in the range = (last multiples of x in the range - first multiples of x in the range)/x + 1
Pythagorean Theorem
định lý pytago
(39897)(0.0096)/(198.76) is approximately
(40000)(0.01)/(200) = 2
common multiple of 2 numbers e.g. least common multiple of 8 and 12 is
/common denominator/ - a number that is a multiple for all the numbers being considered. e.g. Least common multiple of 8 and 12 is 24, meaning there is an integer time 8 that will make 24 and there is an integer time 12 that will make 24. Why 24? Because it is the smallest common multiple of 8 and 12, other than 72, which is another common multiple of 8 and 12.
Area of a trapezoid
A = 1/2*h*(b1+b2)
In 45 - 45 - 90 right triangles, length of sides ratio
1:1:sqrt(2)
In 30 - 60 - 90 right triangles, length of sides ratio
1:sqrt(3):2
2 goes into 6 (evenly) (without a remainder)
2 divides 6
6 is divisible by 2
2 divides 6
6 is divisible by 2
2 goes into 6 (evenly) (without a remainder)
6 is divisible by 2
2 is a divisor of 6
6 is a multiple of 2
2 is a factor of 6
When 24 is divided by the positive integer n, the remainder is 4. Which of the following statements about n must be true? I. n is even. II. n is a multiple of 5. III. n is a factor of 20. A. III only B. I and II only C. I and III only D. II and III only E. I, II and III.
24 = qn + 4 for positive integer q, and n > 4, or qn = 20 and n > 4. It follows that n = 5, n = 10, n = 20 since these are the only factor of 20 that exceed 4. 1. n is not necessarily even. E.g, n could be 5 2. n is necessarily a multiple of 5 since the value of n is either 5, 10, or 20. 3. n is a factor of 20 since 20 = qn for some positiv integer q.
Ratio for right triangle
3:4:5
What is the thousandths digit in the decimal equivalent of 53/5000? A. 0 B. 1 C. 3 D. 5 E. 6
53/5000 = 106/10,000 = 0.0106 and the thoundsandths digit is 0. 53/5000 = (1/5)*(53/1000) = (1/5)(0.053) = 0.0106
For all positive integers m and v, the expression m Θ v represents the remainder when m is divided by v. What is the value of ((98Θ33)Θ17)-(98Θ(33Θ17)) ? A. -10 B. -2 C. 8 D. 13 E. 17
98Θ33 = 32 (remainder) 32Θ17 = 15 33Θ17 = 16 98Θ16 = 2 15 - 2 = 13. D is the right answer.
factorial of positive integer
A factorial is a function applied to natural numbers greater than zero. The symbol for the factorial function is an exclamation mark after a number, like this: 2! E.g. 1! = 1; 2! = 2; 3! = 3*2 = 6; 4! = 4*3*2 = 24; and 5! = 5*4*3*2 = 120.
Parallelogram
A quadrilateral with two pairs of parallel sides
The "prime sum" of an integer n greater than 1 is the sum of all the prime factors of n, including repetitions. For example, the prime sum of 12 is 7, since 12 = 2x2x3 and 2+2+3=7. For which of the following integers is the prime sum greater than 35? (A) 440 (B) 512 (C) 620 (D) 700 (E) 750
A. 440 = 2*2*2*5*11 => sum=21 B. 512 = 2^9 => sum=2*9=18 C. 620 = 2*2*5*31 => sum=40 greater than 35 There can only be one correct answer, D and E need not be checked. The correct answer is C.
If x and y are integers such that 2<x<=8 and 2<y<=9, what is the maximum value of 1/x-x/y? A. -3 1/8 B. 0 C. 1/4 D. 5/18 E. 2
Because x and y are both positive, the maximum value of 1/x - x/y will occur when the value of 1/x is maximum and the value of x/y is minimum. The value of 1/x is maximum when the value of x is minimum or when x = 3. The value of x/y is minimum when the value of x is minimum (x=3) and the value of y is maximum (x=9). Thus, 1/3-3/9=0. Correct answer is B.
For positive integers a and b, the remainder when a is divided by b is equal to the remainder when b is divided by a. Which of the following could be a value of ab? I. 24 II. 30 III. 36 (A) II only (B) III only (C) I and II only (D) II and III only (E) I, II and III
Consider two cases: a = b and a khác b If a = b, then our given condition is trivially satisfied: the remainder when a is divided by a is equal to the remainder when a is divided by b. The condition thus allows that a be equal to b. If a khác b, either a < b or b < a. Supposing that a < b, the remainder when a is divided by b is simply a (e.g. 7/10, r = 7). However, according to our given condition, this remainder a, is also the remainder when b is divided by a, which is impossible. If b is divided by a, then the remainder must be less than a (e.g. any number is divided by 10, the remainder cannot be 10 or greater). Similar reasoning applies if b < a. I. there is no a x a = 24 II. Same with I, there is no a x a = 30 III. a x a = 6 x 6 = 36. The answer is B.
Half of a large pizza is cut into 4 equal-sized pieces, and the other half is cut into 6-equal sized pieces. If a person were to eat 1 of the larger pieces and 2 of the smaller pieces, what fraction of the pizza would remain uneaten? (A) 5/12 (B) 13/24 (C) 7/12 (D) 2/3 (E) 17/24
Each of the 4 equal-sized pieces represents 1/8 of the whole pizza since each slice is 1/4 of 1/2 of the pizza. Each of the 4 equal-sized pieces represents 1/12 of the whole pizza since each slice is 1/6 of 1/2 of the pizza. The fraction of the pizza remaining after a person eats one of the larger pieces and 2 of the smaller pieces is 1 - [1/8 + 2*(1/12)] = 17/24.
1<x<9. What inequality represents this condition? A. |x| < 3 B. |x + 5| < 4 C. |x - 1| < 9 D. |-5 + x| < 4 E. |3 + x| < 5
First of all, we find length (9 - 1) = 8 and center (1+8/2) = 5 of the segment represented 1<x<9. Now let's look at our option. Only B and D has 8/2 = 4 on the right side and D had left side 0 at x = 5. Therefore, answer is D.
To calculate the MEDIAN of n numbers, ...
First order the number from least to greatest; if n is odd, the median is defined as the middle number; if n is even, the median is defined as the average of the two middle numbers.
For any evenly spaced set, median
For any evenly spaced set, median = mean = the average of the first and the last terms = the average of the whole set.
Inscribed vs. Circumscribed
If each vertex of a polygon lies on a circle, then the polygon is inscribed in the circle and the circle is circumscribed about the polygon. If each side of polygon is tangent to a circle, then the polygon is circumscribed about the circle and the circle is inscribed in the polygon.
If the cost of a certain house in 1983 is 300% of its cost in 1970, by what percent did the cost increase?
If n is the cost in 1970, then the percent increase is equal to (3n - n)/n = 2n/n = 2, or 200%.
List S consists of 10 consecutive odd integers, and list T consists of 5 consecutive even integers. If the least integer in S is 7 more than the least integer in T, how much greater is the average (arithmetic mean) of the integers in S than the average of the integers in T? (A) 2 (B) 7 (C) 8 (D) 12 (E) 22
For any evenly spaced set, median = mean = the average of the first and the last terms. So the mean of S will be the average of the first and the last terms: mean = (x + x + 9*2)/2 = x+9, where x is the first term; The mean of T will simply be the median or the third term: mean = (x - 7) + 2*2 = x - 3; The difference will be (x + 9) - (x - 3) = 12. Answer: D.
Multiple of a number... E.g. Multiples of 2 are...
Multiple of a number is that number multiplied by an integer. E.g. Multiples of 2 are 2, 4, 6, 8, 10, etc. To get these numbers, multiplied 2 by 1, 2, 3, 4, 5, which are integers.
What is the sum of odd integers from 35 to 85, inclusive? A) 1,560 B) 1,500 C) 1,240 D) 1,120 E) 1,100
No. of odd integers = (85-35)/2 + 1 = 26 Sum of odd integers = average * no. of integers = ( 35 + 85 ) /2 * 26 = 60 * 26 = 1560 Answer: A
Multiplication rule holds for any independent events A and B:
P ( A and B) = P(A)*P(B)
P (E or F) =
P (E or F) = P(E) + P(F) - P (E and/giao F) P (E or F) = P(E) + P(F) - P(E) * P(F)
Length of arc S
S = r*θ
Surface Area of a Right Cylinder
SA = 2(πr^2) + 2πrh the sum of areas of the two bases plus the area of curved surface => SA = 2πr(r+h)
Each machine at a toy factory assembles a certain kind of toy at a constant rate of one toy every 3 minutes. If 40 percent of the machines at the factory are to be replaced by new machines that assemble this kind of toy at a constant rate of one toy every 2 minutes, what will be the percent increase in the number of toys assembled in one hour by all the machines at the factory, working at their constant rates? A. 20 B. 25 C. 30 D. 40 E. 50
Say the rate is 100 toys by 100 old machines in 1 unit of time (1 toy per 1 machine in 1 unit of time). 40 of the machines are replaced with new ones which are 1.5 times as efficient as the old ones, so they will produce 60 toys in 1 unit of time. Total = 60 + 60 = 120. Percent increase = 20%. Answer: A.
If x^2 - 2x - 15 = 0 and x > 0, which of the following must be equal to 0? I. x^2 - 6x + 9 II. x^2 - 7x + 10 III. x^2 - 10x + 25 (A) I only (B) II only (C) III only (D) II and III only (E) I, II, and III
Since x^2 - 2x + 15 = 0, then (x-5)(x-3)=0, so x = 5 and x = -3. SInce x>0, then x=5. I. (x-3)^2 = (5-3)^2 = 4 (khác 0) II. (x-5)(x-2) = (5-5)(x-2) 0 III. (x-5)^2 = 0 D is the right answer.
prime factor of a number
The prime factors of a number are the prime numbers that must be multiplied together to give the number. E.g. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The prime factors of 30 are 2, 3, and 5
Sum of positive consecutive even integers starting from 2
Sum of positive consecutive even integers starting from 2 is x*(x+1), where x is the number of even integers. E.g. 2+4+6+8+10 = 5*6 = 30
Sum of positive consecutive integers starting from 1
Sum of positive consecutive integers starting from 1 is (x*(x+1))/2, where x is the number of integers. E.g. 1+2+3+4+5+...+10 = (10*11)/2
Sum of positive consecutive odd integers starting from 1
Sum of positive consecutive odd integers starting from 1 is x^2, where x is the number of odd integers. E.g. 1+3+5+7+9+11 = 6^2
The sum of the even numbers between 1 and n is 79*80, where n is an odd number, then n=? (A) 79 (B) 80 (C) 81 (D) 157 (E) 159
Sum of x even integers is x(x+1) = 79*80. So x = 79 i.e. there are 79 even integers. These 79 even integers lie between 1 and n. 2 will be first such even integer, next will be 4, next will be 6 and so on till we reach the last even integer 79*2 = 158. So all even integers from 2 to 158 lie between 1 and 159. So n must be 159.
How many multiples of 4 are there between 12 and 96, inclusive? 21 22 23 24 25
The correct answer is 22. # of multiples of x in the range = (last multiple of x in the range - first multiple off x in the range)/x + 1 = (96−12)/4 + 1 = 22.
The LCM of two numbers
The least common multiple of two numbers A and B is the smallest number that is a multiple of both A and B
When you combine 2 factor trees of x that contain overlapping primes...
When you combine 2 factor trees of x that contain overlapping primes, DROP THE OVERLAP
The price of a certain stock increased by 0.25 of 1% on a certain day. By what fraction did the price of the stock increase that day? A. 1/2500 B. 1/400 C. 1/40 D. 1/25 E. 1/4
a certain stock increased by 0.25 of 1% equivalent to an increase of 1/4 of 1/100 which is (1/4)(1/100)=1/400. B is correct answer.
Trapezoid
a quadrilateral with one pair of parallel sides
Area of a parallelogram
base x height
Hypotenuse
cạnh huyền
rectangular solid
hình hộp chữ nhật có 6 mặt, 12 cạnh, 8 đỉnh. cạnh: edge đỉnh: vertex
Length of arc of a circle
length = θ/360 x 2πr
isoceles triangle
tam giác cân
point of tangency
the point where a tangent intersects a circle
If the sum of the reciprocals of two consecutive odd integers is 12/35, then the greater of the two integers is A. 3 B. 5 C. 7 D. 9 E. 11
1/a + 1/b = (a+b)/ab (a+b)/ab = 12/35 a+b=12 ab=35=5*7 7 is greater.
If A is a positive integer, is A a multiple of 30? (1) A is a mulitple of 20 (2) sqrt(A) is a multiple of 24 A. (1) alone is sufficient, but (2) alone is not sufficient. B. (2) alone is sufficient, but (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but neither statement alone is sufficient. D. EACH statement ALONE is sufficient. E. (1) and (2) TOGETHER are not sufficient.
A multiple of 30 must contain all the prime factors of 30, i.e. {2, 3, 5} (1): Let's pick test values for A that are multiples of 20 to see if A would be consistently a multiple of 30 or not: A = 20 > A is NOT a multiple of 30 A = 60 > A is a multiple of 30. Since A being a multiple of 30 is not definitely yes or no for all A, this statement alone is not sufficient. We can eliminate (A) and (D). The correct answer is either (B), (C), (E). (2): If sqrt(A) is a multiple of 24, then for some positive integer n we have the following: sqrt(A) = 24*n A = (24*n)^2 A = (2^3 * 3 * n)^2 In order for A to be divisible by 30, n would need to be a multiple of 5, as this is the only prime factor of 30 that is not consistently present in A above. Thus, this statement alone is not sufficient. We can eliminate choice (B). The correct answer must be either (C) and (E). (1) & (2): From statement (1). we know A will have at least the prime factors of 2 and 5. Similarly, from statement (2), A will have at least the prime factors of 2 and 3. Putting them together tell us A will at least have the prime factors of 2, 3, and 5, and those are necessary for A to be a multiple of 30. The statements together are sufficient. We can eliminate choice (E). The correct answer is choice (C).
Which of the following lines in the xy-plane does NOT contain any point with integers as both coordinates? A. y = x B. y = x + 1/2 C. y = x + 5 D. y = (1/2)x E. y = (1/2)x + 5
A. y = x, if x is an integer, y is an integer. B. If x is an integer, and if y were an integer then y - x would be an integer, but y - x = 1/2. NOT an integer. Since there is only one correct answer, the lines in C, D, and E need not be checked. C. same like A. D. If x is an even integer, then y is an integer. E. same like D.
How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1, 2, 3, and 6.) (A) 1 (B) 2 (C) 3 (D) 4 (E) 61
Alternatively, if the integer n, where n > 1, has exactly 3 positive integer factors, which include 1 and n, then n has exactly 1 other positive integer factor, say p. Since any factor of p would also be a factor of n, then p is prime, and so p is the only prime factor of n. It follows that n = p^k for some integer k > 1. But if k >= 3, then p^2 is a factor of n in addition to 1, p, and n, which contradicts the fact that n has exactly 3 positive integer factors. Therefore, k = 2, and n = p^2, which means that n is the square of a prime number. Of the integers between 1 and 16, inclusive, only 4 and 9 are the squares of prime numbers.
An integer is divisible by 3 if ...
An integer is divisible by 3 if the sum of its digits is multiple of 3. E.g. adding the digits of the number 2,145 (2+1+4+5) = 12, which is divisible by 3.
An integer is divisible by 9 if ...
An integer is divisible by 9 if the sum of its digits is a multiple of 9. E.g. Take the number 144. 1 + 4 + 4 = 9, so 144 is divisible by 9.
Five machines at a certain factory operate at the same constant rate. If four of these machines, operating simultaneously, take 30 hours to fill a certain production order, how many fewer hours does it take all five machines, operating simultaneously, to fill the same production order? (A) 3 (B) 5 (C) 6 (D) 16 (E) 24
If 4 machines, working simultaneously, each work for 30 hours to fill a production order, it take (4)(30) machine hours to fill the order. If 5 machines are working simultaneously, it will take ((4)(30)/(5)) = 24 hours. Thus, 5 machines working simultaneously will take 30 - 24 = 6 fewer hours to fill the production order than 4 machines working simultaneously. Answer: C
triangle inscribed in a circle
If a triangle is inscribed in a circle so that one of its sides is a diameter of a circle, then the triangle is a right triangle.
At a garage sale, all of the prices of the items sold were different. If the price of a radio sold at the garage sale was both the 15th highest price and the 20th lowest price among the prices of the items sold, how many items were sold at the garage sale? (A) 33 (B) 34 (C) 35 (D) 36 (E) 37
If the price of the radio was the 15th highest price, there were 14 items that sold for prices higher than the price of the radio. If the price of the radio was the 20th lowest price, there were 19 items that sold for prices that were lower than the price of the radio. Therefore, the total number of items sold is 14 + 1 + 19 = 34. The correct answer is B. OR, As the price of an item was BOTH 15th highest price and the 20th lowest price then # of items is: 15+20-1=34.
A total of 5 liters of gasoline is to be poured into two empty containers with capacities of 2 liters and 6 liters, respectively, such that both containers will be filled to the same percent of their respective capacities. What amount of gasoline, in liters, must be poured into the 6-liter container? (A) 4 1/2 (B) 4 (C) 3 3/4 (D) 3 (E) 1 1/4
If x represents the amount, in liters, of gasoline poured into the 6-liter container, then 5 - x represents the amount, in liters, of gasoline poured into the 2-liter container. After the gasoline is poured into the containers, the 6-liter container will be filled to (x/6 x 100)% of its capacity and the 2-liter container will be filled to ((5-x)/2 x 100)% of its capacity. Because these two percents are equal, x/6 = (5-x)/2 2x = 6(5-x) x = 3 3/4 Therefore, 3 3/4 liters of gasoline must be poured into the 6-liter container.
When a subscription to a new magazine was purchased for m months, the publisher offered a discount of 75% off the regular monthly price of the magazine. If the total value of the discount was equivalent to buying the magazine at its regular monthly price for 27 months, what was the value of m? (A) 18 (B) 24 (C) 30 (D) 36 (E) 48
Let P represent the regular monthly price of the magazine. The discounted monthly price is then 0.75P. Paying this price for m months is equavilent to paying the regular price for 27 months. Therefore, 0.75mP = 27P, and so 0.75m = 27. Then m = 36 The correct answer is D.
The average (arithmetic mean) length per film for a group of 21 films is t minutes. If a film that runs for 66 minutes is removed from the group and replaced by one that runs for 52 minutes, what is the average length per film, in minutes, for the new group of films in terms of t? A. t+2/3 B. t-2/3 C. 21t + 14 D. t+3/2 E. t-3/2
Let S denote the sum of the lengths, in minutes, of the 21 films in the original group. Since the average length is t minutes, it follows that S/21 = t. If a 66-minute film is replaced by a 52-minute film, then the sum of the lengths of the 21 films in the resulting group is S - 66 + 52 = S - 14. Therefore, the average length of the resulting 21 films is (S-14)/21 = S/21 - 14/21 = S/21 - 2/3 = t - 2/3.
List S consists of 10 consecutive odd integers, and list T consists of 5 consecutive even integers. If the least integer in S is 7 more than the least integer in T, how much greater is the average (arithmetic mean) of the integers in S than the average of the integers in T? (A) 2 (B) 7 (C) 8 (D) 12 (E) 22
Let the integers in S be s, s+2, s+4, s+6, ..., s+18, where s is odd. Let the integers in T be t, t+2, t+4, t+6, t+8, where t is even. Given that s-t=7 The average of the integers in S is (10s+90)/10 = s+9 The average of the integers in T is (5t+20)/5 = t+4 The difference in these averages is (s+9) - (t+4) = (s-t) + (9-4) = 7 - 5 = 12.
Median of three positive integers is
Median of three positive integers is the value of the middle integer when the integers are in numerical order.
An open box in the shape of a cube measuring 50 cm on each side is constructed from plywood. If the plywood weighs 1.5 gr/square centimeter, which of the following closet to the total weight, in kg, of the plywood used for the box? A. 2 B. 4 C. 8 D. 13 E. 19
Surface area of each side = 50 * 50 Surface area of all sides = 5 * 50 * 50 (since it is an open box, sixth side will be missing) weight of the box in gms = 5 * 50 * 50 * 1.5 = 18750 = 19000(approx) weight of the box in Kgms = 19000 / 1000 = 19
How many multiples of 5 are there between -7 and 35, not inclusive?
The last multiple of 5 IN the range is 30. The first multiple of 5 IN the range is -5. # of multiples of x in the range = (30 - (-5))/5 + 1 = 8
How many multiples of 7 are there between -28 and -1, not inclusive?
The last multiple of 7 IN the range is -7. The first multiple of 7 IN the range is -21. # of multiples of x in the range = ((-7) - (-21))/7 + 1 = 3
If S = {0, 4, 5, 2, 11, 8}, how much greater than the median of the numbers in S is the mean of the numbers in S? (A) 0.5 (B) 1.0 (C) 1.5 (D) 2.0 (E) 2.5
The median of a set with even number of elements is the average of two middle elements, when arranged in ascending/descending order. Thus the median of S = {0, 2, 4, 5, 8, 11} is (4 + 5)/2 = 4.5. The mean of the set = (0 + 4 + 5 + 2 + 11 + 8)/6 = 5. The difference = 5 - 4.5 = 0.5. Answer: A.
Is w equal to the median of the three positive integers w, x, and y? (1) x = y (2) w = x + y A. (1) alone is sufficient, but (2) alone is not sufficient. B. (2) alone is sufficient, but (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but neither statement alone is sufficient. D. EACH statement ALONE is sufficient. E. (1) and (2) TOGETHER are not sufficient.
The median of three positive integers is the value of the middle integer when the integers are in numerical order. We need to know if w has the same value as the middle integer when w, x, and y are in numerical order. (1): It is given that x = y. Let's pick test cases: - If we let x = y = 2 and w = 2, then all three integers are the same and the median would equal w. So the answer to the question would be 'yes.' - If we let x = y = 2 and w = 1, then in numerical order we have 1, 2, 2 so the median is 2 (NOT equal to w). So, the answer to the question would be 'no.' Therefore, we cannot definitively determine if w is equal to the median for all cases. This statement alone is not sufficient. We can eliminate choice (A) and (D). The correct answer is either (B), (C), (E). (2): It is given that w = x + y. Since w is the sum of the other two integers, it follows that w is the GREATEST integer of w, x, and y when in numerical order. Hence, w will never equal the median, and the answer to the question is a DEFINITIVE 'no.' This statement alone is sufficient. We can eliminate answer choices (C) and (E). The correct answer is choice (B).
A set is a collection of numbers, those numbers are called the elements of the set.
The number of elements in a set is denoted b |S| E.g. S = {0, 1, 3} => |S| = 3
If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + s = (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
Using the rule of exponents, (2^r)(4^s) = 16 (2^r)(2^2s) = 2^4 r + 2s = 4 Since r and s are positive integers, s < 2; otherwise, r would not be positive. Therefore, s = 1, and it follows that r + 2*1 = 4, r = 2. The value of 2r + s = 2*2 + 1 = 5. Alternatively, since (2^r)(4^s) = 16 and both r and s are positive, it follows that s < 2; otherwise, 4^s >= 16 and r would not be positive. Therefore, s = 1 amd (2^r)(4) = 16. It follows that 2^r = 4 and r = 2. The value of 2r + s = 2*2 + 1 = 5. Answer: D
standard deviation of n numbers
the more the data spread away from the mean, the greater the standard deviation; (1) find the arithmetic mean; (2) find the differences between the mean and each of the n numbers; (3) square each of the differences; (4) find the average of the squared differences; (5) take the nonnegative square root of this average.
x is divisible by n numbers
x is divisible by the LCM of n numbers and the LCM of n numbers is also the largest number that x is divisible by.
y = xq + r
x, y = positive integers q = quotient r = remainder 0 <= r < x E.g. 28 = 8*3 + 4 28 is divided by 8, the quotient is 3, the remainder is 4
