GMAT Study - Quant
If r and s are positive integers, is r/s an integer? (1) Every Factor of s is also a factor of r. (2) Every prime factor of s is also a prime factor of r.
Arithmetic / Properties of Numbers (1) Sufficient. The integer s is by definition a factor of itself. From this, every factor of S is also a factor of R. Therefore, r/s MUST be an integer. (2) No Sufficient. From this example. If r = 18 and s = 6, then 6 has the prime factors 2 and 3, each of which is also a factor of 18, and r/s = 18/6, which is an integer. However, if r = 18 and s = 8, then r has the prime factors 2 and 3, and s has a prime factor 2, which satisfies this condition. Ever though in this case the prime factor of s is not a prime factor of r, r/s = 18/8, which is not an integer; NOT sufficient.
An investment of $1,000 was made in a certain amount and earned interest that was compounded annually. The annual interest rate was fixed for the duration of the investment, and after 12 years the $1,000 increased to $4,000 by earning interest. In how many ears after the initial investment was made would the $1,000 have increased to $8,000 by earning interest at that rate? a) 16 b) 18 c) 20 d) 24 e) 30
Exponential Growth is constant, so look for the pattern. In 12 years, investment quadruple, meaning it doubled twice, so the rate is doubling every 6 years. As a result, $4,000 (12 years after initial investment) will double to $8,000 in another 6 years....total time since initial investment to get to $8,000 is 18 years (12 + 6)
If 5^x - 5^x - 3 = (124)(5^y), what is y in terms of x? (a) x (b) x - 6 (c) x - 3 (d) 2x + 3 (e) 2x + 6
Factor out the Lowest Power! Pattern Recognition: when you see additions or subtractions (not multiplications or divisions) with the same base but different powers, factor our the lowest power.
How many different 6-letter sequences are there that consist of 1 A, 2 B's, and 3 C's? A) 6 B) 60 C) 120 D) 360 E) 720
First break it down into two parts: Step 1) There are 6 letters, so total combinations are 6! -> 6(5)(4)(3)(2)(1) = 720 sequences However, this is not the answer because the duplicated arrangements of the 2 B's and the 3 C's need to be removed because these arrangements would look exactly the same. Step 2) Figure out the different sequences for the duplicated letters that essentially have the same arrangements: 2 B's = (2)(1) = 2 3 C's = (3)(2)(1) = 6 so the number of difference sequences becomes: 720/(2)(6) = 60 different sequences.
Each of 90 students participated at least 1 of the track tryouts: High jump, long jump, 100 meter dash. If 20 students participated in high jump tryout, 40 students participated in the long jump tryout and 60 students participated in the 100 meter dash tryout, and if 5 students participated in all 3 tryouts, how many students participated in only two of these tryouts? (A) 25 (B) 20 (C) 15 (D) 10 (E) 5
Group A + Group B + Group C + Neither - Doubles - 2(Triples) = Total Memorize the above formula for questions like this! ^
What is the product of all the solutions of X^2 + 4x + 7 = | x + 2 | + 3? A) -6 B) -2 C) 2 D) 6 E) 12
In order to find all of hte solutions of an equation that contains an absolute value, first isolate the absolute value on one-side of the equation and then solve for both the positive and negative of the expression inside the absolute value. Remember, when finding the product of all the solutions, don't double-count the same solution! So, for example, if -2 is a solution that shows up when solving for the positive expression and the negative expression, don't multiple it twice!
If m and n are positive integers and r is the remainder when 5(10^n) + m is divided by 3, what is the value of r? (1) n = 10 (2) m = 1
It's asking about the remainder when some number is divided by 3, so you should be immediately thinking about the divisibility rules for 3! If the sum of the digits add up to a multiple of 3, then you know it is divisible by 3 and there would be no remainder. Or if the number is constant that makes the units digit off by x numbers, then you will always know the remainder.
If an integer n is to be selected at random from the integers 1 to 100, inclusive, what is the probability that n( n+1 ) will be divisible by 4? (a) 1/4 (b) 1/3 (c) 1/2 (d) 2/3 (e) 3/4
Number Properties Patter Recognition! N is either even or odd: N+1 is even if n is odd, and odd if N is even. So, N(N+1) must be even b/c it is always multiplied by at least one even number, thus making the final produce even! Every other even number is divisible by 4, and as a result 1/2 of the numbers will be divisible by 4.
If n is the product of the integers from 1 to 20 inclusive, what is the greatest integer k for which 2^k is a factor of n? A) 10 B) 12 C) 15 D) 18 E) 20
Number Properties! Recognize that only factors of 2 need to be considered! Only even numbers are factors of 2 in this case so only include those! Factors of 2 in the list of integers from 1 to 20 inclusive include: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 2 X 2^2 X 2(3) X 2^3 X 5(2) X (2^2)(3) X 7(2) X 2^4 X 9(2) X 5(2^2) Count the 2's.....you get 2^18, so this is the largest factor!
If x^2y = Z^3, is Z^3 >0? (1) xy^2 > 0 (2) yz > 0
Positive/Negative Question Whenever you notice the >0 (or <0), the multiplications, and the even and odd exponents, you should immediately recognize that this question is essentially testing whether or not you can prove that the variable is negative or positive. Remember this pattern recognition! To solve....stay organized and create a chart to help organize the logic and avoid careless mistakes:
If x does NOT = -1, then [ ( 1 -x^16) / (1+x)(1+x^2)(1+x^4)(1+x^8)] is equal to: a) -1 b) 1 c) x d) 1 - x e) x - 1
Recognize that the numerator is a difference of squares and can be broken out to help simplify the equation
In the storage room of a certain bakery, the ratio of sugar to flour is 5 to 8, and the ratio of flour to baking soda is 10 to 1. If there were 60 more pounds of baking soda in the room, the ratio of flour to baking soda would be 8 to 1. How many pounds of sugar are stored in the room?
Recognize that this is a Ratios & Unknown Multiplier problem! You are given ingredients, with one overlapping, and asked to find the amount of the third ingredient. As a result, there are a couple of steps here Step 1 - setup a chart to visualize the ratios of the ingredients and to see the cross-section. Find a common multiple for the overlapping ingredient. Step 2 - now that you have a common multiple you use the second piece of information to create an algebraic equation. Step 3 - solve the equation and then plug the solution for X into the Multiplier equation for Sugar and there you have the answer! Remember to stay organized with the ingredients and ratios.
A clothing store's revenue last May was 6 percent less than it's revenue last April. The store's revenue last May was greater than its expenses last May, and the store's revenue last April was greater than its expenses last April. Was the store's gross profit last May less than its gross profit last April? (1) The store's revenue last May was $2,400 less than its revenue last April. (2) The store's expenses last May were 4 percent less than its expenses last April.
Recognize that this is difficult b/c of multiple variables, and lots of text to read through, percents, and inequalities. This will be nuisance, so skip!
If n = 20! + 17, then n is divisible by which of the following? I. 15 II. 17 III. 19 A. None B. I only C. II Only D. I and II E. II and III
Remember the Key Divisibility Rules! A multiple of N + a multiple of N is a multiple of N. A multiple of N + a non-multiple of N is NOT a multiple of N. The answer is C, because 20! = 20 X 19 X 18 X 17 X 16 X 15, etc. Then you add 17....so this is a multiple of 17 because it satisfies the Divisibility rule for multiples. It contains a multiple of 17 and then we add a multiple of 17...so it is therefore divisible by 17. 15 and 19 are not divisible....because by adding 17 to the equation....we are turning N into a non-multiple of 15 and 19 per the divisibility rules.
Of the N candies in a bag, some are peppermint and the rest are spearmint. What is the value of N? (1) If 1 peppermint candy were removed from the N Candies, 1/5 of the remaining candies would be peppermint. (2) If 2 spearmint candies were removed from the N candies, 1/4 of the remaining candies would be peppermint.
Remember! This is a Linear Equation question! If you have 2 unknown variables, you need 2 equations to solve. Determine the number, N, of candies in a bag, some of which are peppermint and the rest of which are spearmint. If P and S represent the numbers of peppermint candies and spearmint candies, respectively, in the bag, then N = P + S.(1) If 1 peppermint candy were removed from the bag, there would be N - 1 candies left in the bag and P - 1 of them would be peppermint candies. It is given that of the remaining candies would be peppermint, so (N - 1) = P - 1. Since the value of P can vary, it is not possible to determine the value of N. For example, if P = 11, it follows that N = 51. However, if P = 21, it follows that N = 101; NOT sufficient.(2) If 2 spearmint candies were removed from the bag, there would be N - 2 candies left in the bag, S - 2 of them would be spearmint and P of them would be peppermint. It is given that of the remaining candies would be peppermint, so (N - 2) = P. Since the value of P can vary, it is not possible to determine the value of N. For example, if P = 8, it follows that N = 34. However, if P = 10, it follows that N = 42; NOT sufficient.Taking (1) and (2) together and combining (N - 1) = P - 1 from (1) and (N - 2) = P from (2), it follows that (N - 1) = (N - 2) - 1. This is a linear equation in N and can be solved for a unique value of N; SUFFICIENT.The correct answer is C;both statements together are sufficient.
Is -3x^3 <= -3? (1) -3x < 3 (2) x < 0
Rephrase the question! This is simply a Number Properties question, is X > or < 0! Remember, multiplying or dividing by a negative number reverses the inequality sign! -3x^3 <= -3 Divide both sides by -3 Is x^3 => 1 ? (1) NOT SUFFICIENT. -3x < 3 (simplify by dividing both sides by 3) x > -1 This is not sufficient, because - 1/2 is less than 1, but 2 is greater than 1. (2) SUFFICIENT. This statement says X < 0, so X^3 MUST be less than 1.
What is the value of X? 1) 2x - 5y - 6 = 0 2) 8x - (4x + 10y) +27 = 39
S1 and S2 NOT SUFFICIENT. Answer is E. S1, NOT SUFFICIENT. 2 unknown variables so cannot solve with S1. S2, NOT SUFFICIENT. 2 unknown variables so cannot solve with S2. S1 & S2 together, NOT SUFFICIENT b/c when both statements are simplified we discover that both statements are actually the same equation, so we cannot use both to solve for the unknown variables. THIS IS EASY....whenever the DS question is just asking to solve for the value of X, it is probably a 500 - 600 level question, so try and solve! Dont Skip!
What is the value of x? 1) sqr 4x = 6 2) x^2 = 81
S1 is sufficient because you can sufficient solve for X S2 is NOT sufficient because the even exponent hides the sign of X, so X = 9 or -9 and as a result statement 2 is NOT sufficient. Remember that even exponents always hide the sign of the variable, and this is a trick that GMAT tries to play on test takers!
The Operation ^ is defined for all nonzero x and y by x ^ y = x + x/y. If a > 0, then 1 ^ ( 1 ^ a) = ?
Simplify "Strange Operations" before beginning to solve.
A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?
Special products problem 1) a^2 + b^2 = 200^2 2) a + b = 280 (a + b)^2 = 280^2 a^2 + 2ab + b^2 = (200 + 80)^2 a^2 + b^2 + 2ab= 200^2 + 2*200*80 + 80^2 -a^2 - b^2 = -200^2 --------------------------------------- 2ab = 2*200*80 + 80^2 = 32000 + 6400 = 38400 = 19200 FLASHCARD Tip off: geometry, so draw it to see relationships, and write out eqns. Notice the square and the very similar not squared eqn. Special products - also flashcard (a + b)^2 = a^2 + 2ab + b^2 (a - b)^2 = a^2 - 2ab + b^2 (a + b)(a - b) = a^2 - b^2
A car traveling at a constant speed took 1 minute 48 seconds to travel the distance between a certain road sign and the beginning of a roadwork area. If the distance between the road sign and the beginning of the roadwork area was 2,400 meters, then the car was traveling at what speed, in kilometers per hour? (1 kilometer = 1,000 meters)
Speed = Distance/Time To convert take original speed formula and multiple by 1 kilometer / 1,000 meters * 60 minutes / 1 hour
If 2 different representatives are to be selected at random from a group of 10 employees and if p is the probability that both representatives selected will be women, is p > 1/2? (1) More than 1/2 of the employees are women. (2) The probability that both representatives selected will be men is less than 1/10.
The formula for probaility of an event's happening is: # of ways to get desired outcome / total number of ways event can occur. The probability of two events is found by multiplying the separate probabilities. (1) NOT SUFFICIENT. The number of women employees must be 6, 7, 8, 9, or 10. Using the extremes in this list: 6/10 * 5/9 = 1/3, or a probability of less than 1/2 10/10 * 9/9 = 1, which is greater than 1. So given the choices, there is no way to determine if the probability is > 1/2 (2) NOT SUFFICIENT. From this statement there must be 3 or fewer men in order for the probability to be less than 1/10: For zero or one man: 0 For two men: 2/10 * 1/9 = 2/90 = 0.022 For three men: 3/10 * 2/9 = 6/90 = .067 This means the number of women could be 10, 9, 8, or 7. The probability of two women being chosen is greater than 1/2 if there are 9 women employees 9/10 * 8/9 = 72/90 = 4/5, but less than 1/2 if there are 7 women: 7/10 * 6/9 = 42/90 = 14/30 = 7/15 Given these choices, there is no way to determine if the probability will be less than or greater than 1/2. Statement(2) is a subset of possibilities from statement (1), eliminated only 6 women as a possibility. As such, since (2) i not sufficient, then both statements are also not sufficient.
The ratio of the number of men to the number of children in a room is 5:2:7, respectively. What is the total number of people in the room? (1) The total number of women and children in the room is 12. (2) There are fewer than 4 men in the room.
The number of people in the room is a multiple 14 5x:2x:7x = 14x and x must be an integer <- unknown multiplier problem! 5x + 7x = 12 -> x=1, so sufficient #men = 2x and 2x < 4 -> x < 2 and x is an integer, so x=1, so sufficient FLASHCARD Tip offs: given ratios, asked for overall total or a component total
At the beginning of the year, the ratio of juniors to seniors in high school X was 3 to 4. During the year, 10 juniors and twice as many seniors transferred to another high school, while no new students joined high school X. If, at the end of the year, the ratio of juniors to seniors was 4 to 5, how many seniors were there in high school X at the beginning of the year?
This is a 500 to 600 level question, don't get it wrong! Pretty simple Ratios problem, just set the ratio as follows: j/s = 3/4, so j = 3/4s Then create the equation for the change: J - 10 / S - 10 = 4/5 substitute j for 3/4s and solve for s! Remember you have 2 equations with 2 variables, so you can solve for either variable.