GRE - Incorrect Answer Log (Math only)

If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true? 1. x = y 2. y = z 3. y > z 4. x > z

A: #2, #3, #4 Use PITA here: If x = y, then 3x < 2x, which is incorrect, so #1 is not true If y = z, then 2y < 4y, so #2 is true For y > z: y can be 3 and z can be 2, so #3 is true For x > z: x can be 5 and y can be 4, so #4 is true

An investor placed a total of $6,400 in two accounts for one year. One account earned simple annual interest at 5% and the other at 3%. There were no deposits or withdrawals. If both accounts earned the same amount of interest in the year, what was the total amount of interest earned from both accounts?

A: $240 If one account has $x, the other has 6,400 - x. So we can do the following: (6400 - x) * 0.03 = 0.05x 192 - 0.03x = 0.05x 192 = 0.08x x = $2400 If one account had $2,400 and earned 5% interest, it earned 0.05 * 2400 = $120. Since both accounts earned $120 in interest, both earned a total of $240.

The sale price of a certain radio is 25% less than its list price and 40% greater than its wholesale price. If its wholesale price is $30, what is its list price?

A: $56 Since the sale price is 40% greater than the wholesale price, we get: S = 1.4 * W S = 1.4 * 30 S = 42 Since the Sale price is 25% less than the List price, we get: S = L * 75% 42 = L * 0.75 L = 42 / 0.75 L = $56

Lines x and y intersect to form a 90-degree angle. What is the value of xy?

A: -1 Since the slopes of perpendicular lines are negative reciprocals, their products are always equal to -1. Ex: y = -8x + 4, y = (1/8)x + 7 -8 * (1/8) = -1

If |x + 1| <= 5 and |y - 1| <= 5, what is the least possible value of the product xy?

A: -36 Solving both absolute value equations (which can be done mentally), we get the following: x = 4 or -6 y = 6 or -4 The least possible product of xy is therefore -36.

When the positive integer d is divided by 12, the remainder is 17. What is the remainder when d^2 is divided by 8?

A: 1 Use trial and error. If d = 17, then d^2 = 289. If done using long division, r is found to be 1. If done using a calculator, the remainder will be in decimal form: 0.125. Multiplying this decimal remainder by the divisor, we get: 0.125 * 8 = 1

A rectangular tile has a length between 4 and 6 inches, and a width between 3 and 6 inches. What are possible values of its area in square feet? 1/8, 1/6, 1/2, 4/3

A: 1/8, 1/6 The trick is to first convert the side lengths from inches to feet. Doing so, we get: 1/4 <= L <= 1/2 1/3 <= W <= 1/2 The smallest and largest the area could be are the following: As: 1/4 * 1/3 = 1/12 Al: 1/2 * 1/2 = 1/4 Of the answer choices, 1/8 and 1/6 are within the range, while 1/2 and 4/3 are not.

A jar contains 5 marbles, r of which are red. If the probability of picking 2 red marbles is 1/10, what is r? 1, 2, 3, 4, 5

A: 2 Probability of picking x from n is: x/n * x-1/n-1 * x-2/n-2 * ... * 1/1 Given this, there are two ways to solve. First, algebra: r/5 * (r-1)/4 = 1/10 Solving this, we get r = 2 Second, PITA: Starting with choice B, we get: 2/5 * 1/4 = 1/10, thus r = 2

What is the number of multiples of 9 between 10 and 300?

A: 32 Use the following formula: Let h = highest factor of 9 in the range Let l = lowest factor of 9 in the range Let i = interval of the multiple Total # of multiples = [(h - l) / i] + 1 = [(297 - 18) / 9] + 1 = 32

On a final exam, 75% of the scores were greater than 70, and 60% of the scores were less than 85. What percent of the scores were greater than 70% and less than 85%?

A: 35% There are two ways to consider this problem: 1. Percentages must always add up to 100%. Since we have two percentages that overlap, we can calculate the overlap by setting them equal to 100%: 75% + 60% - 100% = Overlap = 35% 2. Use the percentages not given. If 75% of the scores were above 70, then 25% were below 70. If 60% of the scores were below 85, then 40% were above 85. This sets up the following: 100% - 25% - 40% = Overlap = 35%

For each positive integer n, the nth term of the sequence S is "1 + (-1)^n. Which is larger: 1. The sum of the first 39 terms of S 2. 39

A: 39 Plugging in to find a pattern, we find the following: S1 = 0, S2 = 2, S3 = 0, S4 = 2, etc. The sum of terms from S1 to Sn become the following: S1 = 0, S2 = 2, S3 = 2, S4 = 4, etc. The sum of terms from S1 to Sn equals n when n is even, and equal n-1 when n is odd. Thus, S39 = 38 < 39.

A group of n students split one cake into n equal pieces. They split another cake into another n equal pieces. 3 students were full, so they split a 3rd cake into n - 3 pieces. If Sarah, one of the students, got 1 piece from each cake, the amount of cake she got in total was what fraction of the original cake (in terms of n)?

A: 3n - 6 / n (n - 3) Sarah's 1st piece was 1 / n, her 2nd piece was 1/n, and her 3rd was 1 / (n - 3). So, the total amount of cake she ate relative to one cake is the following: 1/n + 1/n + 1/(n-3) (n-3)/n(n-3) + (n-3)/n(n-3) + n/n(n-3) 2(n-3)+n / n(n-3) 3n-6 / n(n-3) Thus, D is correct.

A certain experiment has three possible outcomes. These outcomes are mutually exclusive and have probabilities: p, p/2, p/4, respectively. What is the value of p?

A: 4/7 Two ways to solve this problem: algebra & PITA Algebra first: Since probabilities must always sum to 1, we can set the following equation: P + p/2 + p/4 = 1 Solving for p, we get p = 4/7 PITA next: Plugging in 3/7 for p, as it's the middle answer choice: (3/7) + (3/7)/2 + (3/7)/4 = 3/4 which is < 1. So A, B, and C are incorrect. Plugging in 4/7 next, we get: (4/7) + (4/7)/2 + (4/7)/4 = 1, so p = 4/7.

What is the degree measurement of an exterior angle of a 9-sided polygon?

A: 40 degrees The equation to find the total sum of all interior angles of a polygon is 180(n-2) where n is the number of sides/angles. Plugging in, we get: sum of interior angles = 180(9-2) = 180(7) = 1,260 There are 9 sides to this polygon, so one single interior angle measures: one interior angle = 1,260 / 9 = 140 The measure of an exterior angle is 180 - the supplementary interior angle. Thus: exterior angle = 180 - 140 = 40 degrees

In the 1st half of last year, a team won 60% of the games it played. In the 2nd half of last year, the team played 20 games, winning 3 of them. If the team won 50% of the games it played last year, what was the total number of games the team played last year?

A: 90 Let g = total games played So, games played in the 1st half of the year: g - 20 Since #W / g = %W, we can set up the below equation: [0.6 (g - 20) + 3] / g = 0.5 Solving for g, we get g = 90.

The volume of a right circular cylinder is 2000π and its height is 16 times its radius. Which is larger: The radius of the cylinder, or 5?

A: Both are equal Volume of a cylinder = π * r^2 * h Plugging into the above formula, we get: 2000π = π * r^2 * (16π) 2000 = 16 * r^3 125 = r^3 r = 5

What is the least integer n such that 1/(2^n) < 0.001?

A: n = 10 The equation can be re-written as follows: 1/(2^n) < 1/1000 2^n > 1000 *note the flipped inequality due to the removal of the reciprocal Since 2^10 = 1,024, n must be at least 10, and so n = 10.

If x and y are integers and x = 50y + 69, which of the following must be odd? 1. xy 2. x + y 3. x + 2y 4. 3x - 1 5. 3x + 1

A: x + 2y There are two ways to solve this problem: even/odd rules reasoning, and plugging in Even/Odd rules first: An odd sum is only possible with one even and one odd numbers, and an odd product is only possible with two odd factors. Thus, "50y" will always be even, so "50y + 69" will always be odd, and so x is always odd. Additionally, notice that if "3x - 1" were odd, then "3x + 1" would also be odd, so neither can be true and so x must be odd. Per the above rules, let's review the answer choices: xy --> y can be e/o, so xy can be e/o, so no x + y --> y can be e/o, so xy can be e/o, so no x + 2y --> 2y is even whether y is even or odd, so x + 2y must be odd x + 2y works, so we stop checking as it is our answer Plugging in next: When y = -1, x = 19 When y = 0, x = 69 When y = 1, x = 119 When y = 2, x = 169 Clearly, x is always odd whether y is even or odd. Of the answer choices, only #3 is odd in all permutations of x and y above.

A developer has land that has x feet of lake frontage. The land is to be subdivided into lots, each of which is to have either 80 feet or 100 feet or lake frontage. If 1/9 of the lots are to have 80 feet of lake frontage each and the remaining 40 lots are to have 100 feet of frontage each, what is the value of x?

A: x = 4,400 If 1/9 of the lots have 80 feet of lake frontage, then the remaining 40 lots represent 8/9 of the total number of lots. Thus, there are (9/8)*40 = 45 lots, and 5 (i.e. 1/9) have 80 feet of lake frontage If 40 lots have 100 feet of lake frontage each, then they have a total of 40*100 = 4,000 feet of lake frontage. If the remaining 5 lots have 80 feet of lake frontage, then there is 80*5 = 400 feet of lake frontage for them. Thus, x = 4,000+400 = 4,400 feet of lake frontage total.

List L consists of the numbers: 1, sq(2), x, x^2, where x > 0, and the range of L is 4. Which is larger: x or 2?

A: x is larger If the range of L is 4, then x or x^2 must either be 5 or -3 (to satisfy L's range criteria). x is positive, so cannot be -3. x^2 must be positive, so cannot be -3. x cannot be 5 because then x^2 would be 25 making the range of L 25-1=24. Thus, x^2 must be 5, and x must be sq(5), which is greater than 2.

A list consists of 3 different positive integers whose sum is 10. Which of the following statements provides sufficient additional information to determine the greatest number in the list? A: The median is 3 B: The range is 5 C: The sum of the largest and smallest numbers is 7

Ans: B only Let x + y + z = 10, where x < y < z A: If the median were 3, then y = 3. So, x + z = 7. However, (x, z) could be (1, 6) or (2, 5). Thus, A is incorrect. B: If the range were 5, then z - x = 5. Substituting into the original equation for z, we get: x + y + (x + 5) = 10 2x + y = 5 This leaves two choices for x and y. However, since x < y, only one choice remains: x = 1, y = 3. Thus, z = 6 and so B is correct. C: If x + z = 7, then substituting into the original equation for z, we get: x + y + (7 - x) = 10 y = 3 Although we'd know y = 3, x and y could still be either (1, 6) or (2, 5). Thus C is incorrect.