Math Workout for the GRE, 3rd Edition (2013)

Chapter 12. Answers and Explanations to Sample Section 2

1. A

If the average of 3 things is 7, the total is (3 × 7) = 21. 21 = a + 2a + 9 = 3a + 9, and a = 4. Therefore, Quantity A is greater.

2. A

Divide both sides of the inequality by 3 to get 4 > x − 3y. Quantity B must be less than 4, and, thus, less than Quantity A.

3. B

Follow the dotted lines. r corresponds to the x-coordinate, −5. s corresponds to the y-coordinate, −3.5. Quantity B is greater than Quantity A.

4. C

Try rearranging the problem. If , then . Hence, AB.A = 78.7, and A + B = 15. The quantities are equal.

5. C

For Quantity A, substitute the numbers the provided in the question for the variables in the function: −|−10 + 7| = −|−3| = −3. Quantity B is −3. The quantities are equal.

6. C

The company sold 300 ÷ 2 = 150 of Product B. The remainder, A + C = 150. Since Quantity B is a number, try plugging it in to the problem for Product A. If A = 100 then C = 50 so the quantities are equal.

7. D

Plug In 1 for z. Quantity A is , and Quantity B is . Quantity B is greater, so eliminate answer choices (A) and (C). Plug In −1 for z. Quantity A is , and Quantity B is . Quantity A is closer to 0, and, thus, is greater. Eliminate choice (B), and select choice (D) since different numbers gave different answers.

8. B

The probability that the flower is red is , so the probability that it is not red is 1− = . Quantity A is . The probability that the flower is not yellow equals the probability that it is red + the probability that it is blue. + 40 percent = . Quantity B is and is greater.

9. B

Redraw the figure so that arc AB looks much smaller than BD. Now, draw radii from the center of the circle to A and to B. If the new angle were 90 degrees, Quantity A would equal . If the new angle were 60 degrees, the three arcs would have equal lengths. Because the length of arc AB is less than that of the other two arcs, the angle must be less than 60 degrees. Quantity A must be less than .

10. E

You are given a value for each of the variables. If you put these values directly into (4 + a)(4 − b), you get (4 + 4)(4 − (−4)) = (8)(4 + 4) = (8)(8) = 64.

11. D

If the average of 2 numbers is 35, the total is (2 × 35) = 70. Plug In for k, and let k = 20. 70 − 20 = 50, the other number and the target answer. Go to the answer choices and Plug In 20 for k. Only answer choice (D) matches your target of 50.

12. A and D

If x is 100, then Alejandro found 25 percent of 100 = 25 instead of increasing 100 by 25 to 125. The question asks what Alejandro could do to turn 25 into 125. If he adds x, this will work because 25 + 100 = 125, or in math terms 0.25x + 1x = 1.25x. Multiplying by x will not work because 25 × 100 = 2,500, not 125. Dividing by 0.25 will not work because 25 ÷ 0.25 = 100, not 125. Multiplying by 5 works because 25 × 5 = 125.

13. D

Use the ratio box. Fill in that the ratio of x to y is 6 to 7. Fill in that the actual value of x is 15. The multiplier is = . Use this multiplier to find y: 7 × = = 17.5.

14. B

Plug In. If b = 2, c = 3, and a = 4, then x = 20. The question asks for the value of b, so the target answer is 2. Plug c = 3, a = 4, and x = 20 into all the answers to find the answer that yields 2. Choice (A) yields a fraction, not 2. Choice (B) yields − 3 = 2, so keep this answer. Estimate that choices (C) and (D) are larger than 2. Choice (E) yields 8, not 2. Choice (B) is the only choice that yielded the target answer.

15. 1

Notice that the exact same numbers appear in the numerator and denominator of the fraction. Since it doesn’t matter in what order you add quantities, the numerator and denominator of the fraction are the same. You don’t even have to add it up. The answer is 1 because a nonzero number divided by itself is 1.

16. E

You could multiply 6 by 17 and start dividing the answer choices by the result, but there’s an easier way. Because 6 = 3 × 2, the correct answer must be a multiple of 2. Eliminate choice (C) because it’s odd. Next, the correct answer is a multiple of 3. To check to see which numbers are multiples of 3, add their digits, and check to see if the sum is a multiple of 3. For example, choice (A) is 6 + 8 = 14, which is not a multiple of 3. Eliminate choice (A). You can also eliminate choice (D) this way. Try dividing choice (B) by 17. Since you don’t get an integer, that leaves only choice (E).

17. D

There are 180 degrees in a triangle, so you know that 2x + 3x + 4x = 180. Solve for x: 9x = 180, and x = 20. The smallest angle is 40 degrees, the middle angle is 60 degrees, and the largest angle is 80 degrees. The sum of the smallest and largest angles is 120, or choice (D).

18. D

You only know the sum of the two prices: (8a + b) + (8b + a) = 135 cents, or 9a + 9b = 135. You could simplify this by dividing by 9 to get a + b = 15. However, you cannot determine the values of a and b individually, nor their values relative to each other. For example, try a = 5 and b = 10, and then try a = 10 and b = 5. Different outcomes are possible, and the answer is choice (D).

19. E

Work with numerical portion of the square root first = 0.6. Eliminate answer choices (A), (B), and (C). Next, work with the variable. If you are struggling with exponent rules you can expand it out. = = x⁴. So y = 0.6 x⁴.

20. E

Notice that x² + 2xy + y² = 25 is a quadratic equation. The question asks for (x + y)³. The first step is to factor x² + 2xy + y² = 25. It becomes (x + y)(x + y) = 25. Therefore, (x + y) = ± 5. Since you know that (x + y) could be 5 or −5, you know (x + y)³ could be 125 or −125. The correct answer is (E).

21. B, C, D, and E

Since the elevator moves 200 pounds of grain every 15 minutes, it moves 800 pounds every hour. Starting at noon and stopping at 3:00, the elevator moves 3 × 800 = 2,400 pounds of grain. Eliminate answer choice (A). At the opposite extreme, starting at noon and stopping at 4:00, the elevator moves 4 × 800 = 3,200 pounds. Eliminate answer choice (F). The upper and lower bounds having been established, any answer choices that fall on or between them—that is, choices (B), (C), (D), and (E)—are correct.

22. E

Remember that a fraction or decimal between 0 and 1 gets smaller when you cube it. So, c > a. Conversely, it gets bigger when you take the cube root, so b > c. Therefore, b > c > a.

23. E, F, G, and H

If students are dropping out of the course, then there had to have been more than 120 students originally. Eliminate answer choice (A). Next, find Professor Quin’s drop-out rate: 20 percent (or ) of 15 is 3, so Professor Quin’s rate is 3 greater than 15 percent, or 18 percent. If 18 percent of the students dropped out over the length of the course, then 82 percent remain on the last day. 82 percent of x is 120: × x = 120. The value of x is 146.34; because you can’t have .34 of a person, the total must have been 147. The rate is at least 18 percent, so there could have been more students originally; hence, any answer choices that are greater than 147 can be correct as well.

24. A

First, find a side of the smaller square. x² + x² = 1 thus 2x² = 1 so x² = and x = . Alternatively, you can apply the ratio for the sides of an isosceles right triangle (s : s : s) to solve for the length of a side. The area of this square is or . Therefore, the area of the larger square must be 1. The formula for the area of a square is a = s². Since s² = 1, that means s = 1. The difference between the larger and smaller sides is .

25. E

Multiply both sides of the second equation by 3 to get 9a + 3b = 93. Add the first and second equations:

If 5a − 3b = −9

Solve to find that a = 6. Plug a = 6 back into 5a − 3b = −9 to find the value of b: 5(6) − 3b = −9, or 30 − 3b = −9, and b = 13. The question asks for a + b, which is 6 + 13 = 19.

26. E

To figure out the probability of being accepted to at least one university, use the following formula: P(at least one) = 1 − P(none). The chance that Claire gets into none of her top five schools is 0.2 × 0.3 × 0.5 × 0.8 × 0.9, which is about 2 percent. Therefore, her chance of getting into at least one of her top five schools is about 98 percent.