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

Chapter 10. Answers and Explanations to Sample Section 1

1. A

Use the Bow Tie to find the value of each quantity. Quantity A = . Quantity B = . Quantity A is greater than Quantity B.

2. C

Vertical angles are equal, so s = v, r = u, and t = w. Therefore, the quantities are equal. To prove it, Plug In. Let u = 50, v = 100, and w = 30; the vertical-angle rule makes s = 100, r = 50, and t = 30, and now both r + t + v and s + u + w total 180.

3. D

First, find the value of x. Multiply both sides by 3 to get 2x < 21. Divide both sides by 2 to find that x < 10.5 as well as being greater than 0. If x = 2, then Quantity B is greater because 2² = 4. Eliminate choices (A) and (C). If x = 10, then the two quantities are equal. Eliminate choice (B), and select choice (D) because different numbers have given different answers.

4. C

Solve for k: 9(3)³ + 4 = 9(27) + 4 = 243 + 4 = 247. So, k = 247. Use trial and error to find the prime factors of 247. They are 13 and 19. The average of 13 and 19 is 16, so the quantities are equal.

5. C

First, divide $90 by 50 to find the bulk rate for a single pound of flour: = $1.80. The amount saved per pound is $.09. The quantities are equal.

6. A

Remember PEMDAS. x = ((4) + 2) × 5 = 6 × 5 = 30; y = (24 ÷ (8)) × 5 = 3 × 5 = 15. Quantity A = 15, and Quantity B = −15.

7. B

The solutions to the equation 32 = |k| are 32 and −32. Of those, only −32 works in the equation 31 = |k + 1|. Quantity A has the value of −32, so Quantity B is greater.

8. B

Plug In. First, try a = 100 and b = 2: Alice weighs 100 kilograms, Bob weighs 98 kilograms, and the sum of their weights is 198 kilograms. Quantity B is 2 × 100, or 200. Quantity B is greater, so eliminate answer choices (A) and (C). Now, try a second set of numbers: If a = 50 and b = 10, then Quantity A is 90 and Quantity B is 100. Quantity B is, again, greater—as it will be with any set of numbers that meets the restriction (ab ≠ 0).

9. D

Plug In for x and y. Since xy must be positive, either both variables are positive, or both are negative. First, let x = 5 and y = 6: Quantity A is , and Quantity B is −10. Quantity A is greater, so eliminate answer choices (B) and (C). Now let x = − and y = −. Quantity A is −23, and Quantity B is −. Quantity B is now greater, so eliminate answer choice (A), and you’re left with answer choice (D) because different numbers gave different answers.

10. A

The weird little symbol (*) gives you a set of directions. 2* = (2)² + 4 = 4 + 4 = 8. Find the answer that also equals 8. (−2)* = (−2)² + 4 = 4 + 4 = 8.

11. B, C, and D

Finding common denominators allows you to compare the values of fractions easily. Because two answer choices are expressed in eighteenths (and two in easily-convertible ninths) start there: = , and = . Eliminate answer choice (A), and select choice (B). Choice (D) converts to , so it’s correct; choice (E), though, converts to and can be eliminated. The answers are listed in ascending order, and so answer choice (C), falling between correct answer choices (B) and (D), must be correct as well.

12. C

Plug In. If the width of the smaller board is 3, then the width of larger board must be 6. If the length of the smaller board is 2, then the length of the larger board must be 12. The area of the smaller board is 3 × 2 = 6. Therefore, N = 6. The area of the larger board is 12 × 6 = 72. The question asks for the difference between the areas: 72 − 6 = 66. The last step is to check which answer choice equals 66 when N = 6. Only choice (C) works.

13. C

Since the answer choices are far apart, you can estimate. If one manual weighs 400 grams, then 48 manuals weigh 48 × 400, or about 20,000 grams. To convert grams to kilograms, divide by 1,000. 20,000 ÷ 1,000 = 20. Look for an answer a little less than 20. Only choice (C) is close.

14. 24

Start by listing the factors of 12 in pairs: 1 and 12, 2 and 6, and 3 and 4. Because the question asks for the sum of the distinct positive even factors, cross off 1 and 3. Now, add up what’s left: 2 + 4 + 6 + 12 = 24.

15. D

The store makes a profit of $300 for each computer sold. Plug In the answer choices, starting with answer choice (C). 16 computers multiplied by $300 yields a profit of $4,800, not enough to cover costs. You need to sell more computers, so eliminate answers choices (A), (B), and (C). Try choice (D). 17 computers multiplied by $300 yields a profit of $5,100, just enough to cover costs.

16. A

Estimate. If x is less than 50 percent of a fraction that is less than of a number that is less than 1, then x must be less than . Therefore, choice (A) is correct.

17. D

Each number in the set occurs only once, and the mode is the most frequently occurring number. To create a mode, x must be same as one of the other numbers in the set. Eliminate choice (C) and choice (E). Plug In the answers. For choice (B), the mode is 5. The median of {1, 3, 5, 5, 7, 11} is 5. The median is equal to the mode. You need a larger number for x. For choice (D), the mode is 7 and the median of {1, 3, 5, 7, 7, 11} is = 6. The mode is now one less than the median. Note that choice (C) is equal to the median and would be a trap answer if you didn’t read the question carefully enough.

18. B

Redraw this figure by adding two descending lines from the upper corners of the figure perpendicular to the base below so that you have a rectangle and two right triangles. The triangles each have a base of 1 and a hypotenuse of 2. Use the Pythagorean theorem or the ratio for 30-60-90 triangles (a : a : 2a) to find the height is . The area of each triangle is . The area of the rectangle is (2)(). The sum of the rectangle and two triangles is = .

19. A

If Mary’s wage is $2 more than Mark’s, and Andy’s wage is $6 more than Mark’s, then Andy’s wage is $4 more than Mary’s. Anne’s wage is $10 more than Mary’s, so Anne’s wage is $6 more than Andy’s. Quantity A is greater than Quantity B.

20. D

Plug In, and let t = 5. M = 5 + 7 + 9 = 21—the sum of the three consecutive odd integers, of which t is the smallest. For the second part of the problem, t is the greatest integer in the series. The sum of the new series is 5 + 3 + 1 = 9. Plug M = 21 into the answers, and answer choice (D) matches the target of 9.

21. A

The bigger circle that includes the garden and the sidewalk has an area of 169π. The radius of the large circle with the garden and sidewalk is 13. The garden plot has area 144π, so the radius of the small circle is 12. Since the circles share a center, the width of the sidewalk, t, is the difference between the radii, or 13 − 12 = 1. Therefore, answer choice (A) is correct.

22. A

Factor the quadratic expression. a² − b² becomes (a + b)(a − b). The average of two numbers is 6, so their sum, (a + b), is 2 × 6 = 12. Substitute the value to get: 12(a − b) = 2. Thus, a − b = .

23. B

Plug In numbers that have a sum of 19, and check to see if their product is 88. For example, the sum of 9 and 10 is 19, but their product is 90. 9 and 10 can’t be the two numbers. Try another pair, such as 8 and 11. Their sum is 19, and their product is 88. These numbers match the information given in the problem, and their difference is 3.

24. E

Plug In. If l = 2, then . The target is . Plug 2 for l into the answer choices. Answer choice (E) is the only answer choice that matches the target.

25. D, E, and F

First, figure out Pierre’s per-glass profit. His per-glass revenue is $2; his per-glass costs are 25 cents × 5 lemons per glass = 125 cents per glass for lemons, and 5 cents × 3 tablespoons of sugar per glass = 15 cents per glass for sugar, for a total of 140 cents per glass. Pierre’s profit on each glass is 60 cents. At a profit of 60 cents per glass, Pierre must sell at least 9 glasses of lemonade to make 5 dollars. Each answer choice greater than or equal to 9 is correct.

26. A

Plug In the answers, starting with choice (C). If the combined total was 220 and A is 150 more than B, then A is 75 more than half of 220, and B is 75 less than half of 220. So, A is 185 and B is 35. Remove 20 from each to get A is 165 and B is 15. Is 165 four times 15? No, so choice (C) is not correct. It’s hard to tell whether choice (C) was too large or too small, so just pick a direction. For choice (A), if the combined total was 290, and A is 150 more than B, then A is 75 more than half of 290, and B is 75 less than half of 290. So, A is 220 and B is 70. Remove 20 from each to get A is 200 and B is 50. Is 200 four times 50? Yes, so choice (A) is correct.