Quantitative Comparison Practice Problems with Solutions, Study Guides, Projects, Research of Mathematics

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Sample Section 2: Answers and Explanations

1. A

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

2. A

Divide both sides of the inequality by 3 to get 4 > x − 3 y. 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, and 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 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; the quantities are equal.

7. D

Plug in 1 for z. Quantity A is , and Quantity B is. Quantity B is greater, so

eliminate (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 (B), and select (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% = + =. Quantity B is and is greater.

9. B

2. Choice (E) yields 8, not 2. Choice (B) is the only choice that yields the target answer.

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 (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, (A) is 6 + 8 = 14, which is not a multiple of 3. Eliminate (A). You can also eliminate (D) this way. Try dividing (B) by 17. Since you don’t get an integer, that leaves only (E).

17. D

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

18. D

You know only the sum of the two prices: (8 a + b ) + (8 b + a ) = 135 cents, or 9 a + 9 b = 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, so the answer is (D).

19. E

Work with numerical portion of the square root first. Eliminate (A), (B), and (C).

Next, work with the variable. If you are struggling with exponent rules, you can expand it out:

= x^4. So y = 0.6 x^4.

20. E

Notice that x^2 + 2 xy + y^2 = 25 is a quadratic equation. The question asks for ( x + y )^3. The first step is to factor x^2 + 2 xy + y^2 = 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 (A). At the opposite extreme, starting at noon and stopping at 4:00, the elevator moves 4 × 800 = 3,200 pounds. Eliminate (F). The upper and lower bounds having been established, any answer choices that fall on or between them—that is, (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 (A). Next, find Professor Quin’s drop-out rate: 20% (or ) of 15 is 3, so Professor Quin’s rate is 3

greater than 15%, or 18%. If 18% of the students dropped out over the length of the course, then 82%

remained on the last day. 82% 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%, 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. x^2 + x^2 = 1. Thus, 2 x^2 = 1 so x^2 = and x =. Alternatively, you

can apply the ratio for the sides of an isosceles right triangle ( x : x : x ) 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 = s^2. Since s^2 = 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 9 a + 3 b = 93. Add the first and second equations:

Solve to find that a = 6. Plug a = 6 back into 5 a − 3 b = −9 to find the value of b : 5(6) − 3 b = −9, or 30 − 3 b = −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%. Therefore, her chance of getting into at least one of her top five schools is about 98%.