Quantitative Comparison Practice Set Answers and Explanations

  1. B
    Comparing the two quantities piece by piece, you find that the only difference is the third piece: −2 in Quantity A and −1 in Quantity B. You don’t know the value of x, but whatever it is, x2 in Quantity A must have the same value as x2 in Quantity B, and 2x in Quantity A must have the same value as 2x in Quantity B. Because any quantity minus 2 must be less than that quantity minus 1, Quantity B is greater than Quantity A. (B) is the correct answer.
  2. C

    Make the quantities look alike. If x = 2y, then −x = −2y. Replace the exponent −x in Quantity B with −2y. Thus, Quantity B becomes  . A value with a negative exponent in the denominator is equivalent to the same value with a positive exponent in the numerator, so Quantity B can be restated as 22y. Since 4 = 22, Quantity A can be written as (22)y = 22y.  The quantities are identical, and (C) is correct.

  3. D

    Do the same thing to both quantities to make them look like the centered information. When you multiply both quantities by 5s, you get qrs in Quantity A and 15 in Quantity B. Because qrs could be any integer greater than 12, qrs could be greater than, equal to, or less than 15. (D) is correct.

  4. D
    Because angle X = angle Y, at least two sides of the triangle are equal. You can draw two diagrams with X and Y as the base angles of a triangle. In one diagram, make the triangle tall and narrow so that angle X and angle Y are very large and angle Z is very small. In this case, Quantity B is greater. In the second diagram, make the triangle short and wide so that angle Z is much larger than angle X and angle Y. In this case, Quantity A is greater. Because more than one relationship between the quantities is possible, the correct answer is (D).
    A tall triangle, XYZ, on the left, and a short triangle, XYZ, on the right.
  5. C

    Since ∠ABC is 45°, ∠BCA is 180° − 90° − 45° = 45°, so ABC is an isosceles right triangle. Because ED = DF, triangle DEF is also an isosceles right triangle. The formula for the area of a triangle is . Since the triangles in this question are isosceles right triangles, the formula for the area of a triangle can be restated as  by substituting the base in for the height, since they are the same value. Given that the area of triangle ABC is four times the area of triangle DEF, set up the equation  . Even though there are two variables but only one equation, you can solve by picking numbers because only the ratio matters, not the actual values. Try 2 for the value of AC to get   , so DF2 = 1 and DF = 1. Given the side ratios of an isosceles right triangle, , side . So Quantity A becomes . Multiply both the numerator and the denominator by  to get  . This is the same as Quantity B, so (C) is correct.

  6. D

    In a set of 35 consecutive integers, the median will be the number in the middle. Of the remaining 34 values, 17 will be less than the median and 17 will be greater.

    If the first integer is odd, there will be 9 odd numbers and 8 even numbers below the median. The median will be even. Then, there will be another 9 odd numbers and 8 even numbers above the median. If the first integer is even, the results are reversed. There will be 9 even numbers and 8 odd numbers below the median, the median will be odd, and there will be 9 even numbers and 8 odd numbers above the median.

    So, if the first integer is odd, there’s an 8 in 35 chance of picking an even number less than the median and a 9 in 35 chance of picking an odd number greater than the median. Quantity B would be greater than Quantity A. However, if the first integer is odd, the odds are reversed, with a 9 in 35 chance of picking an even number less than the median and an 8 in 35 chance of picking an odd number greater than the median. That would make Quantity A greater than Quantity B. Because more than one relationship is possible, (D) is the correct answer. 

  7. A

    The stimulus provides no actual prices for the different tickets, so use A to represent the regular price of an adult’s ticket and C to represent the regular price of a child’s ticket. The tickets are full price on Saturday, so Quantity A could be calculated as: 120A + 100C.

    On Sunday afternoon, there’s a 20% discount on all tickets. That means the tickets will be 80% of their original price. Quantity B would thus be calculated as:

    150(.8A) + 120(.8C) = 120A + 96C

    Comparing the two results, both quantities have an equal total in sales from adults, but Quantity A has 100C while Quantity B has 96C. That means Quantity A is larger, making (A) the correct answer.

  8. A

    Using FOIL, Quantity A can be rewritten as . Quantity B needs to be simplified. In the numerator,   can be factored into . In the denominator,  can be factored into . The fraction can then be simplified by canceling out common factors in the numerator and denominator:

    So, Quantity B is equal to –5. Without knowing what x is, it might seem that Quantity A and Quantity B cannot be compared. However, when x is squared, the result cannot be negative. The smallest value it could have is 0. Subtracting 4, the smallest possible value of Quantity A is –4. That means Quantity A must be greater than or equal to –4. Any such number will always be greater than –5, so Quantity A will always be greater than Quantity B, no matter what x is. That makes (A) the correct answer.

  9. D

    It may be tempting to jump straight into the probability formula here, . Counting each apartment with a 4 in it (those starting with 4 as well as ending in 4) and the number of total apartments would give , which, when simplified, equals Quantity B. However, notice that the centered information gives information about apartments, while Quantity A is based on selecting a random tenant. There’s no information on how many tenants are in each apartment. The above math works if each apartment has an equal number of tenants. However, if each apartment with a 4 in the number has three tenants while other apartments only have one, it would be more likely that a randomly selected tenant lives in an apartment with a 4 in the number. Since more than one relationship is possible, the correct answer is (D).

  10. C

    It’s important to realize that the total distance need not be calculated to compare the two quantities. For this question, let d represent the total distance, in miles, traveled. Quantity A will thus be equal to d.

    Quantity B is the average speed for the entire trip. The average speed, in miles per hour, is calculated by taking the total mileage (d) and dividing that by the total number of hours. The car starts at 30 miles per hour and slows down 5 miles per hour every 10 minutes. That means it will travel first at 30 miles per hour, then 25 miles per hour, then 20, then 15, then 10, then 5. After that, it will stop. It will travel for 10 minutes at each of those 6 speeds, for a total of 60 minutes, which is 1 hour. That means the average speed will be equal to . Regardless of the value of d (which could be calculated, but would only take up valuable time for this question), the two quantities are equal. That makes (C) the correct answer.