Quantitative Comparison Sample Questions

Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.

A symbol that appears more than once in a question has the same meaning throughout the question.


  1. Quantity A Quantity B
    The least prime number greater than 24 The greatest prime number less than 28

    (A) Quantity A is greater.
    (B) Quantity B is greater.
    (C) The two quantities are equal.
    (D) The relationship cannot be determined from the information given.

    Explanation

    For the integers greater than 24, note that 25, 26, 27, and 28 are not prime numbers, but 29 is a prime number, as are 31 and many other greater integers. Thus, 29 is the least prime number greater than 24, and Quantity A is 29. For the integers less than 28, note that 27, 26, 25, and 24 are not prime numbers, but 23 is a prime number, as are 19 and several other lesser integers. Thus, 23 is the greatest prime number less than 28, and Quantity B is 23. Thus, the correct answer is Choice A, Quantity A is greater.


  2. Lionel is younger than Maria.
    Quantity A Quantity B
    Twice Lionel's age Maria's age

    (A) Quantity A is greater.
    (B) Quantity B is greater.
    (C) The two quantities are equal.
    (D) The relationship cannot be determined from the information given.

    Explanation

    If Lionel's age is 6 years and Maria's age is 10 years, then Quantity A is greater, but if Lionel's age is 4 years and Maria's age is 10 years, then Quantity B is greater. Thus, the relationship cannot be determined. The correct answer is Choice D, the relationship cannot be determined from the information given.


  3. Quantity A Quantity B
    54% of 360 150

    (A) Quantity A is greater.
    (B) Quantity B is greater.
    (C) The two quantities are equal.
    (D) The relationship cannot be determined from the information given.

    Explanation

    Without doing the exact computation, you can see that 54 percent of 360 is greater than One half of 360, which is 180, and 180 is greater than Quantity B, 150. Thus, the correct answer is Choice A, Quantity A is greater.


  4. Figure 1
    The figure shows triangle PQR, where P is the leftmost vertex of the horizontal side PR and vertex Q is above PR. Point S lies on horizontal side PR. Point S appears to be the midpoint of PR. Line segment QS is drawn from vertex Q to point S. The lengths of PS and SR appear to be equal.
    Quantity A Quantity B
    PS SR

    (A) Quantity A is greater.
    (B) Quantity B is greater.
    (C) The two quantities are equal.
    (D) The relationship cannot be determined from the information given.

    Explanation

    From Figure 1, you know that PQR is a triangle and that point S is between points P and R, so PS is less than PR. and SR is less than PR. You are also given that PQ is equal to PR. However, this information is not sufficient to compare PS and SR. Furthermore, because the figure is not necessarily drawn to scale, you cannot determine the relative sizes of PS and SR visually from the figure, though they may appear to be equal. The position of S can vary along side PR anywhere between P and R. Below are two possible variations of Figure 1, each of which is drawn to be consistent with the information PQ is equal to PR.

    Figure 2
    Figure 3
    The figure shows triangle PQR, where P is the leftmost vertex of the horizontal side PR and vertex Q is above PR. Point S lies on horizontal side PR. S appears to be closer to R than to P and the length of PS appears to be greater than the length of SR. The figure shows triangle PQR, where P is the leftmost vertex of the horizontal side PR and vertex Q is above PR. Point S lies on horizontal side PR. S appears to be closer to P than to R and the length of PS appears to be less than the length of SR

    Note that Quantity A is greater in Figure 2 and Quantity B is greater in Figure 3. Thus, the correct answer is Choice D, the relationship cannot be determined from the information given.


  5. y = 2 times the square of x + 7x, minus 3
    Quantity A Quantity B
    x y

    (A) Quantity A is greater.
    (B) Quantity B is greater.
    (C) The two quantities are equal.
    (D) The relationship cannot be determined from the information given.

    Explanation

    If x is equal to zero then y = 2 times, open parenthesis, 0 squared close parenthesis, +,7 times open parenthesis 0, close parenthesis, minus 3, which is equal to minus 3. so in this case, x is greater than y but if x is equal to one then y = 2 times, open parenthesis, 1 squared, close parenthesis, + 7 times, open parenthesis, 1 close parenthesis, minus 3, which is equal to 6 so in that case, y is greater than x Thus, the correct answer is Choice D, the relationship cannot be determined from the information given.

    Note that plugging numbers into expressions may not be conclusive. However, it is conclusive if you get different results after plugging in different numbers: the conclusion is that the relationship cannot be determined from the information given. It is also conclusive if there are only a small number of possible numbers to plug in and all of them yield the same result, say, that Quantity B is greater.

    Now suppose there are an infinite number of possible numbers to plug in. If you plug many of them in and each time the result is, for example, that Quantity A is greater, you still cannot conclude that Quantity A is greater for every possible number that could be plugged in. Further analysis would be necessary and should focus on whether Quantity A is greater for all possible numbers or whether there are numbers for which Quantity A is not greater.

    The following sample questions focus on simplifying the comparison.

  6. y is greater than 4
    Quantity A Quantity B
    The fraction with numerator 3y +2 and denominator 5 y

    (A) Quantity A is greater.
    (B) Quantity B is greater.
    (C) The two quantities are equal.
    (D) The relationship cannot be determined from the information given.

    Explanation

    Set up the initial comparison:

    The fraction with numerator 3y + 2 and denominator 5, followed by a question mark symbol, followed by y.

    Then simplify:

    Step 1: Multiply both sides by 5 to get  3y +2, followed by the question mark symbol, followed by 5y

    Step 2: Subtract 3y from both sides to get  2, followed by the question mark symbol, followed by 2y

    Step 3: Divide both sides by 2 to get  1, followed by the question mark symbol, followed by y

    The comparison is now simplified as much as possible. In order to compare 1 and y, note that you are given the information y is greater than 4 (above Quantities A and B). It follows from y is greater than 4 that y is greater than 1 or 1 is less than y so that in the comparison 1, followed by the question mark symbol. the placeholder The question mark symbol. represents less than (<): 1 is less than y..

    However, the problem asks for a comparison between Quantity A and Quantity B, not a comparison between 1 and y. To go from the comparison between 1 and y to a comparison between Quantities A and B, start with the last comparison, 1 is less than y. and carefully consider each simplification step in reverse order to determine what each comparison implies about the preceding comparison, all the way back to the comparison between Quantities A and B if possible. Since step 3 was "divide both sides by 2," multiplying both sides of the comparison 1 is less than y. by 2 implies the preceding comparison 2 is less than 2y thus reversing step 3. Each simplification step can be reversed as follows:

    • Reverse step 3: multiply both sides by 2.
    • Reverse step 2: add 3y to both sides.
    • Reverse step 1: divide both sides by 5.

    When each step is reversed, the relationship remains less than (<), so Quantity A is less than Quantity B. Thus, the correct answer is Choice B, Quantity B is greater.

    While some simplification steps like subtracting 3 from both sides or dividing both sides by 10 are always reversible, it is important to note that some steps, like squaring both sides, may not be reversible.

    Also, note that when you simplify an inequality, the steps of multiplying or dividing both sides by a negative number change the direction of the inequality; for example, if X is less than y. then Negative x is greater than negative y So the relationship in the final, simplified inequality may be the opposite of the relationship between Quantities A and B. This is another reason to consider the impact of each step carefully.


  7. Quantity A Quantity B
    The fraction with numerator 2 to the power 30, minus, 2 to the power 29, and denominator 2 2 to the power 28

    (A) Quantity A is greater.
    (B) Quantity B is greater.
    (C) The two quantities are equal.
    (D) The relationship cannot be determined from the information given.

    Explanation

    Set up the initial comparison:

    The fraction with numerator 2 to the power 30, minus, 2 to the power 29, and denominator 2, followed by a made up question mark symbol, followed by 2 to the power 28.

    Then simplify:

    Step 1: Multiply both sides by 2 to get  2 to the power 30, minus, 2 to the power 29, followed by the question mark symbol, followed by 2 to the power 29.

    Step 2: Add 2 to the power 29 to both sides to get  2 to the power 29, followed by the question mark symbol, followed by 2 to the power 29.

    Step 3: Simplify the right-hand side using the fact that 2 times, open parenthesis, 2 to the power 29, close parenthesis, = 2 to the power 30 to get  2 to the power 30, followed by the question mark symbol, followed by 2 to the power 30.

    The resulting relationship is equal to (=). In reverse order, each simplification step implies equal to in the preceding comparison. So Quantities A and B are also equal. Thus, the correct answer is Choice C, the two quantities are equal.


  8. Quantity A Quantity B
    x squared, + 1 2x, minus 1

    (A) Quantity A is greater.
    (B) Quantity B is greater.
    (C) The two quantities are equal.
    (D) The relationship cannot be determined from the information given.

    Explanation

    Set up the initial comparison:

    x squared + 1, followed by a made up question mark symbol, followed by 2x, minus 1.
    Then simplify by noting that the quadratic polynomial x squared, minus 2x, + 1 can be factored:

    Step 1: Subtract 2x from both sides to get  x squared, minus 2x, + 1, followed by the question mark symbol, followed by negative 1.

    Step 2: Factor the left-hand side to get  Open parenthesis, x squared minus 1, close parenthesis, squared, is greater than negative 1

    The left-hand side of the comparison is the square of a number. Since the square of a number is always greater than or equal to 0, and 0 is greater than Negative one. the simplified comparison is the inequality Open parenthesis, x squared minus 1, close parenthesis, squared, is greater than negative 1 and the resulting relationship is greater than (>). In reverse order, each simplification step implies the inequality greater than (>) in the preceding comparison. Therefore, Quantity A is greater than Quantity B. The correct answer is Choice A, Quantity A is greater.


  9. w is greater than 1.
    Quantity A Quantity B
    7w minus 4 2w plus 5

    (A) Quantity A is greater.
    (B) Quantity B is greater.
    (C) The two quantities are equal.
    (D) The relationship cannot be determined from the information given.

    Explanation

    Set up the initial comparison:

    7w minus 4, followed by a made up question mark symbol, followed by 2w + 5

    Then simplify:

    Step 1: Subtract 2w from both sides and add 4 to both sides to get  5w, followed by the question mark symbol, followed by the number 9.

    Step 2: Divide both sides by 5 to get  w, followed by the question mark symbol, followed by the fraction 9 over 5.

    The comparison cannot be simplified any further. Although you are given that W is greater than 1. you still don't know how w compares to The fraction 9 over 5 or 1.8. For example, if W is equal to 1 point 5. then w is less than 1 point 8 but if w is equal to 2 then w is greater than 1 point 8. In other words, the relationship between w and The fraction 9 over 5 cannot be determined. Note that each of these simplification steps is reversible, so in reverse order, each simplification step implies that the relationship cannot be determined in the preceding comparison. Thus, the relationship between Quantities A and B cannot be determined. The correct answer is Choice D, the relationship cannot be determined from the information given.

    The strategy of simplifying the comparison works most efficiently when you note that a simplification step is reversible while actually taking the step. Here are some common steps that are always reversible:

    • Adding any number or expression to both sides of a comparison
    • Subtracting any number or expression from both sides
    • Multiplying both sides by any nonzero number or expression
    • Dividing both sides by any nonzero number or expression

    Remember that if the relationship is an inequality, multiplying or dividing both sides by any negative number or expression will yield the opposite inequality. Be aware that some common operations like squaring both sides are generally not reversible and may require further analysis using other information given in the question in order to justify reversing such steps.

 

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