GRE Quantitative Comparison Questions Practice Questions with Answers

Quantitative comparison questions account for approximately 35% of the total math section on the GRE General Test. These problems present Two Quantities—Quantity A and Quantity B—and require you to determine the relationship between them based on given information. Because these questions are designed to test your ability to estimate and simplify rather than perform exhaustive calculations, understanding the logical constraints of the four standard answer choices is essential for success. You must decide if Quantity A is greater, Quantity B is greater, the two are equal, or if the relationship cannot be determined from the information provided.

Concept Explanation

GRE Quantitative Comparison Questions are a specific format of mathematical reasoning problems that ask you to compare two mathematical expressions or values. Unlike standard multiple-choice questions, the answer choices for every Quantitative Comparison (QC) question are identical:

• (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.

To excel in this section, you should utilize strategies like "plugging in numbers" (using 0, 1, fractions, and negatives), simplifying both sides simultaneously, and looking for shared components that can be ignored. Since the Educational Testing Service (ETS) focuses on conceptual efficiency, avoid long calculations if a logical shortcut exists. For example, if you are comparing $25 \times 48$ and $24 \times 49$, you don't need the product; you only need to see which side is larger. Comprehensive GRE Prep involves learning to recognize these patterns quickly. Many students use an AI Exam Simulator to practice the timing required for these comparisons, as you generally have less than 90 seconds per question.

Solved Examples

Review these fully worked examples to understand the logic behind choosing the correct relationship between values.

1. Example 1: Algebraic Inequalities
   Given: $x > 0$
   Quantity A: $x^2 + 1$
   Quantity B: $(x + 1)^2$
   Solution:
   1. Expand Quantity B: $(x + 1)^2 = x^2 + 2x + 1$.
   2. Compare the two: Quantity A is $x^2 + 1$; Quantity B is $x^2 + 2x + 1$.
   3. Subtract $x^2 + 1$ from both sides.
   4. Quantity A becomes 0; Quantity B becomes $2x$.
   5. Since the problem states $x > 0$, then $2x$ must be greater than 0.
   6. Therefore, Quantity B is greater. The answer is (B).
2. Example 2: Geometry and Circles
   A circle has a radius $r$.
   Quantity A: The area of the circle.
   Quantity B: The circumference of the circle.
   Solution:
   1. Write the formulas: Area = $\pi r^2$; Circumference = $2\pi r$.
   2. Test different values for $r$.
   3. If $r = 1$: Area = $\pi$, Circumference = $2\pi$. Here, B is greater.
   4. If $r = 2$: Area = $4\pi$, Circumference = $4\pi$. Here, they are equal.
   5. If $r = 3$: Area = $9\pi$, Circumference = $6\pi$. Here, A is greater.
   6. Since the relationship changes depending on the value of $r$, the answer is (D).
3. Example 3: Number Properties
   $n$ is an integer such that $1 < n < 4$.
   Quantity A: $\frac{1}{n}$
   Quantity B: $\frac{n}{4}$
   Solution:
   1. Identify possible values for $n$: Since $n$ is an integer between 1 and 4, $n$ can be 2 or 3.
   2. Test $n = 2$: Quantity A = $1/2 = 0.5$; Quantity B = $2/4 = 0.5$. (Equal)
   3. Test $n = 3$: Quantity A = $1/3 \approx 0.33$; Quantity B = $3/4 = 0.75$. (B is greater)
   4. Since the relationship is not consistent, the answer is (D).

Practice Questions

Test your skills with the following GRE Quantitative Comparison Questions. Remember to consider all possible values for variables unless constraints are provided.

1. $x < 0$
   Quantity A: $|x|$
   Quantity B: $-x$
2. $y = 3x + 2$ and $x > 1$
   Quantity A: $y$
   Quantity B: $5$
3. The average (arithmetic mean) of $a, b,$ and $c$ is 10.
   Quantity A: $a + b + c$
   Quantity B: $30$

4. $k$ is a prime number and $10 < k < 15$.
   Quantity A: $k$
   Quantity B: $13$
5. A rectangular box has dimensions 3, 4, and 12.
   Quantity A: The length of the longest diagonal of the box.
   Quantity B: 13
6. $x^2 = 16$
   Quantity A: $x$
   Quantity B: $4$
7. $0 < p < 1$
   Quantity A: $p^2$
   Quantity B: $\sqrt{p}$
8. The price of an item was increased by 20%, and then the new price was decreased by 20%.
   Quantity A: The original price.
   Quantity B: The final price.

Answers & Explanations

1. Answer: (C). If $x$ is negative, say $x = -5$, then $|x| = 5$ and $-x = -(-5) = 5$. This holds true for all negative numbers.
2. Answer: (A). Since $x > 1$, the smallest value $x$ can approach is 1. If $x = 1.1$, then $y = 3(1.1) + 2 = 5.3$. As $x$ increases, $y$ also increases, always remaining greater than 5.
3. Answer: (C). The formula for the mean is $\frac{a+b+c}{3} = 10$. Multiplying both sides by 3 gives $a+b+c = 30$.
4. Answer: (D). The prime numbers between 10 and 15 are 11 and 13. If $k = 11$, B is greater. If $k = 13$, they are equal.
5. Answer: (C). The diagonal of a rectangular prism is $\sqrt{l^2 + w^2 + h^2}$. Here, $\sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$.
6. Answer: (D). If $x^2 = 16$, then $x$ can be 4 or -4. If $x = 4$, the quantities are equal. If $x = -4$, Quantity B is greater.
7. Answer: (B). For values between 0 and 1, squaring a number makes it smaller, while taking the square root makes it larger. For example, if $p = 0.25$, $p^2 = 0.0625$ and $\sqrt{p} = 0.5$.
8. Answer: (A). Let the original price be $100. After a 20% increase, it is $120. A 20% decrease of $120 is $24, making the final price $96. The original price ($100) is greater than the final price ($96).

1. If \( x + y = 10 \) and \( x > y \), which quantity is larger?

• AQuantity A: \( x \)
• BQuantity B: 5
• CThe two quantities are equal
• DThe relationship cannot be determined

Frequently Asked Questions

What is the most common trap in GRE Quantitative Comparison questions?

The most common trap is failing to consider non-integer values or negative numbers. Students often assume variables are positive integers, but unless specified, variables can be fractions, zero, or negative, which often leads to answer choice D.

Should I always calculate the exact value for both quantities?

No, you should prioritize estimation and simplification over exact calculation. The goal is to determine the relationship, not the specific value, so canceling out identical terms from both quantities is a much faster strategy.

When should I choose answer choice D?

You should choose D if you can find two different scenarios that lead to two different relationships. For example, if one set of numbers makes the quantities equal but another set makes Quantity A greater, the relationship is indeterminate.

How does the GRE define the "relationship cannot be determined"?

This means that with the information provided, it is mathematically possible for more than one of the other three relationships (A, B, or C) to be true depending on the specific values chosen for the variables.

Are geometry figures drawn to scale in these questions?

Geometric figures are generally NOT drawn to scale unless specifically stated. You should rely on geometric theorems and given labels rather than the visual appearance of the lines or angles in the diagram.