Medium GRE Quantitative Comparison Questions Practice Questions

Approximately 35 percent of the GRE Quantitative Reasoning section consists of Quantitative Comparison questions designed to test your ability to evaluate relative magnitudes quickly.

Navigating these problems requires more than just raw calculation; it demands a strategic approach to comparing two quantities based on given information. When working through Medium GRE Quantitative Comparison Questions, you will often encounter scenarios where the answer is not immediately obvious, requiring you to simplify expressions, test numbers, or apply geometric properties. Success in this section is a cornerstone of a high score in GRE Prep, as these questions reward efficiency and logical deduction over tedious arithmetic.

Concept Explanation

Quantitative Comparison (QC) questions ask you to compare two quantities—Quantity A and Quantity B—and determine the relationship between them using four standard answer choices.

The four options are always the same:

• (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 at medium-level questions, you must be comfortable with "The Zone" of numbers: negatives, fractions between 0 and 1, zero, and one. Often, an algebraic expression might look like it favors one side until you plug in a negative number or a decimal, which can flip the inequality or result in equality. This is why testing cases is a vital strategy. Additionally, you should look to simplify both quantities by performing the same operations on each (such as adding or subtracting the same value) to reveal their core relationship. If you find yourself doing heavy calculations, you are likely missing a shortcut or a conceptual property, such as those covered in GRE practice questions with explanations.

Solved Examples

Below are three worked examples demonstrating common tactics for medium-difficulty comparison problems.

1. Example 1: Algebra and Inequalities
   Given: $x > 1$
   Quantity A: $\frac{x^2 - 1}{x - 1}$
   Quantity B: $x + 1$

   1. Simplify Quantity A by factoring the numerator: $x^2 - 1 = (x - 1)(x + 1)$.
   2. Rewrite Quantity A: $\frac{(x - 1)(x + 1)}{x - 1}$.
   3. Since $x > 1$, we know $x - 1 \neq 0$, so we can cancel the term: $x + 1$.
   4. Compare: Quantity A is $x + 1$ and Quantity B is $x + 1$.
   5. The quantities are equal. The answer is (C).

2. Example 2: Number Properties
   Given: $n$ is an integer and $n^2 < 20$
   Quantity A: The greatest possible value of $n$
   Quantity B: 4

   1. Identify the integers whose squares are less than 20: $\{ -4, -3, -2, -1, 0, 1, 2, 3, 4 \}$.
   2. Note that $4^2 = 16$ (which is $< 20$) and $5^2 = 25$ (which is $> 20$).
   3. The greatest integer in this set is 4.
   4. Compare: Quantity A is 4 and Quantity B is 4.
   5. The quantities are equal. The answer is (C).

3. Example 3: Geometry
   Given: A circle has an area of $9\pi$.
   Quantity A: The circumference of the circle.
   Quantity B: $6\pi$

   1. Use the area formula $A = \pi r^2$ to find the radius: $9\pi = \pi r^2 \rightarrow r^2 = 9 \rightarrow r = 3$.
   2. Use the circumference formula $C = 2\pi r$.
   3. Calculate Quantity A: $2 \times \pi \times 3 = 6\pi$.
   4. Compare: Quantity A is $6\pi$ and Quantity B is $6\pi$.
   5. The quantities are equal. The answer is (C).

Practice Questions

1. Given: $x + y = 10$ and $x - y = 4$
Quantity A: $x$
Quantity B: 7

2. Given: $k > 0$
Quantity A: $2k^2$
Quantity B: $(2k)^2$

3. Given: $a$ and $b$ are positive integers such that $a < b$.
Quantity A: $\frac{a}{b}$
Quantity B: $\frac{a + 1}{b + 1}$

4. Given: $x$ is a negative integer.
Quantity A: $|x|$
Quantity B: $-x$

5. Given: A rectangular box has dimensions 3, 4, and 12.
Quantity A: The length of the longest diagonal of the box.
Quantity B: 13

6. Given: $0 < p < 1$
Quantity A: $\sqrt{p}$
Quantity B: $p^2$

7. Given: $n$ is an even integer.
Quantity A: $(-1)^{n}$
Quantity B: $(-1)^{n+2}$

8. Quantity A: The number of distinct prime factors of 60.
Quantity B: The number of distinct prime factors of 90.

9. Given: $y = x^2 + 2x + 1$
Quantity A: The value of $y$ when $x = -1$
Quantity B: 0

10. Given: In a group of 50 people, 30 like tea and 25 like coffee.
Quantity A: The number of people who like both tea and coffee.
Quantity B: 5

Answers & Explanations

1. Answer: (A). Adding the two equations: $(x+y) + (x-y) = 10 + 4 \rightarrow 2x = 14 \rightarrow x = 7$. Wait, if $x=7$, then $7+y=10 \rightarrow y=3$. Quantity A is 7 and Quantity B is 7. Wait, checking the math: $7+3=10$ and $7-3=4$. Both are 7, so the answer is actually (C). (Common trap: miscalculating the sum).
2. Answer: (B). Quantity A is $2k^2$. Quantity B is $(2k)^2 = 4k^2$. Since $k > 0$, $k^2$ is positive. $4k^2$ will always be greater than $2k^2$.
3. Answer: (B). Adding the same positive constant to the numerator and denominator of a proper fraction (where $a < b$) always increases the value of the fraction. For example, if $a=1, b=2$, $1/2 < 2/3$.
4. Answer: (C). By definition, if $x$ is negative, $|x| = -x$. For example, if $x = -5$, $|-5| = 5$ and $-(-5) = 5$.
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: (A). For decimals between 0 and 1, the square root is larger than the original number, and the square is smaller. For example, if $p = 0.25$, $\sqrt{0.25} = 0.5$ and $(0.25)^2 = 0.0625$.
7. Answer: (C). If $n$ is even, $(-1)^n = 1$. If $n$ is even, $n+2$ is also even, so $(-1)^{n+2} = 1$.
8. Answer: (C). Prime factors of 60: $2^2 \times 3 \times 5$ (Distinct: 2, 3, 5). Prime factors of 90: $2 \times 3^2 \times 5$ (Distinct: 2, 3, 5). Both have 3 distinct prime factors.
9. Answer: (C). Substitute $x = -1$: $y = (-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0$. Both quantities are 0.
10. Answer: (D). The minimum overlap is $30 + 25 - 50 = 5$, but the maximum overlap could be 25 (if everyone who likes coffee also likes tea). Since the value could be 5 or higher, the relationship cannot be determined.

1. If \( x > 0 \), which quantity is larger: \( x + 1 \) or \( x - 1 \)?

• AQuantity A: \( x + 1 \)
• BQuantity B: \( x - 1 \)
• CThey are equal
• DCannot be determined

Frequently Asked Questions

What is the best strategy for Quantitative Comparison questions?

The most effective strategy is to simplify both quantities as much as possible and then test "extreme" numbers like -1, 0, 1, and fractions. You can also use AI-powered practice tools to identify patterns in how these questions hide information.

Can I use the calculator on GRE Quantitative Comparison?

Yes, the GRE provides an on-screen calculator, but it is often faster to use logic or algebraic simplification for medium-level questions. Relying too heavily on the calculator can waste valuable time on the exam.

What does it mean if the answer is Choice D?

Choice D means that depending on the values you choose for the variables, Quantity A could be larger in one case and Quantity B larger (or equal) in another. If you can find two different relationships, D is the correct answer.

How do I handle geometry in Quantitative Comparison?

Remember that diagrams are not necessarily drawn to scale unless stated. Use geometric formulas like those found in unlimited GRE practice questions to calculate exact values rather than estimating by sight.

Are these questions harder than multiple-choice math?

They aren't necessarily harder, but they require a different mindset focused on relative size rather than finding a specific numerical solution. Many students find them tricky because they forget to consider negative numbers or zero.