Hard GRE Quantitative Comparison Questions Practice Questions

Concept Explanation

Hard GRE Quantitative Comparison Questions require you to compare two quantities and determine whether one is greater, they are equal, or the relationship cannot be determined. These high-difficulty problems often involve hidden constraints, non-obvious algebraic identities, or complex geometric properties that test your ability to think beyond surface-level calculations. Unlike standard problem-solving, the goal here is not always to find a specific value but to establish a definitive relationship between Quantity A and Quantity B. Success on these items requires a firm grasp of GRE Prep fundamentals, particularly number properties involving negatives, fractions, and zero.

To tackle these challenging questions, you must be disciplined in your approach. Most hard-level comparisons use "traps" where a relationship seems obvious but changes when you plug in non-integer values or negative numbers. A common strategy is to simplify both quantities to their most basic forms before comparing. If the relationship depends on the value of a variable, the answer is usually (D), indicating the relationship is indeterminate. However, on hard questions, the GRE often provides just enough constraint—such as stating a variable is a prime number or a positive integer—to make one quantity consistently larger.

Solved Examples

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 using the difference of squares: $x^2 - 1 = (x - 1)(x + 1)$.
   2. Rewrite Quantity A: $\frac{(x - 1)(x + 1)}{x - 1}$.
   3. Since $x > 1$, the term $(x - 1)$ is not zero, so we can cancel it.
   4. Quantity A simplifies to $x + 1$.
   5. Comparing $x + 1$ to $x + 1$, we find they are equal. The answer is C.
2. Example 2: Number Properties
   Given: $n$ is an integer and $n^2 < 20$.
   Quantity A: The number of possible values for $n$
   Quantity B: 9
   1. Identify the integers whose squares are less than 20.
   2. Test positive integers: $1^2=1, 2^2=4, 3^2=9, 4^2=16$. ($5^2=25$ is too high). So, 1, 2, 3, 4 are 4 values.
   3. Test zero: $0^2=0$, which is less than 20. That is 1 value.
   4. Test negative integers: $(-1)^2=1, (-2)^2=4, (-3)^2=9, (-4)^2=16$. That is 4 more values.
   5. Total count: $4 + 1 + 4 = 9$. Both quantities are 9. The answer is C.
3. Example 3: Geometry and Triangles
   Given: A triangle has side lengths 4 and 10. The third side has length $s$.
   Quantity A: $s$
   Quantity B: 14
   1. Apply the Triangle Inequality Theorem: The sum of any two sides must be greater than the third side.
   2. Calculation: $4 + 10 > s$, which means $14 > s$.
   3. Also, the difference of two sides must be less than the third side: $10 - 4 < s$, so $6 < s$.
   4. Since $s$ must be less than 14, Quantity B is always larger. The answer is B.

Practice Questions

1. Given: $x$ and $y$ are integers such that $xy = 12$.
Quantity A: $x + y$
Quantity B: 7

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

3. Given: The average (arithmetic mean) of five distinct positive integers is 10.
Quantity A: The greatest possible value of one of the integers.
Quantity B: 40

4. Given: $-1 < x < 0$.
Quantity A: $x^2$
Quantity B: $x^3$

5. Given: A circle is inscribed in a square with side length 10.
Quantity A: The area of the circle.
Quantity B: 75

6. Given: $a, b, c$ are consecutive integers such that $a < b < c$.
Quantity A: $\frac{a+c}{2}$
Quantity B: $b$

7. Given: $p$ is a prime number and $p > 2$.
Quantity A: The remainder when $p^2$ is divided by 4.
Quantity B: 1

8. Given: $y = |x - 5|$.
Quantity A: $y$ when $x = 2$
Quantity B: $y$ when $x = 8$

Answers & Explanations

1. Answer: D. If $x=3, y=4$, then $x+y=7$ (Equal). If $x=1, y=12$, then $x+y=13$ (A is greater). If $x=-1, y=-12$, then $x+y=-13$ (B is greater). Since multiple relationships are possible, D is correct.
2. Answer: B. Quantity B is $4k^2$. Since $k > 0$, $k^2$ is positive. Comparing $2k^2$ to $4k^2$, Quantity B is always twice as large as Quantity A.
3. Answer: A. If the average of 5 integers is 10, their sum is 50. To maximize one integer, minimize the others. Since they are distinct positive integers, use 1, 2, 3, and 4. The sum is 10. The 5th integer is $50 - 10 = 40$. However, the question asks for the greatest possible value. If we use 1, 2, 3, 4, the max is 40. But if we use any other set of distinct positive integers, the max will be smaller. Since 40 is achievable, and no higher number is possible while keeping them distinct/positive, they can be equal. Wait—the question asks for the greatest "possible" value. That value is 40. Thus, C is the answer. (Correction: Quantity A is 40, Quantity B is 40. Answer is C).
4. Answer: A. When $x$ is between -1 and 0 (e.g., -0.5), $x^2$ is positive (0.25) and $x^3$ is negative (-0.125). A positive number is always greater than a negative number.
5. Answer: B. If the side of the square is 10, the diameter of the inscribed circle is 10, so the radius is 5. Area = $\pi r^2 = 25\pi$. Since $\pi \approx 3.14$, $25 \times 3.14 = 78.5$. Quantity A (78.5) is greater than 75.
6. Answer: C. For any three consecutive integers, the average of the first and third is always the middle integer. For example, 1, 2, 3: $(1+3)/2 = 2$.
7. Answer: C. Every prime number greater than 2 is odd. Any odd number can be written as $2n+1$. Squaring it gives $4n^2 + 4n + 1$. Factoring out 4 from the first two terms gives $4(n^2+n) + 1$. This always leaves a remainder of 1 when divided by 4.
8. Answer: C. When $x=2, y=|2-5|=|-3|=3$. When $x=8, y=|8-5|=|3|=3$. The values are equal.

1. If \( n \) is an even integer, which of the following must be true about the relationship between Quantity A: \( (-1)^n \) and Quantity B: \( (-1)^{n+2} \)?

• AQuantity A is greater
• BQuantity B is greater
• CThe two quantities are equal
• DThe relationship cannot be determined

Frequently Asked Questions

What is the best strategy for Hard GRE Quantitative Comparison Questions?

The most effective strategy is to try and prove the relationship is indeterminate by testing different types of numbers, such as zero, negatives, and fractions. If you can find one case where A is greater and one case where B is greater, you can immediately select option D. For more extensive practice, using an AI-Powered GRE Practice Questions tool can help identify these patterns.

How do I handle variables without defined constraints?

When variables have no constraints, you must assume they can be any real number, including very large numbers, very small fractions, or negative values. This often leads to an answer of D unless the algebraic structure of the quantities forces a specific relationship regardless of the input. You can refine this skill by working through GRE Practice Questions with Explanations.

Why are geometry comparison questions considered hard?

Geometry comparisons are often difficult because the diagrams provided are not necessarily drawn to scale. Students often make the mistake of assuming a line is longer because it looks longer, whereas the actual geometric properties or theorems might dictate otherwise. Utilizing an AI Exam Simulator can help you practice visualizing these geometric constraints more accurately.

Does the GRE use complex numbers in Quantitative Comparison?

No, the GRE Quantitative section only uses real numbers. You do not need to worry about imaginary numbers or complex analysis; however, you should be very comfortable with radicals and exponents within the real number system.

Should I always simplify the expressions first?

Yes, simplifying both quantities is usually the fastest way to reveal their relationship. By performing the same operations on both sides—such as subtracting the same value or multiplying by a positive constant—you can often turn a complex comparison into a simple one. For more variety in problem types, check out Unlimited GRE Practice Questions.