Estimated study time: 45 minutes
Content:
Quantitative Comparison (QC) questions are unique to the GRE and require a different approach than standard problem-solving. Each QC question presents two quantities — Quantity A and Quantity B — and asks you to choose among four options: (A) Quantity A is greater; (B) Quantity B is greater; (C) both are equal; or (D) the relationship cannot be determined from the information given. The answer is always one of these four choices.
Answer choice D ("cannot be determined") is correct when the comparison depends on information not provided — particularly when the quantities can be greater, less than, or equal depending on different values of an unspecified variable. A key GRE strategy: if you can find one case where A > B and another case where A < B, the answer is D. You only need two different scenarios that produce different comparisons to eliminate A, B, and C.
Strategies for QC questions: First, look for ways to simplify by performing the same operation on both quantities simultaneously. You may add, subtract, multiply (by a positive number), or divide (by a positive number) both sides — just as in algebra, maintaining the comparison direction. Multiplying or dividing by a negative reverses the inequality — always check the sign of any variable before multiplying through. Never multiply or divide by an expression that could be zero.
Picking numbers is a powerful QC strategy when variables are involved. Test at least three types of numbers: positive integers (e.g., 1, 2), fractions (e.g., 1/2), and negative numbers (e.g., −1, −2). Zero is often a crucial test case — many QC traps involve expressions that behave differently at zero. Also test extreme cases when a variable has a range (e.g., x > 0: test a very small positive number like 0.001 and a large number like 1000).
Common QC trap patterns: (1) Assuming a variable is a positive integer when it could be a fraction or negative. (2) Forgetting that x² ≥ 0 always — even when x is negative. (3) Forgetting that √x² = |x|, not x. (4) Confusing equality constraints — "x + y = 10" does not mean x = y = 5. (5) Diagrams that appear to show equal lengths or angles that are not stated to be equal — on the GRE, never assume equality from appearance.
Time management: QC questions are typically faster than problem-solving questions. A QC answer should take 60-90 seconds. If you are algebraically manipulating for more than 2 minutes, try picking numbers instead. Do not over-invest in QC if you are running short on time — move on and return if possible.
Key Terms:
Quiz Questions:
Q1.
Quantity A: x² Quantity B: x
Given: x is a real number
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
Answer: D — Test cases: If x = 2: A = 4 > B = 2 (A greater). If x = 1/2: A = 1/4 < B = 1/2 (B greater). If x = 1: A = 1 = B = 1 (equal). Since all three outcomes are possible, the answer is D.
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Q2.
Quantity A: (a + b)² Quantity B: a² + b²
Given: a and b are positive integers
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
Answer: A — Expand: (a+b)² = a² + 2ab + b². Since a and b are positive integers, 2ab > 0. Therefore (a+b)² = a² + b² + 2ab > a² + b². Quantity A is always greater than Quantity B when a, b > 0.
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Q3.
Quantity A: The number of minutes in 5 hours Quantity B: The number of seconds in 4.5 minutes × 3
A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined
Answer: C — Quantity A: 5 × 60 = 300 minutes. Quantity B: 4.5 minutes × 60 seconds/minute = 270 seconds; × 3 = 810 seconds ÷ 60 = ... wait — let's re-read. "Seconds in 4.5 minutes × 3" = (4.5 × 60) × 3 = 270 × 3 = 810 seconds. Convert to minutes for comparison: 810 seconds ÷ 60 = 13.5 minutes. Hmm, that's B > A. Let me recalculate Quantity A: 5 hours × 60 = 300 minutes. B = 810 seconds = 13.5 minutes. A > B.
Answer: A — Quantity A: 300 minutes. Quantity B: 4.5 × 60 × 3 = 810 seconds = 13.5 minutes. 300 > 13.5. Quantity A is greater.
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Q4.
Given: n is a positive integer greater than 1
Quantity A: n² − n Quantity B: n
A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined
Answer: A — n² − n = n(n−1). Since n > 1, both n and (n−1) are positive, so n(n−1) > 0. We need to compare n(n−1) to n: divide both sides by n (positive): (n−1) vs. 1. Since n > 1, n−1 > 0 — but is it > 1? If n = 2: n−1 = 1, so n(n−1) = 2 = n. Equal for n = 2. If n = 3: n(n−1) = 6 > 3. So for n = 2, equal; for n > 2, A > B. But the problem says n > 1, which includes n = 2. At n = 2, A = 2 = B. So the answer is D (can be equal or A > B).
Answer: D — At n = 2: A = 4 − 2 = 2 = B = 2 (equal). At n = 3: A = 9 − 3 = 6 > B = 3 (A greater). Relationship is not consistent — D.
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Q5.
Quantity A: The probability of rolling at least one 6 in two fair six-sided die rolls Quantity B: 1/3
A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined
Answer: A — P(at least one 6) = 1 − P(no 6) = 1 − (5/6)² = 1 − 25/36 = 11/36 ≈ 0.306. Compare to 1/3 ≈ 0.333. Since 11/36 < 12/36 = 1/3, Quantity B (1/3) is greater.
Correction — Answer: B — 11/36 ≈ 0.306 < 1/3 ≈ 0.333. Quantity B is greater.
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