Quantitative Comparison

What This Module Covers

This module teaches you how to approach Quantitative Comparison (QC) questions on the GRE — a question type that looks simple on the surface but contains some of the most strategic traps on the entire exam. You'll learn the four-choice format, when and how to plug in numbers, when to simplify algebraically, and how to avoid the most common reasoning errors. By the end, you'll have a repeatable decision process for every QC question you face.

Why It Matters on the GRE

Quantitative Comparison questions make up roughly half of the Quant section, so your QC strategy has an outsized impact on your score. Unlike Problem Solving, QC questions never ask you to compute a precise answer — they ask you to compare. That means algebraic reasoning and smart number selection often beat calculation. Understanding the format deeply is the first step to scoring well.

Core Concepts

The QC Format

Every QC question presents two quantities — Quantity A and Quantity B — and always offers the same 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

Choice (D) has a specific meaning: the relationship is not fixed — it can go different ways depending on what values the variables take. To choose (D), you must find two concrete cases where the relationship flips. You cannot choose (D) just because you see a variable; you have to demonstrate instability with actual numbers.

Must Know: (D) means you found two cases with different outcomes. If every case you test gives the same result, (D) is wrong. Verify before you commit.

Simplify Both Quantities First

Before reaching for numbers, try to simplify algebraically. Legal operations on a QC comparison include:

• Adding or subtracting the same value from both sides
• Multiplying both sides by the same positive constant
• Squaring both sides — only if both quantities are guaranteed to be non-negative

You cannot multiply or divide by an unknown variable unless you know its sign. An unknown could be zero or negative, which would reverse or destroy the inequality.

Must Know: Treat the comparison like a balance scale. Any operation you'd apply to both sides of an equation is fair — as long as you know the sign of anything you're multiplying by.

Example — Simplifying a QC:

Suppose you're given:

• Quantity A: 3x + 9
• Quantity B: 3x + 5

Step 1: Subtract 3x from both sides. That's legal because you're subtracting the same thing from both. Step 2: You're left comparing 9 vs. 5. Step 3: 9 > 5 always, regardless of what x is. Step 4: Answer is (A).

Notice: you never needed to plug in a single number. Simplification gave you a clean, definitive answer.

The Plug-In Strategy

When simplification doesn't resolve things cleanly — or when the expression resists algebraic manipulation — plug in numbers. Use this test sequence deliberately:

1. A positive integer greater than 1 (try x = 2 or x = 3)
2. The number 1
3. Zero (x = 0)
4. A negative number (try x = -1 or x = -2)
5. A fraction between 0 and 1 (try x = 1/2)

You're hunting for a case that changes the outcome. The moment you find one result where QA > QB and another where QB > QA (or QA = QB), the answer is (D) and you can stop.

Must Know: Your plug-in sequence should always include 0, a negative, and a fraction — these are the three value types most likely to flip a relationship. Positive integers alone are not enough.

Example — Plug-In in Action:

• Quantity A: x²
• Quantity B: x

Step 1: Try x = 2. QA = 4, QB = 2. QA > QB. This suggests (A). Step 2: Try x = 1/2. QA = 1/4, QB = 1/2. QB > QA. This suggests (B). Step 3: You now have two cases with opposite results. The relationship is not fixed. Step 4: Answer is (D).

One case gave (A), another gave (B). That's all you need.

Variables with Restricted Domains

Pay close attention to what restrictions the problem places on variables. "Let n be a positive integer" limits you to {1, 2, 3, ...} — you cannot test 0 or negatives. "Let x be a real number" with no other restriction opens the door to everything: fractions, zero, negatives, irrationals. The restriction changes which cases are legal to test, and missing it is one of the most common sources of wrong answers.

Must Know: Always read the given information before plugging in. A single constraint like "x > 0" eliminates zero and all negatives from your test set — and might turn a (D) into a definitive (A) or (B).

When the Answer Is Always (C)

If both quantities simplify to the same expression, the answer is (C). Also reach for (C) when the given constraints pin down both quantities to a specific value. For example, if the problem says x = 3 and Quantity A is x + 2 and Quantity B is 5, both equal 5 and the answer is (C). Don't overcomplicate it — when both sides evaluate to the same number, you're done.

Common Traps

• Assuming variables are positive: When no restriction is stated, x could be 0, negative, or a fraction. Skipping these test values is the single most common QC error on hard questions.

• Choosing (D) too quickly: Seeing a variable doesn't mean the answer is (D). Many QC questions with variables simplify to constants — the variable cancels out entirely. Always simplify before assuming the answer is undetermined.

• Illegal squaring: Squaring both sides is only a valid operation when you know both quantities are non-negative. If either side could be negative, squaring changes the relationship and leads to wrong conclusions.

• Not finding a second example for (D): Choosing (D) based on one case that "feels uncertain" is not sufficient. You need two concrete examples: one where QA > QB and one where QB > QA (or one where they're equal and one where they're not).

• Arithmetic errors from plugging in unsimplified expressions: If you plug in before simplifying, you're doing more arithmetic with more room for mistakes. Always simplify first, then plug in if needed.

GRE Strategy

• Simplify first, plug in second. Algebraic simplification is faster, cleaner, and less error-prone than testing numbers. Reserve plug-in for expressions that resist simplification.
• Use the test sequence every time. Don't just try x = 2 and call it a day. One positive integer is not a representative sample. Always include 0, a negative, and a fraction if the domain allows.
• Treat the comparison as a subtraction problem. If you compute QA − QB and the result is always positive (or always negative, or always zero), you have your answer. This is often the fastest algebraic simplification approach.
• Never choose (D) without two explicit counterexamples in hand. Write them down. If you can't produce two specific numbers that give different outcomes, you don't have grounds for (D).
• Watch the given information line carefully. Constraints appear above the quantities in the problem. Many test-takers skim this line and miss the restriction that determines the entire answer.
• On hard QC questions, suspect (A), (B), or (C) when (D) feels obvious. ETS deliberately designs high-difficulty questions to look like (D) when they're actually deterministic. If (D) is screaming at you, slow down and try to simplify.

Worked Example

Question:

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

Solution:

Step 1 — Read the given information. The problem says x is a real number with no further restriction. That means we must test the full range: positives, negatives, zero, and fractions.

Step 2 — Try x = 2 (positive integer). QA = (2)² = 4. QB = 2. QA > QB. This is consistent with choice (A).

Step 3 — Try x = 1/2 (fraction between 0 and 1). QA = (1/2)² = 1/4. QB = 1/2. Now QB > QA. This is consistent with choice (B).

Step 4 — Compare the two results. One case gave QA > QB. Another gave QB > QA. The relationship changes depending on the value of x — it is not fixed.

Step 5 — Answer is (D). We have two explicit cases with opposite outcomes, which is exactly what (D) requires.

Why the other choices fail:

• (A) fails because x = 1/2 gives QB > QA.
• (B) fails because x = 2 gives QA > QB.
• (C) fails because the quantities are not always equal (though they are equal at x = 0 and x = 1 — these don't save (C) because equality must hold for all values, not just some).

Bonus insight: This question is a classic because students who only test x = 2, 3, or 4 always see QA > QB and confidently choose (A). The fraction x = 1/2 is the trap-breaker. When a base is between 0 and 1, squaring it makes it smaller, not larger — the opposite of what intuition says. This is exactly why the plug-in sequence requires you to test a fraction every time.