10.1 Quantitative Comparison: Strategy & Plug-In

10.1 Quantitative Comparison: Strategy & Plug-In

Quick Answer: QC gives you two quantities (A and B) and four FIXED choices: (A) Quantity A is greater, (B) Quantity B is greater, (C) The two quantities are equal, (D) The relationship cannot be determined. Plug in 0, 1, -1, a fraction, and a large value; if any two give different results, pick (D).

Why QC Dominates the GRE Quant Section

Roughly 13-15 of your 40 scored Quant questions are Quantitative Comparison - more than any other single format. ETS uses QC because it tests reasoning, not calculation: many questions are designed to be solved in under 45 seconds with the right strategy, or to trap test-takers who insist on full computation. The on-screen calculator will not save you here; pattern recognition and disciplined plug-in testing will.

The Four Fixed Answer Choices (Memorize Verbatim)

The answer choices for EVERY QC question are identical. Memorize them so you never have to read them during the test:

Because the choices never change, your only job is to decide the relationship - not to compute a final numeric answer.

When Answer (D) Applies - and When It Does NOT

(D) means the relationship genuinely varies depending on the values of the unknowns. Pick (D) ONLY when you can produce two valid cases that yield different relationships.

Valid (D) cases:

• The stem gives variables with no constraints (e.g., "x is a real number") and different inputs produce different results.
• A geometric figure is involved and the problem says nothing about specific lengths, angles, or that the figure is drawn to scale.
• A range or inequality is in play and the boundary values flip the relationship.

NEVER pick (D) when:

• Both quantities contain only numbers (e.g., Quantity A: 3^4 + 2; Quantity B: 80). The relationship is always determinable.
• The stem constrains the variable to a single value (e.g., "x = 7").
• The question says quantities are equal by definition (e.g., Quantity A: the number of prime factors of 12; Quantity B: 2, with 12 = 2 x 2 x 3 - two distinct primes).

The Plug-In Strategy

When a quantity contains a variable, plug in values systematically. Do NOT plug in random numbers; use the standard test set:

1. A simple value (like 2 or 5) - your baseline.
2. 0 - boundary case that often flips inequalities or zeroes out terms.
3. 1 - boundary case where powers and products collapse.
4. -1 or another negative - tests sign-dependent behavior.
5. A fraction between 0 and 1 (like 1/2) - flips exponent and squaring behavior.
6. A large or extreme value (like 100) - tests growth behavior.

If every test value gives the same answer, that is almost certainly the answer (typically A, B, or C). If even two values give different relationships, pick (D).

Worked QC Example 1

Quantity A: x^2. Quantity B: x. (x is a real number.)

• Test x = 2: A = 4, B = 2 -> A is greater.
• Test x = 1/2: A = 1/4, B = 1/2 -> B is greater.

Different relationships -> (D) The relationship cannot be determined.

Worked QC Example 2

Quantity A: 5x + 3. Quantity B: 5x - 7. (x is any real number.)

Subtract 5x from both: A becomes 3, B becomes -7. 3 > -7 always. The variable cancels, so no plug-in needed -> (A) Quantity A is greater.

Matching and Elimination Strategy

Before plugging in, simplify both quantities by performing the same operation to both sides (add, subtract, multiply by a positive, divide by a positive). This is the "matching" strategy: cancel common terms until the comparison is obvious.

Worked QC Example 3:

Quantity A: 2(x + 7) + 3. Quantity B: 2x + 17.

• Expand A: 2x + 14 + 3 = 2x + 17.
• Subtract 2x from both: A = 17, B = 17. -> (C) The two quantities are equal.

Matching often resolves the question with no plug-in at all. Use it first; only fall back to plug-in when matching leaves a genuine comparison.

Common QC Traps

1. Assuming variables are positive integers. If the stem says "x is an integer," do NOT assume positive. Test x = -1. If it says nothing, test fractions too.
2. Assuming geometric figures are drawn to scale. ETS explicitly states geometric figures are NOT drawn to scale. A triangle that looks equilateral might be scalene. Trust labels and given constraints, not appearance.
3. Ignoring "could be" cases. When a question asks which COULD be true, one valid example is enough. When it asks which MUST be true, every valid case must satisfy it.
4. Forgetting that squaring a negative makes it positive. x^2 vs. x: for x = -3, A = 9 > -3 = B; for x = 1/2, A = 1/4 < 1/2 = B.
5. Confusing (D) with "I can't tell." (D) means the relationship genuinely varies, not that you personally are stuck. If both quantities are pure numbers, the answer is never (D).

When NOT to Plug In

• Algebraic equivalence is obvious. If both quantities reduce to the same expression (as in Worked Example 3), the answer is (C) immediately.
• A constraint pins the variable. If "x = 4," there is nothing to test - just evaluate.
• Both quantities are pure numbers. Compute and compare; no (D) is possible.
• The stem gives a clear invariant (e.g., "the sum of two numbers is 10," comparing their product to 25 - by AM-GM the product is at most 25, so A <= B always).

Pacing and Strategic Notes

QC questions should average 60-90 seconds. If a plug-in test gives (D) quickly, mark (D) and move on - do not re-test to find a consistent answer that doesn't exist. If matching resolves the question in 20 seconds, take the time saved to the harder Data Interpretation set later in the section. Remember: the on-screen calculator does not respect order of operations, so for QC it is often faster to reason than to compute.