Quantitative Comparison

Strategy/Review

The QC Mindset: Prove (D), Don't Assume (A) or (B)

Your default posture should be skeptical of simple comparisons. If Quantity A "looks" bigger at first glance, your job is to try to BREAK that conclusion. Can you find a case where they're equal? Can you find a case where B is bigger? If you can find BOTH a case where A > B AND a case where B > A (or A = B), then the answer is (D).

Systematic Testing Protocol

When variables are involved (x, y, n, etc.) and no restrictions are stated, you MUST test across the full range of possible values. Here's the battle-tested testing sequence:

1. Positive integers (easiest, your brain's default): Try x = 1, 2, 5, 100.
2. Negative integers: Try x = −1, −2, −5, −100. Many relationships REVERSE for negatives.
3. Zero: The great neutralizer. Try x = 0.
4. Fractions between 0 and 1: Try x = 0.5, 1/3, 0.1. Squaring makes fractions smaller; square roots make them larger.
5. Fractions between −1 and 0: Try x = −0.5. A weird zone where odd/even powers behave differently.
6. Boundary cases: If a constraint says x ≥ 2, test x = 2 (the boundary). If it says "x is a positive integer," test x = 1 (the minimum).

The critical rule: Do NOT assume variables are positive integers unless the problem explicitly states it. If it says "x is a real number" or says nothing, test all categories.

Example-Driven Strategy

Example 1 — The Negative Trap: Quantity A: x² Quantity B: x

If you assume x is positive (say, x = 3): A = 9, B = 3 → A > B. You'd answer (A). But test x = 1/2: A = 1/4, B = 1/2 → B > A. Now you have a conflict — must be (D). Test x = 0: A = 0, B = 0 → equal. Confirmed (D).

The answer is (D) because the relationship depends on x's value. This is the most common QC answer when variables are unrestricted.

Example 2 — Constrained but Still Tricky: x > 0 Quantity A: x² Quantity B: x

Now with x > 0: if x > 1, then x² > x. If x = 1, then x² = x. If 0 < x < 1, then x² < x. Still (D)! The constraint "positive" isn't enough — you need "x > 1" or "x is an integer > 1" to lock in A > B.

Example 3 — Locked In: x > 1 Quantity A: x² Quantity B: x → Now x² > x always. Answer: (A).

When Quantities Are Expressions Without Variables

If both quantities are pure numbers (no variables), you CAN determine the relationship definitively. The answer will be (A), (B), or (C) — never (D).

Example: Quantity A: 2³ + 3² Quantity B: 4² − 1 → A = 8 + 9 = 17, B = 16 − 1 = 15. A > B. Answer: (A).

Simplification Strategies

Before calculating, try to simplify both quantities:

• Factor out common terms.
• Square both sides (if both quantities are positive — be careful with negatives).
• Subtract the same expression from both.
• Divide both by the same positive quantity.

Example: Quantity A: (x + 3)(x − 3) Quantity B: x² − 9 → Expand A: x² − 9. Both are equal! Answer: (C). No need to test x.

Example: Quantity A: 2⁸ / 2³ Quantity B: 2⁵ → A = 2⁸⁻³ = 2⁵. Equal. Answer: (C).

Geometry in QC Questions

Geometry QC questions often have figures that are "not necessarily drawn to scale." Do NOT trust the visual appearance. A figure may look like a right angle but not be marked as one. An angle may look acute but could be obtuse. Only trust information explicitly given: congruence marks, parallel marks, stated measurements, and labeled right angles.

Example: Triangle ABC is shown. AB = 5, BC = 6. Quantity A: AC Quantity B: 11 → By the triangle inequality, AC < AB + BC = 11. So AC < 11. Answer: (B). But if the question asked for AC's minimum: AC > |AB − BC| = 1. So 1 < AC < 11 — and if quantities were AC vs. 2, the answer would be (D) because AC could be 1.5 or 10.

The "Cannot Be Determined" Decision Rule

(D) is the correct answer when, given the stated constraints:

• There exists at least one valid case where A > B, AND
• There exists at least one valid case where A < B (or A = B when the question allows equality).

You only need ONE counterexample in each direction. If you find A > B and A = B for different valid values, the answer cannot be (A) or (C), and since B > A isn't proven, it might seem like (A) — but (D) is correct because the relationship isn't consistent.

Summary of (D) triggers:

• Unrestricted variables (or only "real number" stated)
• Variables restricted only to "positive" (without "integer" or "> 1")
• Geometry figures without explicit measurements
• Variables that can be zero, negative, or fractional
• Inequalities that flip under certain conditions

Common Traps

Trap 1 — Positive Integer Assumption. The #1 QC trap. Unless the problem says "x is a positive integer," do NOT assume it. x² vs. x is the classic — for x = 2, x² > x; for x = 0.5, x² < x.

Trap 2 — Squaring/Negating Changes Inequality Direction. If both quantities are negative, squaring them reverses the inequality. If you multiply/divide by a negative, the inequality flips. Always check signs before simplifying.

Trap 3 — Trusting the Diagram. QC geometry figures are often deliberately misleading. A line that looks like a diameter may not be labeled as one. An angle that looks like 90° may be 89° or 91°. Only use stated facts.

Trap 4 — Forgetting to Test Boundary Values. If x ≥ 2, test x = 2 specifically. Boundary cases often produce equality (C) while all other values produce inequality (A or B), making the answer (D).

Trap 5 — Overcalculating. If you find yourself doing complex arithmetic to compare two numbers, pause — there's probably a simplification. QC rewards cleverness over brute force.