Quantitative Comparison Strategy • Topic 3 of 4

Plug In Smart Numbers (0, 1, negatives, fractions)

When a QC contains variables, the fastest route is often to test values rather than manipulate symbols. The art is choosing smart numbers — the ones that expose how a relationship can change. A reliable kit is 0, 1, a negative, and a proper fraction between 0 and 1, because each behaves unusually: zero collapses products, one is the multiplicative identity, negatives flip inequalities and reverse under multiplication, and fractions between 0 and 1 get smaller when squared and larger when you take their reciprocal. Here is the decisive logic: if two different legal test values give two different relationships (once A wins, once B wins), the answer must be D — you have proved it cannot be pinned down. If every value you try gives the same relationship, that is strong evidence for A, B or C, and you can usually confirm it algebraically. Always respect the constraints printed above the quantities: only plug in values the problem actually allows.

✅ Solved examples

1. x is a nonzero number. Quantity A: x. Quantity B: x³. Which choice?
Try x = 2: A = 2, B = 8, so B wins. Try x = 1/2: A = 0.5, B = 0.125, so A wins. Two different relationships from allowed values means Choice D.
2. x > 1. Quantity A: x. Quantity B: x². Which choice?
For any x > 1, squaring makes it bigger: try x = 2 (2 vs 4, B) and x = 10 (10 vs 100, B). The constraint x > 1 rules out the fraction case, so Quantity B is always greater — Choice B.
3. 0 < x < 1. Quantity A: x. Quantity B: √x. Which choice?
A proper fraction’s square root is larger than the fraction. Try x = 1/4: A = 1/4, B = √(1/4) = 1/2, so B wins. Try x = 0.81: A = 0.81, B = 0.9, B wins. Consistent, so Choice B.
4. a and b are integers with a < b. Quantity A: a + b. Quantity B: ab. Which choice?
Try a = 1, b = 2: A = 3, B = 2, A wins. Try a = 3, b = 4: A = 7, B = 12, B wins. Different relationships, so Choice D.

✏️ Practice — try these, take hints as needed

1. x is any real number. Quantity A: x². Quantity B: x³. Which choice?
Try x = 2, then x = 1/2, then x = −1.
The order changes across these.
When it flips, the answer is D.
D
2. 0 < x < 1. Quantity A: 1/x. Quantity B: 1. Which choice?
The reciprocal of a proper fraction exceeds 1.
Try x = 1/2: 1/x = 2.
Try x = 0.9: 1/x ≈ 1.11.
A (1/x > 1 for 0 < x < 1)
3. n is a negative integer. Quantity A: n. Quantity B: n². Which choice?
A negative squared is positive.
Try n = −3: −3 vs 9.
Positive always beats negative.
B
4. 0 < x < 1. Quantity A: x². Quantity B: x. Which choice?
Squaring a proper fraction shrinks it.
Try x = 1/2: 1/4 vs 1/2.
Consistent direction.
B (x > x²)
5. x is any real number. Quantity A: x + 2. Quantity B: 2x. Which choice?
Set them equal: x + 2 = 2x gives x = 2.
Try x = 1 (A wins) and x = 3 (B wins).
It flips around x = 2.
D

📝 Topic test — 8 questions

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