Quantitative Comparison Strategy

A Format Unlike Anything Else on the Test

Quantitative Comparison (QC) makes up roughly a third of every GRE Quant section, and it rewards a completely different skill than the multiple-choice problems. You are not asked to compute a value. You are asked to compare two quantities — Quantity A and Quantity B — and decide which is larger, whether they are equal, or whether the relationship can shift depending on unknowns. The four answer choices are always the same, in the same order:

Memorize these cold. On test day you should read "A" as "A is bigger, always" and "D" as "it depends." Because you are only ranking two things, you almost never need the exact value — and doing extra arithmetic is how careless errors creep in.

The Golden Rule of Choice D

Choice D is correct precisely when the relationship changes depending on the values of the unknowns. If you can find one case where A > B and another case where A ≤ B, you are finished — the answer is D. This single idea decides a large fraction of QC questions, and it is the one that separates high scorers from everyone else.

The corollary matters just as much: if both quantities are fully determined constants, D is impossible. A question with no variables — say, comparing $27^{1/3}\cdot 4^{1/2}$ with $6$ — must resolve to A, B, or C. Choosing D there means you gave up on arithmetic you could have finished.

The Plug-In Discipline: Test 0, 1, Negatives, and Fractions

When a QC item contains variables, do not reason in the abstract — substitute strategically chosen numbers. The danger is testing only "nice" values (like 2 and 3) that all point the same way. Train yourself to run through this checklist of boundary-breaking inputs:

• 0 — collapses products and exposes "greater than" claims that fail at zero.
• 1 — the fixed point where $x$, $x^2$, and $\sqrt{x}$ all coincide.
• A negative — flips inequalities and breaks sign assumptions.
• A fraction between 0 and 1 — here squaring makes a number smaller and roots make it larger, the reverse of your intuition.

The classic trap: comparing $x^2$ and $x$. Plug in $x = 2$ and $x^2 = 4 > 2$, so A looks greater. But at $x = \tfrac{1}{2}$, $x^2 = \tfrac{1}{4} < \tfrac{1}{2}$, so B is greater — and at $x = 1$ they are equal. Three different outcomes means the answer is D. A student who tested only $x = 2$ walks straight into the trap.

When Equality Collapses to C

Some QC items look like they contain variables but simplify to an identity. If both quantities reduce to the same expression, the answer is C no matter what the variable is. For example, Quantity A $= 2^{10} - 2^{9}$ factors as $2^{9}(2 - 1) = 2^{9}$, which equals Quantity B $= 2^{9}$: the difference vanishes and the answer is C. Always ask whether algebra makes the two sides identical before you start plugging in.

Efficient Comparison Without Computing

Because you only need the ranking, you can transform both quantities the same way to make them easier to compare — as long as the operation preserves order:

• Add or subtract the same amount from both quantities freely.
• Multiply or divide both by the same positive number freely.
• Never multiply or divide both by a variable of unknown sign — that can flip the inequality and is a top source of wrong answers.

For instance, to compare $\dfrac{a}{7}$ and $\dfrac{b}{5}$ you can cross-multiply only if you know the denominators are positive (they are), giving $5a$ versus $7b$ — simpler, order-preserved.

A Reliable QC Game Plan

1. No variables? Then D is impossible. Simplify both sides to constants and pick A, B, or C.
2. Variables present? Plug in 0, 1, a negative, and a fraction. If two cases disagree, the answer is D immediately.
3. Looks like an identity? Simplify — if both sides match, the answer is C.
4. Comparing directly? Add, subtract, or scale by positives to line the quantities up; never scale by an unknown sign.

Internalize step 2 and you will catch the traps that the GRE builds every QC section around.