Quantitative Comparison Strategy

A QC answer is a claim about every allowed case: choosing "Quantity A is greater" asserts A beats B for all values consistent with the given information, so a single counterexample changes the answer — and one pair of conflicting examples proves "cannot be determined."

Plugging in numbers is a search for counterexamples, which is why the useful test values are the hostile ones: zero kills products, negatives flip inequalities, and fractions between 0 and 1 shrink when squared instead of growing.

Quantitative Comparison questions present two quantities — Quantity A and Quantity B — sometimes with extra given information, and always the same four answer choices: A is greater, B is greater, the two are equal, or the relationship cannot be determined from the information given. Because the choices never change, ETS's first tip is simply to know them cold, including the fact that the last choice is wrong whenever both quantities are concrete numbers that could in principle be computed. A useful mental model is the sign of the difference $A - B$: consistently positive means A is greater, consistently negative means B is greater, always zero means equal, and a sign that varies across allowed values means the relationship cannot be determined.

When the quantities contain variables, plugging in numbers is the core strategy, and its power comes from choosing adversarial values. ETS's own advice is to consider all kinds of appropriate numbers before answering: zero, positive and negative numbers, small and large numbers, fractions and decimals. Squaring, for instance, enlarges numbers greater than 1 but shrinks fractions between 0 and 1 and erases signs — so a comparison of $x^2$ with $x$ swings with the choice of $x$. The moment two allowed values give different comparison results, the answer is settled as "cannot be determined"; there is nothing more to check.

Algebra and arithmetic trade off. Simplifying both quantities the same way — adding or subtracting the same amount from each, or dividing both by a positive number — can reduce a cluttered comparison to an obvious one, and ETS recommends simplifying or estimating over unnecessary computation. But operations that are unsafe on inequalities are equally unsafe here: multiplying or dividing both quantities by a variable that could be negative or zero can silently reverse or destroy the comparison. When the given information pins the variables down (say, $x > 0$ is stated), respect it — plugging in values outside the constraint proves nothing.