GRE Quantitative Comparison: When the Answer Is D

Choice D says the relationship cannot be determined from the information given. It is a claim about the information, not about your ability to compute — and it is correct exactly when the given constraints permit more than one relationship between the two quantities.

That gives you a proof procedure. To establish D, find one case consistent with all the given information where Quantity A is greater, and a second case where it is not. Two examples, and D is proven. You do not need to reason about the general structure.

The mirror of that procedure is what makes A, B, or C safe. If you have tried to break the relationship and cannot, you have some evidence — but only reasoning about why it cannot break gives certainty.

If both quantities consist entirely of specific numbers with no variables and no unknowns, D can never be correct. Two definite numbers have a definite relationship, whether or not you can compute it quickly.

This is the most immediately useful rule in Quantitative Comparison. When you see a comparison of, say, the value of a compound-interest expression against a plain product, and both are fully specified, D is off the table and you are choosing among three options.

Be careful about what counts as fully specified. A quantity involving a variable that has been pinned down by an equation elsewhere — where x is given as satisfying 3x + 2 = 14 — is fully determined, because the equation has one solution. But an equation like x² = 16 does not pin down a single value, and that is often the entire question.

The first two rows account for a large share of D answers. The habit worth building is that whenever a variable appears without a stated restriction, you test a negative value and a fraction between 0 and 1 before concluding anything.

Example 1. Given x > 0. Quantity A: x². Quantity B: x. Test x = 2: A is 4, B is 2, so A is greater. Test x = 1/2: A is 1/4, B is 1/2, so B is greater. Two consistent cases with opposite relationships, so the answer is D. Note that x > 0 does not save it — the fractional range is still open. Had the constraint been x > 1, the answer would be A.

Example 2. Given x² = 25. Quantity A: x. Quantity B: 5. The constraint permits x = 5 and x = −5. In the first case the quantities are equal; in the second, B is greater. Two relationships, so D. The trap here is treating the square root as returning only the positive value — that convention applies to the radical symbol, not to solving an equation.

Example 3. Quantity A: the average of 4, 9, and 14. Quantity B: 9. Both quantities are fully determined: A is 9, and the answer is C. No variables means D was never available, which you could have established before doing any arithmetic.

Example 4. Given that the average of five test scores is 80. Quantity A: the median of the five scores. Quantity B: 80. Consider five scores of 80 each — the median is 80 and the quantities are equal. Now consider 40, 40, 80, 120, 120 — the average is still 80 and the median is 80, so try 60, 60, 60, 100, 120: the average is 80 and the median is 60, making B greater. Two relationships, so D. Averages constrain the sum and nothing else.

The most common mistake is testing only convenient positive integers. Trying x = 1, 2, and 3, finding the same relationship each time, and choosing A is how most missed D answers happen. Zero, negatives, and fractions between 0 and 1 are where relationships break.

The second is choosing D out of uncertainty rather than out of demonstration. D is a specific claim — that the information permits multiple relationships — and "I could not figure it out" is not evidence for it. When you cannot resolve a comparison, guessing D is reasonable as a guess, but it should not feel like a derivation.

The third is forgetting that D is unavailable when both quantities are purely numeric. This is worth checking first on every question, because it converts a four-choice question into a three-choice one at no cost.

The fourth is trusting a geometric figure's appearance. Unless a figure is stated to be drawn to scale, an angle that looks obtuse may not be, and a triangle that looks isosceles may not be. The constraints given in text are the only constraints that hold.

Check whether both quantities are purely numeric. If so, eliminate D and compute or estimate.

If variables are present, note every stated restriction — positive, integer, nonzero, within a range — and then deliberately test outside your defaults but inside those restrictions. A negative value, a fraction between 0 and 1, and zero if permitted.

If two tests give different relationships, select D and move on. There is nothing further to verify; one counterexample pair is a complete proof.

If several tests give the same relationship, look for a structural reason before committing. Simplifying both quantities by the same operation — subtracting a common term, dividing by a positive quantity — often makes the comparison obvious. Note that dividing by a variable is only safe when you know its sign, which is itself frequently the point of the question.