Quantitative Comparison Strategy • Topic 4 of 4

When the Answer is "Cannot Be Determined"

Choice D — "the relationship cannot be determined" — is the choice test-takers most often get wrong in both directions: they pick it out of laziness, or they miss it when it is genuinely correct. Two clear rules cut through the confusion. First, D is impossible whenever both quantities reduce to specific numbers; fixed numbers always have a definite order, so if you have computed both and they are just numbers, cross D off. Second, D is correct exactly when you can produce two allowed scenarios that give opposite relationships — one where A is bigger and one where B is bigger (or one equal and one not). Proving D therefore means finding a single counterexample to "always A" and a single counterexample to "always B". The usual triggers are unrestricted variables, especially those that could be negative, zero or fractional, and geometry figures that are not drawn to scale (the GRE warns you not to trust the picture). When you see a free variable, actively try to break a tentative answer before committing.

✅ Solved examples

1. x is a real number. Quantity A: x². Quantity B: 4. Which choice?
Try x = 3: A = 9 > 4, A wins. Try x = 1: A = 1 < 4, B wins. Two opposite outcomes from allowed values, so Choice D.
2. A rectangle has area 24. Quantity A: its perimeter. Quantity B: 20. Which choice?
Area 24 fixes the product of the sides, not the sides themselves. A 4×6 rectangle has perimeter 2(4+6) = 20 (equal to B). A 2×12 rectangle has perimeter 2(2+12) = 28 (> 20). Different relationships, so Choice D.
3. x < 0. Quantity A: x. Quantity B: 1/x. Which choice?
For any negative x, both x and 1/x are negative. Try x = −2: A = −2, B = −0.5, so B > A. Try x = −0.5: A = −0.5, B = −2, so A > B. Opposite outcomes, so Choice D.
4. x and y are positive with x + y = 10. Quantity A: xy. Quantity B: 24. Which choice?
The product varies: x = 5, y = 5 gives xy = 25 > 24 (A wins); x = 2, y = 8 gives xy = 16 < 24 (B wins). Two outcomes, so Choice D.

✏️ Practice — try these, take hints as needed

1. x is a real number. Quantity A: x. Quantity B: |x|. Which choice?
Try x = 5, then x = −5.
For negatives, |x| is bigger; for positives, equal.
Different relationships appear.
D
2. The product of two integers is 12. Quantity A: their sum. Quantity B: 7. Which choice?
List factor pairs: 3×4, 2×6, 1×12, and negatives.
3 + 4 = 7 but 2 + 6 = 8.
Even more spread with negatives.
D
3. x² = 49. Quantity A: x. Quantity B: 5. Which choice?
x could be 7 or −7.
7 > 5 but −7 < 5.
The square root has two signs.
D
4. A triangle has two sides of length 5 and 8 (figure not to scale). Quantity A: the third side. Quantity B: 10. Which choice?
The third side lies strictly between 3 and 13.
It could be 4 (< 10) or 12 (> 10).
Do not trust the drawing.
D
5. Quantity A: 3 + 8. Quantity B: 12 − 1. Which choice? (Watch the trap.)
These are fixed numbers, so D is impossible.
11 versus 11.
Do not reflexively pick D.
C (both equal 11)

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Plug In Smart Numbers (0, 1, negatives, fractions) Take the chapter test →