If u and v are real numbers, what is the value of u² − v²?
(1) u + v is between 5 and 9.
(2) u² − v² = 3u − 3v, and u + v = 10.
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Answer & Explanation
Correct answer
B
Use the identity u² − v² = (u + v)(u − v).
Statement (1): knowing only that u + v lies between 5 and 9 leaves the difference u − v entirely free, so the product (u + v)(u − v) is wide open. not sufficient.
Statement (2): u² − v² = 3u − 3v = 3(u − v), and also u² − v² = (u + v)(u − v) = 10(u − v). Setting these equal: 10(u − v) = 3(u − v), so 7(u − v) = 0, giving u − v = 0. Then u² − v² = (u + v)(u − v) = 10 × 0 = 0. sufficient.
Statement (2) looks like one tangled equation plus a sum, and solvers may discard it as "not enough to find u and v" (range-not-value) and pick A or E. The asked quantity factors, and the internal equation forces u − v = 0, pinning the value to 0 without solving for u and v individually. Statement (1) gives only a range on the sum, not the value of the asked quantity, so the key is B.