Let x, y be the volumes; m, n be the masses of two metallic cubes P and Q respectively. Each side of Q is two times that of P and mass of Q is two times that of P. Let u = m / x and v = n / y. Which one of the following is correct?

(a) u = 4v ✓ UPSC's answer
(b) u = 2v
(c) v = u
(d) v = 4u

Why the answer is (a)

• Let the side length of cube P be $a$, so its volume $x = a^3$ and mass $m$.

• The side length of cube Q is $2a$, so its volume $y = (2a)^3 = 8a^3$.

• The mass of cube Q is given as $n = 2m$.

• Calculate $u = m/x = m/a^3$.

• Calculate $v = n/y = 2m / 8a^3 = m / 4a^3$.

• Comparing the two, $u = 4 \times (m / 4a^3) = 4v$, which matches option (a).

Why the other options are wrong

(b) u = 2v: Option (b) incorrectly assumes the volume ratio is 2 instead of 8, leading to $u = 2v$.
(c) v = u: Option (c) incorrectly assumes the density or mass-to-volume ratio is identical for both cubes, ignoring the different scaling factors.
(d) v = 4u: Option (d) inverts the relationship, incorrectly concluding that $v$ is four times $u$ instead of $u$ being four times $v$.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.