There are two containers X and Y. X contains 100 ml of milk and Y contains 100 ml of water. 20 ml of milk from X is transferred to Y. After mixing well, 20 ml of the mixture in Y is transferred back to X. If m denotes the proportion of milk in X and n denotes the proportion of water in Y, then which one of the following is correct?

(a) m = n ✓ UPSC's answer
(b) m > n
(c) m < n
(d) Cannot be determined due to insufficient data

Why the answer is (a)

• Initially, X has 100 ml milk and Y has 100 ml water.

• 20 ml of milk is moved from X to Y, so X has 80 ml milk and Y has 20 ml milk + 100 ml water (total 120 ml).

• The mixture in Y is uniform, so 20 ml transferred back to X contains (20/120)*20 = 10/3 ml milk and (100/120)*20 = 50/3 ml water.

• After transfer, X contains 80 + 10/3 = 250/3 ml milk and 50/3 ml water, totaling 100 ml. Thus, proportion of milk in X, m = (250/3)/100 = 5/6.

• Y contains 20 - 10/3 = 50/3 ml milk and 100 - 50/3 = 250/3 ml water, totaling 100 ml. Thus, proportion of water in Y, n = (250/3)/100 = 5/6.

• Since m = 5/6 and n = 5/6, m = n, making option (a) correct.

Why the other options are wrong

(b) m > n: The calculation shows m = 5/6 and n = 5/6, so m is not greater than n.
(c) m < n: The calculation shows m = 5/6 and n = 5/6, so m is not less than n.
(d) Cannot be determined due to insufficient data: The problem provides sufficient data to calculate the exact proportions, so the answer can be determined.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.