A Question is given followed by two Statements I and II. Consider the Question and the Statements. Age of each of P and Q is less than 100 years but more than 10 years. If you interchange the digits of the age of P, the number represents the age of Q. Question : What is the difference of their ages? Statement-I : The age of P is greater than the age of Q. Statement-II : The sum of their ages is 11/6 times their difference. Which one of the following is correct in respect of the above Question and the Statements?
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone ✓ UPSC's answer
- (b) The Question can be answered by using either Statement alone
- (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
- (d) The Question cannot be answered even by using both the Statements together
Why the answer is (a)
• Let P = 10a + b and Q = 10b + a, with digits a, b from 1 to 9 (both ages are between 10 and 100). The difference is 9|a − b| and the sum is 11(a + b).
• Statement I (P older) only says a > b; the difference could be 9, 18, 27, … — not enough.
• Statement II: 11(a + b) = (11/6) × 9|a − b|, so a + b = 1.5|a − b|. Writing |a − b| = 2k gives a + b = 3k, so the larger digit is 2.5k and the smaller 0.5k. Only k = 2 gives digits (5 and 1); k = 4 would need 10.
• The ages are 51 and 15, so the difference is 36 — Statement II alone answers it. One statement is enough, the other is not — option (a).
Why the other options are wrong
- (b) The Question can be answered by using either Statement alone
- Statement I alone is insufficient because it only establishes that P is older than Q without fixing the specific digit difference, allowing for multiple possible age differences.
- (c) The Question can be answered by using both the Statements together, but cannot be…
- The question can be answered using Statement II alone, so it is incorrect to claim that both statements are required together.
- (d) The Question cannot be answered even by using both the Statements together
- The question can be answered using Statement II alone, which uniquely determines the age difference as 36 years, so it is incorrect to claim it cannot be answered.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.