If p, q, r and s are distinct single digit positive numbers, then what is the greatest value of (p + q)(r + s) ?

(a) 230
(b) 225 ✓ UPSC's answer
(c) 224
(d) 221

Why the answer is (b)

• Let A = p + q and B = r + s; then A + B is the sum of four distinct digits from 1 to 9, so A + B ≤ 9 + 8 + 7 + 6 = 30.

• For any positive A and B with fixed sum T, the product AB is maximized when A and B are as close as possible, so AB ≤ (T/2)².

• Since T ≤ 30, AB ≤ (30/2)² = 225.

• The value 225 is attainable by choosing 9, 8, 7, 6 and pairing them as (9 + 6)(8 + 7) = 15 × 15 = 225.

• Therefore the greatest value is 225, which is option (b).

Why the other options are wrong

(a) 230: 230 is impossible because the maximum possible product is bounded by 225.
(c) 224: 224 is possible, for example (9 + 7)(8 + 6) = 16 × 14 = 224, but it is not the greatest value.
(d) 221: 221 is possible, for example (9 + 8)(7 + 6) = 17 × 13 = 221, but it is not the greatest value.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.