Consider the following multiplication problem : (PQ) × 3 = RQQ, where P, Q and R are different digits and R ≠ 0. What is the value of (P + R) ÷ Q ?
- (a) 1
- (b) 2 ✓ UPSC's answer
- (c) 5
- (d) Cannot be determined due to insufficient data
Why the answer is (b)
• (10P + Q) × 3 = 100R + 11Q, so 30P + 3Q = 100R + 11Q, i.e. 30P = 100R + 8Q.
• The product RQQ is at most 99 × 3 = 297, so R is 1 or 2. Dividing by 2: 15P = 50R + 4Q, so 4Q must be a multiple of 5, giving Q = 0 or 5.
• Q = 0 gives 15P = 50R, so P = 10 (R = 3) — not a digit. Q = 5 gives 15P = 50R + 20: R = 2 gives P = 8 (R = 1 gives P = 14/3, not a digit).
• So P = 8, Q = 5, R = 2: 85 × 3 = 255. Then (P + R) ÷ Q = 10 ÷ 5 = 2 — option (b).
Why the other options are wrong
- (a) 1
- With P = 8, Q = 5, R = 2 (85 × 3 = 255) the value is 10 ÷ 5 = 2, not 1.
- (c) 5
- The value is (8 + 2) ÷ 5 = 2, not 5.
- (d) Cannot be determined due to insufficient data
- The data fix a unique solution, 85 × 3 = 255, so the value can be determined.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.