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)

• The equation is (10P + Q) × 3 = 100R + 10Q + Q, which simplifies to 30P + 3Q = 100R + 11Q.

• Rearranging gives 30P - 100R = 8Q, or 15P - 50R = 4Q.

• Since Q is a single digit (0-9), 4Q must be between 0 and 36. Also, 15P - 50R must be non-negative, implying P is sufficiently large relative to R.

• Testing integer values for R (1-9) and P (1-9) that satisfy 15P - 50R = 4Q with Q being an integer digit: If R=1, 15P - 50 = 4Q. For P=4, 60-50=10 (not divisible by 4). For P=5, 75-50=25 (no). For P=6, 90-50=40, so 4Q=40 => Q=10 (invalid, Q must be a digit). For P=7, 105-50=55 (no). For P=8, 120-50=70 (no). For P=9, 135-50=85 (no).

• Let's re-evaluate the carry logic directly. 3 × Q ends in Q. The only digits where 3Q ends in Q are Q=0 (3*0=0) and Q=5 (3*5=15).

• If Q=0, (P0) × 3 = R00. 30P = 100R => 3P = 10R. No integer solution for digits P, R.

• If Q=5, (P5) × 3 = R55. 3 × 5 = 15, write 5 carry 1. 3P + 1 = R5 (ends in 5). So 3P + 1 ends in 5, meaning 3P ends in 4. The only digit P where 3P ends in 4 is P=8 (3*8=24).

• Check: 85 × 3 = 255. Here P=8, Q=5, R=2. All are different digits and R≠0.

• Calculate (P + R) ÷ Q = (8 + 2) ÷ 5 = 10 ÷ 5 = 2.

Why the other options are wrong

(a) 1: The calculated value is 2, not 1.
(c) 5: The calculated value is 2, not 5.
(d) Cannot be determined due to insufficient data: The values of P, Q, and R are uniquely determined as 8, 5, and 2 respectively, so the result is determinable.

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.