If ABC and DEF are both 3-digit numbers such that A, B, C, D, E and F are distinct non-zero digits such that ABC + DEF = 1111, then what is the value of A + B + C + D + E + F ?

(a) 28
(b) 29
(c) 30
(d) 31 ✓ UPSC's answer

Why the answer is (d)

• Since ABC and DEF are 3-digit numbers summing to 1111, the sum of their digits must account for the carries generated in each column.

• In the units column, C + F must equal 11 (to produce a digit 1 and carry 1 to the tens place), so C + F = 11.

• In the tens column, B + E + 1 (carry) must equal 11 (to produce a digit 1 and carry 1 to the hundreds place), so B + E = 10.

• In the hundreds column, A + D + 1 (carry) must equal 11 (to produce a digit 1 and carry 1 to the thousands place), so A + D = 10.

• The total sum A + B + C + D + E + F is therefore (A + D) + (B + E) + (C + F) = 10 + 10 + 11 = 31.

• Thus, the value is 31, which corresponds to option (d).

Why the other options are wrong

(a) 28: The sum 28 is incorrect because it fails to account for the necessary carries in the addition, specifically underestimating the sum of the units digits.
(b) 29: The sum 29 is incorrect because it does not satisfy the arithmetic constraints imposed by the column-wise addition resulting in 1111.
(c) 30: The sum 30 is incorrect because it miscalculates the contribution of the carries, leading to a total that is one less than the required 31.

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.