If the sum of the two-digit numbers AB and CD is the three-digit number 1CE, where the letters A, B, C, D, E denote distinct digits, then what is the value of A?

(a) 9 ✓ UPSC's answer
(b) 8
(c) 7
(d) Cannot be determined due to insufficient data

Why the answer is (a)

• The equation is AB + CD = 1CE, which translates to (10A + B) + (10C + D) = 100 + 10C + E.

• Simplifying the equation gives 10A + B + D = 100 + E.

• Since A, B, C, D, and E are distinct digits, the maximum possible value for B + D is 9 + 8 = 17 (assuming A is not 9 or 8 to keep them distinct, or generally max sum of two distinct digits is 17).

• For 10A + (B + D) to equal 100 + E, 10A must be close to 100. If A were 8, 10A = 80, requiring B + D = 20 + E, which is impossible since max B+D is 17.

• Therefore, A must be 9. If A = 9, the equation becomes 90 + B + D = 100 + E, or B + D = 10 + E.

• This is solvable with distinct digits (e.g., if E=1, B+D=11; if E=2, B+D=12, etc.), confirming A is uniquely 9.

Why the other options are wrong

(b) 8: If A were 8, the sum 10A + B + D would be at most 80 + 17 = 97, which cannot reach the 100s place required for the result 1CE.
(c) 7: If A were 7, the sum 10A + B + D would be at most 70 + 17 = 87, which is far too small to produce a three-digit number starting with 1.
(d) Cannot be determined due to insufficient data: The value of A is uniquely determined to be 9 by the constraints of the addition and the distinctness of the digits.

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.