Q. 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?

Explanation : 

We are given:
– Two two-digit numbers: AB and CD
– Their sum is a three-digit number: 1CE
– All letters represent distinct digits.

Step 1: Set Up the Addition

   A B
+ C D
———-
1 C E

Step 2: Analyze the Units Place (Rightmost Digit)

Adding the units digits:
– B + D = E (with possible carryover to the tens place)

Let k be the carryover from the units place addition (k can be 0 or 1 because the sum of two digits can’t produce a carryover greater than 1 in decimal addition).

So:
1. B + D = E + 10k
2. Equation (1): B + D = E + 10k

Step 3: Analyze the Tens Place

Adding the tens digits, plus any carryover from the units place:
– A + C + k = C + 10
(since the tens digit in the result is C, and there’s a carryover to make it the same C)

Simplify:
1. A + k = 10
2. Equation (2): A + k = 10

Step 4: Solve for A and k

From Equation (2):
– Since A is a single digit (0-9), and k is 0 or 1:
– If k = 0: A = 10 (invalid, as A must be a single digit)
– If k = 1: A = 9 (valid, as A is a single-digit number)

Therefore, A = 9 and k = 1.

Step 5: Conclusion

Using the carryover method simplifies the problem and directly leads us to the value of A:
– A = 9