# 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?

a) 9

b) 8

c) 7

d) Cannot be determined due to insufficient data

Correct answer: a) 9

#### Question from UPSC Prelims 2024 CSAT

**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