In a two digit number the unit's digit is twice the ten's digit. If 27 is added to the number, the digits interchange their places, the number is :

Question

Accepted Answer

To solve the problem, let's break down the information given and verify why Option 3 (36) is the correct answer.

1. **Understanding the Problem:**
   - We have a two-digit number where the unit's digit is twice the ten's digit.
   - If 27 is added to this number, the digits interchange their places.

2. **Setting Up the Equation:**
   - Let the ten's digit be $ x $.
   - Then, the unit's digit will be $ 2x $.
   - Therefore, the original number can be expressed as $ 10x + 2x = 12x $.

3. **Condition After Adding 27:**
   - When 27 is added to the number, the digits interchange their places. This means the new number becomes $ 10(2x) + x = 20x + x = 21x $.

4. **Equation Formation:**
   - According to the problem, adding 27 to the original number results in the new number:
     $$
     12x + 27 = 21x
     $$

5. **Solving the Equation:**
   - Rearrange the equation:
     $$
     21x - 12x = 27
     $$
     $$
     9x = 27
     $$
     $$
     x = 3
     $$

6. **Finding the Number:**
   - If $ x = 3 $, the ten's digit is 3, and the unit's digit is $ 2 \times 3 = 6 $.
   - Therefore, the original number is $ 36 $.

7. **Verification:**
   - Add 27 to 36: $ 36 + 27 = 63 $.
   - The digits of 63 are indeed the reverse of the digits of 36, confirming the condition.

8. **Explanation of Incorrect Options:**
   - **Option 1 (24):** If the number were 24, adding 27 would result in 51, which does not have reversed digits.
   - **Option 2 (12):** If the number were 12, adding 27 would result in 39, which does not have reversed digits.
   - **Option 4 (48):** If the number were 48, adding 27 would result in 75, which does not have reversed digits.

Thus, Option 3 (36) is the correct answer, as it satisfies all the conditions given in the problem.