In a two digit no. ten's digit is 5 more than unit digit. After subtracting 5 times the sum of digits from the number, the digits are interchanged. Find out the sum of both digits.

Options

A. a)9
B. b)11
C. c)7
D. d)13

Answer

To solve the problem step by step, we will define the digits of the two-digit number and set up equations based on the information provided.

### Step-by-Step Solution:

1. **Define the Digits**:
Let the unit digit be \( x \). According to the problem, the ten's digit is 5 more than the unit digit. Therefore, the ten's digit can be expressed as:
\[ \text{Ten's digit} = x + 5 \]

2. **Formulate the Two-Digit Number**:
The two-digit number can be represented as:
\[ \text{Number} = 10 \times (\text{Ten's digit}) + (\text{Unit digit}) = 10(x + 5) + x \]
Simplifying this gives:
\[ \text{Number} = 10x + 50 + x = 11x + 50 \]

3. **Calculate the Sum of the Digits**:
The sum of the digits is:
\[ \text{Sum of digits} = (\text{Ten's digit}) + (\text{Unit digit}) = (x + 5) + x = 2x + 5 \]

4. **Set Up the Equation Based on the Problem Statement**:
According to the problem, after subtracting 5 times the sum of the digits from the number, the digits are interchanged. Therefore, we can write:
\[ \text{New number} = \text{Number} - 5 \times (\text{Sum of digits}) \]
This can be expressed as:
\[ 11x + 50 - 5(2x + 5) = 10x + x + 5 \]
Simplifying the left side:
\[ 11x + 50 - (10x + 25) = 10x + x + 5 \]
This simplifies to:
\[ 11x + 50 - 10x - 25 = 11x + 5 \]
Which further simplifies to:
\[ x + 25 = 11x + 5 \]

5. **Solve for \( x \)**:
Rearranging the equation gives:
\[ 25 - 5 = 11x - x \]
Thus:
\[ 20 = 10x \]
Dividing both sides by 10:
\[ x = 2 \]

6. **Find the Ten's Digit**:
Now that we have \( x \), we can find the ten's digit:
\[ \text{Ten's digit} = x + 5 = 2 + 5 = 7 \]

7. **Calculate the Sum of Both Digits**:
Finally, the sum of both digits is:
\[ \text{Sum} = \text{Ten's digit} + \text{Unit digit} = 7 + 2 = 9 \]

### Final Answer:
The sum of both digits is \( 9 \).