A number consists of two digits such that the digit in the ten's place is less by 2 than the digit In the unit's place. Three times the number added to `(6)/(7)` times the number obtained by reversing the digits equals 108. The sum of digits in the number is:

Options

A. 8
B. 9
C. 6
D. 7

Answer

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

### Step 1: Define the digits
Let the unit's digit be \( x \). According to the problem, the digit in the ten's place is less by 2 than the digit in the unit's place. Therefore, the ten's digit can be expressed as:
\[ \text{Ten's digit} = x - 2 \]

### Step 2: Form the original number
The original two-digit number can be represented as:
\[ \text{Number} = 10 \times (\text{Ten's digit}) + (\text{Unit's digit}) = 10(x - 2) + x = 10x - 20 + x = 11x - 20 \]

### Step 3: Form the reversed number
When the digits are reversed, the unit's digit becomes the ten's digit and vice versa. Thus, the reversed number is:
\[ \text{Reversed number} = 10 \times (\text{Unit's digit}) + (\text{Ten's digit}) = 10x + (x - 2) = 10x + x - 2 = 11x - 2 \]

### Step 4: Set up the equation
According to the problem, three times the original number added to \(\frac{6}{7}\) times the reversed number equals 108. We can write this as:
\[ 3(11x - 20) + \frac{6}{7}(11x - 2) = 108 \]

### Step 5: Simplify the equation
Now, let's simplify the equation:
\[ 33x - 60 + \frac{6}{7}(11x - 2) = 108 \]
To eliminate the fraction, multiply the entire equation by 7:
\[ 7(33x - 60) + 6(11x - 2) = 7 \times 108 \]
This simplifies to:
\[ 231x - 420 + 66x - 12 = 756 \]
Combining like terms gives:
\[ 297x - 432 = 756 \]

### Step 6: Solve for \( x \)
Now, add 432 to both sides:
\[ 297x = 756 + 432 \]
\[ 297x = 1188 \]
Now, divide by 297:
\[ x = \frac{1188}{297} = 4 \]

### Step 7: Find the ten's digit
Since \( x \) is the unit's digit, the ten's digit is:
\[ \text{Ten's digit} = x - 2 = 4 - 2 = 2 \]

### Step 8: Calculate the sum of the digits
The sum of the digits in the number is:
\[ \text{Sum of digits} = \text{Ten's digit} + \text{Unit's digit} = 2 + 4 = 6 \]

### Final Answer
Thus, the sum of the digits in the number is:
\[ \boxed{6} \]