A number consists of two digits and the digit in the ten's place exceeds that in the unit's place by 5. If 5 times the sum of the digits be subtracted from the number, the digits of the number are reversed. Then the sum of digits of the number is

Options

A. 11
B. 7
C. 9
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 given.

### Step 1: Define the digits
Let the digit in the unit's place be \( x \). According to the problem, the digit in the ten's place exceeds that in the unit's place by 5. Therefore, the digit in the ten's place can be expressed as:
\[ \text{Ten's place} = x + 5 \]

### Step 2: Form the number
The two-digit number can be represented as:
\[ \text{Number} = 10 \times (\text{Ten's place}) + (\text{Unit's place}) = 10(x + 5) + x = 10x + 50 + x = 11x + 50 \]

### Step 3: Calculate the sum of the digits
The sum of the digits is:
\[ \text{Sum of digits} = (\text{Ten's place}) + (\text{Unit's place}) = (x + 5) + x = 2x + 5 \]

### Step 4: Set up the equation based on the problem statement
According to the problem, if 5 times the sum of the digits is subtracted from the number, the digits of the number are reversed. Thus, we can write the equation:
\[ \text{Number} - 5 \times (\text{Sum of digits}) = \text{Reversed number} \]

Substituting the expressions we derived:
\[ (11x + 50) - 5(2x + 5) = 10x + x + 5 \]

### Step 5: Simplify the equation
Now, simplifying the left-hand side:
\[ 11x + 50 - (10x + 25) = 11x + 50 - 10x - 25 = x + 25 \]
So, we have:
\[ x + 25 = 11x + 5 \]

### Step 6: Solve for \( x \)
Now, rearranging the equation:
\[ x + 25 - 5 = 11x \]
\[ x + 20 = 11x \]
\[ 20 = 11x - x \]
\[ 20 = 10x \]
\[ x = 2 \]

### Step 7: Find the digits
Now that we have \( x \), we can find the digits:
- Unit's place \( = x = 2 \)
- Ten's place \( = x + 5 = 2 + 5 = 7 \)

### Step 8: Calculate the sum of the digits
Finally, the sum of the digits is:
\[ \text{Sum of digits} = 2 + 7 = 9 \]

### Conclusion
The sum of the digits of the number is \( 9 \).

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