If the two digits of the age of Mr Manoj are reversed then the new age so obtained is the age of his wife. `(1)/(11)` of the sum of their ages is equal to the difference between their ages. If Mr Manoj is elder than his wife then find the difference between their ages

Options

A. Cannot be determined
B. 10 years
C. 9 years
D. 7 years

Answer

To solve the problem step by step, let's denote Mr. Manoj's age as a two-digit number where the digits are represented as follows:

- Let the digit in the tens place be \( y \).
- Let the digit in the units place be \( x \).

Thus, Mr. Manoj's age can be expressed as:
\[ \text{Age of Mr. Manoj} = 10y + x \]

When the digits are reversed, his wife's age becomes:
\[ \text{Age of Mrs. Manoj} = 10x + y \]

According to the problem, we have the following condition:
\[ \frac{1}{11} \times (\text{Age of Mr. Manoj} + \text{Age of Mrs. Manoj}) = |\text{Age of Mr. Manoj} - \text{Age of Mrs. Manoj}| \]

Substituting the expressions for their ages:
\[ \frac{1}{11} \times ((10y + x) + (10x + y)) = |(10y + x) - (10x + y)| \]

This simplifies to:
\[ \frac{1}{11} \times (11y + 11x) = |(10y + x) - (10x + y)| \]
\[ \frac{1}{11} \times 11(y + x) = |(10y - 10x + x - y)| \]
\[ y + x = |9y - 9x| \]

Since Mr. Manoj is older than his wife, we can drop the absolute value:
\[ y + x = 9y - 9x \]

Rearranging gives:
\[ y + x + 9x = 9y \]
\[ y + 10x = 9y \]
\[ 10x = 8y \]
\[ \frac{x}{y} = \frac{8}{10} = \frac{4}{5} \]

From this ratio, we can set:
\[ x = 4k \quad \text{and} \quad y = 5k \]
for some integer \( k \).

Since \( x \) and \( y \) must be single digits (0 to 9), we can find \( k \):
- The maximum value for \( y = 5k \) must be less than or equal to 9, hence \( k \) can be at most 1.
- If \( k = 1 \), then \( x = 4 \) and \( y = 5 \).

Now substituting back to find their ages:
\[ \text{Age of Mr. Manoj} = 10y + x = 10(5) + 4 = 54 \]
\[ \text{Age of Mrs. Manoj} = 10x + y = 10(4) + 5 = 45 \]

Now, we can find the difference between their ages:
\[ \text{Difference} = \text{Age of Mr. Manoj} - \text{Age of Mrs. Manoj} = 54 - 45 = 9 \]

Thus, the difference between their ages is:
\[ \boxed{9} \]