The ratio of the ages of the husband and the wife five years ago was 11:9 where as at the same ages of the husband and his son was 5:1 . Five years hence the sum of the three persons of the family ?

Options

A. 141 years
B. 127 years
C. 126 years
D. Data inadeuate

Answer

To solve the problem step by step, we will first establish the ages of the husband, wife, and son based on the given ratios.

### Step 1: Establish the ratios
Five years ago, the ratio of the ages of the husband (H) and wife (W) was 11:9. This can be expressed as:
\[ H - 5 = 11x \]
\[ W - 5 = 9x \]
where \( x \) is a common multiplier.

### Step 2: Express the husband's and son's ages
At the same time (five years ago), the ratio of the ages of the husband (H) and son (S) was 5:1. This can be expressed as:
\[ H - 5 = 5y \]
\[ S - 5 = y \]
where \( y \) is another common multiplier.

### Step 3: Set the equations equal
From the two equations for the husband's age:
\[ 11x = 5y \]
This implies:
\[ y = \frac{11x}{5} \]

### Step 4: Substitute y in the son's equation
Now substitute \( y \) in the son's age equation:
\[ S - 5 = \frac{11x}{5} \]
Thus,
\[ S = \frac{11x}{5} + 5 \]

### Step 5: Calculate the husband's and wife's current ages
From the wife's equation:
\[ W - 5 = 9x \]
Thus,
\[ W = 9x + 5 \]

From the husband's equation:
\[ H - 5 = 11x \]
Thus,
\[ H = 11x + 5 \]

### Step 6: Determine the ages in terms of x
Now we have:
- Husband's current age: \( H = 11x + 5 \)
- Wife's current age: \( W = 9x + 5 \)
- Son's current age: \( S = \frac{11x}{5} + 5 \)

### Step 7: Calculate the sum of their ages five years hence
Five years hence, their ages will be:
- Husband: \( H + 5 = 11x + 10 \)
- Wife: \( W + 5 = 9x + 10 \)
- Son: \( S + 5 = \frac{11x}{5} + 10 \)

Now, sum these ages:
\[ \text{Sum} = (11x + 10) + (9x + 10) + \left(\frac{11x}{5} + 10\right) \]
\[ = 20x + 30 + \frac{11x}{5} \]

### Step 8: Combine the terms
To combine \( 20x \) and \( \frac{11x}{5} \), convert \( 20x \) into a fraction:
\[ 20x = \frac{100x}{5} \]
So,
\[ \text{Sum} = \frac{100x}{5} + \frac{11x}{5} + 30 = \frac{111x}{5} + 30 \]

### Step 9: Find a suitable value for x
To find a suitable integer value for \( x \), we can assume a value that makes the ages reasonable. Let's assume \( x = 5 \):
\[ \text{Sum} = \frac{111 \times 5}{5} + 30 = 111 + 30 = 141 \]

### Final Answer
The sum of the ages of the husband, wife, and son five years hence is **141 years**.