There are only five people in the Aman verma's family aman his wife a son and two daughters . The younger daughter's age . The younger daughter's age is (4/5)th of the elder's daughter's age. The age of eldest daughter is 3/8 times that of her father Aman and the age of the son is (1/5)th that of his father Aman.4 years a go the age of her wife was 8 times that of his son and now the sum of the ages of the younger daughter and wife is same as the sum of the ages of Aman and his son. The average age of the family is

Options

A. 22.22 years
B. 25.4 years
C. 21.2 years
D. none of these

Answer

To solve the problem step by step, we will define the ages of each family member in terms of a variable and then set up equations based on the information given.

### Step 1: Define Variables
Let:
- A = Age of Aman (father)
- W = Age of Wife
- S = Age of Son
- ED = Age of Elder Daughter
- YD = Age of Younger Daughter

### Step 2: Set Up Relationships
From the problem statement, we have the following relationships:
1. The younger daughter's age is (4/5)th of the elder daughter's age:
\[ YD = \frac{4}{5} \times ED \]

2. The age of the elder daughter is (3/8) times that of her father:
\[ ED = \frac{3}{8} \times A \]

3. The age of the son is (1/5)th that of his father:
\[ S = \frac{1}{5} \times A \]

4. Four years ago, the age of his wife was 8 times that of his son:
\[ W - 4 = 8 \times (S - 4) \]

5. Now, the sum of the ages of the younger daughter and wife is the same as the sum of the ages of Aman and his son:
\[ YD + W = A + S \]

### Step 3: Substitute Variables
Now, let's substitute \( S \) and \( ED \) in terms of \( A \) into the equations.

From \( S = \frac{1}{5}A \):
- Four years ago, \( S - 4 = \frac{1}{5}A - 4 \)

Substituting into the equation for the wife's age:
\[ W - 4 = 8 \left( \frac{1}{5}A - 4 \right) \]
\[ W - 4 = \frac{8}{5}A - 32 \]
\[ W = \frac{8}{5}A - 28 \]

Next, substitute \( ED \) into \( YD \):
\[ ED = \frac{3}{8}A \implies YD = \frac{4}{5} \times \frac{3}{8}A = \frac{12}{40}A = \frac{3}{10}A \]

### Step 4: Set Up the Age Sum Equation
Now substitute \( YD \) and \( W \) into the equation:
\[ \frac{3}{10}A + \left( \frac{8}{5}A - 28 \right) = A + \frac{1}{5}A \]
Combine like terms:
\[ \frac{3}{10}A + \frac{8}{5}A - 28 = \frac{6}{5}A \]
Convert \( \frac{8}{5}A \) to tenths:
\[ \frac{3}{10}A + \frac{16}{10}A - 28 = \frac{12}{10}A \]
Combine:
\[ \frac{19}{10}A - 28 = \frac{12}{10}A \]
Rearranging gives:
\[ \frac{19}{10}A - \frac{12}{10}A = 28 \]
\[ \frac{7}{10}A = 28 \]
Multiply both sides by \(\frac{10}{7}\):
\[ A = 40 \]

### Step 5: Calculate Other Ages
Now that we have \( A \):
- \( S = \frac{1}{5} \times 40 = 8 \)
- \( ED = \frac{3}{8} \times 40 = 15 \)
- \( YD = \frac{4}{5} \times 15 = 12 \)
- \( W = \frac{8}{5} \times 40 - 28 = 32 \)

### Step 6: Calculate Average Age
Now, we can find the average age of the family:
\[ \text{Total Age} = A + W + S + ED + YD = 40 + 32 + 8 + 15 + 12 = 107 \]
\[ \text{Average Age} = \frac{107}{5} = 21.4 \]

### Final Answer
The average age of the family is **21.4 years**.