Average age of Sohan, Mohini and their twin's daughter is 22.5 years and average age of Mohini, and her twin's daughter is `52/3` years. If average age of Sohan and Mohini is 37 years, then find average age of Sohan and a his twin's daughter?

Options

A. A)24 years
B. B)16 years
C. C)19 years
D. D)18 years

Answer

To solve the problem step by step, we will use the information provided about the average ages of Sohan, Mohini, and their twin daughters.

### Step 1: Calculate the sum of ages of Sohan, Mohini, and their twin daughters.
Given that the average age of Sohan, Mohini, and their twin daughters is 22.5 years, we can find the sum of their ages.

\[ \text{Average age} = \frac{\text{Sum of ages}}{\text{Number of people}} \]

Let the ages of Sohan, Mohini, and the twin daughters be \( S \), \( M \), \( D_1 \), and \( D_2 \) respectively.

\[ \text{Average age} = 22.5 = \frac{S + M + D_1 + D_2}{4} \]

Multiplying both sides by 4 gives:

\[ S + M + D_1 + D_2 = 22.5 \times 4 = 90 \quad \text{(1)} \]

### Step 2: Calculate the sum of ages of Mohini and her twin daughters.
Next, we know that the average age of Mohini and her twin daughters is \( \frac{52}{3} \).

\[ \text{Average age} = \frac{M + D_1 + D_2}{3} = \frac{52}{3} \]

Multiplying both sides by 3 gives:

\[ M + D_1 + D_2 = 52 \quad \text{(2)} \]

### Step 3: Calculate the age of Sohan.
Now, we can find Sohan's age by subtracting equation (2) from equation (1):

\[ S + M + D_1 + D_2 - (M + D_1 + D_2) = 90 - 52 \]

This simplifies to:

\[ S = 90 - 52 = 38 \quad \text{(3)} \]

### Step 4: Calculate the average age of Sohan and Mohini.
We are given that the average age of Sohan and Mohini is 37 years.

\[ \text{Average age} = \frac{S + M}{2} = 37 \]

Multiplying both sides by 2 gives:

\[ S + M = 74 \quad \text{(4)} \]

### Step 5: Calculate Mohini's age.
Now, we can substitute \( S \) from equation (3) into equation (4):

\[ 38 + M = 74 \]

Solving for \( M \):

\[ M = 74 - 38 = 36 \quad \text{(5)} \]

### Step 6: Calculate the sum of ages of the twin daughters.
Using equation (2), we can find the sum of the ages of the twin daughters:

\[ M + D_1 + D_2 = 52 \]

Substituting \( M \) from equation (5):

\[ 36 + D_1 + D_2 = 52 \]

Solving for \( D_1 + D_2 \):

\[ D_1 + D_2 = 52 - 36 = 16 \quad \text{(6)} \]

### Step 7: Calculate the average age of Sohan and his twin daughters.
Finally, we need to find the average age of Sohan and his twin daughters:

\[ \text{Average age} = \frac{S + D_1 + D_2}{3} \]

Substituting \( S \) from equation (3) and \( D_1 + D_2 \) from equation (6):

\[ \text{Average age} = \frac{38 + 16}{3} = \frac{54}{3} = 18 \]

Thus, the average age of Sohan and his twin daughters is **18 years**.