Ram's age is (3x+2y) years and his only son's age is 'x' years while his only daughter's age is 'y' years. Ram's son is 3 years elder than her sister. Find average age of family of Ram, if Ram's wife age who is 5 years younger than her husband is 29 years in years).

Options

A. `14.5`
B. `19.0`
C. `16.5`
D. `22.5`

Answer

To solve the problem step by step, we will first define the ages of Ram, his son, daughter, and wife based on the information given in the question.

### Step 1: Define the ages
- Let Ram's age be \( A_R = 3x + 2y \) years.
- Let his son's age be \( A_S = x \) years.
- Let his daughter's age be \( A_D = y \) years.
- Ram's wife is 5 years younger than Ram, so her age \( A_W = A_R - 5 \) years.

### Step 2: Establish the relationship between son's and daughter's age
According to the problem, Ram's son is 3 years older than his daughter:
\[ A_S = A_D + 3 \]
This translates to:
\[ x = y + 3 \]

### Step 3: Calculate Ram's wife's age
We know that Ram's wife is 29 years old:
\[ A_W = 29 \]
Since she is 5 years younger than Ram, we can express Ram's age as:
\[ A_R = A_W + 5 = 29 + 5 = 34 \]

### Step 4: Set up the equation for Ram's age
Now, we can set up the equation based on Ram's age:
\[ 34 = 3x + 2y \]

### Step 5: Substitute the value of x
From the relationship \( x = y + 3 \), we can substitute \( x \) in the equation:
\[ 34 = 3(y + 3) + 2y \]
Expanding this gives:
\[ 34 = 3y + 9 + 2y \]
Combining like terms:
\[ 34 = 5y + 9 \]

### Step 6: Solve for y
Now, we can isolate \( y \):
\[ 34 - 9 = 5y \]
\[ 25 = 5y \]
\[ y = 5 \]

### Step 7: Find the value of x
Now that we have \( y \), we can find \( x \):
\[ x = y + 3 = 5 + 3 = 8 \]

### Step 8: Calculate the ages of all family members
Now we can find the ages of all family members:
- Ram's age: \( A_R = 34 \) years
- Son's age: \( A_S = x = 8 \) years
- Daughter's age: \( A_D = y = 5 \) years
- Wife's age: \( A_W = 29 \) years

### Step 9: Calculate the average age of the family
The average age of the family can be calculated as follows:
\[ \text{Average Age} = \frac{A_R + A_S + A_D + A_W}{4} = \frac{34 + 8 + 5 + 29}{4} \]
Calculating the sum:
\[ 34 + 8 + 5 + 29 = 76 \]
Now, divide by 4:
\[ \text{Average Age} = \frac{76}{4} = 19 \]

### Final Answer
The average age of Ram's family is **19 years**.

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