The ratio of the father’s and son’s age is 7 : 4. The product of their ages is 1008. The ratio of their ages after 6 years hence will be-

Options

A. 5:3
B. 8:5
C. 7:4
D. 5:8

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define the Variables
Let the present age of the father be \( 7x \) and the present age of the son be \( 4x \), based on the given ratio of their ages.

### Step 2: Set Up the Equation
According to the problem, the product of their ages is given as 1008. Therefore, we can set up the equation:
\[ (7x) \times (4x) = 1008 \]

### Step 3: Simplify the Equation
Now, simplify the left side of the equation:
\[ 28x^2 = 1008 \]

### Step 4: Solve for \( x^2 \)
Next, divide both sides by 28 to isolate \( x^2 \):
\[ x^2 = \frac{1008}{28} \]
Calculating the right side:
\[ x^2 = 36 \]

### Step 5: Solve for \( x \)
Now, take the square root of both sides to find \( x \):
\[ x = 6 \]

### Step 6: Find the Present Ages
Now that we have \( x \), we can find the present ages of the father and the son:
- Father's age: \( 7x = 7 \times 6 = 42 \)
- Son's age: \( 4x = 4 \times 6 = 24 \)

### Step 7: Calculate Their Ages After 6 Years
Now, we need to find their ages after 6 years:
- Father's age after 6 years: \( 42 + 6 = 48 \)
- Son's age after 6 years: \( 24 + 6 = 30 \)

### Step 8: Find the Ratio of Their Ages After 6 Years
Now, we can find the ratio of their ages after 6 years:
\[ \text{Ratio} = \frac{48}{30} = \frac{8}{5} \]

### Final Answer
The ratio of the father's and son's ages after 6 years will be \( 8:5 \).

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