Father is aged three times more than his son Ramu. After 8 years, he would is `2(1)/(2)` times of Ramu's age. After further 8 years, how many times, would he be of Ramu's age?

Options

A. 2 times
B. `2(1)/(2)` times
C. `2(3)/(4)` times
D. 3 times

Explanation

To solve the problem step by step, let's denote Ramu's current age as \( x \).

### Step 1: Define the Father's Age
According to the problem, the father is aged three times more than his son Ramu. This means:
- Ramu's age = \( x \)
- Father's age = \( 3x + x = 4x \)

### Step 2: Set Up the Age After 8 Years
After 8 years:
- Ramu's age will be \( x + 8 \)
- Father's age will be \( 4x + 8 \)

According to the problem, after 8 years, the father's age will be \( 2\frac{1}{2} \) times Ramu's age. This can be expressed as:
\[ 4x + 8 = \frac{5}{2}(x + 8) \]

### Step 3: Solve the Equation
Now, let's solve the equation:
1. Start by multiplying both sides by 2 to eliminate the fraction:
\[ 2(4x + 8) = 5(x + 8) \]
\[ 8x + 16 = 5x + 40 \]

2. Rearranging the equation gives:
\[ 8x - 5x = 40 - 16 \]
\[ 3x = 24 \]

3. Dividing both sides by 3:
\[ x = 8 \]

### Step 4: Find Ramu's and Father's Current Age
Now that we have Ramu's age:
- Ramu's current age \( x = 8 \)
- Father's current age \( 4x = 4 \times 8 = 32 \)

### Step 5: Calculate Their Ages After Another 8 Years
After another 8 years (16 years from now):
- Ramu's age will be \( 8 + 16 = 24 \)
- Father's age will be \( 32 + 16 = 48 \)

### Step 6: Find the Ratio of Their Ages
Now, we need to find how many times the father's age will be of Ramu's age:
\[ \text{Ratio} = \frac{\text{Father's Age}}{\text{Ramu's Age}} = \frac{48}{24} = 2 \]

### Conclusion
After further 8 years, the father will be **2 times** Ramu's age.

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