The present age of a man is twice that of his son. Eight years hence, their ages will be in the ratio `7:4`. Find their present ages.

Options

A. `56` years and `58` years
B. `55` years and `24` years
C. `48` years and `34` years
D. `48` years and `24` years

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let the present age of the son be \( x \). According to the problem, the present age of the man (father) is twice that of his son, so we can express the father's age as \( 2x \).

### Step 2: Set up the equation based on future ages
The problem states that eight years hence, their ages will be in the ratio \( 7:4 \). After 8 years, the son's age will be \( x + 8 \) and the father's age will be \( 2x + 8 \). We can set up the following equation based on the given ratio:

\[ \frac{2x + 8}{x + 8} = \frac{7}{4} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ 4(2x + 8) = 7(x + 8) \]

### Step 4: Expand both sides of the equation
Expanding both sides, we have:

\[ 8x + 32 = 7x + 56 \]

### Step 5: Rearrange the equation to isolate \( x \)
Now, we will rearrange the equation to isolate \( x \):

\[ 8x - 7x = 56 - 32 \]

This simplifies to:

\[ x = 24 \]

### Step 6: Calculate the present ages
Now that we have the value of \( x \), we can find the present ages:
- The son's age is \( x = 24 \) years.
- The father's age is \( 2x = 2 \times 24 = 48 \) years.

### Conclusion
Thus, the present ages are:
- Son's age: 24 years
- Father's age: 48 years

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