Sonam's age after eight years will be half of her father's age. Eight years ago the ratio of their ages was 1:3. Find the present age of Sonam's father.
A. 48
B. 56
C. 36
D. 65

Options

A. B
B. D
C. C
D. A

Answer

To solve the problem, we will set up equations based on the information provided about Sonam's age and her father's age.

### Step 1: Define the variables
Let Sonam's current age be \( x \) and her father's current age be \( y \).

### Step 2: Set up the first equation
According to the problem, eight years ago, the ratio of Sonam's age to her father's age was 1:3. Therefore, we can express this as:
\[ \frac{x - 8}{y - 8} = \frac{1}{3} \]
Cross-multiplying gives us:
\[ 3(x - 8) = 1(y - 8) \]
Expanding this, we get:
\[ 3x - 24 = y - 8 \]
Rearranging this equation gives us:
\[ 3x - y = 16 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation
The problem also states that Sonam's age after eight years will be half of her father's age at that time. This can be expressed as:
\[ x + 8 = \frac{1}{2}(y + 8) \]
Multiplying both sides by 2 to eliminate the fraction gives us:
\[ 2(x + 8) = y + 8 \]
Expanding this, we have:
\[ 2x + 16 = y + 8 \]
Rearranging this equation gives us:
\[ 2x - y = -8 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have a system of two equations:
1. \( 3x - y = 16 \) (Equation 1)
2. \( 2x - y = -8 \) (Equation 2)

We can subtract Equation 2 from Equation 1:
\[ (3x - y) - (2x - y) = 16 - (-8) \]
This simplifies to:
\[ 3x - 2x = 16 + 8 \]
\[ x = 24 \]

### Step 5: Find the father's age
Now that we have Sonam's age \( x = 24 \), we can substitute this value back into either equation to find \( y \). We'll use Equation 2:
\[ 2(24) - y = -8 \]
\[ 48 - y = -8 \]
Rearranging gives:
\[ y = 48 + 8 = 56 \]

### Conclusion
Thus, the present age of Sonam's father is \( y = 56 \).

### Final Answer
The present age of Sonam's father is **56** (Option B).

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