The sum of the present ages of father and son is 68 years. 8 years ago, the ratio of their ages was `12 : 1`. The ratio of their ages after 4 years hence is:

Options

A. `15 : 4`
B. `14 : 3`
C. `16 : 5`
D. `13 : 2`

Answer

To solve the problem step by step, let's define the variables and set up the equations based on the information provided.

### Step 1: Define the Variables
Let:
- \( x \) = present age of the father
- \( y \) = present age of the son

### Step 2: Set Up the First Equation
According to the problem, the sum of their present ages is 68 years:
\[ x + y = 68 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem states that 8 years ago, the ratio of their ages was \( 12:1 \). Therefore, 8 years ago:
- Father's age = \( x - 8 \)
- Son's age = \( y - 8 \)

The ratio can be expressed as:
\[ \frac{x - 8}{y - 8} = \frac{12}{1} \]

Cross-multiplying gives:
\[ x - 8 = 12(y - 8) \]
Expanding this, we get:
\[ x - 8 = 12y - 96 \]
Rearranging the equation, we have:
\[ x - 12y = -88 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( x + y = 68 \)
2. \( x - 12y = -88 \)

We can solve these equations simultaneously. Let's express \( x \) from Equation 1:
\[ x = 68 - y \]

Now substitute \( x \) in Equation 2:
\[ (68 - y) - 12y = -88 \]
Simplifying this gives:
\[ 68 - y - 12y = -88 \]
\[ 68 - 13y = -88 \]
\[ -13y = -88 - 68 \]
\[ -13y = -156 \]
\[ y = \frac{-156}{-13} = 12 \]

### Step 5: Find the Father's Age
Now, substitute \( y \) back into Equation 1 to find \( x \):
\[ x + 12 = 68 \]
\[ x = 68 - 12 = 56 \]

### Step 6: Calculate Ages After 4 Years
After 4 years, the ages will be:
- Father's age = \( 56 + 4 = 60 \)
- Son's age = \( 12 + 4 = 16 \)

### Step 7: Find the Ratio of Their Ages After 4 Years
The ratio of their ages after 4 years is:
\[ \frac{60}{16} = \frac{15}{4} \]

### Final Answer
The ratio of their ages after 4 years hence is \( 15:4 \).

---