The sum of the present ages of a father and his son is 78 years. After six years, the ratio of their ages becomes 6 : 3. What is the present age (in years) of the father?

Options

A. 45
B. 54
C. 55
D. 50

Answer

To solve the problem, we need to find the present age of the father given the conditions in the question. Let's break it down step by step.

### Step 1: Define Variables
Let the present age of the father be \( F \) years and the present age of the son be \( S \) years.

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of their present ages is 78 years:
\[ F + S = 78 \quad \text{(Equation 1)} \]

2. After 6 years, the ratio of their ages will be 6:3 (which simplifies to 2:1):
\[ \frac{F + 6}{S + 6} = \frac{6}{3} \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
From Equation 2, we can cross-multiply to eliminate the fraction:
\[ 3(F + 6) = 6(S + 6) \]
Expanding both sides gives:
\[ 3F + 18 = 6S + 36 \]
Rearranging this equation, we get:
\[ 3F - 6S = 36 - 18 \]
\[ 3F - 6S = 18 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( F + S = 78 \) (Equation 1)
2. \( 3F - 6S = 18 \) (Equation 3)

From Equation 1, we can express \( S \) in terms of \( F \):
\[ S = 78 - F \]

Substituting \( S \) into Equation 3:
\[ 3F - 6(78 - F) = 18 \]
Expanding this gives:
\[ 3F - 468 + 6F = 18 \]
Combining like terms:
\[ 9F - 468 = 18 \]
Adding 468 to both sides:
\[ 9F = 486 \]
Dividing by 9:
\[ F = 54 \]

### Step 5: Find the Present Age of the Son
Now, substituting \( F \) back into Equation 1 to find \( S \):
\[ 54 + S = 78 \]
\[ S = 78 - 54 = 24 \]

### Conclusion
The present age of the father is \( \boxed{54} \) years.