The sum of the present ages of a father and son is 39 years. Four years hence, the son's age will be `(1)/(4)` that of the father's age. Find the ratio of the ages of the father and son?

Options

A. `73:9`
B. `56:9`
C. `56:11`
D. `65:12`

Answer

To solve the problem step by step, we will define the present ages of the father and son, set up equations based on the information given, and then solve for their ages and the ratio.

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

### Step 2: Set Up the First Equation
According to the problem, the sum of their present ages is 39 years. Therefore, we can write the first equation as:
\[ S + F = 39 \quad \text{(1)} \]

### Step 3: Set Up the Second Equation
The problem states that four years from now, the son's age will be \( \frac{1}{4} \) of the father's age. Four years hence, the son's age will be \( S + 4 \) and the father's age will be \( F + 4 \). Thus, we can write the second equation as:
\[ S + 4 = \frac{1}{4}(F + 4) \quad \text{(2)} \]

### Step 4: Simplify the Second Equation
To eliminate the fraction, we can multiply both sides of equation (2) by 4:
\[ 4(S + 4) = F + 4 \]
Expanding this gives:
\[ 4S + 16 = F + 4 \]
Rearranging it leads to:
\[ F - 4S = 12 \quad \text{(3)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( S + F = 39 \) (equation 1)
2. \( F - 4S = 12 \) (equation 3)

We can solve these equations simultaneously. From equation (1), we can express \( F \) in terms of \( S \):
\[ F = 39 - S \]
Substituting this expression for \( F \) into equation (3):
\[ (39 - S) - 4S = 12 \]
This simplifies to:
\[ 39 - 5S = 12 \]
Rearranging gives:
\[ 5S = 39 - 12 \]
\[ 5S = 27 \]
Dividing both sides by 5:
\[ S = \frac{27}{5} \quad \text{(4)} \]

### Step 6: Find the Father's Age
Now substitute \( S \) back into equation (1) to find \( F \):
\[ F = 39 - S = 39 - \frac{27}{5} \]
To perform this subtraction, convert 39 to a fraction:
\[ F = \frac{195}{5} - \frac{27}{5} = \frac{195 - 27}{5} = \frac{168}{5} \quad \text{(5)} \]

### Step 7: Find the Ratio of Their Ages
Now we have both ages:
- Son's age \( S = \frac{27}{5} \)
- Father's age \( F = \frac{168}{5} \)

The ratio of the father's age to the son's age is:
\[ \text{Ratio} = \frac{F}{S} = \frac{\frac{168}{5}}{\frac{27}{5}} = \frac{168}{27} \]
To simplify this ratio, we can divide both the numerator and the denominator by 27:
\[ \frac{168 \div 27}{27 \div 27} = \frac{56}{9} \]

### Final Answer
The ratio of the ages of the father and son is:
\[ \text{Ratio} = 56 : 9 \]