A father's age was 5 times his son's age 5 years ago and will be 3 times son's age after 2 years the ratio of their present ages is

Options

A. 5 : 2
B. 5 : 3
C. 10 : 3
D. 11 : 5

Answer

To solve the problem, we need to find the present ages of the father and son based on the information given. Let's denote:

- Father's present age = \( x \)
- Son's present age = \( y \)

### Step 1: Set up the equations based on the problem statement.

1. **Five years ago:** The father's age was 5 times his son's age.
- Equation: \( x - 5 = 5(y - 5) \)

2. **Two years from now:** The father's age will be 3 times his son's age.
- Equation: \( x + 2 = 3(y + 2) \)

### Step 2: Simplify the equations.

1. From the first equation:
\[ x - 5 = 5(y - 5) \]
Expanding this gives:
\[ x - 5 = 5y - 25 \]
Rearranging leads to:
\[ x = 5y - 20 \quad \text{(Equation 1)} \]

2. From the second equation:
\[ x + 2 = 3(y + 2) \]
Expanding this gives:
\[ x + 2 = 3y + 6 \]
Rearranging leads to:
\[ x = 3y + 4 \quad \text{(Equation 2)} \]

### Step 3: Set the two equations equal to each other.

From Equation 1 and Equation 2, we have:
\[ 5y - 20 = 3y + 4 \]

### Step 4: Solve for \( y \).

1. Rearranging gives:
\[ 5y - 3y = 4 + 20 \]
Simplifying this results in:
\[ 2y = 24 \]
Therefore:
\[ y = 12 \]

### Step 5: Substitute \( y \) back to find \( x \).

Using \( y = 12 \) in Equation 1:
\[ x = 5(12) - 20 \]
Calculating this gives:
\[ x = 60 - 20 = 40 \]

### Step 6: Find the ratio of their present ages.

The present ages are:
- Father's age \( x = 40 \)
- Son's age \( y = 12 \)

Thus, the ratio of their present ages is:
\[ \text{Ratio} = \frac{x}{y} = \frac{40}{12} = \frac{10}{3} \]

### Final Answer:
The ratio of the father's age to the son's age is \( \frac{10}{3} \).

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