The ratio of the present ages of a son and his father is `1:5` and that of his mother and father is `4:5` After 2 years the ratio of the age of the son to that of his mother becomes `3:10`. What is the present age of the father ?

Options

A. 30 years
B. 28 years
C. 37 years
D. None of these

Explanation

To find the present age of the father based on the given ratios of ages, we can follow these steps:

### Step 1: Define Variables
Let:
- The present age of the son be \( S \).
- The present age of the father be \( F \).
- The present age of the mother be \( M \).

### Step 2: Set Up the Ratios
From the problem, we have the following ratios:
1. The ratio of the present ages of the son and his father is \( 1:5 \).
\[ \frac{S}{F} = \frac{1}{5} \implies F = 5S \]

2. The ratio of the present ages of the mother and father is \( 4:5 \).
\[ \frac{M}{F} = \frac{4}{5} \implies M = \frac{4}{5}F \]

### Step 3: Set Up the Future Age Ratio
After 2 years, the ratio of the age of the son to that of his mother becomes \( 3:10 \).
This can be expressed as:
\[ \frac{S + 2}{M + 2} = \frac{3}{10} \]

### Step 4: Substitute for M
Using the expression for \( M \) from Step 2:
\[ \frac{S + 2}{\frac{4}{5}F + 2} = \frac{3}{10} \]

### Step 5: Cross Multiply
Cross multiplying gives:
\[ 10(S + 2) = 3\left(\frac{4}{5}F + 2\right) \]

### Step 6: Substitute for F
Substituting \( F = 5S \) into the equation:
\[ 10(S + 2) = 3\left(\frac{4}{5}(5S) + 2\right) \]
This simplifies to:
\[ 10(S + 2) = 3(4S + 2) \]

### Step 7: Expand Both Sides
Expanding both sides:
\[ 10S + 20 = 12S + 6 \]

### Step 8: Rearrange the Equation
Rearranging gives:
\[ 10S + 20 - 6 = 12S \]
\[ 14 = 12S - 10S \]
\[ 2S = 14 \]

### Step 9: Solve for S
Dividing both sides by 2:
\[ S = 7 \]

### Step 10: Find the Father's Age
Using \( F = 5S \):
\[ F = 5 \times 7 = 35 \]

### Conclusion
The present age of the father is \( 35 \) years.

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