The ratio of the ages of a father and his son 10 years hence will be `5 : 3`, while 10 years ago, it was `3:1`. The ratio of the age of the son to that of the father today, is

Options

A. `1:2`
B. `1:3`
C. `2:3`
D. `2:5`

Answer

To solve the problem, we need to set up equations based on the information provided about the ages of the father and son at different times. Let's denote the present age of the father as \( F \) and the present age of the son as \( S \).

### Step 1: Set up equations based on the information given

1. **10 years hence (future)**: The ratio of their ages will be \( 5:3 \).
\[ \frac{F + 10}{S + 10} = \frac{5}{3} \]
Cross-multiplying gives:
\[ 3(F + 10) = 5(S + 10) \]
Simplifying this:
\[ 3F + 30 = 5S + 50 \]
Rearranging gives us:
\[ 3F - 5S = 20 \quad \text{(Equation 1)} \]

2. **10 years ago (past)**: The ratio of their ages was \( 3:1 \).
\[ \frac{F - 10}{S - 10} = \frac{3}{1} \]
Cross-multiplying gives:
\[ 1(F - 10) = 3(S - 10) \]
Simplifying this:
\[ F - 10 = 3S - 30 \]
Rearranging gives us:
\[ F - 3S = -20 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations

We now have two equations:
1. \( 3F - 5S = 20 \) (Equation 1)
2. \( F - 3S = -20 \) (Equation 2)

We can solve these equations simultaneously. First, we can express \( F \) from Equation 2:
\[ F = 3S - 20 \]

Now, substitute \( F \) in Equation 1:
\[ 3(3S - 20) - 5S = 20 \]
Expanding this gives:
\[ 9S - 60 - 5S = 20 \]
Combining like terms results in:
\[ 4S - 60 = 20 \]
Adding 60 to both sides:
\[ 4S = 80 \]
Dividing by 4:
\[ S = 20 \]

Now, substitute \( S \) back into the expression for \( F \):
\[ F = 3(20) - 20 = 60 - 20 = 40 \]

### Step 3: Find the ratio of the son's age to the father's age today

Now that we have the present ages:
- Father's age \( F = 40 \)
- Son's age \( S = 20 \)

The ratio of the son's age to the father's age today is:
\[ \frac{S}{F} = \frac{20}{40} = \frac{1}{2} \]

### Final Answer
The ratio of the age of the son to that of the father today is \( \frac{1}{2} \) or \( 1:2 \).

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