Solving Father Son Age Ratio Problem

This problem involves setting up equations based on the given information about the ages of a father and his son at different points in time.

Setting up the Equations

Let's use variables to represent the ages:

• Let the present age of the son be \(S\) years.
• Let the present age of the father be \(F\) years.

According to the first statement, the father is four times as old as his son today. We can write this as an equation:

\(F = 4S\)

Next, we consider the ages after 24 years:

• Son's age after 24 years will be \(S + 24\) years.
• Father's age after 24 years will be \(F + 24\) years.

The problem states that after 24 years, the ratio of the ages of the son and father will be 1 : 3. This gives us another equation:

\(\frac{S + 24}{F + 24} = \frac{1}{3}\)

Solving the Equations

Now we have a system of two equations:

1. \(F = 4S\)
2. \(\frac{S + 24}{F + 24} = \frac{1}{3}\)

We can substitute the first equation into the second equation to eliminate \(F\). Replace \(F\) with \(4S\) in the second equation:

\(\frac{S + 24}{4S + 24} = \frac{1}{3}\)

Now, we solve this equation for \(S\). We can cross-multiply:

\(3 \times (S + 24) = 1 \times (4S + 24)\)

Distribute the numbers:

\(3S + 72 = 4S + 24\)

To isolate \(S\), subtract \(3S\) from both sides of the equation:

\(72 = 4S - 3S + 24\)

\(72 = S + 24\)

Now, subtract 24 from both sides to find the value of \(S\):

\(72 - 24 = S\)

\(S = 48\)

So, the present age of the son is 48 years.

Verification

Let's check if this answer fits the conditions:

• Son's present age = 48 years.
• Father's present age = \(4 \times 48 = 192\) years. (Father is four times as old as the son)
• After 24 years, son's age = \(48 + 24 = 72\) years.
• After 24 years, father's age = \(192 + 24 = 216\) years.
• Ratio of son's age to father's age after 24 years = \(\frac{72}{216}\).

Let's simplify the ratio \(\frac{72}{216}\):

\(\frac{72}{216} = \frac{72 \div 72}{216 \div 72} = \frac{1}{3}\)

The ratio is 1:3, which matches the condition given in the problem. Therefore, the present age of the son is 48 years.

The final answer is \(48\).