Solving Father and Son Age Problem

Let's break down this age problem step by step to find the present age of the son.

We need to set up equations based on the information given in the problem. Let:

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

Formulating Equations from the Problem

The problem gives us two conditions related to their ages at different points in time:

Condition 1: Five years ago

Five years ago, the father's age was \(F - 5\) and the son's age was \(S - 5\). The problem states that the father's age was 5 times that of his son's age at that time.

So, we can write the equation:

\(F - 5 = 5 \times (S - 5)\)

\(F - 5 = 5S - 25\)

Rearranging this equation to isolate F:

\(F = 5S - 25 + 5\)

\(F = 5S - 20 \quad \text{(Equation 1)}\)

Condition 2: After 3 years

After 3 years from now, the father's age will be \(F + 3\) and the son's age will be \(S + 3\). The problem states that the father will be 3 times as old as his son at that time.

So, we can write the equation:

\(F + 3 = 3 \times (S + 3)\)

\(F + 3 = 3S + 9\)

Rearranging this equation to isolate F:

\(F = 3S + 9 - 3\)

\(F = 3S + 6 \quad \text{(Equation 2)}\)

Solving the System of Equations

Now we have two equations for \(F\). We can set them equal to each other to solve for \(S\):

\(5S - 20 = 3S + 6\)

Subtract \(3S\) from both sides:

\(5S - 3S - 20 = 6\)

\(2S - 20 = 6\)

Add 20 to both sides:

\(2S = 6 + 20\)

\(2S = 26\)

Divide by 2 to find \(S\):

\(S = \frac{26}{2}\)

\(S = 13\)

So, the present age of the son is 13 years.

We can also find the father's present age using either Equation 1 or Equation 2. Using Equation 2:

\(F = 3S + 6\)

\(F = 3(13) + 6\)

\(F = 39 + 6\)

\(F = 45\)

The father's present age is 45 years.

Verification

Let's check if these ages satisfy the original conditions:

• Five years ago: Son's age was \(13 - 5 = 8\) years. Father's age was \(45 - 5 = 40\) years. Is 40 equal to 5 times 8? \(5 \times 8 = 40\). Yes, the first condition is satisfied.
• After 3 years: Son's age will be \(13 + 3 = 16\) years. Father's age will be \(45 + 3 = 48\) years. Is 48 equal to 3 times 16? \(3 \times 16 = 48\). Yes, the second condition is satisfied.

The calculated ages are correct.

Final Answer

The present age of the son is 13 years.