Father and Son Age Problem

Video Explanation

Question

A father is three times as old as his son. After 12 years, his age will be twice that of his son. Find their present ages using linear equations in two variables.

Solution

Step 1: Let the Variables

Let father’s present age = \(x\) years

Let son’s present age = \(y\) years

Step 2: Form the Equations

Father is three times son:

\[ x = 3y \]

\[ x – 3y = 0 \quad (1) \]

After 12 years:

\[ x + 12 = 2(y + 12) \]

\[ x + 12 = 2y + 24 \]

\[ x – 2y = 12 \quad (2) \]

Step 3: Solve by Elimination

Subtract equation (1) from equation (2):

\[ (x – 2y) – (x – 3y) = 12 – 0 \]

\[ x – 2y – x + 3y = 12 \]

\[ y = 12 \]

Step 4: Find x

Substitute \(y = 12\) in equation (1):

\[ x – 3(12) = 0 \]

\[ x = 36 \]

Conclusion

Father’s present age:

\[ \boxed{36 \text{ years}} \]

Son’s present age:

\[ \boxed{12 \text{ years}} \]