Finding the Ages of Ani and Biju

Video Explanation

Question

The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani. Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.

Solution

Step 1: Let the Variables

Let age of Ani = \(x\) years

Let age of Biju = \(y\) years

Step 2: Form the First Equation

Ages differ by 3 years:

\[ x – y = 3 \quad (1) \]

Step 3: Express Other Ages

Father Dharam:

\[ \text{Dharam} = 2x \]

Cathy:

\[ y = 2 \times \text{Cathy} \]

\[ \text{Cathy} = \frac{y}{2} \]

Given difference between Dharam and Cathy:

\[ 2x – \frac{y}{2} = 30 \quad (2) \]

Step 4: Simplify Equation (2)

Multiply by 2:

\[ 4x – y = 60 \quad (3) \]

Step 5: Solve the Equations

We have:

\[ x – y = 3 \]

\[ 4x – y = 60 \]

Subtract first from second:

\[ (4x – y) – (x – y) = 60 – 3 \]

\[ 3x = 57 \]

\[ x = 19 \]

Step 6: Find Age of Biju

\[ 19 – y = 3 \]

\[ y = 16 \]

Conclusion

Age of Ani:

\[ \boxed{19 \text{ years}} \]

Age of Biju:

\[ \boxed{16 \text{ years}} \]