Question

The ages of two friends Ani and Biju differ by $3$year.

Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy.

The ages of Cathy and Dharam differ by $30$year.

Find the ages of Ani and Biju.

Answer 1

Let the present age of Ani $=a$years

Let the present age of Biju $=b$years

Given that the age of Ani and Biju differ by $3$ years

Step 1 - When Ani is older than Biju

$\Rightarrow a-b=3-----\left(i\right)$

Also,

Ani's father Dharam is twice as old as Ani

The present age of Dharam $=2a$

And, Biju is twice as old as Cathy

The present age of Biju $=2\times$Present age of Cathy

$\Rightarrow$Present age of Cathy $\mathbf{=}\frac{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{s}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{}\mathbf{a}\mathbf{g}\mathbf{e}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{B}\mathbf{i}\mathbf{j}\mathbf{u}}{\mathbf{2}}$

$\Rightarrow$Present age of Cathy $=\frac{b}{2}$

Also, The age of Dharam and Cathy differ by $30$years

$\Rightarrow 2a-\frac{b}{2}=30------\left(ii\right)$

From equation(i) $a=b+3$substituting the value in equation(ii)

$\begin{array}{rcl}& \Rightarrow & 2\left(b+3\right)-\frac{b}{2}=30\\ & \Rightarrow & \frac{4\left(b+3\right)-b}{2}=30\\ & \Rightarrow & 4b+12-b=60\\ & \Rightarrow & 3b+12=60\\ & \Rightarrow & 3b=60-12\\ & \Rightarrow & 3b=48\\ & \Rightarrow & b=\frac{48}{3}\\ & \Rightarrow & b=16years\end{array}$

$$\begin{gathered} \because a=b+3 \\ \Rightarrow a=16+3 \\ \Rightarrow a=19years \end{gathered}$$

Hence, the age of Biju is $16$ years

the age of Ani is $19$ years

Step-2-When Biju is older than Ani

$$\begin{gathered} \Rightarrow b-a=3 \\ \Rightarrow b=a+3---------\left(iii\right) \end{gathered}$$

substituting the value of $b$ from eq(iii) in eq(ii)

$\begin{array}{rcl}& \Rightarrow & 2a-\frac{a+3}{2}=30\\ & \Rightarrow & \frac{4a-(a+3)}{2}=30\\ & \Rightarrow & \frac{4a-a-3}{2}=30\\ & \Rightarrow & \frac{3a-3}{2}=30\\ & \Rightarrow & 3a-3=60\\ & \Rightarrow & 3a=63\\ & \Rightarrow & a=\frac{63}{3}\\ & \Rightarrow & a=21years\\ & & \end{array}$

$$\begin{gathered} \because b=3+a \\ \Rightarrow b=21+3 \\ \Rightarrow b=24years \end{gathered}$$

Hence, The age of Ani is $21$ yrs

The age of Biju is $24$yrs.