**Question:**

The ages of two friends Ani and Biju differ by 3 years. Ani's father Dharma is twice as old as Ani and Biju as 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:**

Let the present ages of Ani, Biju, Dharam* *and Cathy be *x, y, z *and *t *years respectively.

The ages of Ani and Biju differ by 3 years. Thus, we have

$x-y=\pm 3$

$\Rightarrow x=y \pm 3$

Dharam is twice as old as Ani. Thus, we have $z=2 x$

Biju is twice as old as Cathy. Thus, we have $y=2 t$

The ages of Cathy and Dharam differ by 30 years. Clearly, Dharam is older than Cathy. Thus, we have $z-t=30$

So, we have two systems of simultaneous equations

(i) $x=y+3$,

$z=2 x$,

$y=2 t$,

$z-t=30$

(ii) $x=y-3$,

$z=2 x$

$y=2 t$

$z-t=30$

Here *x, y, z* and *t* are unknowns. We have to find the value of *x *and *y*.

(i) By using the third equation, the first equation becomes $x=2 t+3$

From the fourth equation, we have

$t=z-30$

Hence, we have

$x=2(z-30)+3$

$=2 z-60+3$

$=2 z-57$

Using the second equation, we have

$x=2 \times 2 x-57$

$\Rightarrow x=4 x-57$

$\Rightarrow 4 x-x=57$

$\Rightarrow 3 x=57$

$\Rightarrow x=\frac{57}{3}$

$\Rightarrow x=19$

From the first equation, we have

$x=y+3$

$\Rightarrow y=x-3$

$\Rightarrow y=19-3$

$\Rightarrow y=16$

Hence, the age of Ani is 19 years and the age of Biju is 16 years.

(ii) By using the third equation, the first equation becomes $x=2 t-3$

From the fourth equation, we have

$t=z-30$

Hence, we have

$x=2(z-30)-3$

$=2 z-60-3$

$=2 z-63$

Using the second equation, we have

$x=2 \times 2 x-63$

$\Rightarrow x=4 x-63$

$\Rightarrow 4 x-x=63$

$\Rightarrow 3 x=63$

$\Rightarrow x=\frac{63}{3}$

$\Rightarrow x=21$

From the first equation, we have

$x=y-3$

$\Rightarrow y=x+3$

$\Rightarrow y=21+3$

$\Rightarrow y=24$

Hence, the age of Ani is 21 years and the age of Biju is 24 years.

Note that there are two possibilities.

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