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

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.