Question

The sum of the reciprocals of Rehman's ages (in years) $3$ years ago and $5$ years from now is $\frac{1}{3}$. Find his present age.

Answer 1

Let the present age of Rahman be $x$.

Three years ago, Rehman's age $=x-3$

Five years hence, Rehman's age $=x+5$

It is given that, sum of the reciprocals of Rehman's ages $3$ years ago and $5$ years from now is $\frac{1}{3}$.

$\Rightarrow \frac{1}{x-3}+\frac{1}{x+5}=\frac{1}{3}$
$\Rightarrow \frac{x+5+x-3}{(x-3)(x+5)}=\frac{1}{3}$
$\Rightarrow \frac{2x+2}{(x-3)(x+5)}=\frac{1}{3}$
$\Rightarrow 3(2x+2)=(x-3)(x+5)$
$\Rightarrow 6x+6=x^2+2x-15$
$\Rightarrow x^2+2x-6x-15-6=0$
$\Rightarrow x^2-4x-21=0$
$\Rightarrow x^2-7x+3x-21=0$
$\Rightarrow (x-7)(x+3)=0$
$\Rightarrow x-7=0\text{or}x+3=0$
$\Rightarrow x=7\text{or}x=-3$

So, $x$ can be $7$ or $-3$.

Since, the age of a person cannot be negative.

Hence, Rehman's present age is $7$ years.