Question

The sum of the reciprocal of Rehman's age (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 Rehman be $x$ years
So, $3$ years ago, Rehman's age $=(x-3)$ years
And $5$ years from now, Rehman's age $=(x+5)$years
Now according to question, we have
$\frac{1}{x-3}+\frac{1}{x+5}=\frac{1}{3}$
$\Rightarrow$ $\frac{x+5+x-3}{(x-3)(x+5)}=\frac{1}{3}\Rightarrow$ $\frac{2x+2}{(x-3)(x+5)}=\frac{1}{3}$
$\Rightarrow 6x+6=(x-3)(x+5)\Rightarrow 6x+6=x^2+5x-3x-15$
$\Rightarrow x^2+2x-15-6x-6=0\Rightarrow x^2-4x-21=0$
$\Rightarrow$ $x^2-7x+3x-21=0$
$\Rightarrow$ $x(x-7)+3(x-7)=0$
$\Rightarrow$ $(x-7)(x+3)=0$
$\Rightarrow$ $x=7$ or $x=-3$
But $x\ne -3$ (age cannot be negative)
Therefore, present age of Rehman $=7$ years