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 Rehman's current age $=x$
Rehman's age $3$ years ago $=x-3$
Rehman's age $5$ years from now $=x+5$
According to given condition in the question:
$\frac{1}{x-3}+\frac{1}{x+5}=\frac{1}{3}$
$\Rightarrow \frac{x+5+x-3}{(x-3)(x+5)}=\frac{1}{3}$
$\Rightarrow \frac{2x+2}{(x-3)(x+5)}=\frac{1}{3}$
$\Rightarrow 3\times (2x+2)=(x-3)(x+5)$
$\Rightarrow 6x+6=x^2+5x-3x-15$
$\Rightarrow x^2+2x-15-6x-6=0$
$\Rightarrow x^2-4x-21=0$
$\Rightarrow x^2-7x+3x-21=0$
$\Rightarrow x(x-7)+3(x-7)=0$
$\Rightarrow (x-7)(x+3)=0$
$\therefore x=7,x=-3$
Since, age can't be negative.
Therefore, the Rehman's present age is $7$ years.