Question

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

Answer 1

Let Rehman's present age $=x$ years.

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

$\Rightarrow$ Reciprocal of $x-3$ is $\frac{1}{x-3}$---(1)

$5$ yrs hence, his age $=x+5$ years

$\Rightarrow$ Reciprocal of $x+5$ is $\frac{1}{x+5}$ ---(2)

It is given that, Sum of the reciprocals according to given question is $\frac{1}{3}$.

$\Rightarrow \frac{1}{(x-3)}+\frac{1}{(x+5)}=\frac{1}{3}\Rightarrow \frac{(x+5+x-3)}{(x+5)(x-3)}=\frac{1}{3}\Rightarrow \frac{2x+2}{x^2+2x-15}=\frac{1}{3}$

$\Rightarrow x^2+2x-15=6x+6\Rightarrow x^2-4x-21=0\Rightarrow x^2-7x+3x-21=0\Rightarrow x(x-7)+3(x-7)=0\Rightarrow (x+3)(x-7)=0$

$\Rightarrow x+3=0,x-7=0$

$\Rightarrow x=-3,7$

Since, age can't be negative.

Rehman's present age is $7$ years.