Question

Question 4
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 Rehman be x years.
Three years ago, his age was (x - 3) years.
Five years hence, his age will be (x + 5) years.
It is given that the sum of the reciprocals of Rehman's ages 3 years ago and 5 years from now is $\frac{1}{3}$.
$\therefore \frac{1}{x-3}+\frac{1}{x+5}=\frac{1}{3}$
$\frac{x+5+x-3}{(x-3)(x+5)}=\frac{1}{3}$
$\frac{2x+2}{(x-3)(x+5)}=\frac{1}{3}$
$\Rightarrow 3(2x+2)=(x-3)(x+5)$
$\Rightarrow 6x+6=x^2+2x-15$
$\Rightarrow x^2-4x-21=0$
$\Rightarrow x^2-7x+3x-21=0$
$\Rightarrow x(x-7)+3(x-7)=0$
$\Rightarrow (x-7)(x+3)=0$
$\Rightarrow x=7,-3$
However, age cannot be negative.
Therefore, Rehman's present age is 7 years.