# The sum of the reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is 1/3.

Question.

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

Solution:

Let the present age of Rehman be x years.

We are given that $\frac{\mathbf{1}}{\mathbf{x}-\mathbf{3}}+\frac{\mathbf{1}}{\mathbf{x}+\mathbf{5}}=\frac{\mathbf{1}}{\mathbf{3}}$

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

$\Rightarrow 3(2 x+2)=(x-3)(x+5)$

$\Rightarrow 6 x+6=x^{2}+2 x-15$

$\Rightarrow x^{2}-4 x-21=0$

$a=1, b=-4, c=-21$

$D=(-4)^{2}-4(1)(-21)=16+84=100$

$\Rightarrow \sqrt{\mathbf{D}}=\sqrt{\mathbf{1 0 0}}=\mathbf{1 0}$

Then $x=\frac{-b \pm \sqrt{\mathbf{D}}}{\mathbf{2 a}}=\frac{\mathbf{4} \pm \mathbf{1 0}}{\mathbf{2}}$, i.e., $x=7,-3$

We reiect $x=-3$

$(\because x$ cannot be negative)

$\Rightarrow x=7$

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