The sum of the reciprocals of Rehman's ages

Solution:

Let the present age of Rehman be $x$ years

Then, 8 years later, age of her $=(x+5)$ years

Five years ago, her age $=(x-3)$ years

Then according to question,

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

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

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

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

$x^{2}+2 x-15-6 x-6=0$

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

$x^{2}-7 x+3 x-21=0$

$x(x-7)+3(x-7)=0$

$(x-7)(x+3)=0$

So, either 

$(x-7)=0$

$x=7$

Or

$(x+3)=0$

$x=-3$

But the age never be negative

 

Hence, the present age of Rehman be $=7$ years