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

Step 1: Let us assume, The present age of Rahman is $x$ years.

Three years ago, Rehman’s age was $(x–3)$ years.

Five years after, his age will be $(x+5)$ years.

Given that,

The sum of the reciprocals of Rehman’s ages $3$ years ago and after $5$ years is equal to $\frac{1}{3}$.

$\begin{array}{rcl}\frac{1}{x-3}+\frac{1}{x+5}& =& \frac{1}{3}\\ \frac{(x+5+x-3)}{(x-3)(x+5)}& =& \frac{1}{3}\\ \frac{(2x+2)}{(x-3)(x+5)}& =& \frac{1}{3}\end{array}$

Step 2: Simplify the equation

$\begin{array}{rcl}3(2x+2)& =& (x-3)(x+5)\\ 6x+6& =& x^2+2x–15\\ x^2–4x–21& =& 0\\ x^2–7x+3x–21& =& 0\\ x(x–7)+3(x–7)& =& 0\\ (x–7)(x+3)& =& 0\\ x& =& 7,-3\end{array}$

As we know, age cannot be negative.

Hence, Rahman’s present age is $7$ years.