Question

The sum of the reciprocals of Meena's ages (in years) $3$ years ago and $5$ years hence is $\frac{1}{3}$. Find her present age.

Answer 1

Let the present age of Meena be $x$ years.
Meena’s age $3$ years ago $=(x–3)$ years

$\Rightarrow$ Reciprocal of $(x–3)=\frac{1}{x-3}$ ---(1)
Meena’s age $5$ years hence $=(x+5)$ years

$\Rightarrow$ Reciprocal of $(x+5)=\frac{1}{x+5}$ ---(2)
It is given that, sum of the reciprocals of Meena's ages (in years) $3$ years ago and $5$ years hence is $\frac{1}{3}$

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

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

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

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

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

$\Rightarrow x^2+2x-15-6x-6=0$

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

$\Rightarrow x^2-7x+3x-21=0$

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

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

$\Rightarrow x-7=0$ (OR) $x+3=0$

$\Rightarrow x=7$ (OR) $x=-3$

Since, age cannot be negative.

So, $x=7$
Hence, the present age of Meena is $7$ years.