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

Meena's age 5 years hence = (x + 5) years

According to the given condition,
$$\begin{gathered} \frac{1}{x-3}+\frac{1}{x+5}=\frac{1}{3} \\ \Rightarrow \frac{x+5+x-3}{\left(x-3\right)\left(x+5\right)}=\frac{1}{3} \\ \Rightarrow \frac{2x+2}{x^2+2x-15}=\frac{1}{3} \\ \Rightarrow x^2+2x-15=6x+6 \end{gathered}$$
$$\begin{gathered} \Rightarrow x^2-4x-21=0 \\ \Rightarrow x^2-7x+3x-21=0 \\ \Rightarrow x\left(x-7\right)+3\left(x-7\right)=0 \\ \Rightarrow \left(x-7\right)\left(x+3\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow x-7=0\mathrm{or}x+3=0 \\ \Rightarrow x=7\mathrm{or}x=-3 \end{gathered}$$
∴ x = 7 (Age cannot be negative)

Hence, the present age of Meena is 7 years.