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.

Given:

The sum of the reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is $\frac{1}{3}$.

To do:

We have to find his present age.

Solution:

Let the present age of Rehman be $x$ years.

This implies,

Age of Rehman 3 years ago$=x-3$ years

Reciprocal of Rehman's age 3 years ago$=\frac{1}{x-3}$

Age of Rehman 5 years later $=x+5$ years

Reciprocal of Rehman's age 5 years later$=\frac{1}{x+5}$

According to the question,

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

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

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

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

$3(2x+2)=1(x^2+2x-15)$    (On cross multiplication)

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

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

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

Solving for $x$ by factorization method, we get,

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

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

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

$x+3=0$ or $x-7=0$

$x=-3$ or $x=7$

Age cannot be negative. Therefore, the value of $x$ is $7$.

The present age of Rehman is $7$ years.

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Updated on: 10-Oct-2022

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