#### Question

The sum of the reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is 1/3. Find his present age.

#### Solution

Let the present age of Rehman be x years.

Three years ago, his age was (x - 3) years.

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

It is given that the sum of the reciprocals of Rehman's ages 3 years ago and 5 years from now is 1/3.

`∴ 1/(x-3) + 1/(x-5) = 1/3`

`(x+5+x-3)/((x-3)(x+5)) = 1/3`

`(2x+2)/((x-3)(x+5)) = 1/3`

⇒ 3(2x + 2) = (x-3)(x+5)

⇒ 6x + 6 = x2 + 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

However, age cannot be negative.

Therefore, Rehman's present age is 7 years.

Is there an error in this question or solution?

Solution The sum of the reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is 1/3. Find his present age. Concept: Solutions of Quadratic Equations by Completing the Square.