Question

Vivek is older than Kishor by 5 years. The sum of the reciprocals of their ages is \[\frac{1}{6}\] Find their present ages. 

Answer 1

Let the present age of Kishor be x years.

Vivek is older than Kishor by 5 years.

So, Vivek's age will be x + 5.

The sum of the reciprocals of their ages is  \[\frac{1}{6}\] 

\[\frac{1}{x} + \frac{1}{x + 5} = \frac{1}{6}\]

\[ \Rightarrow \frac{x + 5 + x}{x\left( x + 5 \right)} = \frac{1}{6}\]

\[ \Rightarrow \frac{5 + 2x}{x^2 + 5x} = \frac{1}{6}\]

⇒ 6 (5 + 2x) = 1(x² + 5x)

⇒ 30 + 12x = x² + 5x

⇒ x² + 5x - 12x - 30 = 0 

⇒ x² - 7x - 30 = 0

⇒ x² - 10x + 3x - 30 = 0

⇒ (x - 10) + 3(x - 10) = 0

⇒ (x + 3) (x - 10) = 0

⇒ x = -3 or x = 10

But age cannot be negative so, age of Kishore is 10 years and that of Vivek is 10 + 5 = 15 years.