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  \[\frac{1}{6}\] 

\[\frac{1}{x} + \frac{1}{x + 5} = \frac{1}{6}\]
\[ \Rightarrow \frac{x + 5 + x}{x\left( x + 5 \right)} = \frac{1}{6}\]
\[ \Rightarrow \frac{5 + 2x}{x^2 + 5x} = \frac{1}{6}\]
\[ \Rightarrow 30 + 12x = x^2 + 5x\]
\[ \Rightarrow x^2 - 7x - 30 = 0\]
\[ \Rightarrow x^2 - 10x + 3x - 30 = 0\]
\[ \Rightarrow x\left( x - 10 \right) + 3\left( x - 10 \right) = 0\]
\[ \Rightarrow \left( x + 3 \right)\left( x - 10 \right) = 0\]
\[ \Rightarrow x = - 3, 10\] 

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