Question

Vivek is older than Kishor by 5 years. The sum of the reciprocal of their ages is $\dfrac{1}{6}$. Find their present ages.

Answer 1

Hint: We assume that the present age of Kishor is $x$ and then the present age of Vivek will be $x+5$. The reciprocal of the present age of Kishor is $\dfrac{1}{x}$ and the present age of Vivek is $\dfrac{1}{x+5}$. We make the equation $\dfrac{1}{x}+\dfrac{1}{x+5}=\dfrac{1}{6}$ from the given data in the question and solve to get $x$ and then $x+5$.

Complete step-by-step solution:
We know that the reciprocal of a non-zero real number $a$ is $\dfrac{1}{a}$. We are given in the question that Vivek is older than Kishor by 5 years. Let us assume that the present age of Kishor is $x$ and then the present age of Vivek will be $x+5$. We are also given in the question that sum of reciprocals of their ages is $\dfrac{1}{6}$. The reciprocal of present age of Kishor is $\dfrac{1}{x}$ and the present age of Vivek is $\dfrac{1}{x+5}$. Now we have,
\[\dfrac{1}{x}+\dfrac{1}{x+5}=\dfrac{1}{6}\]
We add the fractions at the left hand side and get,
\[\begin{align}
  & \dfrac{\left( x+5 \right)+x}{x\left( x+5 \right)}=\dfrac{1}{6} \\
 & \Rightarrow \dfrac{2x+5}{{{x}^{2}}+5x}=\dfrac{1}{6} \\
\end{align}\]
We cross multiply and get ,
\[\begin{align}
  & \Rightarrow {{x}^{2}}+5x=12x+30 \\
 & \Rightarrow {{x}^{2}}-7x-30=0 \\
\end{align}\]
The above obtained equation is a quadratic equation with a single variable which we solve by factorization using splitting the middle term method . So we have
\[\begin{align}
  & \Rightarrow {{x}^{2}}-7x-30=0 \\
 & \Rightarrow {{x}^{2}}-10x+3x-30=0 \\
 & \Rightarrow x\left( x-10 \right)+3\left( x-10 \right)=0 \\
 & \Rightarrow \left( x-10 \right)\left( x+3 \right)=0 \\
 & \Rightarrow x-10=0,x-3=0 \\
\end{align}\]
So we have two values for $x$ which are $x=10$ and $x=-3$. We reject the negative value$x=-3$ because age cannot be negative. So the present age of Kishor is $x=10$ and the present age of Vivek is $x+5=10+5=15$.

Note: We note that the reciprocals $\dfrac{1}{x}$ exist because the present age zeroes only at the time of birth. We can also solve the problem taking the present of Vivek as $x$ and then the present age of Kishor at $x-5$. We can also solve the quadratic equation using the quadratic formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.