Question

A man can do a job in h hours alone and his son can do the job in 2h hours alone. Together in how many hours can they finish the job?
[a] 3h
[b] $\dfrac{h}{3}$
[c] $\dfrac{3h}{2}$
[d] $\dfrac{2h}{3}$
[e] $\dfrac{h}{2}$

Answer 1

Hint: Assume that the number of hours it takes them to complete the work together be x. Find the fraction of the total work that the man does in 1 hour and hence find the fraction of the total works the man does in x hours. Similarly find the fraction of work his son does in 1 hour and hence find the fraction of work his son does in x hours. The sum of these fractions should be 1. Hence form an equation x. Solve for x and hence find the number of hours it takes them together to complete the work.

Complete step by step solution:
Let the number of hours it takes them to complete the work be x.
Now, we have
In h hours the man does the complete work.
Hence in 1 hour the man does ${{\left( \dfrac{1}{h} \right)}^{th}}$ of the total work.
Hence in x hours the man does ${{\left( \dfrac{x}{h} \right)}^{th}}$ of total work.
Also, we have
In 2h hours the son does the complete work.
Hence in 1 hour the son does ${{\left( \dfrac{1}{2h} \right)}^{th}}$ of the total work.
Hence in x hours the son does ${{\left( \dfrac{x}{2h} \right)}^{th}}$ of the total work.
Hence, we have
$\dfrac{x}{h}+\dfrac{x}{2h}=1$
Multiplying both sides by 2h, we get
$\begin{align}
  & 2x+x=2h \\
 & \Rightarrow 3x=2h \\
\end{align}$
Dividing both sides by 3, we get
$x=\dfrac{2h}{3}$
Hence it takes the $\dfrac{2h}{3}$ hours to complete the work together.
Hence option [d] is correct.

Note: Alternatively, we can use the fact that if A can do a job in x hours and B can do the job in y hours and they together can do the job in z hours, then $\dfrac{1}{z}=\dfrac{1}{x}+\dfrac{1}{y}$(This can be extended to any number of workers).
Hence, for this question, we have
$\dfrac{1}{x}=\dfrac{1}{h}+\dfrac{1}{2h}=\dfrac{2+1}{2h}=\dfrac{3}{2h}$
Taking reciprocals on both sides, we get
$x=\dfrac{2h}{3}$, which is the same as obtained above.
Hence option [d] is correct.