Question

An intern can do a job in $15$ days. The manager and senior manager are busy with other priorities and thus take $25$ and $40$ days respectively to complete the task.How long will they take to finish the task if all of them work together?

Answer 1

Hint: Here we have been given numbers of days taken by an intern, manager and senior manager for completing a task and we have to find how long it will take if they work together. Firstly we will find one day's work for each individual. Then we will add the one day's work of all of them and find the number of days they take to complete the work together and get our desired answer.

Complete step by step answer:
It is given that the intern can do the job in $15$ days so,
Intern one day work$=\dfrac{1}{15}$….$\left( 1 \right)$
Next the manager and senior manager take $25$ and $40$ days respectively so,
Manager one day work $=\dfrac{1}{25}$…..$\left( 2 \right)$
Senior manager one day work $=\dfrac{1}{40}$…..$\left( 3 \right)$
Now we will find the total work done by intern, manager and senior manager by adding equation (1), (2) and (3) as follows,
$\Rightarrow \dfrac{1}{15}+\dfrac{1}{25}+\dfrac{1}{40}$…$\left( 4 \right)$

Now we will find the LCM of the denominator as follows,
$\begin{align}
  & 2\left| \!{\underline {\,
  15,25,40 \,}} \right. \\
 & 2\left| \!{\underline {\,
  15,25,20 \,}} \right. \\
 & 2\left| \!{\underline {\,
  15,25,10 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15,25,5 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5,25,5 \,}} \right. \\
 & 5\left| \!{\underline {\,
  1,5,1 \,}} \right. \\
 & \,\,\left| \!{\underline {\,
  1,1,1 \,}} \right. \\
\end{align}$

We get the LCM as,
$\Rightarrow 2\times 2\times 2\times 3\times 5\times 5$
$\Rightarrow 600$
Taking the LCM of equation (4) as $600$ we get,
$\Rightarrow \dfrac{40+24+15}{600}$
$\Rightarrow \dfrac{79}{600}$
Now the reciprocal of the above value will give us the time taken by them together,
$\Rightarrow \dfrac{600}{79}$
$\therefore 7\dfrac{47}{79}$
So they will together take $7\dfrac{47}{79}$ days.

Hence the intern, manager and senior manager will take $7\dfrac{47}{79}$ days for completing the work.

Note: This type of question is known as Time and work problems which deal with the time taken by an individual or a group of individuals for completing any work. There are many formulas for finding different questions on this topic. When the total time taken while working together is asked we always find the work done in one day by every individual and then add it to get the work done together in one day as efficiency and time are inversely proportional to each other.