Question

A man, a woman and a boy can complete a job in 3, 4 and 12 days respectively. How many boys must assist 1 man and 1 woman to complete the job in $ \dfrac{1}{4} $ of a day?
\[\begin{align}
  & A.1 \\
 & B.4 \\
 & C.19 \\
 & D.41 \\
\end{align}\]

Answer 1

Hint: Given is the work or a job done by 1 man, 1 woman and 1 boy in 3, 4 and 12 days respectively. Find the rate of work being done by them individually that is everyone's one day work. According to the question add the 1 days’ work of 1 man, 1 woman and x boys and equate it with the total number of jobs/works can be done in 1 day.

Complete step-by-step answer:
A complete job can be done in a total of 3 days by a man, 4 days by a woman and 12 days by a boy.Given is a job which is done by 1 man and 1 woman and some boys, being done or being completed in $ \dfrac{1}{4} $ of a day.
We know, 1 man completes a job in 3 days, so, the rate of doing work in 1 day of a man will be \[\Rightarrow \dfrac{1}{3}\text{(per day)}\]
 $ \dfrac{1}{3} $ is 1 man's 1 day work.
Similarly, 1 woman’s 1 day work $ \dfrac{1}{4} $
And 1 boys 1 day work $ \dfrac{1}{12} $
Given, 1 man, 1 woman and let's say x boys together complete a work/job in $ \dfrac{1}{4} $ of day.
\[\begin{align}
  & \Rightarrow \text{1 work = }\dfrac{1}{4}\text{ day} \\
 & \Rightarrow \text{in 1 day = 4 works/jobs can be completed}\text{.} \\
\end{align}\]
Now, add the 1 day work of 1 man, 1 woman and x boys together.
\[\begin{align}
  & \Rightarrow \text{1 men }\!\!'\!\!\text{ s day work + 1 women }\!\!'\!\!\text{ s day work + x boy }\!\!'\!\!\text{ s 1 day work = total work in 1 day} \\
 & \Rightarrow \dfrac{1}{3}+\dfrac{1}{4}+x\times \dfrac{1}{12}=4 \\
\end{align}\]
Since, in 1 day total 4 jobs can be completed.
\[\begin{align}
  & \Rightarrow \dfrac{1}{3}+\dfrac{1}{4}+x\times \dfrac{1}{12}=4 \\
 & \Rightarrow \dfrac{8+6+2x}{24}=4 \\
 & \Rightarrow 14+2x=96 \\
 & \Rightarrow 2x=82 \\
 & \Rightarrow x=41 \\
\end{align}\]
So, the number of boys required to assist 1 man and 1 woman to complete 1 job in $ \dfrac{1}{4} $ of day.
So, the correct answer is “Option D”.

Note: There are also some other methods to do this question, like by converting the work in units of $ \dfrac{1}{4} $ days and then adding the total work of 1 man, 1 woman and x boys. We must be careful while adding the fractions and take the LCM properly to get the right answer. Here we just have to add all the work done per day by man, woman, boy and equate it to 4, there is no need of taking reciprocal like in similar type of questions.