Question

A and B can do a piece of work in 12 days, B and C in 15 days, C and A in 20 days then the number of days taken by A, B and C together to finish the work are:(a) 10 days(b) 12 days(c) 14 days(d) 15 days

Hint: First, before proceeding for this, we must know the following condition that the work done is always directly proportional to the number of days required. Then, we add all the above cases, we can see that A, B, C all the coming two times which indicate that the addition of these three equations gives the amount of work done by A, B and C in two days. Then, to get the amount of work done in one day by A, B and C, divide it by 2.

In this question, we are supposed to find the number of days taken by A, B and C together to finish the work when the conditions are given that A and B can do a piece of work in 12 days, B and C in 15 days, C and A in 20 days.
So, before proceeding for this, we must know the following condition that the work done is always directly proportional to the number of days required.
Then, by using this condition, we get the following condition for the work done by A and B as:
$\dfrac{1}{12}$
Similarly, we get the following condition for the work done by B and C as:
$\dfrac{1}{15}$
Again, we get the following condition for the work done by C and A as:
$\dfrac{1}{20}$
So, if we add all the above cases, we can see that A, B, C all the coming two times which indicate that the addition of these three equations gives the amount of work done by A, B and C in two days.
So, the two days work of A, B and C is given by:
$\dfrac{1}{12}+\dfrac{1}{15}+\dfrac{1}{20}$
Then, solve the above expression for getting the total value of work done by A, B and C in two days as:
\begin{align} & \dfrac{5+4+3}{60}=\dfrac{12}{60} \\ & \Rightarrow \dfrac{1}{5} \\ \end{align}
So, the amount of work done by A, B and C in two days is $\dfrac{1}{5}$ .
Then to get the amount of work done in one day by A, B and C is given by:
$\dfrac{1}{5}\times \dfrac{1}{2}=\dfrac{1}{10}$
So, A, B and C all together complete the work in 10 days.
So, the correct answer is “Option A”.

Note: Now, to solve these types of questions we need to know some of the basics of the work and time concept as we can make mistakes in the fact that more days are required to complete more work but less number of people are required to complete more work. So, the above fact defines that work is directly proportional to number of days but inversely proportional to the men required.