Question

A can do a job in 10 days and B in 20 days. They work together, but 4 days before the finish of the job, A left. How many days did they work together?
(A) 6
(B) \[\dfrac{16}{3}\]
(C) 8
(D) 14

Answer 1

Hint: It is given that A can do a job in 10 days and B in 20 days. The part of the work done by A and B in 1 day is \[\dfrac{1}{10}\] and \[\dfrac{1}{20}\] respectively. Now, calculate the part of the work done by B in 4 days. The part of the work done when A and B worked together is \[\dfrac{4}{5}\] . Now, calculate the part of work done in 1 day when A and B worked together. Then, calculate one part of the work done when A and B worked together. Now, get the number of days to finish \[\dfrac{4}{5}\] part of the work.

Complete step-by-step answer:
According to the question, we have been given that,
The time taken by A to finish the whole part of a work = 10 days.
Using the unitary method, calculating,
The part of work done by A in 1 day = \[\dfrac{1}{10}\] ……………………………..(1)
We have also been given the time taken by B to finish the whole part of a work = 20 days.
Using the unitary method, calculating,
The part of work done by B in 1 day = \[\dfrac{1}{20}\] ……………………………..(2)
We have also been given that A left the job 4 days before the finish of the job.
It means that B alone worked for 4 days.
From equation (2), we have the part of the work done by B in 1 day.
Using the unitary method, calculating,
The part of the work done by B in 4 days = \[4\times \dfrac{1}{20}=\dfrac{1}{5}\] ………………………………..(3)
In the whole part of work, some part of work is done by alone B while some part of the work is one by A and B together.
The remaining part of the work that was done when A and B worked together = \[1-\dfrac{1}{5}=\dfrac{5-1}{5}=\dfrac{4}{5}\] ……………………………(4)
The part of work done in 1 day when A and B worked together = \[\dfrac{1}{10}+\dfrac{1}{20}=\dfrac{2+1}{20}=\dfrac{3}{20}\] …………………………….(5)
It means that \[\dfrac{3}{20}\] part of the work is done 1 day when A and B work together.
Now, using the unitary method, calculating,
The time required to finish the work when A and B worked together = \[\dfrac{20}{3}\] days ……………………………(6)
From equation (4), we have the part of the work done when A and B worked together.
Now, using the unitary method, calculating,
The number of days taken to finish \[\dfrac{4}{5}\] part of the work when A and B worked together = \[\dfrac{4}{5}\times \dfrac{20}{3}=\dfrac{16}{3}\] …………………………….(7)
Hence, the number of days when A and B worked together is \[\dfrac{16}{3}\] days.

Note: In this question, one might make a silly mistake while calculating the part of work done in 1 day when A and B worked together and calculate it by the product of the part of work done by A in 1 day and the part of work done by B. This is wrong. The correct way to calculate the part of the work done in 1 day when A and B worked together is the summation of the part of work done by A in 1 day and the part of work done by B.