Question

A can do a piece of work in 14 days while B can do it in 21 days. They begin together and work at it for 7 days. Then A fell ill and B had to complete the work. In how many ways, the work can be completed?

Answer 1

Hint: We use the unitary method for direct variation and find the amount of work A and B can complete separately in 1 day and 7 days. We find the amount of work A and B together can complete in 7 days. We find how many days B will take after A fell ill at the ${{7}^{\text{th}}}$ day and add it 7 to get the number of days required to complete the work.

Complete step-by-step solution
We are given the question that A can do a piece of work in 14 days while B can do it in 21 days. We assume the complete piece of work as 1. \[\]
We use the unitary method of direct variation to find the value of a single unit as the amount of work in one day.
So A can finish in 1 day is $\dfrac{1}{14}$ of total work. Similarly, B can finish in 1 day is $\dfrac{1}{21}$ of the total piece of work. \[\]
We are further given the question that they begin together and work at it for 7 days. Then A and B had to complete the work. Now we multiply the value of a single unit and find the work they can do in 7 days. So A can finish $7\times \dfrac{1}{4}=\dfrac{7}{4}$ of the total work in 7 days and B can finish $7\times \dfrac{1}{21}=\dfrac{7}{21}$of total work in 7 days. \[\]
So the part of work that can be done by A and B together is the sum of their part of work they can finish in 7 days that is $\dfrac{7}{14}+\dfrac{7}{21}=\dfrac{1}{2}+\dfrac{1}{3}=\dfrac{2+3}{6}=\dfrac{5}{6}$ of the total work. \[\]
Now after A left the work because of illness, the left amount of work has to be done by B. So the part of the work left now is $1-\dfrac{5}{6}=\dfrac{1}{6}$ of total work. \[\]
We are given that B can complete total work means 1 work in 21 days. So B can complete $\dfrac{1}{6}$ part of total work in $21\times \dfrac{1}{6}=\dfrac{21}{6}=3.5$ days. \[\]
So the number of days that are required to complete the work is $7+3.5=10.5$days.

Note: We know that the unitary method is a technique for solving a problem by first finding the value of a single unit by dividing indirect variation or multiplying in indirect variation. We must be careful that men and work problems are an indirect variation problem but time and work problems like we solved here are direct variation problems. If a person can complete the work in $n$ days and the efficiency in percentage is given by $\dfrac{100}{n}$.