07 Time & Work

In the following example, we will discuss how to find the total time taken to complete the work, if the two or more workers are not working together but they are working on alternate days. Suppose \(A\) and \(B\) are the two workers working on a project such that \(A\) and \(B\) can complete a work working alone in 20 and 12 days respectively. Now they are working on alternate day, now to find the total time required to complete the work, there can be two cases:

(a) Starting with \(‘A’\) 

(b) Starting with \(‘B’\)  

In these types of questions work done on the first day is not same as the work done on the second day but work done in first two days is same as the work done in the next two days  and that is same as the work done in the nest two days and so on. 

AB     AB      AB  ………

Work done in 2 days = \(\frac{1}{{20}} + \frac{1}{{12}} = \frac{{3 + 5}}{{60}} = \frac{8}{{60}} = \frac{2}{{15}}\)

If we assume 2 days to be one cycle and total work is always considered as 1 unit, then approximate number of cycles required to complete the work 

= \(\frac{{15}}{2} = 7\) (integral number)

Total work done after 7 complete cycles = \(7 \times \frac{2}{{15}}\) 

= \(\frac{{14}}{{15}}\). Hence the remaining work is \(\frac{1}{{15}}\)

(1) A started the work: In one day \(A\) completes \(\frac{1}{{20}}th\) of the work, hence  \(\frac{1}{{15}}th\) of the work will be completed in more than one day. \(A\) will complete \(\frac{1}{{20}}th\) part and the remaining work will be done by \(B\). 

Remaining work =\(\frac{1}{{15}} - \frac{1}{{20}} = \frac{{4 - 3}}{{60}} = \frac{1}{{60}}\)

Now \(B\) will complete the remaining work in \(\frac{1}{{60}} \times 12 = \frac{1}{5}\) days. Total time taken will be: \({\rm{14 }} + {\rm{ 1 }} + \frac{1}{5} = 15\frac{1}{5}\)

(2) \(B\) started the work: \(B\) will complete the remaining work \(\left( {\frac{1}{{15}}th} \right)\) of the complete work in \(\frac{1}{{15}} \times 12 = \frac{4}{5}\) days

Example 12: \(A, B, C\) can complete a work working alone in 20, 12 and 30 days. If they work on alternate days starting with \(‘A’\) find total time required to complete the work.

Solution: In this question work done on the first day is not same as work done on the second day but work done in the first 3 days = work done in the next 3 days       

ABC         ABC ……………….. 

Suppose 3 days mean one cycle, hence work done in one cycle 

=\(\frac{1}{{20}} + \frac{1}{{12}} + \frac{1}{{30}}\, = \frac{{3 + 5 + 2}}{{60}} = \frac{{10}}{{60}} = \frac{1}{6}\)

Hence total cycles required = 6

Total cycles required =  \(6 \times 3{\rm{ }} = {\rm{ }}18\) days