Anushkaathak
A certain amount of work can be completed by 20 people in 100 days, working 12 hours a day. 5 people started the work, and starting from the second day, 7 people joined at the beginning of each day. Every alternate day, starting from the third day, two people left at the beginning of that day. If each person worked for two hours per day, then what fraction of the work was not completed after 60 days? (Assume efficiency of each person is the same)
Given that 20 people can complete the work in 100 days working 12 hours per day
So the total work = 20 * 100 * 12 = 24,000 person-hours
Initially 5 people started working and they worked 2 hours per day each
So in the first day they worked = 5 * 2 = 10 hours
From the second day, 7 more people joined each day
So from day 2 onwards, number of people = 5 + 7 = 12
They can work 2 hours each, so work done = 12 * 2 = 24 hours
From the third day, 2 people leave each alternate day while 7 people join every day
So on day 3, number of people = 12 + 7 - 2 = 17
They work 2 hours each, so work done = 17 * 2 = 34 hours
On day 4, number of people = 17 + 7 = 24
Work done = 24 * 2 = 48 hours
On day 5, number of people = 24 + 7 - 2 = 29
They work 2 hours each, so work done = 29 * 2 = 58 hours... On day 6, number of people = 29 + 7 = 36
Work done = 24 * 2 = 72 hours........ On analysing the pattern there are two Arithmetic Progressions here:
One with sequence as 10, 34, 58, ... with number of terms as 30 and common difference as 24... Thus Last term is = 10 + (30-1)*24 = 706; Hence Sum of these 30 terms = (10 + 706)*15 = 10740
Another series is 24, 48, 72, ... with number of terms as 30 and common difference as 24... Thus Last term is = 24 + (30-1)*24 = 720; Hence Sum of these 30 terms = (24 + 720)*15 = 11,160
Thus total work done = 10,740 + 11,160 person-hours = 21,900 person-hours
Hence fraction of work remaining after 60 days = (24000 - 21900)/24000 = 2100/24000 = 7/80