From a group of three people, pairs have to be formed. This can only be done in 3 ways. PQ, PR, QR.
So in this question, we have to solve for the least time taken by using only these 3 pairs.
Times taken by each and times taken by each pair are:
P = \(\frac{1}{10}\)
Q = \(\frac{1}{12}\)
R = \(\frac{1}{15}\)
PQ = \(\frac{1}{10}\) + \(\frac{1}{12}\) = \(\frac{11}{60}\)
QR = \(\frac{1}{12}\) + \(\frac{1}{15}\) = \(\frac{9}{60}\)
PR = \(\frac{1}{10}\) + \(\frac{1}{15}\) = \(\frac{10}{60}\)
In order to find the least possible time to complete the work, we have to use a couple who can do maximum work in an hour. So out of 3 pairs above, arranging the teams in decreasing order of their ability to do work in an hour: PQ>PR>QR.
So PR is the couple who can complete maximum work in an hour and QR is the couple with the least amount of work done in an hour.
So let's start the work with PQ now.
1st hour: PQ
Now PQ together cannot work in the second hour, so out of PR and QR, any team can work. As PR completes next maximum possible work in an hour after PQ we will consider PR to work in 2nd hour.
2nd hour: PR
Now PR cannot work together in the third hour. And also P cannot work in the third hour as it already worked for 2 continuous hours. So the remaining choice is only QR.
3rd hour: QR
So work done in 3 hours is \(\frac{11}{60}\) + \(\frac{10}{60}\) + \(\frac{9}{60}\) = \(\frac{30}{60}\)
Half of the work is done in 3 hours.
Now in fourth hour R cannot work as R already worked for 2 continuous hours. So the only possible choice is PQ.
4th hour: PQ
Now in fifth hour Q cannot work as Q already worked for 2 continuous hours. So the only possible choice is PR.
5th hour: PR
Now in sixth hour P cannot work as P already worked for 2 continuous hours. So the only possible choice is QR.
6th hour: QR
Work done in 4th, 5th and 6th hours is \(\frac{11}{60}\) + \(\frac{10}{60}\) + \(\frac{9}{60}\) = \(\frac{30}{60}\)
Remaining Half of the work is done in another 3 hours.
So the least possible time to complete the work is 6 hours.
OPTION:
C
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Regards,
Chaitanya
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