Correct Option (3)
To determine the minimum time required to complete the work, an optimal work allocation strategy must be employed, adhering to the given constraints:
- Only one person works per hour.
- No person works for two consecutive hours.
The individual work rates are:
- X: 1/6 of the work per hour
- Y: 1/8 of the work per hour
- Z: 1/8 of the work per hour
To minimize the total time, the fastest worker (X) should be utilized as frequently as possible. This implies X works every alternate hour, while Y and Z alternate in the remaining slots. An optimal sequence for the first six hours would involve X working for three hours and Y and Z (or Y and Y, or Z and Z) working for three hours, ensuring no one works consecutively.
Work completed in the first 6 hours:
- Work done by X (3 hours): 3×61=21 of the work.
- Work done by Y and Z (3 hours combined): 3×81=83 of the work.
Total work done in 6 hours =21+83=84+83=87.
Remaining work =1−87=81.
To complete the remaining work in the minimum possible time, the fastest worker, X, should be engaged. X's work rate is 1/6 of the work per hour.
Time taken by X to complete the remaining 1/8 work =6181=81×6=86=43 hours.
Converting 3/4 hours to minutes: 43×60=45 minutes.
Therefore, the total minimum time to finish the work is 6 hours 45 minutes.
Incorrect Options:
Options 1 (6 hours 15 minutes) and 2 (6 hours 30 minutes) are incorrect because the calculated minimum time required to complete the work under the given constraints is 6 hours 45 minutes. These options represent durations insufficient to complete the entire work, even with optimal worker allocation.
Option 4 (7 hours) is incorrect because it exceeds the minimum calculated time of 6 hours 45 minutes. The work can be completed more efficiently by adhering to the optimal strategy of utilizing the fastest worker (X) for the remaining task, leading to a shorter overall duration than 7 hours.