X, Y and Z can complete a piece of work individually in 6 hours, 8 hours and 8 hours respectively. However, only one person at a time can work in each hour and nobody can work for two consecutive hours. All are engaged to finish the work. What is the minimum amount of time that they will take to finish the work?

EXPLANATION

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): $3 \times \frac{1}{6} = \frac{1}{2}$ of the work.
Work done by Y and Z (3 hours combined): $3 \times \frac{1}{8} = \frac{3}{8}$ of the work.

Total work done in 6 hours $= \frac{1}{2} + \frac{3}{8} = \frac{4}{8} + \frac{3}{8} = \frac{7}{8}$ .

Remaining work $= 1 - \frac{7}{8} = \frac{1}{8}$ .

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 $= \frac{\frac{1}{8}}{\frac{1}{6}} = \frac{1}{8} \times 6 = \frac{6}{8} = \frac{3}{4}$ hours.

Converting 3/4 hours to minutes: $\frac{3}{4} \times 60 = 45$ 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.