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…

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}$3×61​=21​ of the work.
• Work done by Y and Z (3 hours combined): $3 \times \frac{1}{8} = \frac{3}{8}$3×81​=83​ of the work.

Total work done in 6 hours $= \frac{1}{2} + \frac{3}{8} = \frac{4}{8} + \frac{3}{8} = \frac{7}{8}$=21​+83​=84​+83​=87​.

Remaining work $= 1 - \frac{7}{8} = \frac{1}{8}$=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 $= \frac{\frac{1}{8}}{\frac{1}{6}} = \frac{1}{8} \times 6 = \frac{6}{8} = \frac{3}{4}$=61​81​​=81​×6=86​=43​ hours.

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