8. 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.

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Accepted Answer

The correct answer is option C. X's work rate = 1/6 per hour. Y's work rate = 1/8 per hour. Z's work rate = 1/8 per hour. To minimize time, the most efficient worker (X) should work as much as possible, but not for two consecutive hours. The other two (Y and Z) have equal efficiency. So, the optimal sequence would be X, Y, X, Z, X, Y, and so on. Let's calculate work done in cycles: In 2 hours (X + Y/Z), work done = (1/6) + (1/8) = 7/24. Let's try a sequence of 6 hours: X, Y, X, Z, X, Y. Work done by X (3 hours) = 3 * (1/6) = 1/2. Work done by Y (2 hours) = 2 * (1/8) = 1/4. Work done by Z (1 hour) = 1 * (1/8) = 1/8. Total work done in 6 hours = 1/2 + 1/4 + 1/8 = 7/8. Remaining work = 1 - 7/8 = 1/8. The next turn is X's. X can do 1/6 of the work in 1 hour. To complete 1/8 of the work, X will take (1/8) / (1/6) = 6/8 = 3/4 hours. 3/4 hours = 45 minutes. So, the total time taken is 6 hours and 45 minutes. This question requires careful sequencing and calculation of work rates under specific constraints.

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