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?

Correct Option: (c) 6 hours 45 minutes

Let the total work be the Least Common Multiple (LCM) of the individual times taken by X, Y, and Z, which are 6 hours, 8 hours, and 8 hours. LCM(6, 8, 8) = 24 units. The individual efficiencies (work done per hour) are: X = 24 / 6 = 4 units/hour, Y = 24 / 8 = 3 units/hour, and Z = 24 / 8 = 3 units/hour. To complete the work in the minimum time, we must maximize the work done in each hour by utilizing the most efficient worker, X, as much as possible, subject to the condition that nobody works for two consecutive hours. Thus, X must work every alternate hour. To achieve this, we schedule X in odd hours and Y and Z in even hours: Hour 1: X (4 units), Hour 2: Y (3 units), Hour 3: X (4 units), Hour 4: Z (3 units), Hour 5: X (4 units), Hour 6: Y (3 units). Total work done in 6 hours = 4 + 3 + 4 + 3 + 4 + 3 = 21 units. Remaining work = 24 - 21 = 3 units. In Hour 7, X works next. Since X's efficiency is 4 units/hour, X will take 3/4 of an hour (which is 45 minutes) to complete the remaining 3 units. Thus, the minimum time taken is 6 hours and 45 minutes. The correct option is (c).