Q8. 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?

‍

A. 6 hours 15 minutes
B. 6 hours 30 minutes
C. 6 hours 45 minutes
D. 7 hours

‍

Answer: C. 6 hours 45 minutes

‍

Explanation:

X, Y, and Z can complete the work individually in 6 hours, 8 hours, and 8 hours, respectively. To minimize the total time required to finish the work, we need to utilize the most efficient worker, X, as much as possible. Thus, we will alternate between X and either Y or Z, starting with X.

‍

**Work Rates:**

  • X completes the work in 6 hours, so in one hour, X completes 1/6 of the work.
  • Y (or Z) completes the work in 8 hours, so in one hour, Y (or Z) completes 1/8 of the work.

‍

**Work Distribution:**

We will alternate between X and Y (or Z) in the following pattern:

  • First hour: X works, completing 1/6 of the work.
  • Second hour: Y (or Z) works, completing 1/8 of the work.
  • Third hour: X works, completing 1/6 of the work.
  • Fourth hour: Y (or Z) works, completing 1/8 of the work.
  • Fifth hour: X works, completing 1/6 of the work.

‍

After 6 hours, the total work completed is: 

3X + 3Y = 3 × 1/6 + 3 × 1/8 = 1/2 + 3/8 = 7/8

‍

**Remaining Work:**

After 6 hours, 7/8 of the work is completed, leaving 1/8 of the work remaining. Since it’s now X’s turn to work, we calculate how much time X will need to complete the remaining 1/8 of the work:

  • X completes 1/6 of the work in 1 hour,
  • So to complete 1/8, the time required is: (1/8 ÷ 1/6) hours = 3/4 hours = 3/4 × 60 minutes = 45 minutes

‍

**Total Time:** Thus, the total time to complete the work is:

6 hours (for the first 7/8 of the work) + 45 minutes (for the remaining 1/8) = 6 hours 45 minutes.

‍

Thus, the correct answer is **C. 6 hours 45 minutes**.

‍

Read more blogs