# Q. 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

Correct answer: c) 6 hours 45 minutes

#### Question from UPSC Prelims 2024 CSAT

**Explanation : **

## Minimum amount of time to complete work

Let’s approach this problem step by step:

1) First, let’s calculate how much work each person can do in one hour:

X: 1/6 of the work

Y: 1/8 of the work

Z: 1/8 of the work

2) Now, let’s consider the constraints:

– Only one person can work each hour

– Nobody can work for two consecutive hours

3) To minimize time, we should use the most efficient worker (X) as much as possible, alternating with Y and Z.

4) Let’s see how much work can be done in 6 hours:

Hour 1: X does 1/6

Hour 2: Y does 1/8

Hour 3: X does 1/6

Hour 4: Z does 1/8

Hour 5: X does 1/6

Hour 6: Y does 1/8

5) After 6 hours, the total work done is:

(3 * 1/6) + (2 * 1/8) + (1 * 1/8) = 3/6 + 3/8 = 1/2 + 3/8 = 7/8

6) This means 7/8 of the work is done after 6 hours. We need to find how long it takes to do the remaining 1/8.

7) X is the fastest worker and can do 1/6 of the work in an hour. To do 1/8 of the work, X would need:

(1/8) / (1/6) = 3/4 of an hour = 45 minutes

Therefore, the minimum time to complete the work is 6 hours 45 minutes.

The correct answer is c) 6 hours 45 minutes.