Let's break down the problem step by step.

Given:

• A group of men can complete a job in $M$ hours if they work together continuously.
• Instead of working continuously, after every 8 hours, half of the men leave.
• The job is completed in 40 hours.

Step 1: Understanding the problem

Let's assume the total number of men is $N$.

• Initial phase (first 8 hours): All $N$ men are working.
• Second phase (next 8 hours): Half of the men leave, so $\frac{N}{2}$ men are working.
• Third phase (next 8 hours): Half of the remaining men leave, so $\frac{N}{4}$ men are working.
• Fourth phase (next 8 hours): Half of the remaining men leave, so $\frac{N}{8}$ men are working.
• Fifth phase (final 8 hours): Half of the remaining men leave, so $\frac{N}{16}$ men are working.

Step 2: Calculate the work done in each phase

Since all men work with equal efficiency, the work done is proportional to the number of men and the time they work.

• Work done in the first 8 hours: $W_1 = N \times 8$
• Work done in the second 8 hours: $W_2 = \frac{N}{2} \times 8 = \frac{8N}{2} = 4N$
• Work done in the third 8 hours: $W_3 = \frac{N}{4} \times 8 = \frac{8N}{4} = 2N$
• Work done in the fourth 8 hours: $W_4 = \frac{N}{8} \times 8 = \frac{8N}{8} = N$
• Work done in the fifth 8 hours: $W_5 = \frac{N}{16} \times 8 = \frac{8N}{16} = \frac{N}{2}$

Step 3: Total work done

The total work $W$ done to complete the job is the sum of the work done in each phase: $W = W_1 + W_2 + W_3 + W_4 + W_5$ $W = 8N + 4N + 2N + N + \frac{N}{2} = 15.5N$

Step 4: Relating to $M$

If all $N$ men work together for $M$ hours, the work done is $W = N \times M$.

Since the total work $W$ done is the same in both scenarios: $N \times M = 15.5N$ $M = 15.5$

Final Answer:

The value of $M$ is 15.5 hours.

Would you like more details on any part of this solution?

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Relative questions:

1. What happens if the men leave after every 10 hours instead of 8 hours?
2. How would the value of $M$ change if a quarter of the men left after every 8 hours?
3. What if instead of half leaving, 75% of the men left after every 8 hours?
4. How would the calculation change if the efficiency of the men decreased by 10% every 8 hours?
5. What is the effect on $M$ if the total job took 50 hours instead of 40?

Tip: When faced with a problem involving work or efficiency, it's helpful to break down the process step by step to see how each component contributes to the total outcome.