A group of men (all work with equal efficiency) working together can complete a job in M hours. Instead of working together, after every 8 hours, half of the men of the group working at that point of time leave the job. Continuing this way, the job is finished in 40 hours.

What is the value of M?

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.

A group of men (all work with equal efficiency) working together can complete a job in M hours. Instead of working together, after every 8 hours, half of the men of the group working at that point of time leave the job. Continuing this way, the job is finished in 40 hours. What is the value of M?

Explore how to determine the time M in which a group of men can complete a job when half of the workforce leaves every 8 hours, leading to job completion in 40 hours. This problem involves key concepts in work and time as well as geometric progression, making it suitable for Grades 10-12.

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