Fifty men can complete a piece of work in 28 days. They all start together, but at the end of every 10th day, 10 men leave the job. In how many days will the entire work be completed?

Introduction / Context:
This time and work problem involves changing manpower over time. The total number of men working on a job decreases periodically, so the overall speed of completing the work changes in stages. The question tests the ability to calculate work done in successive intervals with different workforce sizes and then to determine the total time taken to complete the job.

Given Data / Assumptions:
- Fifty men can complete the entire work in 28 days. - All 50 men start the work together. - At the end of every 10th day, 10 men leave the job. - Work rate of each man is constant over time. - We treat the total work as one complete unit.

Concept / Approach:
First we find the total amount of work in equivalent “man-days”. Total work = number of men * number of days when they alone could finish the job. Then we consider the work done in each 10 day block with a different number of men. At each stage, we subtract the work completed so far from the total work. Once the cumulative work equals the total required, we know the total number of days taken. This approach uses the idea that work = rate * time and that rates add linearly when multiple workers contribute.

Step-by-Step Solution:
Step 1: Total work in man-days = 50 men * 28 days = 1400 man-days. Step 2: Days 1 to 10: 50 men work for 10 days, work done = 50 * 10 = 500 man-days. Remaining = 1400 - 500 = 900 man-days. Step 3: Days 11 to 20: 10 men leave, so 40 men work for the next 10 days. Work done = 40 * 10 = 400 man-days. Remaining = 900 - 400 = 500 man-days. Step 4: Days 21 to 30: Again 10 men leave, so 30 men work for the next 10 days. Work done = 30 * 10 = 300 man-days. Remaining = 500 - 300 = 200 man-days. Step 5: Days 31 onward: Another 10 men leave, leaving 20 men to finish the remaining work. Step 6: Time needed for 20 men to finish 200 man-days of work = 200 / 20 = 10 days. Step 7: Total time = 10 + 10 + 10 + 10 = 40 days.

Verification / Alternative check:
We can check by adding all work segments: 500 + 400 + 300 + 200 = 1400 man-days, exactly equal to the total work required. The corresponding durations are 10, 10, 10 and 10 days, which add up to 40 days. Since all work is accounted for and there is no remaining work after 40 days, the answer is consistent.

Why Other Options Are Wrong:
36 days or 38 days: These are too small; the total man-days contributed in those times would be less than 1400, so the work would remain incomplete.
45 days or 42 days: These are larger than necessary. By 40 days the entire work is already finished, so any extra days are not required. Hence these values do not match the calculated schedule.

Common Pitfalls:
Students sometimes assume that men leave every 10 working days from the start without carefully calculating the work done in each segment. Another common mistake is to average the total number of men over the entire duration, which does not correctly handle piecewise changes. Always convert to total work in man-days and then subtract work done interval by interval when workforce changes occur.

Final Answer:
The work will be completed in 40 days.