Twenty-four men can complete a piece of work in 17 days when working together. If the same work has to be completed in 51 days instead, how many men will be required, assuming each man works at the same constant rate?

Introduction / Context:
This question illustrates the direct inverse relationship between the number of workers and the time taken to complete a job when the total amount of work stays constant. You are given how many men can finish the work in a certain number of days, and you must find how many men are needed to finish the same work in a different number of days at the same rate of working.

Given Data / Assumptions:

Twenty-four men can complete the work in 17 days.
We want to complete the same work in 51 days instead.
All men work at identical constant rates, and total work remains unchanged.
No man joins or leaves once the new group size is decided.

Concept / Approach:
We use the concept of man-days, where Work = Number of men * Number of days * Rate per man. Since the rate per man is constant and the work is fixed, the product of men and days must remain the same. So, men1 * days1 = men2 * days2. We use this proportional relationship to solve for the unknown number of men required to finish the work in the new time frame.

Step-by-Step Solution:
Let total work be W man-days. From the initial situation: 24 men * 17 days = W. So W = 24 * 17 = 408 man-days. Now, let the required number of men for finishing the work in 51 days be x. Then x men * 51 days = W = 408 man-days. So 51x = 408. Solve for x: x = 408 / 51. 408 divided by 51 = 8. Therefore, x = 8 men.

Verification / Alternative check:
Check the man-days: 8 men working for 51 days will contribute 8 * 51 = 408 man-days, which matches the original total of 408 man-days contributed by 24 men in 17 days. This confirms that the work done is the same and that using 8 men for 51 days will indeed complete the identical job.

Why Other Options Are Wrong:
Option 10 men would produce 10 * 51 = 510 man-days, which is more work than needed and would finish the job earlier than 51 days.
Option 12 men would give 12 * 51 = 612 man-days, which is even more, clearly not matching the original workload.
Option 6 men would provide 6 * 51 = 306 man-days, which is insufficient to complete the job that requires 408 man-days.
Option 16 men would provide 16 * 51 = 816 man-days, more than double the needed work, so the time would have to be shorter than 51 days to keep the workload constant.

Common Pitfalls:
Students sometimes confuse direct and inverse proportionality and may incorrectly multiply or divide the wrong way around. Another mistake is to ignore the total man-days concept and attempt ad-hoc reasoning. Remember that when the amount of work is fixed and worker efficiency is constant, the number of workers and the time taken are inversely proportional. Multiplying number of men and days to check equality is a reliable method.

Final Answer:
Hence, to complete the same work in 51 days, 8 men are required, so the correct option is 8 men.