A group of men can complete a certain job in K hours when all remain working. Instead, after every 4 hours of work, half of the men currently working leave the job. The work is finished in 16 hours under this pattern. Assuming constant individual efficiency, what is the value of K?

Introduction / Context:
This problem explores how varying the number of workers over time affects the total time to finish a fixed job. Initially, a group of men can complete the work in K hours if all continue working. However, in the actual scenario, half the workers leave after every 4 hour block, and we are told the job then finishes in 16 hours. We must find K for the original full team.

Given Data / Assumptions:
• Initial number of men = N.
• If all N men work without leaving, they finish the job in K hours.
• In reality, all N men work for the first 4 hours.
• After 4 hours, half leave, so N / 2 men work for the next 4 hours.
• After 8 hours total, again half of the remaining leave, and so on.
• The pattern continues until the job is completed in a total of 16 hours.

Concept / Approach:
We treat total work as a fixed quantity equal to N multiplied by K and the per man hourly rate. Under the leaving pattern, we compute how much work is done during each 4 hour segment with the reduced workforce. Adding these contributions and equating them to the total work allows us to solve for K in terms of the known time 16 hours.

Step-by-Step Solution:
Let total work = W and each man hourly work rate = r. If all N men work for K hours, W = N × r × K. Under the actual pattern, divide the 16 hours into four segments of 4 hours each. First 4 hours: N men work, so work done = N × r × 4. Second 4 hours: N / 2 men work, so work done = (N / 2) × r × 4 = 2N × r. Third 4 hours: N / 4 men work, so work done = (N / 4) × r × 4 = N × r. Fourth 4 hours: N / 8 men work, so work done = (N / 8) × r × 4 = 0.5N × r. Total work under the leaving pattern = (4N r) + (2N r) + (N r) + (0.5N r) = 7.5N r. This must equal W = N × r × K. So N r K = 7.5N r, which implies K = 7.5 hours.

Verification / Alternative check:
We can think of 7.5 hours as the effective equivalent of full manpower. In 16 hours with decreasing team sizes, they deliver 7.5 hours worth of work at full strength. This is a reasonable interpretation, because 4 + 2 + 1 + 0.5 equals 7.5, reflecting how the total workforce equivalent is distributed over time.

Why Other Options Are Wrong:
Values like 7 or 8 hours would require different total effective man hours than those implied by the four segments. For example, if K were 7, N r K would be 7N r, while we computed 7.5N r as the actual work, causing a mismatch. Only 7.5 hours satisfies equality between the planned and actual work.

Common Pitfalls:
Learners sometimes incorrectly assume the same number of men leave each time, rather than half of the then current workforce. Others forget to separate the work into four distinct time segments and simply multiply the final number of men by total time, which does not reflect the changing manpower.

Final Answer:
The value of K is 7.5 hours.