A contractor undertakes to complete a piece of work in 40 days. He engages 100 men at the beginning and, after 35 days, engages 100 more men so that the work is finished in time. If he had not engaged the additional men, how many days behind schedule would the work have been finished?

Introduction / Context:
This time and work question explores how adding more workers partway through a project affects completion time. We are asked to compute how much delay would have occurred if these extra workers had not been engaged. Such problems illustrate the concept of total man days and show how to compare different workforce allocations for the same total work.

Given Data / Assumptions:

• The contractor plans to finish the work in 40 days.
• Initially, he employs 100 men from day 1.
• After 35 days, he hires 100 additional men, making 200 men in total for the last 5 days.
• With this arrangement, the work finishes exactly in 40 days.
• We need to find how many days late the work would have been if the additional 100 men had never been hired.

Concept / Approach:
The central idea is that total work can be measured in man days, which equal the number of men multiplied by the number of days they work. We first compute the total man days actually used with the extra men. That gives the total amount of work required. Next, we find how many days 100 men alone would need to provide that many man days. The difference between this time and the scheduled 40 days gives the delay.

Step-by-Step Solution:
Step 1: Calculate man days contributed in the first 35 days: 100 men * 35 days = 3500 man days.Step 2: For the remaining 5 days, 200 men work, so their contribution is 200 * 5 = 1000 man days.Step 3: Total man days required to complete the work are 3500 + 1000 = 4500 man days.Step 4: If only 100 men were employed from the beginning to the end, let T be the total number of days needed. Then 100 * T = 4500, so T = 4500 / 100 = 45 days.Step 5: The planned duration was 40 days. With only 100 men, the work would take 45 days, which is 5 days longer than scheduled.
Verification / Alternative check:
We can sanity check by comparing average workforce. For the first 35 days, there are 100 men, and for the last 5 days, 200 men. The effective average over 40 days is higher than 100 men, so using only 100 men throughout should naturally take more than 40 days. A difference of 5 days is reasonable. Also, if we think numerically, 100 men working 45 days provide 4500 man days, exactly matching the total work already computed, verifying our algebraic solution.

Why Other Options Are Wrong:
Three days would correspond to a total of 43 days with 100 men, giving only 4300 man days, which is insufficient to complete the work that actually required 4500 man days.Six days would mean 46 days with 100 men, implying 4600 man days, which is more than required for the project.Nine days would lead to 49 days of work with 100 men, or 4900 man days, which is much greater than the required 4500 man days and clearly not minimal.
Common Pitfalls:
Many learners forget to add the contributions of both phases correctly or misinterpret the question and try to compute the completion time with 200 men from the start. Another error is failing to convert the question into total man days and instead attempting to apply chain rule mechanically. Using total man days keeps the reasoning organised and helps avoid these mistakes.

Final Answer:
If no additional men had been engaged, the work would have been finished 5 days behind schedule.