A contractor agrees to complete a piece of work in 40 days. He initially engages 100 men. After 35 days, he adds 100 more men (so 200 men now) and just manages to complete the work in the originally stipulated 40 days. If he had not engaged the additional men, by how many days would he have missed the scheduled completion time?

Introduction / Context:
This is a practical work and manpower planning question. It examines how changes in workforce size at different stages affect total time, and asks you to work backwards to see what would happen without extra help. Such questions are common in aptitude tests under the topic of time and work or chain rule.

Given Data / Assumptions:
- Contract time: 40 days.
- Initially, 100 men work for the first 35 days.
- After 35 days, 100 more men are added, so 200 men work for the last 5 days.
- With this pattern, the work is completed exactly in 40 days.
- All men work with the same constant efficiency.
- We must find the delay if only the initial 100 men worked until completion.

Concept / Approach:
We calculate the total work in man-days based on the actual schedule (with extra men). Then we compute how many days 100 men alone would need for the same amount of work. The difference between this time and the contracted 40 days gives the number of days the contractor would be behind schedule if he had not hired the additional 100 men.

Step-by-Step Solution:
Step 1: Compute work done by 100 men in the first 35 days: 100 * 35 = 3500 man-days.Step 2: In the last 5 days, 200 men work: 200 * 5 = 1000 man-days.Step 3: Total work W = 3500 + 1000 = 4500 man-days.Step 4: If only 100 men were employed throughout, let T be the number of days required.Step 5: Then 100 * T = 4500 → T = 4500 / 100 = 45 days.Step 6: The scheduled time was 40 days, so the delay would be 45 - 40 = 5 days.

Verification / Alternative check:
You can test the result by imagining the original plan: 100 men for 40 days provide only 100 * 40 = 4000 man-days, which is less than the needed 4500 man-days. The shortfall of 500 man-days would require an extra 500 / 100 = 5 more days with 100 men, confirming a total of 45 days and a 5-day delay.

Why Other Options Are Wrong:
Values like 3, 6, 7 or 9 days do not match the actual shortfall in man-days. They result from miscalculating total work (e.g., forgetting the last 5 days with 200 men) or incorrectly dividing by the number of men when converting man-days back to days.

Common Pitfalls:
Students often confuse the timeframe and might incorrectly multiply 200 men by the full 40 days, or forget to separate the work phases. Another common error is to assume work is proportional only to men or only to days. Always compute man-days for each phase separately and then add them to find the total work.

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