Ten men can complete a certain piece of work in 15 days. If thirty seven men are employed instead, approximately how many days will they take to complete the same work, assuming all men work at the same constant rate?

Introduction / Context:
This question is about direct proportion between the number of workers and the time taken to complete a fixed amount of work. When all workers have the same efficiency and the total work is fixed, time is inversely proportional to the number of workers. Such questions appear very frequently in basic quantitative aptitude sections and are usually considered straightforward, though sometimes fractional results need to be interpreted carefully with respect to the options given.

Given Data / Assumptions:

• Ten men complete the work in 15 days.
• All men have identical and constant working efficiency.
• We now employ thirty seven men to do the same job.
• We are asked how many days the thirty seven men will require to complete the same work.

Concept / Approach:
If a fixed job requires W man days of effort, then W equals the product of the number of men and the number of days. When the number of men changes, the time adjusts so that the product remains constant. We first compute the total amount of work in man days using the first scenario. Then we divide this total work by the new number of men to obtain the time needed in days when more men are employed. Since the exact value is slightly above an integer, we choose the closest available option.

Step-by-Step Solution:
Step 1: Total work in man days is 10 men times 15 days which equals 150 man days. Step 2: When thirty seven men are working, their daily total capacity is 37 man days per day. Step 3: Time required equals total work divided by daily total capacity which is 150 divided by 37 days. Step 4: 150 divided by 37 is approximately 4.054 days. Step 5: From the given options, the nearest reasonable whole number of days is 4 days.

Verification / Alternative check:
To confirm, check how much work is done in 4 days by thirty seven men. That is 37 times 4 which equals 148 man days, slightly less than the total 150 man days. In a real world situation, the team would need a little more than 4 days to finish exactly. Given that the options are rounded to whole days, 4 days is clearly the only reasonable choice. No other listed option is closer to the exact computed value of about 4.05 days.

Why Other Options Are Wrong:

• 3 days would give only 111 man days, far less than the required 150 man days.
• 5 days would give 185 man days, which is significantly more than needed and not minimal.
• 6 days would produce 222 man days, much larger than required and clearly not the correct minimal time.

Common Pitfalls:
A common mistake is to think that if the number of men increases by around four times, the time must reduce exactly to one quarter. Here the increase is from 10 to 37, which is not an integer multiple, so the resulting time is not a neat whole number. Another pitfall is not recognizing that exam options are sometimes rounded, and thus the closest option to the computed value must be chosen. Always calculate the exact value first and then match it to the nearest option as needed.

Final Answer:
Thirty seven men will require approximately 4 days to complete the work.