A group of men can finish a piece of work in 100 days. If there were 10 men fewer, it would take 10 days more to finish the work. How many men were there originally?

Introduction / Context:
This is a classic time and work problem using the concept that total work equals the product of the number of workers and the duration when each worker has a constant rate. The question compares two situations with different numbers of men and different durations for the same total work. It is an excellent example of how to set up and solve simple equations based on proportional reasoning.

Given Data / Assumptions:

• A certain number of men, say M, can finish a piece of work in 100 days.
• If there are 10 fewer men, that is M - 10 men, the work takes 110 days.
• The nature of work does not change between the two scenarios.
• Each man works at a constant rate throughout the job.

Concept / Approach:
The total work can be written as men * days * individual rate. Since the kind of work remains the same, the total amount of work is constant across both scenarios. Thus, we can equate the work expressions for the two cases. The individual rate cancels out, leaving an equation involving M, the original number of men, and the given durations. Solving this equation yields M.

Step-by-Step Solution:
Step 1: Let the original number of men be M.Step 2: In the first situation, the total work W can be expressed as W = M * 100 * r, where r is the rate of one man.Step 3: In the second situation, the same work W is done by M - 10 men in 110 days, so W = (M - 10) * 110 * r.Step 4: Equate the two expressions for W and cancel r: M * 100 = (M - 10) * 110.Step 5: Expand the right hand side: M * 100 = 110M - 1100. Rearranging gives 110M - 100M = 1100, so 10M = 1100 and M = 110.
Verification / Alternative check:
We can verify by plugging M = 110 back into both scenarios. With 110 men, the work takes 100 days, corresponding to 110 * 100 = 11000 man days. With 100 men, because 10 have been removed, the work takes 110 days, corresponding to 100 * 110 = 11000 man days again. Both give the same total man days, confirming that the number of men originally was 110.

Why Other Options Are Wrong:
Seventy five men would give 75 * 100 = 7500 man days in the first case and 65 * 110 = 7150 in the second, which do not match.Eighty two men would produce 82 * 100 = 8200 and 72 * 110 = 7920 man days, which are unequal.One hundred men would give 100 * 100 = 10000 and 90 * 110 = 9900 man days, again inconsistent with a fixed total work.
Common Pitfalls:
Many students misplace the terms and end up with an incorrect equation such as M * 110 = (M - 10) * 100, which reverses the given condition. Others forget to cancel the common individual rate r, making the algebra seem more complicated than it is. Carefully translating the language into an equation where total work is equal in both cases is the key skill in solving this problem correctly.

Final Answer:
The original number of men was 110.