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

Introduction / Context:
This work and time question tests your ability to model the relationship between number of workers, duration, and total work using simple algebra. By comparing two scenarios with different numbers of men and days, you can set up an equation to find the unknown original workforce strength.

Given Data / Assumptions:
- Let the original number of men be x.
- With x men, the work is finished in 100 days.
- With (x - 10) men, the same work takes 110 days (10 days more).
- Work done is directly proportional to (number of men) * (number of days).
- All men work at the same constant rate.

Concept / Approach:
Total work is fixed. If W is the total work and each man works at the same rate, then:
- W = x * 100 (man-days) in the first case.
- W = (x - 10) * 110 (man-days) in the second case.
Since both expressions represent the same total work, we can equate them and solve for x, the original number of men.

Step-by-Step Solution:
Step 1: Express total work using the first situation: W = x * 100.Step 2: Express total work using the second situation: W = (x - 10) * 110.Step 3: Set the two expressions equal: x * 100 = (x - 10) * 110.Step 4: Expand the right-hand side: 100x = 110x - 1100.Step 5: Rearrange: 1100 = 110x - 100x = 10x.Step 6: Solve for x: x = 1100 / 10 = 110.

Verification / Alternative check:
With 110 men, total work W = 110 * 100 = 11,000 man-days. If there are 10 fewer men, i.e. 100 men, to do the same work, days needed = 11,000 / 100 = 110 days. This matches the condition given in the problem, confirming that 110 is correct.

Why Other Options Are Wrong:
If you choose 100 or 120, the resulting man-days in the second scenario will not equal the first scenario's total, breaking the equality of work. Values like 75 or 82 arise from algebraic or arithmetic mistakes, such as mis-distributing 110 or misplacing terms when equating the two expressions.

Common Pitfalls:
Common errors include forgetting that total work must be equal in both cases, mixing up which scenario has more days, or incorrectly simplifying the equation (for example, subtracting 110 from both sides instead of handling the x terms correctly). Writing the basic equation of man-days carefully and solving stepwise avoids these mistakes.

Final Answer:
The original number of men was 110.