A canal of a village can be cleaned by 24 villagers in 12 days. In how many days will 36 villagers be able to clean the same canal, assuming everyone works at the same constant rate?

Introduction / Context:
This problem is based on the concept of man-days in time and work. It is a direct application of the inverse relationship between the number of workers and the time required, given a fixed total amount of work. We know how many days 24 villagers take, and we must find the corresponding time if 36 villagers work instead.

Given Data / Assumptions:
- 24 villagers can clean the canal in 12 days.
- We need the number of days required for 36 villagers to clean the same canal.
- Every villager works at the same constant rate.
- The total amount of cleaning work remains the same.

Concept / Approach:
The total work can be measured in man-days, which is the product of the number of workers and the days they work. First, we compute the total work in man-days using the original group of villagers. Then, we divide this total man-day value by the new number of workers (36) to obtain the required time with the larger group.

Step-by-Step Solution:
Step 1: Compute total work in man-days using the initial scenario. Step 2: Total work = 24 villagers * 12 days = 288 man-days. Step 3: When 36 villagers work, let the required number of days be D. Step 4: Then 36 * D = 288 man-days, because total work remains the same. Step 5: Solve for D: D = 288 / 36 = 8 days.

Verification / Alternative check:
We can verify this by using proportional reasoning. Time is inversely proportional to the number of workers when work is fixed. So required days = 12 * (24 / 36) = 12 * (2 / 3) = 8 days. This matches the man-day method and confirms that the answer is correct and consistent.

Why Other Options Are Wrong:
- 18 and 16 days: These are greater than the original 12 days, which does not make sense because more workers should reduce, not increase, the time required.
- 72 days: This is far too large and corresponds to a misunderstanding of the relationship between workers and time.
- 6 days: This would require even more total man-days than available or a higher per-worker efficiency than given by the data.

Common Pitfalls:
A frequent mistake is failing to use the man-day concept and simply guessing based on rough intuition. Some learners also incorrectly treat the relationship as directly proportional instead of inversely proportional. Always remember that for a fixed job, doubling the number of workers roughly halves the time, provided all other conditions remain the same.

Final Answer:
36 villagers can clean the canal in 8 days.