Thirty-six men can complete a certain piece of work in 18 days. Assuming all men work at the same constant rate, in how many days will 27 men complete the same work?

Introduction / Context:
This is another basic time and work problem illustrating inverse variation between the number of workers and the time required to finish a fixed amount of work. It reinforces the idea of total work measured in man-days and how changes in workforce size affect completion time.

Given Data / Assumptions:
- 36 men complete the work in 18 days.
- We need the number of days required if 27 men do the same work.
- Total work remains unchanged.
- All men have the same constant efficiency.

Concept / Approach:
Total work can be expressed in man-days. If W is the total work, then:
- W = men * days.
Once we compute W using the first scenario, we divide by the new number of workers (27) to find the new time. Because work is fixed, the time and the number of workers are inversely proportional.

Step-by-Step Solution:
Step 1: Calculate total man-days in the first case: W = 36 * 18.Step 2: Compute: 36 * 18 = 648 man-days.Step 3: Let T be the number of days required for 27 men.Step 4: Then 27 * T must equal the same total work: 27 * T = 648.Step 5: Solve for T: T = 648 / 27.Step 6: Calculate 648 / 27: 27 * 24 = 648, so T = 24 days.

Verification / Alternative check:
Use inverse proportion: T2 / T1 = M1 / M2 = 36 / 27 = 4 / 3. So T2 = T1 * (4 / 3) = 18 * (4 / 3) = 18 * 4 / 3 = 24 days. This agrees perfectly with the man-day method and confirms the answer.

Why Other Options Are Wrong:
28, 34 or 35 days are larger than necessary and would imply more man-days than required to complete the same work. 20 days is too small and would give fewer man-days (27 * 20 = 540) than the original 648 man-days, so the work would remain incomplete.

Common Pitfalls:
Some learners mistakenly use direct proportion (assuming more men require more time) or perform division incorrectly, such as mixing up 36 / 27 with 27 / 36. Another common mistake is miscalculating 36 * 18. Writing out the intermediate products and checking the proportion logic helps avoid these issues.

Final Answer:
27 men will complete the work in 24 days.