36 men can complete a piece of work in 18 days. If all men work at the same constant rate, in how many days will 27 men complete the same work?

Introduction / Context:
This question tests the basic concept of work and time when the amount of work is fixed and the number of workers changes. It is a standard direct and inverse proportion or chain rule problem that appears frequently in aptitude exams, project planning, and productivity based questions.

Given Data / Assumptions:

• 36 men can complete the entire work in 18 days.
• All men work at the same constant rate.
• The total work remains the same.
• We need the time taken by 27 men to complete the same work.

Concept / Approach:
The total amount of work can be represented in man days. This is the product of the number of men and the number of days. When more men work, the task finishes faster, and when fewer men work, it takes longer. Mathematically, for a fixed work, number of men and number of days are inversely proportional.

Step-by-Step Solution:
Step 1: Compute total work in man days: Work = 36 * 18.Step 2: 36 * 18 = 648 man days.Step 3: Let the required number of days for 27 men be D.Step 4: Since work is constant, 27 * D = 648.Step 5: Solve for D: D = 648 / 27 = 24 days.

Verification / Alternative check:
We can also use the inverse proportion shortcut. Time is inversely proportional to number of men, so D2 = D1 * (M1 / M2) = 18 * (36 / 27). The ratio 36 / 27 simplifies to 4 / 3. Thus D2 = 18 * (4 / 3) = 18 * 4 / 3 = 24 days, which matches the detailed calculation.

Why Other Options Are Wrong:
18 days would mean that reducing the number of men has no effect, which contradicts inverse proportion. 20 and 22 days are less than 24 and would imply that fewer men finish faster, which is impossible for identical workers. 27 days is greater than 24 but does not satisfy the correct proportional relationship obtained from man day calculations.

Common Pitfalls:

• Multiplying instead of dividing when using the ratio of men, leading to incorrect time.
• Forgetting that work is constant and trying to adjust both men and days independently.
• Doing mental arithmetic incorrectly when dividing 648 by 27.

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