Eight men can reap 80 hectares of a field in 24 days. Assuming all men work at the same constant rate, how many hectares can 36 men reap in 30 days under similar conditions?

Introduction / Context:
This is a multi-step chain rule problem based on work and time. It examines your understanding of how work (here measured in hectares reaped) changes when both the number of workers and the number of days change. The situation uses direct proportion among work, men, and days.

Given Data / Assumptions:
- 8 men reap 80 hectares in 24 days.
- All men work at the same constant rate.
- We need to find how many hectares 36 men can reap in 30 days.
- Work is directly proportional to (number of men) * (number of days).

Concept / Approach:
We can calculate the work rate per man-day in terms of hectares per man-day. Then, using that rate, we compute the total work (in hectares) done by 36 men over 30 days. This method ensures we handle both changes (men and days) correctly.

Step-by-Step Solution:
Step 1: Compute total man-days in the initial scenario: 8 men * 24 days = 192 man-days.Step 2: Total work done is 80 hectares, so the rate per man-day = 80 / 192 hectares.Step 3: Simplify 80 / 192: divide numerator and denominator by 16 → (80 / 16) / (192 / 16) = 5 / 12 hectares per man-day.Step 4: In the new scenario, we have 36 men working 30 days, so total man-days = 36 * 30 = 1080 man-days.Step 5: Total hectares reaped = 1080 * (5 / 12).Step 6: Compute 1080 / 12 = 90, so total hectares = 90 * 5 = 450 hectares.

Verification / Alternative check:
Use direct chain rule: hectares ∝ men * days. Factor change in men = 36 / 8 = 4.5, factor change in days = 30 / 24 = 1.25. Combined factor = 4.5 * 1.25 = 5.625. Therefore, new work = 80 * 5.625 = 450 hectares, confirming the previous calculation.

Why Other Options Are Wrong:
Values like 350, 400, 425 or 500 hectares come from incorrect scaling, such as adjusting only for extra men or only for extra days, or miscalculating the factor 1080 * (5 / 12). Only 450 hectares is consistent with the proportional change in both men and days.

Common Pitfalls:
Students often forget to divide 80 by the original man-days and instead try to adjust directly, leading to arithmetic errors. Another mistake is to treat men and days inconsistently, for example using a factor of 36 / 8 but forgetting to also include 30 / 24. Always compute the per man-day rate or carefully apply the chain rule with both ratios.

Final Answer:
36 men can reap 450 hectares in 30 days.