Ten men, working 6 hours a day, can complete a piece of work in 18 days. How many hours a day must 15 men work to complete the same work in 12 days?

Introduction / Context:
This question is a multi factor work problem involving number of men, hours per day and number of days. It tests the chain rule concept where total work is proportional to men * hours per day * days. Several quantities change, and you must adjust the remaining variable to keep the total work constant.

Given Data / Assumptions:

• 10 men working 6 hours per day complete the work in 18 days.
• All men work at a constant rate.
• New scenario: 15 men must complete the same work in 12 days.
• We must find required working hours per day in the new situation.

Concept / Approach:
Total work can be expressed as the product of men, hours per day and days. Since the work does not change between scenarios, the total man hours in both cases must be equal. We compute total man hours from the first scenario, then divide by the product of the new number of men and days to obtain the required hours per day.

Step-by-Step Solution:
Step 1: Total work in man hours for the first case = 10 men * 6 hours per day * 18 days.Step 2: Compute: 10 * 6 * 18 = 1080 man hours.Step 3: In the new case, number of men = 15, number of days = 12.Step 4: Let required hours per day be H.Step 5: Total man hours in the new case = 15 * H * 12.Step 6: Since work is the same, 15 * H * 12 = 1080.Step 7: Simplify: 15 * 12 = 180, so 180H = 1080, giving H = 1080 / 180 = 6 hours per day.

Verification / Alternative check:
Check with proportions. The number of men increases from 10 to 15, a factor of 1.5. The days decrease from 18 to 12, a factor of 12 / 18 = 2 / 3. To keep total man hours unchanged, hours per day must adjust so that (men factor) * (day factor) * (hour factor) = 1. So 1.5 * (2 / 3) * hour factor = 1, giving hour factor = 1. Therefore, the hours per day remain the same at 6, which matches the detailed solution.

Why Other Options Are Wrong:
4 or 5 hours per day would reduce the total man hours below 1080. 7 or 8 hours per day would give more man hours than needed, meaning the work would finish earlier than 12 days, contradicting the requirement to exactly match the original work.

Common Pitfalls:

• Multiplying or dividing the factors incorrectly when many quantities change at once.
• Trying to solve only with ratios and losing track of which factors are in the numerator or denominator.
• Forgetting that all three elements (men, hours per day, days) contribute equally to total man hours.

Final Answer:
The 15 men must work 6 hours a day to finish the work in 12 days.