Ten men, working 6 hours a day, can complete a certain piece of work in 18 days. At the same efficiency, how many hours per day must fifteen men work to complete the same work in 12 days?

Introduction / Context:
This is a time and work question involving three factors: number of men, hours per day, and days. It tests your ability to set up and use the relationship that total work is proportional to men * hours per day * number of days when everyone works at a constant rate.

Given Data / Assumptions:
- 10 men working 6 hours per day complete the work in 18 days.
- The work rate per man per hour is constant.
- We must find the number of hours per day, h, that 15 men must work to complete the same work in 12 days.

Concept / Approach:
We represent total work as the product of men, hours per day and number of days. The total work is the same in both situations. So, we equate the expressions for work in the two cases and solve for the unknown h. This is a direct extension of the man-day concept to man-hours.

Step-by-Step Solution:
Step 1: Express total work W using the first scenario.Step 2: W = number of men * hours per day * days = 10 * 6 * 18.Step 3: Compute: 10 * 6 * 18 = 10 * 108 = 1080 man-hours.Step 4: For the second scenario, we have 15 men working h hours per day for 12 days.Step 5: Work done in the second scenario = 15 * h * 12 man-hours.Step 6: Set total work equal: 15 * h * 12 = 1080.Step 7: Simplify: 15 * 12 = 180, so 180h = 1080.Step 8: Solve for h: h = 1080 / 180 = 6 hours per day.

Verification / Alternative check:
Check the total man-hours in the second scenario with h = 6 hours. 15 men * 6 hours/day * 12 days = 15 * 72 = 1080 man-hours, matching the original total. Thus the work equality holds, confirming that 6 hours per day is correct.

Why Other Options Are Wrong:
4 or 5 hours per day would give fewer total man-hours than 1080, so the work would remain incomplete. 7 or 8 hours per day would exceed 1080 man-hours, implying the work would finish before 12 days at that effort level. Only 6 hours per day maintains exactly the same total man-hours.

Common Pitfalls:
Some students forget to include all three factors (men, hours, days) and mistakenly equate only two. Others miscalculate 10 * 6 * 18 or 15 * 12. Keeping track of units (man-hours) and performing multiplications step by step will greatly reduce these errors.

Final Answer:
Fifteen men must work 6 hours a day to complete the work in 12 days.