Working 7 hours per day, 4 men can complete a certain piece of work in 8 days. If the men instead work 8 hours per day, how many men are required to complete the same work in 4 days?

Introduction / Context:

This question is a direct application of the concept of man hours in time and work problems. It tests the understanding that total work is proportional to the product of the number of workers, the number of days, and the number of hours worked per day, assuming constant efficiency. Problems like this regularly appear in competitive exams to check proportional reasoning skills.

Given Data / Assumptions:

• In the first scenario, 4 men work 7 hours per day for 8 days to finish the work.
• In the second scenario, men work 8 hours per day and must complete the work in 4 days.
• Efficiency of each man remains constant in both scenarios.
• Total work in both scenarios is exactly the same.

Concept / Approach:

We use the man hours approach: Work is proportional to number of men multiplied by hours per day multiplied by number of days. Since the work is the same in both cases, the total man hours in the first case should be equal to the total man hours in the second case. We set up an equation equating these two products and solve for the unknown number of men in the second scenario.

Step-by-Step Solution:

Verification / Alternative Check:

We can verify by substituting back. In the second scenario with 7 men: 7 * 8 * 4 = 224 man hours, which matches the 224 man hours of the first scenario. Since total man hours are the same and efficiency is constant, the work completed must be the same. Hence the calculation is consistent and confirms that 7 men are sufficient.

Why Other Options Are Wrong:

If 8 men worked, total man hours would be 8 * 8 * 4 = 256, which is more than needed and implies the work would be done earlier. If 4 men worked, total man hours would be 4 * 8 * 4 = 128, which is less than required, so the work would not finish in 4 days. Similarly, 9 men would give 288 man hours, far exceeding the required amount. Only 7 men produce the exact required total of 224 man hours.

Common Pitfalls:

Many learners confuse the relationship between time and number of workers, forgetting that hours per day and number of days also affect total work. Another common error is to adjust days and men but ignore the change in working hours per day. The safest method is always to compute total man hours in each scenario and equate them when the work is the same.

Final Answer:

The number of men required to complete the work in the new scenario is 7 men.