Determining the Optimal Number of Men for Enhanced Work Output

In the context of project management and labor efficiency, understanding the relationship between the number of workers, the duration of work, and the total man-hours is crucial. This article explores a specific problem: how to calculate the number of men required to complete a task that is ten times more extensive. We will break down the steps involved and apply the principle of man-hours to achieve a more efficient and scalable solution.

Step-by-Step Breakdown

To solve this problem, we will follow a structured approach, considering the original scenario, scaling it up to the new requirements, and then applying the principles of man-hours to find the optimal number of men needed.

Calculating the Initial Man-Hours

Let's begin with the initial scenario, where 6 men can complete a piece of work in 30 days, working 9 hours each day. To determine the total man-hours, we use the formula:

[ text{Total work} text{Number of men} times text{Number of days} times text{Hours per day} ]

Substituting the given values:

[ text{Total work} 6 times 30 times 9 1620 text{ man-hours} ]

Scaling Up the Work

Now, let's determine the total amount of work needed for the new scenario, which is ten times the original work:

[ text{Total work needed} 10 times 1620 16200 text{ man-hours} ]

Calculating Available Man-Hours for the New Scenario

In the new scenario, we need to distribute the 10 times the work among the men working for 25 days, but with 8 hours per day. The total man-hours available can be calculated as follows:

[ text{Total man-hours available} x times 25 times 8 ]

Where ( x ) is the number of men required for the new task.

Setting Up the Equation

Set up the equation to find the number of men, ( x ), needed:

[ 25 times 8 times x 16200 ]

[ 20 16200 ]

[ x frac{16200}{200} 81 ]

Therefore, 81 men are required to complete 10 times the amount of work if they work for 25 days of 8 hours each.

Alternative Approaches for Scaling Up Work

Let's explore a couple of additional approaches to solve a similar problem but for tasks of varying scales.

Approach 1: Man-Hours Calculation for a Different Scenario

Consider the scenario where we need to perform a task comparable to the original one, but the number of men required is 6.75. To find the total man-hours needed:

[ 6.75 times 9 times 30 1921.5 text{ man-hours} ]

For 8 times the work:

[ 8 times 1921.5 15372 text{ man-hours} ]

With 5 men working 8 hours a day for 25 days:

[ 5 times 8 times 25 1000 text{ man-hours} ]

Thus, the number of men required is:

[ 15372 div 1000 15.372 ]

Rounding up, we need 16 men for 8 times the work.

Approach 2: Using Proportional Reasoning

Utilizing the principle of proportional reasoning, we have:

[ M1H1D1 M2H2D2 ]

Substituting the given values:

[ 5 times 9 times 30 M2 times 8 times 25 ]

[ 1350 M2 times 200 ]

[ M2 frac{1350}{200} 6.75 ]

This is for one unit of work. Therefore, for 8 times the work:

[ 6.75 times 8 54 ]

Thus, 54 men are required.

Conclusion

The optimal number of men to complete a task ten times more extensive, working for 25 days, 8 hours per day, is 81. This solution emphasizes the importance of precise calculations and the efficient use of man-hours to optimize work productivity.

Keywords: man-hours, work efficiency, labor productivity