Effect of Changes in Workforce on Project Duration: A Detailed Analysis

In many industries, the productivity of a team is closely tied to the number of workers involved in a project. This relationship is especially important in fields like construction, where the completion of a task is often contingent on the availability and efficiency of the labor force. Consider the scenario where three people can paint and finish a house in 4 days. What would happen if one person did not come to work? How many extra days will it take for the remaining two people to complete the job?

1. Determining Work Rate and Workforce Impact

First, let’s break down the problem step by step. We need to understand the impact of changing the workforce on project duration. The initial information given is that three people can paint and finish a house in 4 days. This provides us with the work rate of the combined workforce.

Step 1: Calculate the Work Rate of Three People

The work rate of the three-person team can be calculated as follows:

Work rate of 3 people frac{1text{ house}}{4text{ days}} frac{1}{4}text{ houses per day}

Step 2: Determine the Work Rate of One Person

Since the three people can complete the house together, the work rate of one person would be one-third of the work rate of the team:

Work rate of 1 person frac{1}{3} times frac{1}{4} frac{1}{12}text{ houses per day}

Step 3: Calculate the Work Rate of Two People

When one person is absent, the work rate of the remaining two people will be double the work rate of one person:

Work rate of 2 people 2 times frac{1}{12} frac{1}{6}text{ houses per day}

Step 4: Calculate the Time Required for Two People to Complete the House

To determine how long it will take for two people to finish the house, we use the formula:

Time frac{1text{ house}}{text{Work rate of 2 people}} frac{1}{frac{1}{6}} 6text{ days}

Step 5: Calculate the Extra Days Needed

Originally, the three-person team could complete the house in 4 days. With two people, it takes 6 days. Therefore, the extra days needed are:

Extra days 6 - 4 2text{ days}

2. Alternative Calculations and Theoretical Insights

Let's explore a few alternative methods to confirm our calculations.

3/4x 1

Using the equation (frac{3}{4}x 1) where (x) represents the original time in days:

x frac{4}{3} approx 1.33text{ days}

Since we have 2 people working, the equation becomes (frac{2}{4}x 1), where (x) represents the new time in days:

x 2text{ days}

Theoretical analysis suggests that it would take one additional day for the remaining two people to complete the work.

2x/4 1

Here, we use the equation (frac{2x}{4} 1) to find (x):

2x 4 Rightarrow x 2text{ days}

Man-Days Calculation

Assuming the painting work requires 12 man-days (since (3text{ workers} cdot 4text{ days} 12text{ man-days})), with only 2 people working, it will take:

frac{12text{ man-days}}{2text{ workers}} 6text{ days}

Thus, the extra days needed are:

Extra days 6 - 4 2text{ days}

3. Conclusion

The analysis confirms that if one person does not come to work, it will take an additional 2 days to complete the painting and finishing of the house. This underscores the impact of worker availability and productivity on project duration in the construction industry.