Introduction

Efficiently managing resources to complete tasks within specified time frames is a critical skill in project management. This article delves into a real-world problem where determining the number of additional workers needed to complete a job within a given time frame is central. We will use mathematical modeling and logical reasoning to find the solution, ensuring optimal workforce utilization.

Problem Statement

Given the scenario that 6 workers can paint 4 houses in 8 days, how many additional workers are required to paint 8 houses in 4 days?

Step-by-Step Analysis

Step 1: Calculate the Current Work Rate

We start by analyzing the initial condition where 6 workers can paint 4 houses in 8 days. We need to determine the work rate, expressed in worker-days per house.

Formula: Total worker-days Number of workers times; Number of days

Calculation:

Total worker-days needed to paint 4 houses:

Total worker-days 6 workers times; 8 days 48 worker-days

Worker-days per house:

Worker-days per house frac{48 worker-days}{4 houses} 12 worker-days per house

Step 2: Calculate the Total Worker-Days Needed for 8 Houses

Now, we need to find out how many worker-days are required to paint 8 houses, given the previous calculation.

Total worker-days for 8 houses 12 worker-days per house times; 8 houses 96 worker-days

Step 3: Determine the Number of Workers Needed to Finish in 4 Days

Next, we aim to determine the number of workers required to complete the work in 4 days. We set up the following equation to find the number of workers needed:

Workers needed times; 4 days 96 worker-days

Solving for workers needed:

Workers needed frac{96 worker-days}{4 days} 24 workers

Rewritten using proportional reasoning:

Houses ………………… …………………………Workers

4 ……………………… …………………………6

8 ……………………… …………………………x

x 8/4 * 6 12 current workers needed, but we need:

Total: 24 workers - 6 current workers 18 additional workers

Step 4: Conclusion and Solution

In conclusion, to finish painting 8 houses in 4 days, 18 additional workers are needed.

Rewritten Solution:

To calculate the number of workers needed, we can use the original work rate equation:

$frac{4}{6 times 8} frac{8}{w times 4}$

By solving for w (the total workers required), we get:

align'center'>#34;(1/12) (2/w)
#34;w 2 times; 12 24 workers
#34;ANSWER: 18 more workers are needed.

Pertinent Equations and Formulas

Original Work Rate
$frac{4}{6 times 8} frac{8}{w times 4}$

Proportional Reasoning
Houses ………………… …………………………Workers
4 ……………………… …………………………6
8 ……………………… …………………………_
8/4 * 6 12 current workers needed, but 24 total workers - 6 current workers 18 additional workers

Conclusion

The problem of determining the optimal number of workers needed to complete a project within a specified time frame is a practical application of mathematical analysis. By carefully breaking down the problem and using step-by-step logical reasoning, we can determine the exact number of additional workers needed, ensuring efficient use of resources.

The solution shows that 18 additional workers are necessary to complete painting 8 houses in 4 days, leveraging the initial work rate and proportional reasoning.