Efficient Work Rate Calculation for Multiple Workers

Introduction

Efficiency in task completion is crucial in various fields, particularly in project management and task delegation. When dealing with multiple workers completing a task, it is essential to determine the combined effort and the time required to finish the work. This article will explore how to calculate the time taken by three individuals, A, B, and C, to complete a task when they work together, based on their individual working speeds.

Work Rate Calculation for Multiple Workers

Suppose three individuals A, B, and C are working on a task. A can complete the work in 20 days, B in 30 days, and C in 60 days. Our goal is to determine the time required for them to complete the work together.

Step 1: Determine Individual Work Rates

The work rate of each individual can be expressed as the fraction of the task completed per day. Let's calculate their individual work rates:

A's work rate: (frac{1}{20}) B's work rate: (frac{1}{30}) C's work rate: (frac{1}{60})

Step 2: Add the Work Rates Together

The combined work rate is the sum of the individual work rates:

Combined work rate (frac{1}{20} frac{1}{30} frac{1}{60})

To add these fractions, we need a common denominator. The least common multiple (LCM) of 20, 30, and 60 is 60. We then convert each fraction:

(frac{1}{20} frac{3}{60})

(frac{1}{30} frac{2}{60})

(frac{1}{60} frac{1}{60})

Adding these fractions:

Combined work rate (frac{3}{60} frac{2}{60} frac{1}{60} frac{6}{60} frac{1}{10})

This means that together, A, B, and C can complete (frac{1}{10}) of the work in one day.

The total time required to complete the work is the reciprocal of the combined work rate:

Total days (frac{1}{frac{1}{10}} 10) days

Conclusion

By adding the individual work rates and finding the reciprocal, we have determined that A, B, and C will complete the work together in 10 days. This method is useful for project management and task allocation in various industries, ensuring optimal resource utilization and efficient task completion.

Formula Summary

To find the combined work rate of multiple workers, calculate the individual work rates, add them, and then take the reciprocal of the sum.

Further Reading

To further explore similar problems and concepts, you can refer to the following resources:

Math is Fun: Rates Math Warehouse: Simplify Rational Expressions Khan Academy: Rational Expressions