Collaborative Work Rate: How Machines A, B, and C Can Work Together More Efficiently

When machines A, B, and C work individually, they require 3, 6, and 9 hours, respectively, to finish a particular task. By understanding their work rates and combining them, we can determine how quickly they can complete the task together. This post will walk you through the process of calculating the collaborative work rate of these machines.

Understanding the Problem

Let's revisit the scenario: Machine A can complete a job in 3 hours, Machine B can complete the same job in 6 hours, and Machine C can do it in 9 hours. Our goal is to determine how long it will take for all three machines to finish the job together.

Calculating Individual Work Rates

The first step is to express the work rate of each machine as a fraction of the job completed per unit time:

Machine A: 1/3 of the job per hour Machine B: 1/6 of the job per hour Machine C: 1/9 of the job per hour

The combined work rate is the sum of the individual work rates. To add these fractions, we need a common denominator.

Using the Least Common Multiple (LCM)

The least common multiple (LCM) of 3, 6, and 9 is 18. We convert each fraction so that they share this common denominator:

1/3 6/18 1/6 3/18 1/9 2/18

Now we can add the fractions:

combined_work_rate  6/18   3/18   2/18  11/18

This means machines A, B, and C, working together, can complete 11/18 of the job in one hour.

Calculating the Total Time

To find the total time, T, required to complete the entire job, we use the equation:

Select the total work done per unit time from the combined work rate. Divide the total work (1 unit of work) by the combined work rate.

T  1 / (11/18)  18/11 hours

Calculating 18/11 gives approximately:

18/11 ≈ 1.636 hours ≈ 1 hour and 38.18 minutes

Therefore, it will take approximately 1 hour and 38 minutes for machines A, B, and C to complete the job when working together.

Additional Insights

Together, A, B, and C can complete 11/18 of the job in one hour. In 18 hours, A, B, and C will complete 6 3 2 11 jobs, each taking 1/11 of the original job.

Conclusion

Understanding the collaborative work rate is crucial for optimizing resource allocation and improving efficiency. By calculating the combined work rate and determining the total time required, we can effectively manage and allocate tasks among multiple machines.

Key Takeaways

Learn how to use the least common multiple (LCM) to combine work rates. Understand the concept of work rates and their significance in collaborative tasks. Calculate the total time required to complete a job using the combined work rate.