Working Together: How Efficiently Can A and B Complete a Task?

When faced with complex tasks, understanding the combined efforts of multiple individuals can provide valuable insights into project timelines and resource optimization. This article explores how two individuals, A and B, can collaborate to complete a task more efficiently than when working alone. We'll dissect the problem of A and B completing a task that takes each of them 8 and 12 hours individually.

Understanding Individual Work Rates

First, let's understand the concept of work rates. A work rate is a measure of the amount of work an individual can complete in a unit of time. For this problem, we need to determine the individual work rates of A and B.

Calculating Individual Work Rates

A can complete the task in 8 hours, so A's work rate is:

Work rate of A 1/8 of the task per hour.

B can complete the task in 12 hours, so B's work rate is:

Work rate of B 1/12 of the task per hour.

Combining Work Rates

To find out how long it will take A and B to complete the task together, we need to combine their work rates:

Combined work rate (1/8) (1/12)

To perform the addition, we need a common denominator. The least common multiple (LCM) of 8 and 12 is 24. Therefore, we convert the fractions:

(1/8) 3/24

(1/12) 2/24

Now, add the fractions:

Combined work rate (3/24) (2/24) 5/24

Calculating the Time Taken Together

When the combined work rate of A and B is 5/24 of the task per hour, we can determine the time taken to complete 1 whole task by taking the reciprocal of the combined work rate:

Time taken 1 / (5/24) 24/5 hours 4.8 hours.

Thus, A and B working together will take 4.8 hours to complete the task.

Alternative Solutions: Additional Context and Calculations

For a different scenario, consider the following:

Scenario 2: A and B's Task Completion in Different Conditions

Let's assume A can complete the task in 10 hours, and B can complete the task in 15 hours. This means:

A completes 1/10 of the task per hour, and B completes 1/15 of the task per hour.

To find their combined work rate:

Combined work rate 1/10 1/15 3/30 2/30 5/30 1/6 of the task per hour.

Therefore, they can complete the task in:

Time 1 / (1/6) 6 hours.

Scenario 3: B's Incomplete Contribution

In another scenario, if B can only complete 2/3 of the task in 10 hours, he would take 15 hours to complete it alone. The work rates are:

A's work rate: 1/10 of the task per hour.

B's work rate: 2/3 of the task in 10 hours, so 1/15 of the task per hour.

Combined work rate:

Combined work rate 1/10 1/15 3/30 2/30 5/30 1/6 of the task per hour.

Hence, the task will be completed in:

Time 1 / (1/6) 6 hours.

Conclusion

Whether you approach the problem through different timeframes or under varying conditions, the key takeaway is that A and B working together can significantly reduce the time needed to complete a task. This is a practical application of work rates and combined work rates, useful in project management and real-world problem-solving.

Additional Resources

For more on work rate problems and other mathematical concepts, explore the following resources:

Work Rate Calculator Khan Academy Work Rate Problems Math Planet Work Rate Problem

Feel free to provide comments and further insights for a deeper understanding of these concepts.