Collaboration and Efficiency: How Long Would It Take for Mary and Mike to Complete Their Task Together?

When two individuals with different work rates collaborate, the outcome can often be surprising and effective. Let's explore this scenario as we delve into a practical example involving Mary and Mike. We'll determine how long it would take for them to complete their task together, and how their efficiency changes when working as a team.

Introduction to Work Rates

In many real-world scenarios, understanding the concept of work rate is crucial. A work rate is the amount of work a person can complete in a given amount of time. In this article, we'll calculate the combined work rate of Mary and Mike and determine the time it takes for them to complete their task together.

Calculating Work Rates

Let's start by calculating the individual work rates of Mary and Mike. We know that:

It takes Mary 6 days to complete a task. Therefore, her work rate is (frac{1}{6}) of the task per day. It takes Mike 8 days to complete the same task. Therefore, his work rate is (frac{1}{8}) of the task per day.

Combining Work Rates

The combined work rate of Mary and Mike can be found by adding their individual work rates. To add fractions, we need a common denominator. The least common multiple of 6 and 8 is 24. Thus, we convert the fractions:

(frac{1}{6} frac{4}{24}) (frac{1}{8} frac{3}{24})

Now, we add these fractions:

(frac{4}{24} frac{3}{24} frac{7}{24})

So, their combined work rate is (frac{7}{24}) of the task per day. To find the time it takes for them to complete one whole task, we take the reciprocal of their combined work rate:

(t frac{1}{frac{7}{24}} frac{24}{7} approx 3.43) days or (3) days and (frac{3}{7}) of a day.

A Second Perspective on Combined Efficiency

Let's consider another example to solidify our understanding of combined efficiency:

A can complete a task in 6 days. Therefore, A completes (frac{1}{6}) of the work in a day. B in one day completes (frac{1}{12}) of the work.

When they work together, in one day they complete:

(frac{1}{6} frac{1}{12} frac{2}{12} frac{1}{12} frac{3}{12} frac{1}{4}) of the task.

Thus, together they complete the task in:

(t frac{1}{frac{1}{4}} 4) days.

Understanding Collaborative Efficiency

A skilled practical approach to solving this problem involves the following steps:

A completes (frac{1}{6}) of the work in a day. In a day, B completes half of what A does, which is (frac{1}{12}) of the work. The total work done in one day is (frac{1}{6} frac{1}{12} frac{1}{4}) of the task.

Therefore, the time required to complete the task is:

(t frac{1}{frac{1}{4}} 4) days.

This method shows that when working together, individuals can complete tasks more efficiently than when working alone. It's important to note that while A alone can complete the task in 6 days, B seems to take longer, so their combined effort requires more days than the individual's shortest completion time.

Conclusion

In conclusion, understanding the concept of work rate and combined efficiency can significantly enhance the effectiveness of collaborative workflows. By analyzing individual work rates and combining them, we can determine the optimal time to complete tasks. Whether it's Mary and Mike or A and B, the key is to recognize that collaboration often leads to better outcomes than working in isolation.