How to Collaborate Efficiently: A Case Study of Huong and Molly Cleaning an Attic

The efficiency of collaboration can be illustrated vividly through a practical scenario involving Huong and Molly, who are adept at cleaning attics. Specifically, it takes Huong 11 hours to complete the task alone, while Molly can finish it in 9 hours. By understanding the individual and combined work rates of Huong and Molly, we can derive a more efficient methodology for such tasks.

Understanding Individual Work Rates

We begin by calculating the individual work rates of Huong and Molly. Huong's rate of work is defined as 1/11 of the attic per hour, indicating that for every hour, he completes 1/11th of the job. Similarly, Molly's work rate is 1/9 of the attic per hour, meaning she accomplishes 1/9th of the task in an hour. This understanding forms the foundation for our analysis.

Combined Work Rate

When Huong and Molly work together, their combined work rate becomes the sum of their individual rates. Mathematically, this is calculated as follows:

Combined Rate (1/11) (1/9) (9 11) / 99 20/99

The combined work rate of 20/99 indicates that together, they can complete 20/99 of the attic in one hour. To calculate the time required for them to complete the job together, we take the reciprocal of the combined rate:

Time Taken 1 / (20/99) 99/20 4.95 hours

This conversion reveals that with collaborative efforts, the two can complete the attic cleaning in approximately 4 hours and 57 minutes, which is a 50% reduction in time from their individual efforts.

Calculating Through Least Common Multiple (LCM)

For a more detailed analysis, let's use the least common multiple (LCM) method. Assigning the total work as the LCM of 11 and 9 (which is 99 units), we can establish:

Huong's work rate: 9 units per hour Molly's work rate: 11 units per hour

When combined, their rate is 20 units per hour. Thus, the time taken to complete the 99 units of work is:

Time Taken 99 / 20 4.95 hours 4 hours 57 minutes

Comparison with Another Scenario

To further solidify our understanding, let's consider the case of Amanda and Imani, a similar pair tasked with cleaning another attic. Amanda can clean 1/16 of the attic in one hour, while Imani can clean 1/11 of the attic in one hour. Together, they can clean 1/16 1/11 27/176 of the attic in one hour. Thus, the time required for both to complete the job together is:

Time Taken 176 / 27 6.51 hours ≈ 6 hours 31 minutes

Both examples emphasize the significant benefits of collaboration in achieving tasks more efficiently. For Huong and Molly, working together results in a time reduction, while Amanda and Imani showcase a different time frame but similar principles of combined effort.

Conclusion

In summary, understanding the individual and combined work rates of team members is crucial in determining the efficiency of collaboration. By applying the principles of work rates and using methods like the least common multiple, we can effectively estimate how long it will take a pair to complete a task together. This approach not only optimizes time but also highlights the value of working collaboratively.