Lea And Marie Are Washing All The Windows In Their House. It Would Take Lea 80 Minutes By Herself, And It Would Take Marie 60 Minutes To Wash Them By Herself. Lea Creates A Table To Find How Long It Will Take To Wash The Windows If They Work Together.
Introduction
In the world of mathematics, problems often arise when individuals work together to achieve a common goal. Lea and Marie, two friends, are faced with the task of washing all the windows in their house. While they could work separately, they decide to collaborate and create a table to determine how long it will take to complete the task together. In this article, we will explore the mathematical concept of work rates and how Lea and Marie's collaboration can lead to a more efficient outcome.
Work Rates and Time
Before we dive into Lea and Marie's table, let's understand the concept of work rates and time. Work rate is the amount of work an individual can complete in a given time. In this case, Lea can wash the windows in 80 minutes, while Marie can do it in 60 minutes. To find their combined work rate, we need to calculate the fraction of work they can complete in one minute.
Calculating Work Rates
Let's calculate Lea's work rate first. Since she can wash the windows in 80 minutes, her work rate is 1/80 of the work per minute.
Lea's Work Rate
- Work rate = 1/80 of the work per minute
Now, let's calculate Marie's work rate. Since she can wash the windows in 60 minutes, her work rate is 1/60 of the work per minute.
Marie's Work Rate
- Work rate = 1/60 of the work per minute
Combined Work Rate
When Lea and Marie work together, their combined work rate is the sum of their individual work rates.
Combined Work Rate Formula
Combined work rate = Lea's work rate + Marie's work rate
Combined Work Rate Calculation
Combined work rate = 1/80 + 1/60
To add these fractions, we need to find a common denominator, which is 240.
Combined Work Rate Calculation (continued)
Combined work rate = (3/240) + (4/240) = 7/240 of the work per minute
Lea and Marie's Table
Now that we have the combined work rate, let's create a table to determine how long it will take for Lea and Marie to wash the windows together.
| Time (minutes) | Work Done |
|---|---|
| 1 | 7/240 |
| 2 | 14/240 |
| 3 | 21/240 |
| 4 | 28/240 |
| 5 | 35/240 |
| 6 | 42/240 |
| 7 | 49/240 |
| 8 | 56/240 |
| 9 | 63/240 |
| 10 | 70/240 |
| 11 | 77/240 |
| 12 | 84/240 |
| 13 | 91/240 |
| 14 | 98/240 |
| 15 | 105/240 |
| 16 | 112/240 |
| 17 | 119/240 |
| 18 | 126/240 |
| 19 | 133/240 |
| 20 | 140/240 |
| 21 | 147/240 |
| 22 | 154/240 |
| 23 | 161/240 |
| 24 | 168/240 |
| 25 | 175/240 |
| 26 | 182/240 |
| 27 | 189/240 |
| 28 | 196/240 |
| 29 | 203/240 |
| 30 | 210/240 |
| 31 | 217/240 |
| 32 | 224/240 |
| 33 | 231/240 |
| 34 | 238/240 |
| 35 | 245/240 |
| 36 | 252/240 |
| 37 | 259/240 |
| 38 | 266/240 |
| 39 | 273/240 |
| 40 | 280/240 |
| 41 | 287/240 |
| 42 | 294/240 |
| 43 | 301/240 |
| 44 | 308/240 |
| 45 | 315/240 |
| 46 | 322/240 |
| 47 | 329/240 |
| 48 | 336/240 |
| 49 | 343/240 |
| 50 | 350/240 |
| 51 | 357/240 |
| 52 | 364/240 |
| 53 | 371/240 |
| 54 | 378/240 |
| 55 | 385/240 |
| 56 | 392/240 |
| 57 | 399/240 |
| 58 | 406/240 |
| 59 | 413/240 |
| 60 | 420/240 |
| 61 | 427/240 |
| 62 | 434/240 |
| 63 | 441/240 |
| 64 | 448/240 |
| 65 | 455/240 |
| 66 | 462/240 |
| 67 | 469/240 |
| 68 | 476/240 |
| 69 | 483/240 |
| 70 | 490/240 |
| 71 | 497/240 |
| 72 | 504/240 |
| 73 | 511/240 |
| 74 | 518/240 |
| 75 | 525/240 |
| 76 | 532/240 |
| 77 | 539/240 |
| 78 | 546/240 |
| 79 | 553/240 |
| 80 | 560/240 |
| 81 | 567/240 |
| 82 | 574/240 |
| 83 | 581/240 |
| 84 | 588/240 |
| 85 | 595/240 |
| 86 | 602/240 |
| 87 | 609/240 |
| 88 | 616/240 |
| 89 | 623/240 |
| 90 | 630/240 |
| 91 | 637/240 |
| 92 | 644/240 |
| 93 | 651/240 |
| 94 | 658/240 |
| 95 | 665/240 |
| 96 | 672/240 |
| 97 | 679/240 |
| 98 | 686/240 |
| 99 | 693/240 |
| 100 | 700/240 |
Conclusion
In conclusion, Lea and Marie's collaboration has led to a more efficient outcome. By working together, they can complete the task in less time than if they were to work separately. The table created by Lea shows that it will take approximately 100 minutes for them to wash the windows together. This is a significant reduction in time compared to Lea's individual time of 80 minutes or Marie's individual time of 60 minutes.
Real-World Applications
The concept of work rates and collaboration is not limited to window washing. It can be applied to various real-world scenarios, such as:
- Team projects: When working on a team project, each member's work rate contributes to the overall project's completion time.
- Business operations: Companies can optimize their operations by assigning tasks to employees based on their work rates, leading to increased productivity and efficiency.
- Personal productivity: Individuals can apply the concept of work rates to their daily tasks, allocating time and resources accordingly to achieve their goals.
Final Thoughts
Introduction
In our previous article, we explored the concept of work rates and collaboration through the story of Lea and Marie, two friends who are washing all the windows in their house. Lea created a table to determine how long it will take for them to complete the task together. In this article, we will answer some frequently asked questions (FAQs) related to the concept of work rates and collaboration.
Q&A
Q: What is work rate?
A: Work rate is the amount of work an individual can complete in a given time. It is usually expressed as a fraction of the work per unit of time.
Q: How do you calculate work rate?
A: To calculate work rate, you need to divide the total work by the time it takes to complete the work. For example, if it takes 80 minutes to wash all the windows, Lea's work rate is 1/80 of the work per minute.
Q: What is the difference between individual work rate and combined work rate?
A: Individual work rate refers to the amount of work an individual can complete in a given time, while combined work rate refers to the amount of work two or more individuals can complete together in a given time.
Q: How do you calculate combined work rate?
A: To calculate combined work rate, you need to add the individual work rates of the two or more individuals working together. For example, if Lea's work rate is 1/80 and Marie's work rate is 1/60, their combined work rate is 1/80 + 1/60 = 7/240 of the work per minute.
Q: What is the significance of the table created by Lea?
A: The table created by Lea shows the amount of work completed by Lea and Marie together in a given time. It helps to visualize how their combined work rate affects the time it takes to complete the task.
Q: How can I apply the concept of work rates and collaboration to real-world scenarios?
A: You can apply the concept of work rates and collaboration to various real-world scenarios, such as team projects, business operations, and personal productivity. By understanding work rates and collaboration, you can optimize your time and resources to achieve your goals more efficiently.
Q: What are some common mistakes to avoid when working with others?
A: Some common mistakes to avoid when working with others include:
- Underestimating individual work rates: Make sure to accurately calculate individual work rates to avoid underestimating the time it takes to complete a task.
- Overestimating combined work rate: Be cautious not to overestimate the combined work rate, as this can lead to unrealistic expectations and disappointment.
- Failing to communicate: Communication is key when working with others. Make sure to clearly communicate your work rates, goals, and expectations to avoid misunderstandings.
Q: How can I improve my collaboration skills?
A: To improve your collaboration skills, focus on:
- Active listening: Listen carefully to others and ask questions to clarify their work rates and goals.
- Clear communication: Clearly communicate your work rates, goals, and expectations to avoid misunderstandings.
- Flexibility: Be flexible and adapt to changing circumstances and work rates.
- Respect: Respect others' work rates and goals, and be willing to compromise and adjust your approach as needed.
Conclusion
In conclusion, the concept of work rates and collaboration is essential in various real-world scenarios. By understanding work rates and collaboration, you can optimize your time and resources to achieve your goals more efficiently. Remember to avoid common mistakes, improve your collaboration skills, and apply the concept of work rates and collaboration to achieve success.
Final Thoughts
Lea and Marie's window washing adventure demonstrates the power of collaboration and the importance of understanding work rates. By working together, they can achieve a common goal more efficiently. As we continue to navigate the complexities of mathematics and real-world applications, it is essential to remember the value of collaboration and the impact it can have on our daily lives.