Calculating the Work Rate Together: Tom and Huck Painting the Fence

In this article, we explore the process of determining how long it will take Tom and Huck to paint Mr. Thatcher's fence together. By breaking down the task into manageable steps and applying the concept of combined work rate, we will arrive at a precise answer.

Introduction

Let's consider a scenario where Tom can complete the painting of Mr. Thatcher's fence in 6 hours, and Huck can complete the same task in 5 hours. The question is: how long will it take them to paint the fence if they work together?

Understanding Individual Work Rates

First, we need to determine the individual work rates of Tom and Huck.

Tom's rate: Tom can paint 1}{6} of the fence per hour.

Huck's rate: Huck can paint 1}{5} of the fence per hour.

Combining the Work Rates

To find out their combined work rate, we add their individual rates together.

Step 1: Find a Common Denominator

The least common multiple (LCM) of 6 and 5 is 30. This is our common denominator.

Step 2: Convert Fractions

Convert 1}{6} to a fraction with a denominator of 30: 1}{6} 5}{30}frac> Convert 1}{5} to a fraction with a denominator of 30: 1}{5} 6}{30}frac>

Step 3: Add the Fractions

Add the two fractions: 5}{30} 6}{30} 11}{30}frac> This means their combined rate is 11}{30} of the fence per hour.

Calculating the Total Time Together

Now, to find out how long it will take to complete the entire fence, we take the reciprocal of the combined rate.

Time 1}{frac{11}{30}} 30}{11}frac> hours.

Converting to Hours and Minutes

30}{11} hours is approximately 2.727 hours. Convert 2.727 hours to hours and minutes: 2 hours 0.727 hours × 60 minutes/hour ≈ 2 hours and 43.62 minutes.

Rounded to the nearest minute, it will take them approximately 2 hours and 44 minutes to paint the fence together.

Educational Context and Practical Applications

In terms of educational context, this problem helps students understand the concept of combined work rates, which is an important part of arithmetic and algebra. Practically, it can be applied to real-world scenarios such as estimating the completion time for collaborative projects, understanding efficiency, and planning tasks.

Man Hours and Task Completion

For a more practical approach, we can also use the concept of man hours. The painting task requires 6 man-hours in total. If two people are doing the job, it will take them 3 days to complete (6 man-hours ÷ 2 people 3 days).

Assumption and Real-World Considerations

When solving such problems, it's important to make assumptions. For instance, we assumed that both Tom and Huck can work simultaneously and without any overlap or delay. In the real world, factors like communication, collaboration, and task-specific challenges can affect the actual time. However, using the combined work rate method, we can make a good estimate. Other factors, like the difficulty of the fence or any obstacles they might face, can be considered in a more detailed scenario analysis.

Conclusion

In conclusion, by using the combined work rate method, we determined that it will take Tom and Huck approximately 2 hours and 44 minutes to paint Mr. Thatcher's fence together. Understanding work rates and their applications can greatly enhance problem-solving skills in both educational and professional settings.