How to Determine Work Rates: A Real-Life Example of Mowing the Lawn

In this article, we will explore how to calculate work rates through a real-life example involving mowing the lawn. This problem not only helps in understanding basic arithmetic but also demonstrates practical problem-solving techniques that can be applied in various scenarios.

Problem Introduction

Let's start with a scenario: John can mow the lawn in 4 hours by himself. When John and his younger brother Michael work together, it takes only 3 hours to mow the same lawn. The question is, how long would it take Michael to mow the lawn by himself, if he were to work solo?

Understanding Work Rates

Work rates are a powerful tool in solving scenarios where tasks are divided among individuals. The work rate is the inverse of the time taken to complete a task. This means if a task takes 4 hours, the work rate is 1/4 task per hour.

John's Work Rate

To begin, let's calculate John's work rate. Since John can complete the task (mow the lawn) in 4 hours, his work rate is:

John's Rate 1 lawn / 4 hours 1/4 lawn per hour.

Combined Work Rate

When John and Michael work together, they complete the task in 3 hours. Therefore, their combined work rate is:

Combined Rate 1 lawn / 3 hours 1/3 lawn per hour.

Michael's Work Rate

Now, let's establish Michael's work rate. We know the combined rate is 1/3 lawn per hour, and John's rate is 1/4 lawn per hour. Using the equation for their combined work rate, we can write:

John's rate Michael's rate Combined rate

1/4 Michael's rate 1/3

To find Michael's rate, we need to solve for Michael's rate:

Micheal's rate 1/3 - 1/4

Find a common denominator (12) and perform the subtraction:

Micheal's rate (4/12) - (3/12) 1/12 lawn per hour

This means Michael can complete 1/12 of the lawn in one hour.

Time for Michael to Mow the Lawn Solo

With Michael's work rate known, we can calculate the time it would take him to mow the entire lawn by himself:

Time for Michael 1 lawn / (1/12 lawn per hour) 12 hours

Thus, it would take Michael 12 hours to mow the lawn by himself.

Additional Scenarios

In a more complex scenario, consider the situation where John's enthusiasm for using a faster mower leads to a 30-minute argument. During these 30 minutes, both John and Michael do some mowing. If John can complete 10% of the lawn in 30 minutes and Michael can mow 25% in 15 minutes, we can still solve for their individual work rates.

Adjusted Work Rates

Let's assume John and Michael continue their mowing after the argument. In this case, the total time taken

To solve the adjusted scenario, we need to add the partial work done during the argument to the time taken to finish the lawn mowing. Given the constraints, the actual time taken would be:

Time 3 hours 30 minutes (john's mowing) 15 minutes (michael's mowing) 3.5 hours 10% (john's) 25% (michael's)

Precisely, in this case, the time is 5 2/25 hours, which simplifies to 10/7 hours.

Therefore, Michael would still take 12 hours to mow the lawn by himself, despite the distractions and partial work done during the argument.

Conclusion

Understanding and applying work rates can help solve a variety of real-life problems, from scheduling tasks to optimizing team productivity. This problem not only helps in improving arithmetic skills but also teaches careful consideration of all factors involved in a task.