Equation For Bethany And Colin Mowing Lawn Together

#h1 Bethany can mow her family's lawn in 4 hours. Her brother Colin can mow the lawn in 3 hours. Which equation can be used to find the number of hours, x, it would take for Bethany and Colin to mow the lawn together?

Understanding how to solve problems involving rates of work is a fundamental skill in mathematics. This problem presents a classic scenario where two individuals, Bethany and Colin, are working together to complete a task – mowing the lawn. Bethany can mow the lawn in 4 hours, while Colin can do it in 3 hours. The question asks us to determine the equation that would help us find the time it takes for them to mow the lawn together. This involves understanding the concept of work rate and how individual rates combine when people work together. To solve this, we need to consider the fraction of the lawn each person can mow in an hour and then combine these fractions to find their combined work rate. This combined rate will then help us formulate the equation to solve for the time x they take to mow the lawn together. This problem not only tests our mathematical skills but also our ability to apply these skills to real-world scenarios.

Setting up the Problem: Work Rate

The cornerstone of solving this problem lies in understanding the concept of work rate. Work rate is the amount of work done per unit of time. In this scenario, the 'work' is mowing the entire lawn, and the unit of time is hours. Bethany's work rate is the fraction of the lawn she can mow in one hour, and similarly, Colin's work rate is the fraction of the lawn he can mow in one hour. Let's delve deeper into how we calculate individual work rates and how they contribute to the overall solution.

Calculating Individual Work Rates

To determine Bethany's work rate, we consider that she completes the entire lawn in 4 hours. This means that in one hour, she mows 1/4 of the lawn. Therefore, Bethany's work rate is 1/4 lawns per hour. Similarly, Colin completes the lawn in 3 hours, so in one hour, he mows 1/3 of the lawn. Thus, Colin's work rate is 1/3 lawns per hour. Understanding these individual rates is crucial because when they work together, their rates combine to complete the task faster. The next step is to figure out how these rates add up and how we can use this combined rate to find the time they take to mow the lawn together.

Combining Work Rates

When Bethany and Colin work together, their individual work rates combine. This means that the fraction of the lawn Bethany mows in an hour is added to the fraction of the lawn Colin mows in an hour. Mathematically, this is represented as the sum of their individual work rates. If we let x be the time in hours it takes for them to mow the lawn together, then their combined work rate is the fraction of the lawn they complete together in one hour. This combined rate is essential for formulating the equation that will solve for x. The combined effort leads to a faster completion time, and understanding how to quantify this is a critical step in solving this problem.

Formulating the Equation

To find the equation, we need to express the combined work rate in terms of the time x it takes for Bethany and Colin to mow the lawn together. The fundamental principle here is that the combined work rate multiplied by the time spent working together should equal the total work done, which in this case is mowing one entire lawn. This principle allows us to bridge the gap between the individual work rates we calculated earlier and the unknown time x.

Expressing Combined Work Rate

We know Bethany's work rate is 1/4 lawns per hour and Colin's work rate is 1/3 lawns per hour. When they work together, their work rates add up. Therefore, their combined work rate is (1/4 + 1/3) lawns per hour. To add these fractions, we need a common denominator, which is 12. So, we rewrite the fractions as 3/12 and 4/12, respectively. Adding these together, we get a combined work rate of 7/12 lawns per hour. This means that together, Bethany and Colin can mow 7/12 of the lawn in one hour. Now, we can use this combined work rate to formulate the equation.

Constructing the Equation

Since their combined work rate is 7/12 lawns per hour, and they work together for x hours, the total work done is (7/12) * x lawns. We know that the total work done is mowing one entire lawn, so we can set up the equation: (7/12) * x = 1. This equation represents the relationship between their combined work rate, the time they work together, and the completion of the task. Solving this equation for x will give us the number of hours it takes for Bethany and Colin to mow the lawn together. However, the question asks for the equation itself, not the solution. Therefore, our focus is on ensuring the equation accurately reflects the problem's conditions.

Alternative Equation Forms

While the equation (7/12) * x = 1 is a valid representation of the problem, it's important to recognize that there are other equivalent forms of the equation that could also be used. These alternative forms are derived from the same fundamental principles but may be expressed differently. Understanding these variations is crucial for a comprehensive grasp of the problem and its solution.

Considering Individual Contributions

Another way to think about the problem is to consider the fraction of the lawn each person mows when working together for x hours. Bethany mows at a rate of 1/4 lawns per hour, so in x hours, she mows x/4 of the lawn. Similarly, Colin mows at a rate of 1/3 lawns per hour, so in x hours, he mows x/3 of the lawn. Together, they mow the entire lawn, which is represented as 1. This leads to the equation: x/4 + x/3 = 1. This equation directly represents the sum of the fractions of the lawn each person mows, equaling the entire lawn.

Comparing Equation Structures

The equation x/4 + x/3 = 1 is algebraically equivalent to (7/12) * x = 1. To see this, we can find a common denominator for the fractions in the first equation and combine them. This process demonstrates that different approaches to setting up the equation can lead to equivalent expressions. The key is to ensure that the equation accurately represents the relationships between the work rates, time, and total work done. Understanding these different perspectives can enhance problem-solving skills and provide a deeper understanding of the underlying mathematical concepts.

Conclusion: Equation to Find Combined Mowing Time

In conclusion, we explored the problem of Bethany and Colin mowing their family's lawn together by breaking it down into manageable parts. We began by understanding the concept of work rate, calculated individual work rates for Bethany and Colin, and then combined these rates to find their combined work rate. We formulated the equation x/4 + x/3 = 1, which represents the sum of the fractions of the lawn each person mows, equaling the entire lawn. This equation accurately captures the relationships between their individual work rates, the time they work together, and the total work done. We also discussed an alternative form of the equation, (7/12) * x = 1, and demonstrated how both equations are equivalent and represent the same problem from different perspectives. Understanding how to set up and interpret these equations is a valuable skill in problem-solving and demonstrates a strong grasp of mathematical principles.

Therefore, the equation that can be used to find the number of hours, x, it would take for Bethany and Colin to mow the lawn together is x/4 + x/3 = 1. This equation not only provides a solution to the specific problem but also illustrates a broader concept of combined work rates, which is applicable in various real-world scenarios. By mastering these concepts, individuals can confidently tackle similar problems involving rates and time, making it a valuable skill in mathematics and beyond.