Planting Trees: Finding Combined Time Equation
Introduction
Hey guys! Ever wondered how to calculate the time it takes for two people to complete a task together? This is a classic problem, and in this article, we're diving into a scenario where Bill and Rose are teaming up to plant 100 trees. Bill can plant all 100 trees in 45 hours, while Rose can do it in 40 hours. The big question is: How do we figure out the equation to find the time it takes for them to plant the trees together? Let's break it down step by step, making it super easy to understand. We'll explore the concepts of work rate, combined effort, and how to translate these into a mathematical equation. So, grab your thinking caps, and let's get started on this tree-planting puzzle!
When we talk about work rate, we're essentially looking at how much of the job someone can do in one unit of time (in this case, an hour). Think of it like this: if Bill can plant 100 trees in 45 hours, he plants a fraction of those trees every hour. Similarly, Rose has her own work rate. When they work together, their work rates combine, allowing them to complete the task faster than either could individually. To form the equation, we'll need to express each person's work rate mathematically and then add them up to find their combined work rate. The combined work rate will then help us determine the total time it takes for them to plant all 100 trees together. This involves a bit of algebraic manipulation, but don't worry, we'll guide you through it. The key is to understand the relationship between work, rate, and time, and how these elements interact when people work collaboratively. By the end of this article, you'll not only know the equation but also the reasoning behind it, making you a pro at solving similar problems. So, let's get those mental shovels ready and dig into the solution!
Understanding Individual Work Rates
Okay, first things first, let's figure out how much work Bill and Rose can do individually in one hour. This is what we call their individual work rates. For Bill, if he takes 45 hours to plant 100 trees, then in one hour, he plants 1/45 of the total job. Think of it like dividing the entire task into 45 equal parts, and Bill completes one part every hour. Similarly, Rose can plant 100 trees in 40 hours, so in one hour, she plants 1/40 of the total job. See? It's all about fractions! These fractions represent their efficiency or speed at planting trees.
To put it simply, Bill's work rate is 1/45 (of the job per hour), and Rose's work rate is 1/40 (of the job per hour). Now, why is this important? Well, when they work together, their individual efforts combine. It's like having two people pushing a car – the combined force gets the job done faster. In our case, their combined work rate will tell us how much of the tree-planting they accomplish together in one hour. This is a crucial piece of the puzzle because it directly relates to how quickly they can finish the entire task. By understanding their individual contributions, we can then calculate their combined efficiency and ultimately determine the time it takes for them to plant all 100 trees as a team. So, with their individual work rates in hand, we're one step closer to cracking the code of this problem. Let's move on to see how their efforts merge when they work side-by-side!
Combining Work Rates
Alright, now for the exciting part: figuring out how Bill and Rose's work rates combine when they work together! Remember, Bill plants 1/45 of the trees in an hour, and Rose plants 1/40 in the same time. So, when they team up, we simply add their work rates to find their combined work rate. This means we need to add the fractions 1/45 and 1/40. But hold on, we can't just add them directly because they have different denominators. We need to find a common denominator first. The least common multiple of 45 and 40 is 360. So, we convert the fractions: 1/45 becomes 8/360, and 1/40 becomes 9/360.
Now we can easily add them: 8/360 + 9/360 = 17/360. This means that together, Bill and Rose plant 17/360 of the trees in one hour. Think of it as if they've formed a super-planting team, accomplishing more work together than they could alone. This combined work rate is the key to unlocking the final piece of the puzzle: the total time it takes them to plant all 100 trees. We now know how much of the job they complete in one hour, but we need to flip this information to find out how many hours it takes them to complete the entire job (planting all 100 trees). This involves a simple but crucial mathematical step, and once we've done that, we'll have the equation we're looking for. So, let's move on and see how we can use this combined work rate to calculate the total time!
Formulating the Equation
Okay, guys, we're in the home stretch now! We know that Bill and Rose together plant 17/360 of the trees in one hour. Let's say it takes them 'p' hours to plant all 100 trees together. In 'p' hours, they would complete the entire job, which we can represent as 1 (or 100% of the job). So, we can set up an equation that relates their combined work rate, the time 'p', and the completion of the job.
The equation looks like this: (Combined Work Rate) * (Time) = (Total Work Done). In our case, this translates to (17/360) * p = 1. This equation is the heart of the solution! It tells us that the fraction of work they do per hour (17/360), multiplied by the number of hours they work (p), equals the entire job (1). Now, this equation might look simple, but it encapsulates the entire problem-solving process we've gone through. We've taken individual work rates, combined them, and used that information to create a mathematical statement that allows us to find the unknown time 'p'. To solve for 'p', we would simply multiply both sides of the equation by the reciprocal of 17/360, which is 360/17. This would isolate 'p' on one side of the equation and give us the numerical answer. However, the question only asks for the equation, and we've nailed it! This equation (17/360) * p = 1 is the key to finding the time it takes for Bill and Rose to plant all 100 trees together. So, we've successfully translated a real-world problem into a mathematical model, and that's a pretty awesome feeling!
Identifying the Correct Equation Format
Now, let's think about how this equation might look in different formats. The core idea is (17/360) * p = 1, but there are other ways to represent this mathematically. Remember, 17/360 came from adding Bill's and Rose's individual work rates, which were 1/45 and 1/40 respectively. So, we can rewrite 17/360 as (1/45 + 1/40). This means our equation can also be written as (1/45 + 1/40) * p = 1. This form of the equation highlights how the individual work rates contribute to the combined effort. It's like showing the building blocks that make up the final equation.
Another way to think about it is to distribute the 'p' across the terms inside the parentheses. This would give us (1/45) * p + (1/40) * p = 1. This version of the equation is also perfectly valid and might even be easier for some people to understand. It breaks down the problem into how much of the job Bill does in 'p' hours plus how much of the job Rose does in 'p' hours, which together equals the entire job (1). So, while the equation (17/360) * p = 1 is the most concise form, recognizing these alternative representations is crucial. Depending on the options presented in a multiple-choice question, you might need to identify one of these equivalent forms. The key is to understand the underlying concept – that the combined work rate multiplied by the time equals the total work done – and be flexible in how you express it mathematically. With this understanding, you'll be able to spot the correct equation, no matter how it's disguised!
Conclusion
So, there you have it, guys! We've successfully navigated the world of combined work rates and tree planting. We started with the individual efforts of Bill and Rose, combined their work rates, and formulated an equation to find the time it takes for them to plant 100 trees together. The key takeaway here is the relationship between work rate, time, and the total work done. By understanding this connection, you can tackle similar problems with confidence. Whether it's painting a house, mowing a lawn, or, in this case, planting trees, the principles remain the same.
Remember, breaking down the problem into smaller, manageable steps is crucial. First, find the individual work rates. Second, combine them. And third, use the combined work rate to form an equation that relates to the total time taken. With practice, these steps will become second nature, and you'll be solving these types of problems like a pro. And that's it for this problem! I hope you found this explanation helpful and engaging. Keep practicing, keep exploring, and keep those mathematical gears turning. Until next time, happy problem-solving!
