Pine Vs. Fir: Solving Tree Sales At Taylor's Farm

Hey guys! Let's dive into a fun math problem about Taylor's Tree Farm. They had a busy November selling trees, and we're going to figure out exactly how many of each type they sold. So, grab your thinking caps, and let's get started!

Understanding the Tree Sales Scenario

In this section, we'll break down the information we have about Taylor's Tree Farm. Taylor's Tree Farm had a fantastic November, selling a grand total of 110 trees. That's a lot of trees! But what's really interesting is the mix of trees they sold. They weren't all the same; there were two main types: pine trees and fir trees. Now, here comes the juicy part: they sold a lot more pine trees than fir trees. Specifically, they sold 4.5 times as many pine trees as fir trees. This piece of information is crucial because it gives us a relationship between the number of pine trees and the number of fir trees. To make things a bit more mathematical, we can use letters to represent these quantities. Let's say 'p' stands for the number of pine trees sold and 'f' represents the number of fir trees sold. This allows us to translate the word problem into a mathematical equation, which makes it much easier to solve. So far, we know that the total number of trees sold is 110, which means that the number of pine trees plus the number of fir trees must equal 110. We can write this as our first equation: p + f = 110. This is a fundamental equation for solving our problem. It tells us the total quantity we're working with. But we also have another key piece of information: the relationship between the number of pine trees and fir trees. We know that the number of pine trees is 4.5 times the number of fir trees. We can write this as a second equation: p = 4.5f. This equation is equally important because it links the two variables we're trying to find. Now, we have a system of two equations with two variables. This is a classic setup for solving simultaneous equations, which is exactly what we'll do in the next section. These two equations give us a framework to find out precisely how many pine and fir trees Taylor's Tree Farm sold in November. By understanding these equations, we're one step closer to solving the mystery of the tree sales! Remember, math problems are just puzzles, and we've got all the pieces we need to put this one together. So, let's move on and see how we can use these equations to get our answer!

Setting Up the Equations

Now, let's translate the tree tale into math! We've got two key pieces of information that we can turn into equations. First, we know that Taylor's Tree Farm sold a combined total of 110 trees. To break it down, some were pine trees, and some were fir trees. If we use 'p' to represent the number of pine trees and 'f' to represent the number of fir trees, we can write our first equation: p + f = 110. This equation simply states that the sum of pine trees and fir trees equals the total number of trees sold. It's like saying, "Hey, every tree sold was either a pine or a fir, so if we add them up, we get 110!" The second piece of information is the relationship between the number of pine and fir trees. We know that they sold 4.5 times as many pine trees as fir trees. This means that the number of pine trees ('p') is 4.5 times the number of fir trees ('f'). We can write this as our second equation: p = 4.5f. This equation is super useful because it tells us exactly how the number of pine trees relates to the number of fir trees. It's like having a secret code that links the two variables together. Now, here's where the magic happens. We've got two equations, and two unknowns (p and f). This means we can use a method called substitution to solve for these variables. Basically, we're going to use one equation to express one variable in terms of the other and then plug that expression into the other equation. This will give us an equation with just one variable, which we can easily solve. Think of it like a puzzle where we're fitting pieces together until we find the solution. Our two equations are like the puzzle pieces, and we're about to fit them together perfectly. The next step is to use these equations to figure out the actual numbers of pine and fir trees sold. We've set up the foundation, and now we're ready to start solving. So, let's jump into the next section where we'll use these equations to find our answers!

Solving for Pine and Fir Trees

Alright, let's get down to the nitty-gritty and solve these equations! We've got our system of equations: p + f = 110 and p = 4.5f. The easiest way to tackle this is using substitution. Since we already know that p = 4.5f, we can substitute this into our first equation. This means we replace 'p' in the equation p + f = 110 with 4.5f. So, our new equation looks like this: 4. 5f + f = 110. See what we did there? We've eliminated one variable and now have an equation with just 'f'. This is fantastic because we can now solve for the number of fir trees. Let's simplify the equation by combining the 'f' terms. We have 4.5f plus 1f (remember, if there's no number in front of a variable, it's understood to be 1), which gives us 5. 5f = 110. Now, to isolate 'f', we need to divide both sides of the equation by 5.5. So, we get: f = 110 / 5.5. Doing the math, we find that f = 20. Hooray! We've found the number of fir trees sold. Taylor's Tree Farm sold 20 fir trees in November. But we're not done yet! We still need to find the number of pine trees. Remember our equation p = 4.5f? Now that we know 'f', we can plug it in to find 'p'. So, p = 4.5 * 20. Multiplying that out, we get p = 90. Awesome! We've solved for both variables. Taylor's Tree Farm sold 90 pine trees. To make sure our answer is correct, let's quickly check it against our original equations. We know that p + f = 110, so 90 + 20 = 110, which is correct. We also know that p = 4.5f, so 90 = 4.5 * 20, which is also correct. We've nailed it! We've successfully solved the system of equations and found that Taylor's Tree Farm sold 90 pine trees and 20 fir trees in November. Math can be fun, right? Now you know how to solve a problem like this using substitution. Great job, guys!

Conclusion: Tree Sales Solved!

Woo-hoo! We did it, guys! We've successfully navigated the world of Taylor's Tree Farm and figured out exactly how many pine and fir trees they sold in November. Just to recap, we found that they sold a whopping 90 pine trees and a solid 20 fir trees. This was all thanks to setting up and solving a system of equations. Remember, we started with the information that Taylor's Tree Farm sold a total of 110 trees and that they sold 4.5 times as many pine trees as fir trees. We turned these facts into two equations: p + f = 110 and p = 4.5f. Then, we used the magic of substitution to solve for the unknowns. By substituting the second equation into the first, we were able to find the number of fir trees. Once we knew that, we could easily calculate the number of pine trees. This whole process shows how powerful math can be in solving real-world problems. It might seem like a simple tree-selling scenario, but the principles we used here can be applied to all sorts of situations. From calculating business profits to figuring out the best route for a road trip, math is all around us, helping us make sense of the world. So, what's the takeaway here? Don't be intimidated by word problems! Break them down into smaller pieces, identify the key information, and translate that information into equations. Once you have your equations, you're well on your way to finding the solution. And remember, practice makes perfect. The more you solve problems like this, the better you'll become at it. Keep your thinking caps on, guys, and keep exploring the world of math! You never know what amazing things you'll discover. And who knows, maybe one day you'll be running your own tree farm and using these very equations to manage your sales! Until then, happy problem-solving!