Finding The Equation For Stu's Hike A Math Adventure

Hey guys! Let's dive into a fun math problem that involves Stu, a trail, and some hiking and running. This is a classic problem that uses the concepts of distance, rate, and time, so get ready to put on your thinking caps! We're going to break down the problem step-by-step to figure out which equation helps us find the time Stu spent hiking. So, buckle up, and let’s embark on this mathematical journey together!

Understanding the Problem: Stu's Hiking Adventure

First, let's understand the core question. The main goal here is to identify the equation that accurately represents the time Stu spent hiking the trail. We know Stu hiked at an average rate of 3 miles per hour and ran back at 5 miles per hour. The total trip took 3 hours. To solve this, we need to connect these pieces of information using a mathematical equation. The key here is recognizing that the distance Stu hiked is the same as the distance he ran back. This is a crucial point because it allows us to set up an equation that relates the time spent hiking and running. Many times in word problems, there’s a hidden connection or piece of information that is not explicitly stated but is vital to solving the problem. In this case, it’s the equal distances.

Let's break down the known information. Stu's hiking rate is 3 miles per hour, his running rate is 5 miles per hour, and the total time for the trip is 3 hours. We need to find the equation that represents the time it took Stu to hike. The first step is to define our variables. Let's use 't' to represent the time Stu spent hiking. Since the total trip took 3 hours, the time Stu spent running would be (3 - t) hours. Remember, it's all about making these connections and converting the word problem into manageable mathematical expressions. The beauty of algebra is its ability to represent real-world scenarios with simple letters and symbols, making complex problems solvable with basic operations.

Now, let’s connect the dots using the formula: distance = rate × time. For the hiking part, the distance is 3t (3 miles per hour multiplied by t hours). For the running part, the distance is 5(3 - t) (5 miles per hour multiplied by (3 - t) hours). Since the distance is the same in both directions, we can set these two expressions equal to each other. This gives us the equation 3t = 5(3 - t). This equation is the heart of the solution, as it encapsulates all the given information and the crucial relationship between distance, rate, and time. By solving this equation, we can find the exact time Stu spent hiking. But for now, the question is simply asking us to identify the correct equation, and we've nailed it!

Setting Up the Equation: Distance, Rate, and Time

To identify the correct equation, we need to understand the relationship between distance, rate, and time. The fundamental formula we'll use is: Distance = Rate × Time. This formula is the cornerstone of solving problems involving motion and is something you’ll use time and time again in physics and mathematics. Think of it this way: if you’re driving at a certain speed (rate) for a certain amount of time, you’ll cover a certain distance. Now, let’s apply this concept to Stu’s hike. He hiked at 3 miles per hour and ran back at 5 miles per hour. The total trip took 3 hours. Let's use 't' to represent the time Stu spent hiking. This means the time he spent running is (3 - t) hours because the total time is 3 hours.

Here's where it gets interesting. The distance Stu hiked is the same as the distance he ran back. This is a critical piece of information. We can express the distance he hiked as 3t (3 miles per hour × t hours). Similarly, the distance he ran is 5(3 - t) (5 miles per hour × (3 - t) hours). Since these distances are equal, we can set up the equation: 3t = 5(3 - t). This equation beautifully captures the essence of the problem. It states that the distance covered while hiking is equal to the distance covered while running. Solving this equation will give us the time Stu spent hiking. But for this problem, we are focusing on identifying the correct equation, not solving it.

Let's consider why this approach works so well. By setting the distances equal to each other, we've created an equation with one variable, 't'. This is a powerful technique in algebra. It allows us to take a complex word problem and break it down into a manageable equation. The equation 3t = 5(3 - t) is a linear equation, which means it has a single solution. This solution will tell us the exact time Stu spent hiking. Remember, the art of problem-solving often lies in identifying the right relationships and translating them into mathematical expressions. In this case, the relationship between distance, rate, and time, and the fact that the distances are equal, were the keys to setting up the correct equation.

Analyzing the Options: Finding the Right Match

Now that we understand how to set up the equation, let's look at the options provided and see which one matches our analysis. We're looking for an equation that represents the relationship we established: 3t = 5(3 - t). This equation equates the distance Stu hiked with the distance he ran, considering their respective rates and times. When presented with multiple choices, it's often helpful to eliminate the incorrect ones first. This narrows down your options and increases your chances of selecting the correct answer. Sometimes, the incorrect options are deliberately designed to mislead you, so it's essential to understand the logic behind each choice.

Let’s go through the thought process. We know the distance Stu hiked is represented by 3t, and the distance he ran is represented by 5(3 - t). The key is that these distances are equal. Therefore, the correct equation should have these two expressions set equal to each other. Other options might involve adding the times or using incorrect rates, but they won't accurately represent the scenario. For example, an equation like 3 + t = 5 might seem tempting at first glance, but it doesn't consider the distances or the relationship between rate and time. It's crucial to remember that in this problem, distance is the common thread that ties everything together.

When analyzing the options, pay close attention to the structure of each equation. Does it make logical sense in the context of the problem? Does it correctly incorporate the rates, times, and the key fact that the distances are equal? By carefully examining each option and comparing it to our derived equation, we can confidently identify the correct answer. This process of elimination and logical deduction is a valuable skill in problem-solving, not just in math, but in many areas of life. It's about breaking down complex situations into smaller, more manageable parts and then systematically analyzing each part.

The Correct Equation: 3t = 5(3 - t)

After our detailed analysis, we've arrived at the equation that accurately represents the scenario: 3t = 5(3 - t). This equation is the key to unlocking the time Stu spent hiking. It encapsulates all the critical information given in the problem – Stu's hiking rate, running rate, total travel time, and the crucial fact that the distance hiked is equal to the distance ran. This equation isn't just a random collection of numbers and symbols; it's a powerful statement that mathematically describes a real-world situation. The beauty of mathematics is its ability to model and explain the world around us, and this problem is a perfect example of that.

Why is this equation correct? Let's break it down one more time. On one side, we have 3t, which represents the distance Stu hiked. This is his rate (3 miles per hour) multiplied by the time he spent hiking (t hours). On the other side, we have 5(3 - t), which represents the distance Stu ran. This is his running rate (5 miles per hour) multiplied by the time he spent running (3 - t hours). The (3 - t) term is crucial because it represents the remaining time after hiking is accounted for. Setting these two expressions equal to each other makes perfect sense because the distances are the same.

This equation is a testament to the power of algebraic thinking. By translating a word problem into a mathematical equation, we've transformed a complex situation into a solvable problem. The equation 3t = 5(3 - t) is a linear equation, meaning it can be solved using basic algebraic techniques. Once solved, it will provide us with the exact value of 't', which is the time Stu spent hiking. But for this particular question, identifying the equation is the goal, and we've successfully done that! Remember, problem-solving in mathematics is like building a puzzle. Each piece of information is a clue, and the equation is the framework that holds it all together.

Conclusion: Math is an Adventure!

So, there you have it! We've successfully navigated this mathematical trail and identified the equation that helps us find the time Stu spent hiking. Remember, the key to solving these kinds of problems is to break them down into smaller parts, understand the relationships between the variables, and translate the words into mathematical expressions. It’s like learning a new language, where each symbol and equation is a word or sentence that tells a story. Math isn't just about numbers; it's about logic, problem-solving, and seeing the world in a new way.

We started by understanding the problem, identifying the knowns and unknowns. Then, we used the formula Distance = Rate × Time to set up the equation. We analyzed the options and matched them to our derived equation. Finally, we celebrated our success in finding the correct equation! This journey through Stu's hiking adventure shows us that math can be engaging and even fun. It's not just about memorizing formulas; it's about applying them to real-world scenarios and using them to solve problems.

So, next time you encounter a math problem, don't be intimidated! Think of it as an adventure, a puzzle waiting to be solved. Use the tools and techniques we've discussed, and you'll be amazed at what you can achieve. Keep exploring, keep learning, and most importantly, keep having fun with math! Math is more than just a subject; it’s a way of thinking, a way of understanding, and a way of making sense of the world around us. So, embrace the challenge, and let's continue our mathematical adventures together!