Solving The Biciclist's Route: A Math Problem

Hey guys! Let's dive into a classic math problem about a cyclist's journey. This is the kind of problem you might find in an algebra textbook, and it's a great way to practice setting up and solving equations. We'll break down the problem step-by-step so it's super clear, even if math isn't your favorite subject. Get ready to flex those brain muscles!

Understanding the Problem: The Biciclist's Journey

Okay, so here's the deal: A cyclist rides a certain route over four days. We're given some clues about how the distances of each day relate to each other. The goal? To figure out the total distance the cyclist traveled. Here’s the breakdown:

• Day 1: The cyclist rides 30 km more than on Day 2.
• Day 1: The cyclist rides 40 km less than on Day 3.
• Day 4: The cyclist rides half the distance of Day 1.

Sounds like a fun challenge, right? The key to solving this is to translate those wordy descriptions into mathematical equations. Don't worry, it's not as scary as it sounds. We'll start by assigning a variable to represent an unknown quantity, in this case, the distance covered on the second day. Then we'll use that variable to express the distances of the other days. Let's get started. This kind of problem is typical in basic algebra, focusing on understanding relationships between variables and applying fundamental arithmetic operations. The beauty of these problems is that they encourage logical thinking and the ability to break down complex information into manageable parts. By the time we're done, you'll feel confident tackling similar challenges. This approach not only helps you find the solution but also builds a solid foundation for more complex mathematical concepts later on.

Let's get into the details of the problem and understand how we can approach the solution. We will use a variable to denote the distance covered each day and create equations based on the information provided. These equations will then be solved to find the distance covered on each day. This is a common approach in solving word problems, which helps in translating real-world scenarios into mathematical models. Remember, the ability to do this is a valuable skill, not just for academics, but for everyday life as well. The problem also touches upon critical thinking skills. We will work to carefully analyze the information presented and look for the relationships that exist between the variables. This step is a critical part of the problem-solving process and will help us avoid making unnecessary assumptions or missing important details. Let's make sure we've got all the information we need to solve the problem and that we understand what we're being asked to do. Alright, are you ready to jump in and solve it?

Setting up the Equations: Translating Words to Math

Alright, let's turn those descriptions into math. This is where the magic happens! We'll use a variable to represent an unknown distance, and then write equations based on the relationships described in the problem.

Let's say:

• x = distance on Day 2.

Now, let's express the distances for the other days in terms of x:

• Day 1: Since the cyclist rode 30 km more than on Day 2, Day 1's distance is x + 30.
• Day 3: The cyclist rode 40 km more on Day 1 than on Day 3. Since Day 1 is x + 30, Day 3 is (x + 30) + 40 which simplifies to x + 70.
• Day 4: The cyclist rode half the distance of Day 1. So, Day 4's distance is (x + 30) / 2.

Now that we've got the equations, we need one more piece of information. The problem states that the distance on day four is half the distance of day one. The equation for day one is already expressed, so we just divide that equation by two to derive the equation for day four. Now, let's recap where we are at. We've defined our variable x, and used it to write expressions for the distances covered each day. We're well on our way to solving the problem! Keep in mind that we are working step by step to find the total distance covered by the cyclist. The equations we made serve as the stepping stones to solve the problem. Also, this approach of converting word problems into math problems is a foundational skill in algebra. The ability to do this is critical for solving more complex problems. You can use these steps for a wide variety of problem types, so it's a valuable skill.

Next, we'll need to figure out how to put all these pieces together to solve for x. The goal is to find the value of x, which represents the distance on Day 2. Once we know that, we can plug that value back into our equations for each day and get all the distances. From there, we'll easily calculate the total distance. So, let's move forward and get our answer!

Solving for the Unknown: Finding the Distance on Day 2

Okay, here's where we bring it all together. To find the value of x, we need to use the fact that the problem asks to solve the route. We know that the total distance of Day 4 is half the distance of Day 1. We know from our previous equations that Day 1 is x + 30, and Day 4 is (x + 30)/2. Let's get to work!

We know that Day 4 distance is half the distance of Day 1: (x + 30) / 2. To make things easier, we're going to set up an equation, so let's set up the equation like this: (x + 30) / 2 = x + 30. That means, we multiply both sides of the equation by 2, and we have x + 30 = 2x. Let's isolate x and subtract x from both sides of the equation. We have: 30 = x, which is the distance traveled on Day 2.

Now that we know x = 30, we can find the distance for each day:

• Day 1: x + 30 = 30 + 30 = 60 km
• Day 2: x = 30 km
• Day 3: x + 70 = 30 + 70 = 100 km
• Day 4: (x + 30) / 2 = (30 + 30) / 2 = 30 km

So there you have it, guys. We solved for x and we know the distance traveled on all four days. You've successfully navigated a word problem! Feel proud of yourselves. Let's add them up and calculate the total distance traveled!

Calculating the Total Distance: The Final Step

Alright, we're at the finish line! Now that we know the distance the cyclist traveled each day, we can easily calculate the total distance for the entire journey. This is the final step, and it's super simple: just add up the distances from each of the four days. It's like collecting all the pieces of a puzzle to see the whole picture.

Here's how it breaks down:

• Day 1: 60 km
• Day 2: 30 km
• Day 3: 100 km
• Day 4: 30 km

Total distance = 60 km + 30 km + 100 km + 30 km = 220 km

So, the cyclist traveled a total of 220 kilometers over the four days. Boom! We did it! You took a complex problem, broke it down into smaller parts, and solved it systematically. You've shown that you can translate word problems into equations, solve for unknowns, and arrive at a correct answer. Give yourselves a pat on the back! You can apply these same techniques to other types of problems, making you a more confident problem-solver in general.

Conclusion: You Did It!

Awesome work, everyone! We successfully solved the biciclist route problem. We started with a word problem, broke it down into manageable parts, and used our algebra skills to find the solution. Remember that the key to tackling these problems is to take it slow, write things down, and don't be afraid to ask for help if you need it. By practicing these techniques, you'll become a pro at solving these types of problems. Now that you've got this one down, try some other problems, and keep those math skills sharp! You guys are awesome. Keep up the great work! And don't forget to celebrate your victories, no matter how big or small. You've earned it!