Cyclist's Journey: Solving A Distance Problem Step-by-Step
Hey everyone, let's dive into a classic math problem! We're going to break down the journey of a cyclist who traveled a certain distance over three days. The problem involves fractions and figuring out how far the cyclist went each day. It's a great exercise in understanding how to approach word problems and use the information given to find the solution. Ready? Let's get started!
Understanding the Problem: The Cyclist's Adventure
Alright, so here's the deal: A cyclist is on a three-day trip. On day one, they knock out a quarter of the total distance. Day two sees them covering a third of what's left. And finally, on day three, they cycle 15 km more than they did on the first day. Our mission? To calculate exactly how many kilometers the cyclist traveled each day. This kind of problem is super common in math and helps us sharpen our skills in fractions, algebra, and logical thinking. We'll break it down step-by-step so you can easily follow along and understand how to solve similar problems in the future. Remember, the key is to take it one piece at a time and visualize what's happening. Ready to unravel this cycling mystery? Let's go!
So, the main focus here is figuring out the distances for each of the three days. We're given information about the proportions of the total distance covered each day and some details that link the distances together. The first thing we can identify is that the distance covered on the third day is connected to the distance on the first day. This is a very important hint and will help us solve the problem. Also, the second day's distance depends on what's left after day one, which is another clue. Keep in mind that solving this problem requires careful attention to detail. We need to keep track of the fractions, percentages, and what each part of the distance represents. Let's make sure we're on the right track and understand the question clearly. The goal is to calculate the distance for each day and make sure everything adds up correctly. Are you ready to dive deeper?
This problem is a great example of a fractional word problem, commonly seen in middle school math. Solving this type of problem requires skills in understanding fractions, algebraic thinking, and careful attention to detail. The cyclist's journey provides an excellent context for practicing these skills. Now, before we start solving it, let's break down the key elements again. First, we know the cyclist's traveled distance on the first day, is a quarter. Then, on the second day, the cyclist covered a third of the remaining distance. The last piece of information is that on the third day the cyclist traveled 15 km more than on the first day. With these key points in mind, we can set up the equations and solve the problem systematically. Each step we take will get us closer to our goal: determining the distance covered on each day. Therefore, understanding the concepts is the most important part.
Setting Up the Equations: Breaking Down the Journey
Alright, let's get down to the nitty-gritty and start setting up the equations to solve this problem. We'll start by defining some variables. Let's say 'x' represents the total distance the cyclist traveled. So, on day one, the cyclist covered 1/4 of x, which we can write as (1/4)x. Now, what about day two? After day one, there's (3/4)x of the distance left (because the cyclist has already completed 1/4). On day two, the cyclist covers 1/3 of that remaining distance, so that's (1/3) * (3/4)x, or (1/4)x. On day three, the cyclist travels 15 km more than on day one. So, the distance on day three is (1/4)x + 15. The total distance covered over the three days is the sum of the distances for each day. And that must equal the total distance x. In mathematical terms:
(1/4)x + (1/4)x + (1/4)x + 15 = x
This equation is the core of our problem. Each term represents the distance covered on a specific day, and the sum of all distances should equal the total distance. Now that we have the equation, the next step is to simplify and solve for x. This will tell us the total distance traveled by the cyclist. From there, we can easily calculate the distance for each individual day. We must make sure that we correctly set up the equation, to get to the correct result. This step is super crucial for getting the right answers.
Now we're diving into the meat of the math! Solving for x means we're figuring out the total distance. To simplify the equation, let's combine the fractions. We have (1/4)x + (1/4)x + (1/4)x, which adds up to (3/4)x. So, our equation becomes:
(3/4)x + 15 = x
Next, to isolate x, let's subtract (3/4)x from both sides of the equation. This gives us:
15 = x - (3/4)x
Which simplifies to:
15 = (1/4)x
Now, to solve for x, we multiply both sides of the equation by 4. This gives us:
60 = x
So, x equals 60 km. This is the total distance the cyclist traveled. Great! Now that we know the total distance, we can figure out the distance for each day.
Calculating Daily Distances: Putting the Pieces Together
Okay, guys, we've found the total distance: 60 km! Now, it's time to figure out how far the cyclist went each day. Remember, on day one, the cyclist covered (1/4) of the total distance. So, that's (1/4) * 60 km = 15 km. On day two, we found that the cyclist covered (1/4) of the total distance again, which is also 15 km. Day three was a bit trickier, as the cyclist traveled 15 km more than on day one. Since day one was 15 km, day three's distance is 15 km + 15 km = 30 km. And there we have it! Let's sum it up: on day one, the cyclist traveled 15 km; on day two, 15 km; and on day three, 30 km. Does this add up to the total distance of 60 km? Yes, it does! That's how we know we did it correctly. This step is all about making sure we understand what the problem requires and using our previously found data to create the proper results.
To make this clearer, let's recap the distances for each day:
- Day 1: (1/4) * 60 km = 15 km
- Day 2: (1/3) * (3/4) * 60 km = 15 km
- Day 3: 15 km (from day 1) + 15 km = 30 km
Let's check our answers. Does 15 km (day 1) + 15 km (day 2) + 30 km (day 3) = 60 km (total distance)? Absolutely! It does. That means our calculations are spot on. Remember, problem-solving is about taking it one step at a time, being careful, and checking your work. Well done, guys! You've successfully solved this problem.
Conclusion: The Cyclist's Journey Complete!
Alright, folks, we've successfully navigated the cyclist's journey and found out how far they traveled each day! We started by understanding the problem, then setting up our equations, solving for the total distance, and finally calculating the distances for each day. We used fractions, simple algebra, and a bit of logical thinking to solve this problem. This problem helps demonstrate how math can be applied in real-life scenarios, even to something as simple as a bike ride. The key takeaway here is to break down complex problems into smaller, manageable steps. By doing so, you can solve almost any problem. Remember to take it one step at a time, and always double-check your work.
If you enjoyed this problem, keep practicing! Try creating your own word problems and solving them. The more you practice, the better you'll become. And if you have any questions, don't hesitate to ask! Thanks for joining me on this mathematical adventure! Until next time, keep those math skills sharp!
