Speed Difference Calculation Between Cyclist And Motorcyclist
Hey guys! Let's dive into a classic math problem involving speed, distance, and time. We've got a cyclist and a motorcyclist traveling between two points, and we need to figure out how much faster the motorcyclist is. This is a super common type of problem you'll see in algebra, so let's break it down step by step.
Problem Statement
Imagine a scenario where a cyclist travels from one village to a city, covering a distance of 39 kilometers. The cyclist completes this journey in 1.5 hours. Now, a motorcyclist makes the same trip, but they do it in less time – specifically, one hour less than the cyclist. The big question we need to answer is: By how many kilometers per hour is the motorcyclist's speed greater than the cyclist's speed?
Breaking Down the Problem: Speed, Distance, and Time
To solve this, we need to remember the fundamental relationship between speed, distance, and time. It's a simple formula, but it's the key to unlocking this problem:
Speed = Distance / Time
This formula tells us that speed is calculated by dividing the distance traveled by the time it took to travel that distance. We'll use this formula to calculate the speeds of both the cyclist and the motorcyclist.
Before we jump into the calculations, let's recap what we know:
- Distance: 39 kilometers (for both cyclist and motorcyclist)
- Cyclist's Time: 1.5 hours
- Motorcyclist's Time: 1.5 hours - 1 hour = 0.5 hours
Now we have all the pieces we need to find their speeds.
Calculating the Cyclist's Speed
Alright, let's start with the cyclist. We know the distance they traveled (39 kilometers) and the time it took them (1.5 hours). Using our formula, we can calculate their speed:
Cyclist's Speed = 39 kilometers / 1.5 hours
To make the calculation easier, we can convert 1.5 hours into a fraction: 1.5 = 3/2.
Cyclist's Speed = 39 kilometers / (3/2) hours
Dividing by a fraction is the same as multiplying by its reciprocal, so:
Cyclist's Speed = 39 kilometers * (2/3) hours
Now we can simplify:
Cyclist's Speed = (39 * 2) / 3 kilometers per hour
Cyclist's Speed = 78 / 3 kilometers per hour
Cyclist's Speed = 26 kilometers per hour
So, the cyclist was traveling at a speed of 26 kilometers per hour. We've got the first piece of the puzzle!
Calculating the Motorcyclist's Speed
Now let's move on to the motorcyclist. We know they covered the same distance (39 kilometers), but they did it in a shorter amount of time (0.5 hours). Let's plug these values into our formula:
Motorcyclist's Speed = 39 kilometers / 0.5 hours
Again, we can convert 0.5 into a fraction: 0.5 = 1/2
Motorcyclist's Speed = 39 kilometers / (1/2) hours
Dividing by a fraction means multiplying by its reciprocal:
Motorcyclist's Speed = 39 kilometers * (2/1) hours
Motorcyclist's Speed = 39 * 2 kilometers per hour
Motorcyclist's Speed = 78 kilometers per hour
Wow! The motorcyclist was cruising at a speed of 78 kilometers per hour. That's significantly faster than the cyclist.
Finding the Speed Difference
We're almost there! The final step is to find the difference between the motorcyclist's speed and the cyclist's speed. This will tell us exactly how much faster the motorcyclist was.
Speed Difference = Motorcyclist's Speed - Cyclist's Speed
Speed Difference = 78 kilometers per hour - 26 kilometers per hour
Speed Difference = 52 kilometers per hour
Therefore, the motorcyclist's speed is 52 kilometers per hour greater than the cyclist's speed. That's our answer!
Conclusion
So, there you have it! We've successfully solved the problem by breaking it down into smaller steps. We used the formula Speed = Distance / Time to calculate the speeds of both the cyclist and the motorcyclist, and then we found the difference between their speeds. The motorcyclist was a whopping 52 kilometers per hour faster than the cyclist.
This type of problem is a great example of how algebra can be used to solve real-world scenarios. By understanding the relationship between speed, distance, and time, we can figure out how fast things are moving and compare their speeds. Keep practicing these types of problems, and you'll become a speed-calculating pro in no time! Remember, the key is to break down the problem into smaller, manageable steps, and you'll be able to tackle even the trickiest math challenges. You've got this!
Let's tackle a classic algebra problem that involves calculating the speed difference between a cyclist and a motorcyclist traveling the same distance. This problem perfectly illustrates the relationship between distance, time, and speed, and it's a fantastic way to sharpen your problem-solving skills. So, buckle up, and let's dive in!
Understanding the Scenario: Distance, Time, and Two Travelers
In this scenario, we have two individuals, a cyclist and a motorcyclist, traveling from a village to a city. The distance between the village and the city is 39 kilometers. The cyclist completes the journey in 1.5 hours, while the motorcyclist covers the same distance in a shorter time, specifically, one hour less than the cyclist. Our mission is to determine how much faster, in kilometers per hour, the motorcyclist's speed is compared to the cyclist's speed. This problem emphasizes understanding the relationships between speed, distance and time. We have the cyclist and motorcyclist traveling 39 kilometers. The cyclist takes 1.5 hours while the motorcyclist is faster, by exactly one hour. Understanding how each variable impacts the other is key to finding a solution. The crucial point is recognizing that speed is derived from the relationship between distance and time. The faster the speed, the shorter the time it takes to cover the same distance. This problem tests your ability to translate a word problem into mathematical expressions and your understanding of how time affects speed over a fixed distance. We'll need to first calculate individual speeds and then determine the difference. This exercise not only reinforces algebraic principles but also enhances analytical thinking, a valuable skill in various fields. Remember, clear problem breakdown and systematic approach are the foundation of any successful problem-solving strategy. Let's get into solving this one, by first focusing on the speeds individually before comparing them to reach the final answer about how much faster the motorcyclist is. This will give us a clear picture of how speed, time, and distance are interacting in this specific situation, leading us to a firm understanding of the solution.
The Formula is Key: Speed = Distance / Time
To solve this problem, we need to recall the fundamental formula that connects speed, distance, and time. This formula is the cornerstone of our calculations:
Speed = Distance / Time
This simple yet powerful formula states that the speed of an object is equal to the distance it travels divided by the time it takes to travel that distance. To apply this, let's begin by summarizing the information provided. The total distance is consistent at 39 kilometers for both the cyclist and motorcyclist. This is our constant variable. The cyclist's time is 1.5 hours, a fixed point we can use in our initial speed calculation. The motorcyclist's time is critical because it's given relative to the cyclist's. It's 1 hour less, meaning the motorcyclist traveled the same 39 kilometers in 1.5 hours - 1 hour = 0.5 hours. Now we have two sets of values to work with, each representing a different scenario. Before we rush into calculations, let's appreciate what this formula implies. Speed and time are inversely proportional when the distance is constant. This means, for the same journey, if the time is less, the speed must be greater, and vice versa. Thinking this way helps us anticipate the motorcyclist's speed will be higher, which is intuitive. The formula also helps to avoid errors by providing a clear mathematical structure. Without this, we might fall into the trap of guessing or making incorrect assumptions. We can also rearrange this basic formula to solve for distance (Distance = Speed * Time) or time (Time = Distance / Speed), depending on what we're asked to find in a problem. For now, let's focus on speed, since that's our primary goal in this problem. By keeping this foundation in mind, we can methodically apply the formula to each traveler's information, which is a practical demonstration of how fundamental principles serve as the building blocks for solving complex problems.
Cyclist's Speed: Calculating the Pace of Pedaling
Let's first calculate the cyclist's speed. We know the cyclist covered a distance of 39 kilometers in 1.5 hours. Using the formula Speed = Distance / Time, we can plug in these values:
Cyclist's Speed = 39 kilometers / 1.5 hours
To simplify this calculation, let's convert the decimal 1.5 into a fraction. 1.5 is equivalent to 3/2. So, we can rewrite the equation as:
Cyclist's Speed = 39 kilometers / (3/2) hours
Now, remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 3/2 is 2/3. Therefore, we can rewrite the equation as:
Cyclist's Speed = 39 kilometers * (2/3)
Multiplying 39 by 2 gives us 78, so the equation becomes:
Cyclist's Speed = 78/3 kilometers per hour
Finally, dividing 78 by 3, we get:
Cyclist's Speed = 26 kilometers per hour
Therefore, the cyclist's speed is 26 kilometers per hour. This step gives us a benchmark. We now have the cyclist's speed firmly established, allowing us to meaningfully compare it with the motorcyclist's speed. The important thing here is the systematic application of the formula. We converted the decimal into a fraction to make it easier to divide and used the principle of reciprocating when dividing by a fraction. Each of these steps is not just a mathematical operation but also a strategy to ensure accuracy. The cyclist's speed of 26 kilometers per hour gives us a clear sense of how much distance is covered in a given time. Now we can contrast this with how the motorcyclist performs, given the significantly reduced travel time. Before we move on, it's beneficial to double-check. Does 26 kilometers per hour make sense for a cyclist over a 39-kilometer distance in 1.5 hours? Yes, it does. This quick reasonableness check helps prevent gross errors, a useful habit in problem-solving.
Motorcyclist's Speed: Zooming to the Finish Line
Next, let's determine the motorcyclist's speed. We know the motorcyclist also traveled 39 kilometers, but in a shorter time of 0.5 hours (which is 1.5 hours - 1 hour). Applying the same formula, Speed = Distance / Time, we have:
Motorcyclist's Speed = 39 kilometers / 0.5 hours
Again, let's convert the decimal 0.5 into a fraction, which is 1/2. So, the equation becomes:
Motorcyclist's Speed = 39 kilometers / (1/2) hours
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 1/2 is 2. Therefore:
Motorcyclist's Speed = 39 kilometers * 2
Multiplying 39 by 2, we get:
Motorcyclist's Speed = 78 kilometers per hour
So, the motorcyclist's speed is 78 kilometers per hour. Now we have a clear understanding of both speeds: the cyclist at 26 km/h and the motorcyclist at 78 km/h. The calculation here was quite straightforward after converting the time to a fraction. The key takeaway is how much faster the motorcyclist is compared to the cyclist for the same distance. This speed demonstrates the impact of reduced travel time. Intuitively, this result makes sense. If it takes the motorcyclist half the time to cover the same distance as the cyclist, we should expect the motorcyclist's speed to be roughly double, which is indeed what we observed. Before moving forward, it's always a good practice to perform a reasonableness check. Does 78 kilometers per hour seem feasible for a motorcyclist traveling between a village and a city? Absolutely. So, we can confidently move on to the final step: calculating the difference in speeds. What we've established so far is the individual speeds. The next step brings us closer to the answer we are looking for, which is the comparison of these speeds.
The Speed Difference: The Final Calculation
Finally, to answer the question, we need to find the difference between the motorcyclist's speed and the cyclist's speed. This will tell us exactly how much faster the motorcyclist was.
The difference in speed is calculated by subtracting the cyclist's speed from the motorcyclist's speed:
Speed Difference = Motorcyclist's Speed - Cyclist's Speed
We already calculated that the motorcyclist's speed is 78 kilometers per hour, and the cyclist's speed is 26 kilometers per hour. Plugging these values into the equation, we get:
Speed Difference = 78 kilometers per hour - 26 kilometers per hour
Subtracting 26 from 78, we find:
Speed Difference = 52 kilometers per hour
Therefore, the motorcyclist's speed is 52 kilometers per hour greater than the cyclist's speed. This is our final answer! By calculating this difference, we've provided the specific answer the problem required. The unit, kilometers per hour, is crucial to include because it clarifies that we're talking about a difference in speed, not just distance or time. Reviewing the whole process, we: (1) established the fundamental speed formula, (2) calculated the speed for each traveler individually, and then (3) found the difference to address the core question. This systematic approach highlights the power of breaking down a word problem into smaller, manageable steps. Before we conclude, we can perform one final reasonableness check: 52 kilometers per hour, the difference in speed, seems significant. Given the differences in travel time, a considerable speed difference is sensible. We've not only solved the math but also made a common-sense verification. We can confidently say we understand not just the numbers but the scenario itself. Congrats, we have reached our conclusion and thoroughly validated it!
Conclusion: Math in Motion
In conclusion, we've successfully solved this algebra problem by carefully analyzing the given information, applying the formula for speed, and performing the necessary calculations. The motorcyclist's speed is 52 kilometers per hour greater than the cyclist's speed. This problem illustrates a real-world application of algebra and demonstrates how understanding the relationship between distance, time, and speed can help us solve practical problems. Remember, the key to solving such problems is to break them down into manageable steps and apply the appropriate formulas and concepts. With practice, you'll become a pro at solving these types of problems! This exercise also emphasizes the significance of clear problem definition and methodical problem-solving. We didn't just rush to an answer. We first understood the scenario, identified the key variables, applied the appropriate formula, and then validated the result. This approach is transferable to countless other problem-solving situations, both in math and beyond. So, take the confidence from this success and apply the same systematic thinking to future challenges. The world is full of puzzles waiting to be solved, and with these techniques, you are better equipped to tackle them. Keep practicing, keep questioning, and you will become more confident in your abilities to solve problems creatively and effectively.
