Finding A Point On The Number Line 1/5th Of The Way Between Two Points

Hey guys! Ever found yourself staring at a number line, trying to pinpoint a specific spot between two numbers? It can feel a bit like navigating uncharted territory, right? Well, today we're diving deep into a fun little mathematical journey. We're going to figure out exactly where 1/5th of the distance lies between -7 and 17 on the number line. This isn't just about getting the right answer; it's about understanding the process, the logic, and the why behind it all. So, buckle up, and let's get started!

Understanding the Problem: Visualizing the Number Line

Before we jump into calculations, let's take a moment to visualize what we're dealing with. Imagine a straight line stretching out infinitely in both directions – that's our number line. On this line, we have two key points: -7 and 17. Our mission, should we choose to accept it, is to find the point that sits exactly 1/5th of the way from -7 towards 17.

Visualizing the Distance: Think of the space between -7 and 17 as a road trip. The total distance of the trip is the difference between the two points. To find that, we subtract the starting point (-7) from the ending point (17). So, the total distance is 17 - (-7), which equals 24 units. Now, we're not going the whole way; we're just going 1/5th of the way. This is where fractions come into play. We need to find 1/5th of this total distance.

Why Visualization Matters: Visualizing the problem is a powerful tool in mathematics. It helps us build a mental model, making the problem less abstract and more tangible. When we can "see" the problem, we're better equipped to understand the steps involved in solving it. It also helps us estimate the answer before we even start calculating, which can be a great way to check our work later. So, always try to visualize the problem first – it's like having a map for our mathematical journey!

Breaking it down: So, to recap, we've established that we need to find a point 1/5th of the distance between -7 and 17. We've visualized the number line and the distance between these points. Now, let's move on to the next step: calculating that 1/5th distance.

Calculating the Distance: Finding 1/5th of the Total

Now that we've got a good mental picture of our problem, it's time to crunch some numbers. We know the total distance between -7 and 17 is 24 units. Our goal is to find 1/5th of this distance. This is where our fraction skills come in handy. Finding a fraction of a whole number is a fundamental concept in math, and it's something we use all the time in everyday life, from splitting a pizza to calculating discounts.

The Magic of Multiplication: To find 1/5th of 24, we simply multiply 24 by 1/5. Mathematically, this looks like (1/5) * 24. Remember, multiplying by a fraction is the same as dividing by the denominator. So, we're essentially dividing 24 by 5. When we do this, we get 4.8. This means that 1/5th of the distance between -7 and 17 is 4.8 units. This is a crucial piece of information, so let's hold onto it tightly!

What Does 4.8 Mean? It's essential to understand what this 4.8 represents in the context of our problem. It's not the final answer; it's the distance we need to move from our starting point, which is -7. Think of it as the length of the first leg of our journey. We've traveled 4.8 units along the number line, but we need to know where we landed. This brings us to the final step: adding this distance to our starting point.

Double-Checking Our Understanding: Before we move on, let's make sure we're all on the same page. We've calculated 1/5th of the distance between -7 and 17, which is 4.8 units. This means that the point we're looking for is 4.8 units away from -7, in the direction of 17. We're almost there, guys! Just one more step to pinpoint the exact location.

Pinpointing the Final Point: Adding the Distance to the Starting Point

Alright, we've done the heavy lifting! We know the total distance, we've calculated 1/5th of that distance, and now it's time to put it all together and find our final destination on the number line. Remember, we started at -7, and we're moving 4.8 units in the positive direction (towards 17). This means we need to add 4.8 to -7. This step is where we bring together our understanding of negative numbers and addition.

The Dance of Negative and Positive: Adding a positive number to a negative number can sometimes feel a bit tricky. Think of it like climbing up a ladder from an underground floor. Our starting point is -7, which we can imagine as being 7 floors below ground level. We then climb up 4.8 floors. This means we're getting closer to the surface (zero), but we might not reach it yet. To find our exact location, we perform the addition: -7 + 4.8.

The Calculation: When we add -7 and 4.8, we get -2.2. This is our final answer! This means the point that is 1/5th of the way from -7 to 17 is located at -2.2 on the number line. This is it! We've successfully navigated the number line and found our target point.

Connecting the Dots: Let's take a moment to connect all the steps we've taken. We visualized the problem, calculated the total distance, found 1/5th of that distance, and then added it to our starting point. This process highlights the interconnectedness of mathematical concepts. Each step builds upon the previous one, leading us to the solution. This is the beauty of math – it's a logical journey where each step makes sense.

Checking Our Answer: Ensuring Accuracy and Understanding

In mathematics, getting to an answer is only half the battle. The other half is ensuring that our answer is correct and makes sense in the context of the problem. This is where checking our work becomes super important. It's like proofreading a paper or testing a recipe – it's the final step that guarantees success. There are several ways we can check our answer in this case.

Estimation as a Sanity Check: Before we even start calculating, we can make a rough estimate of where 1/5th of the way between -7 and 17 might be. Since the total distance is 24, 1/5th of the way should be less than half. Also, since we're starting from a negative number and moving towards a positive number, our answer should still be negative but closer to zero. -2.2 fits this description, which gives us some initial confidence.

Working Backwards: Another way to check is to work backwards. If -2.2 is indeed 1/5th of the way from -7 to 17, then the distance between -7 and -2.2 should be 1/5th of the total distance. The distance between -7 and -2.2 is 4.8, which we already calculated as 1/5th of 24. This confirms our answer.

Visualizing on the Number Line Again: We can also revisit our visualization of the number line. Does -2.2 look like it's approximately 1/5th of the way between -7 and 17? It's always good to use our visual sense to confirm our numerical calculations.

The Importance of Checking: Checking our answers isn't just about getting a correct grade; it's about developing a deeper understanding of the concepts. When we take the time to check, we're reinforcing the steps we took and solidifying our knowledge. It also helps us catch any errors we might have made, which is a valuable skill in mathematics and in life.

The Answer and Its Significance: What We've Learned

So, after our mathematical adventure, we've arrived at our final destination: the point -2.2 on the number line. This point is exactly 1/5th of the way from -7 to 17. But what does this mean, beyond just a number on a line? It signifies our understanding of several key mathematical concepts, including distance, fractions, negative numbers, and the number line itself. We've not just found an answer; we've also strengthened our problem-solving skills.

The Power of Problem-Solving: This problem, while seemingly simple, encapsulates the essence of problem-solving in mathematics. We broke down a complex question into smaller, manageable steps. We visualized the problem, identified the relevant information, applied the appropriate operations, and checked our answer. This is a process that can be applied to a wide range of mathematical challenges and even to real-world situations.

Beyond the Number Line: The skills we've honed in this exercise extend far beyond the number line. Understanding fractions, distances, and the interplay of positive and negative numbers are fundamental to many areas of mathematics, including algebra, geometry, and calculus. By mastering these basics, we're building a strong foundation for future learning.

Congratulations, Math Explorers! Give yourselves a pat on the back, guys! We've successfully navigated the number line, conquered the fractions, and pinpointed our target point. More importantly, we've learned valuable problem-solving skills and reinforced our understanding of fundamental mathematical concepts. Keep exploring, keep questioning, and keep enjoying the journey of mathematics!

Therefore, the correct answer is A. -2.2