Fraction Of The Journey: Paula's Mountain Expedition
Hey guys! Let's break down this math problem about Paula's mountain expedition. It's all about fractions, and we're going to make it super easy to understand. We'll go through each step together, so you can see exactly how to solve this type of problem. So, Paula went on an expedition covering 4/5 of the way to a mountain, and she paused for pics at the 3/4 mark of her journey. The big question is: what fraction of the total path had she traveled before her photo break? Stick with me, and we'll figure it out!
Understanding the Problem
Okay, so let's dive deep into understanding the problem before we even think about calculations. This is crucial, guys! Our main keyword here is fraction of the journey, and it's important we grasp what the question is really asking. Paula didn't just walk a random distance; she completed a portion of a larger journey, and we need to pinpoint exactly what part that was. Think of it like slicing a cake – we're trying to figure out the size of the slice Paula took compared to the whole cake (or, in this case, the entire path to the mountain).
Now, let's break down the key information we've got. Paula traveled 4/5 of the total distance. This means if you split the entire path into five equal parts, she covered four of those parts. Then, she stopped at 3/4 of that distance to snap some photos. This is where it gets a little tricky, but we can handle it! The 3/4 isn't of the whole journey, but of the 4/5 she already traveled. Visualizing this can help a lot. Imagine that path split into sections, and Paula's only gone partway, then stopped within that partway distance. The question boils down to figuring out what fraction 3/4 of 4/5 represents in relation to the entire journey. To put it simply, the problem wants us to calculate a fraction of a fraction. We need to find a single fraction that represents Paula’s progress when she paused for those snapshots. This kind of problem is common in math, and once you get the hang of it, you'll be solving them like a pro! Remember, the secret is to carefully dissect what the question is asking and identify the pieces of information you need. We've got those pieces, so let's move on to how we're going to put them together.
Setting up the Calculation
Alright, let's get into the nitty-gritty of setting up the calculation. Remember, our main goal is to find out what fraction of the total journey Paula had covered when she stopped for photos. We've already established that she traveled 4/5 of the way, and then stopped at 3/4 of that distance. So, how do we translate this into a mathematical operation? Well, the key word here is "of." In math, "of" often indicates multiplication. So, what we need to do is multiply these two fractions together: 3/4 multiplied by 4/5.
Now, before we jump right into multiplying, let's think about why multiplication works in this scenario. When we multiply fractions, we're essentially finding a part of a part. In this case, we're finding 3/4 of the 4/5 that Paula traveled. Multiplication helps us determine what fraction of the whole journey that 3/4 of 4/5 represents. Think of it like this: if you have half a pizza (1/2) and you eat half of that half (1/2 of 1/2), you've eaten 1/4 of the whole pizza. We're doing the same thing here, just with different fractions representing the journey. So, the core calculation we need to perform is (3/4) * (4/5). This setup is crucial because it directly reflects the problem's wording and the relationships between the distances. Make sure you understand why we're multiplying before we actually do the multiplication. It’ll make solving similar problems much easier in the future. Once you're comfortable with the setup, the actual calculation is the next step, and it's surprisingly straightforward.
Performing the Multiplication
Okay, guys, now for the fun part – performing the multiplication! We've already set up our equation: (3/4) * (4/5). Multiplying fractions is actually pretty straightforward. You just multiply the numerators (the top numbers) together and then multiply the denominators (the bottom numbers) together. So, let's break it down.
First, we multiply the numerators: 3 * 4. That gives us 12. This 12 represents the numerator of our answer. Next, we multiply the denominators: 4 * 5. That gives us 20. This 20 represents the denominator of our answer. So, after multiplying, we get the fraction 12/20. That means Paula traveled 12/20 of the total journey before she stopped for photos. But hold on a sec! We're not quite done yet. Fractions often need to be simplified, and in this case, 12/20 can definitely be made simpler. Think of it like this: it’s usually best to express your answer in its most basic form. Imagine telling someone you ate 12/20 of a pizza – they might look at you a little funny. But saying you ate 3/5 sounds much clearer, right? Simplifying fractions is all about making them easier to understand. So, let's simplify 12/20 in the next step.
Simplifying the Fraction
Alright, let's simplify the fraction we got from our multiplication. We ended up with 12/20, which tells us Paula had traveled twelve-twentieths of the way before stopping for photos. But, just like we discussed, it's always best to express fractions in their simplest form. This makes them easier to understand and compare. To simplify a fraction, we need to find the greatest common factor (GCF) of the numerator (12) and the denominator (20). The GCF is the largest number that divides evenly into both numbers.
So, what's the GCF of 12 and 20? Let's list the factors of each number. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 20 are 1, 2, 4, 5, 10, and 20. Looking at these lists, we can see that the largest number they have in common is 4. So, 4 is our GCF. Now, to simplify the fraction, we divide both the numerator and the denominator by the GCF. That means we'll divide 12 by 4, which gives us 3, and we'll divide 20 by 4, which gives us 5. This leaves us with the simplified fraction 3/5. This means that 12/20 is equivalent to 3/5, but 3/5 is in its simplest form. So, Paula traveled 3/5 of the total journey before she stopped to take pictures. See how much cleaner and easier to understand 3/5 is compared to 12/20? That’s why simplifying fractions is so important!
Stating the Answer
Okay, we've done all the hard work, and now it's time for the super satisfying part: stating the answer! We've gone through understanding the problem, setting up the calculation, performing the multiplication, and simplifying the fraction. We know that Paula traveled 3/5 of the total journey before she stopped to take pictures. But just stating the fraction isn't quite enough. We need to clearly answer the original question in a complete sentence. This makes sure that anyone reading our solution understands exactly what we found.
So, how do we do that? We go back to the original question: "What fraction of the road had Paula traveled until she stopped to take pictures?" Now, we take our simplified fraction, 3/5, and weave it into a clear and concise answer. A great way to answer is: "Paula had traveled 3/5 of the road until she stopped to take pictures." See how that answers the question directly and leaves no room for confusion? Always remember to state your answer clearly and completely. It’s the final touch that shows you not only did the math correctly but also understand what the answer means in the context of the problem. This skill is super important, not just in math, but in all areas of life. Clear communication is key! So, there you have it! We've successfully solved the problem. Paula traveled 3/5 of the way before snapping those photos. Awesome job, guys!