Calculating Ramiro's Distance In The ESSALUD Marathon

Hey guys! Today, we're diving into a cool math problem set in the middle of the ESSALUD marathon. Picture this: Raúl's been running for an hour and has covered 2/5 of the total marathon distance. His buddy Ramiro has run 3/4 of the distance Raúl has covered. The big question is: What fraction of the total marathon distance has Ramiro run by this point? Let's break it down and solve it together!

Understanding the Marathon Scenario

Okay, so to really nail this problem, we need to get a clear picture of what's happening in the marathon. First off, Raúl has run 2/5 of the total distance. This is our starting point. Now, Ramiro hasn't run 3/4 of the total distance; he's run 3/4 of Raúl's distance. This is a crucial detail because it means we're dealing with a fraction of a fraction. Think of it like this: if the total marathon distance were, say, 10 kilometers, Raúl would have run 4 kilometers (2/5 of 10 km). Then, Ramiro would have run 3/4 of those 4 kilometers. To figure out the exact fraction of the total distance Ramiro has run, we need to multiply these fractions. We're essentially asking, "What is 3/4 of 2/5?" and in math language, "of" often means multiplication. This is where we start to transition from visualizing the problem to actually crunching the numbers. To visualize this better, imagine dividing the marathon route into five equal parts because Raúl ran two of those five parts. Now, consider Ramiro's run. He covered three-quarters of Raúl’s two parts. It's like slicing those two parts into quarters and taking three of those slices. This visual representation really helps to clarify the relationship between the distances covered by Raúl and Ramiro, and how it relates back to the total marathon distance. It's all about breaking down the problem into smaller, manageable chunks and then putting it back together to get the final answer. So, with this understanding, we're ready to move on to the next step: setting up the equation. We'll take the fractions we've identified, put them in the correct order for multiplication, and then we'll be well on our way to solving this marathon mystery!

Setting Up the Equation

Alright, guys, let's get into the nitty-gritty of setting up the equation for this problem. Remember, Ramiro has covered 3/4 of the distance Raúl has run, and Raúl has run 2/5 of the total distance. So, to find out what fraction of the total distance Ramiro has covered, we need to find 3/4 of 2/5. In mathematical terms, this translates to multiplying the two fractions together. So, the equation we need to solve is: (3/4) * (2/5) = ?. This simple equation is the key to unlocking the answer. It clearly represents the relationship between Ramiro's distance and Raúl's distance in terms of fractions of the total marathon length. When you look at it, you can see how the fractions are directly linked – Ramiro's distance is calculated as a fraction of Raúl's distance, which in turn is a fraction of the total distance. It’s like a chain reaction, where each step is dependent on the previous one. Now, why do we multiply fractions in this case? Think of it like this: when you take a fraction of a fraction, you're essentially reducing a part of a part. Multiplication is the operation that allows us to do this precisely. It combines the two fractions into a single fraction that represents the portion of the whole that Ramiro has covered. This setup is not only crucial for solving this particular problem but also for understanding how to handle similar problems involving fractions in the future. It's a fundamental concept in math that has wide-ranging applications. So, now that we have our equation all set up and ready to go, the next step is to actually do the multiplication. We'll take a look at the rules for multiplying fractions and then apply them to our equation. Get ready, because we're about to find out exactly how far Ramiro has run in this marathon!

Multiplying the Fractions

Okay, let's get down to business and multiply these fractions! Remember our equation: (3/4) * (2/5). When we multiply fractions, it's actually pretty straightforward. We multiply the numerators (the top numbers) together and then multiply the denominators (the bottom numbers) together. So, in this case, we have: Numerators: 3 * 2 = 6 Denominators: 4 * 5 = 20 Putting these together, we get the fraction 6/20. This means that Ramiro has run 6/20 of the total marathon distance. But hold up, we're not quite done yet! In math, it's always a good practice to simplify fractions to their lowest terms. This makes the fraction easier to understand and work with in the future. When we simplify a fraction, we're looking for the greatest common factor (GCF) of the numerator and the denominator, and then we divide both by that number. So, why is simplifying important? Well, think of it this way: 6/20 and its simplified form represent the same amount, but the simplified form uses smaller numbers, which can make it easier to visualize and compare to other fractions. Plus, it's just good mathematical etiquette to present your answer in its simplest form! Now, before we dive into simplifying 6/20, let’s just recap why we multiplied the fractions the way we did. Multiplying numerators and denominators is the direct way to calculate a fraction of a fraction. It's a rule that's been proven time and again in mathematics, and it’s the cornerstone of fraction multiplication. So, with our result of 6/20 in hand, we're ready to move on to the next step: finding that GCF and simplifying the fraction to get our final, neat answer. Let's make this fraction as sleek and simple as it can be!

Simplifying the Fraction

Alright, let's simplify that fraction we got, 6/20. To simplify a fraction, we need to find the greatest common factor (GCF) of the numerator (6) and the denominator (20). The GCF is the largest number that divides evenly into both numbers. So, let's think about the factors of 6 and 20. The factors of 6 are 1, 2, 3, and 6. The factors of 20 are 1, 2, 4, 5, 10, and 20. Looking at these lists, we can see that the greatest common factor of 6 and 20 is 2. This means that 2 is the largest number that can divide both 6 and 2 without leaving a remainder. Now that we've found the GCF, we can simplify the fraction by dividing both the numerator and the denominator by 2. So, we have: 6 ÷ 2 = 3 20 ÷ 2 = 10 This gives us the simplified fraction 3/10. So, 6/20 simplified to its lowest terms is 3/10. This means that 3/10 represents the same amount as 6/20, but it's expressed in a more concise way. Think of it like saying