Luis's Road Trip Calculating The First Stop

by Scholario Team 44 views

Introduction

Hey guys! Let's dive into a cool mathematical problem today. Imagine Luis embarking on a road trip from a terminal where he finds a phone, a restaurant, and a gas station all conveniently located together. Along his route, there are phones every 4 kilometers, restaurants every 10,000 meters, and gas stations every 250,000 meters. Luis decides he'll only stop at kilometers where all three amenities are available. The challenge? To figure out at which kilometer mark Luis will make his first stop.

This problem isn't just about numbers; it's a practical scenario that requires us to think about units of measurement, common multiples, and planning. We're essentially trying to find the least common multiple (LCM) of the distances between these amenities, but before we jump into calculations, let's break down each component. This involves converting all distances into the same unit, understanding the significance of the LCM in this context, and then applying the concept to solve the problem. So, buckle up as we embark on this mathematical journey to help Luis plan his perfect pit stop!

Converting Units to Kilometers

The first step in solving this problem is ensuring all measurements are in the same unit. We have distances in kilometers (km) and meters (m), so let's convert everything to kilometers since that's what Luis is using to mark his stops. We know that:

  • 1 kilometer (km) = 1000 meters (m)

So, let's convert the distances for the restaurant and gas station into kilometers:

  • Restaurants: Every 10,000 meters
    • 10,000 m * (1 km / 1000 m) = 10 km
  • Gas Stations: Every 250,000 meters
    • 250,000 m * (1 km / 1000 m) = 250 km

Now we have all distances in kilometers:

  • Phones: Every 4 km
  • Restaurants: Every 10 km
  • Gas Stations: Every 250 km

Having all our measurements in the same unit is crucial for accurately comparing and calculating the distances. This conversion sets the stage for finding the common points where Luis can find all three amenities. It simplifies the problem and makes it easier to identify the kilometers at which Luis can make a stop. In the next section, we'll explore how to find the least common multiple (LCM) of these distances, which will lead us to the solution. Stay tuned, guys!

Finding the Least Common Multiple (LCM)

Okay, with all our distances now in kilometers, the next step is to find the least common multiple (LCM) of 4 km (phones), 10 km (restaurants), and 250 km (gas stations). The LCM is the smallest multiple that all these numbers share. In our road trip scenario, the LCM will tell us the shortest distance after which all three amenities—a phone, a restaurant, and a gas station—will be available at the same kilometer mark.

There are a couple of ways to find the LCM, but let's use the prime factorization method, which is super reliable and helps us understand what's going on behind the scenes. Here’s how it works:

  1. Prime Factorization: We break down each number into its prime factors:
    • 4 = 2 * 2 = 2²
    • 10 = 2 * 5
    • 250 = 2 * 5 * 5 * 5 = 2 * 5³
  2. Identify Highest Powers: For each prime number, we identify the highest power that appears in any of the factorizations:
    • The highest power of 2 is 2²
    • The highest power of 5 is 5³
  3. Multiply the Highest Powers: We multiply these highest powers together to get the LCM:
    • LCM(4, 10, 250) = 2² * 5³ = 4 * 125 = 500

So, the LCM of 4, 10, and 250 is 500. This means that every 500 kilometers, Luis will find a phone, a restaurant, and a gas station all at the same location. This is a key piece of information for planning his stops! In the next section, we'll see how this LCM helps us answer the original question: At which kilometer mark will Luis make his first stop?

Determining Luis's First Stop

Alright, guys, we've done the groundwork! We've converted all distances to kilometers and found that the least common multiple (LCM) of the distances between phones, restaurants, and gas stations is 500 km. This means that the facilities coincide every 500 kilometers. Now, let's circle back to our original question: At which kilometer mark will Luis make his first stop where he can access a phone, a restaurant, and a gas station?

Since Luis is starting from a terminal where all three amenities are available, and we've determined that they coincide every 500 kilometers, his first stop where all three are present will be at the first multiple of the LCM. In this case, that's simply the LCM itself.

  • First Stop: 500 km

So, Luis will make his first stop at the 500-kilometer mark. At this point, he'll find a phone, a restaurant, and a gas station, allowing him to rest, refuel, and communicate if needed. This is a crucial piece of information for his road trip planning. He knows that after leaving the terminal, he can confidently travel 500 kilometers before needing to stop for these essential services.

This problem highlights the practical application of mathematical concepts like LCM in real-world scenarios. By understanding and applying these concepts, we can solve problems related to planning, logistics, and resource management. In the next section, we'll recap our journey and discuss the importance of these mathematical tools.

Conclusion

So, there you have it, guys! We've successfully navigated Luis's road trip dilemma using some cool mathematical skills. We started with the problem of finding when Luis could make his first stop where he'd have access to a phone, a restaurant, and a gas station. By converting all the distances to the same unit (kilometers) and calculating the least common multiple (LCM) of the distances between these amenities, we pinpointed that Luis would make his first stop at the 500-kilometer mark.

This journey underscores the importance of mathematical concepts in everyday situations. The LCM, in particular, is a powerful tool for solving problems that involve recurring intervals or cycles. Whether it's planning a road trip, scheduling events, or managing resources, understanding and applying the LCM can lead to efficient and effective solutions. The ability to convert units, perform prime factorization, and compute the LCM are valuable skills that extend beyond the classroom.

By breaking down the problem into manageable steps, we made it easier to understand and solve. We first converted the distances into a common unit, then found the LCM, and finally applied that information to answer the specific question. This step-by-step approach is a great strategy for tackling any complex problem, whether it's mathematical or otherwise. Always remember to break things down, identify the key components, and use the appropriate tools to find the solution.

I hope you enjoyed this mathematical adventure as much as I did! Understanding these concepts not only helps in solving problems but also in appreciating the logical and interconnected nature of mathematics. Keep exploring, keep questioning, and keep applying these skills in your everyday lives. You never know when a little bit of math can make a big difference!

Repair Input Keyword

  • Original Question: Sobre una ruta, cada 4 km hay un telefono, cada 10 000 m un restaurante y cada 250,000 om una estación de gasolina. Luis parte de la terminal en donde hay un teléfono, un restaurante y una estación de gasolina y decide que sólo parará en el kilómetro
  • Rewritten Question: If there is a phone every 4 km, a restaurant every 10,000 m, and a gas station every 250,000 m along a route, and Luis starts from a terminal with all three, at what kilometer mark will Luis make his first stop if he only stops where all three are available?