How Many Kilometers From The Terminal Will Bus Lines A, B, And C Stop At The Same Place, Given That Line A Stops Every 15 Km, Line B Every 24 Km, And Line C Every 30 Km?

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In this mathematical problem, we explore the concept of least common multiples (LCM) in a real-world scenario. Imagine three bus lines – A, B, and C – all departing from the same terminal and following the same route. Each line has its own distinct stopping pattern: Line A stops every 15 kilometers, Line B every 24 kilometers, and Line C every 30 kilometers. Our primary objective is to determine the distance from the terminal at which all three bus lines will simultaneously make a stop at the same location. This problem highlights the practical application of LCM in understanding recurring events or meeting points within different cycles.

To solve this problem effectively, we need to identify the common multiples of the distances at which each bus line stops. This involves finding the smallest number that is divisible by 15, 24, and 30. The least common multiple will represent the distance from the terminal where all three lines will coincide at a stop. Understanding the LCM not only helps in solving this specific problem but also provides a framework for addressing similar scenarios involving cycles, schedules, and recurring events. In the following sections, we will delve into the step-by-step process of calculating the LCM and arriving at the solution, thereby demonstrating the importance of this mathematical concept in everyday applications.

To solve this problem, we need to find the least common multiple (LCM) of 15, 24, and 30. The LCM is the smallest positive integer that is divisible by all the given numbers. There are several methods to calculate the LCM, but one of the most common is the prime factorization method. This method involves breaking down each number into its prime factors and then identifying the highest power of each prime factor present in any of the numbers. Let's break down the steps:

  1. Prime Factorization:

    • First, we find the prime factorization of each number:
      • 15 = 3 * 5
      • 24 = 2 * 2 * 2 * 3 = 2^3 * 3
      • 30 = 2 * 3 * 5

  2. Identify Highest Powers:

    • Next, we identify the highest power of each prime factor that appears in any of the factorizations:
      • The highest power of 2 is 2^3 (from 24).
      • The highest power of 3 is 3^1 (present in all three numbers).
      • The highest power of 5 is 5^1 (present in 15 and 30).

  3. Calculate the LCM:

    • Now, we multiply these highest powers together to get the LCM:
      • LCM (15, 24, 30) = 2^3 * 3 * 5 = 8 * 3 * 5 = 120

Therefore, the least common multiple of 15, 24, and 30 is 120. This means that the three bus lines will stop at the same location 120 kilometers from the terminal. This step-by-step approach not only helps in finding the solution but also provides a clear understanding of the underlying mathematical principles. Understanding prime factorization and its application in finding LCM is crucial for solving various mathematical problems related to cycles, schedules, and recurring events. In the subsequent sections, we will explore the implications of this solution and its real-world significance.

The solution to this problem, which is 120 kilometers, has significant implications in understanding the dynamics of transportation systems and logistical planning. The least common multiple (LCM) of the distances at which the bus lines stop provides a crucial point of convergence. At 120 kilometers from the terminal, all three bus lines – A, B, and C – will simultaneously make a stop. This convergence point can be vital for various logistical and operational considerations.

Firstly, from a passenger perspective, this common stop could serve as a key transfer point. Passengers wishing to switch between bus lines can do so efficiently at this location, minimizing wait times and streamlining their journeys. This is particularly important for passengers traveling long distances or those needing to reach specific destinations served by different lines. By understanding the meeting points of these bus lines, transit authorities can enhance the convenience and efficiency of public transportation networks.

Secondly, from a logistical standpoint, this common stop can be utilized for coordinating services such as maintenance, refueling, or driver shift changes. Having a designated point where multiple lines converge allows for better resource allocation and scheduling. For instance, maintenance crews can be stationed at this location to service buses from all three lines, reducing downtime and optimizing the use of resources. Similarly, refueling operations can be coordinated to ensure that buses from different lines can efficiently refuel without causing delays.

Moreover, understanding the convergence points of different routes is crucial for emergency planning. In case of an incident or disruption on one line, the common stop can serve as a central point for rerouting buses and assisting passengers. This ensures that passengers can continue their journeys with minimal disruption and that emergency services can be effectively coordinated.

In conclusion, the solution to this mathematical problem has practical implications beyond just finding the LCM. It highlights the importance of mathematical concepts in optimizing transportation systems, enhancing passenger convenience, and improving logistical efficiency. Understanding these convergence points allows for better planning and coordination, ultimately leading to a more reliable and efficient transportation network. The principles applied in this scenario can be extended to various other fields, such as scheduling, supply chain management, and resource allocation.

While the prime factorization method is widely used for calculating the least common multiple (LCM), there are alternative methods that can be employed, each with its own advantages and suitability depending on the numbers involved. Exploring these methods can provide a more comprehensive understanding of LCM calculation and offer flexibility in problem-solving. Here, we will discuss two alternative methods: the listing multiples method and the division method.

  1. Listing Multiples Method:

    • The listing multiples method involves listing out the multiples of each number until a common multiple is found. This method is particularly useful when dealing with smaller numbers, as it is straightforward and easy to understand. For the bus lines problem (15, 24, and 30), we would list the multiples as follows:
      • Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150...
      • Multiples of 24: 24, 48, 72, 96, 120, 144, 168...
      • Multiples of 30: 30, 60, 90, 120, 150, 180...
    • By examining the lists, we can see that the smallest common multiple is 120. While this method is intuitive, it can become cumbersome and time-consuming when dealing with larger numbers or numbers with no obvious common multiples within the first few multiples.

  2. Division Method:

    • The division method, also known as the ladder method, is a more systematic approach that involves dividing the numbers by their common prime factors until all the quotients are 1. This method is particularly efficient for finding the LCM of multiple numbers simultaneously. To apply this method to the bus lines problem, we would follow these steps:

      1. Write the numbers (15, 24, and 30) in a row.
      2. Divide the numbers by a common prime factor. If a number is not divisible, carry it down to the next row.
      3. Repeat the process until all the quotients are 1.
      4. Multiply all the divisors to get the LCM.

    • Here’s how it would look:

    15 24 30
    2 15 12 15
    2 15 6 15
    2 15 3 15
    3 5 1 5
    5 1 1 1

    • LCM = 2 * 2 * 2 * 3 * 5 = 120

    • The division method is advantageous for its structured approach and efficiency, especially when dealing with three or more numbers. It reduces the chances of errors and provides a clear pathway to the solution.

In conclusion, while the prime factorization method is a reliable way to calculate LCM, exploring alternative methods like listing multiples and the division method can enhance problem-solving skills and provide flexibility in different scenarios. Each method offers a unique approach, and the choice of method often depends on the specific numbers involved and the individual's preference. Understanding these methods broadens the mathematical toolkit and allows for more efficient and effective problem-solving.

In conclusion, the problem of determining the meeting point of three bus lines – A, B, and C – serves as an excellent illustration of the practical application of the least common multiple (LCM) in real-world scenarios. By identifying the LCM of the distances at which each bus line makes a stop (15, 24, and 30 kilometers, respectively), we were able to pinpoint the location where all three lines would coincide: 120 kilometers from the terminal.

This solution underscores the importance of mathematical concepts in everyday life. The LCM is not merely an abstract mathematical idea but a valuable tool for solving practical problems related to scheduling, logistics, and coordination. In the context of transportation, understanding the LCM can lead to more efficient route planning, optimized transfer points, and improved resource allocation. For instance, transit authorities can leverage this information to create convenient interchange locations for passengers, coordinate maintenance schedules, and plan for emergency scenarios.

Moreover, the application of LCM extends beyond transportation. It is relevant in various other fields, including:

  1. Scheduling and Time Management: Determining when recurring events will coincide, such as meetings, project deadlines, or periodic maintenance tasks.
  2. Manufacturing and Production: Optimizing production cycles to minimize waste and maximize efficiency.
  3. Computer Science: Calculating the timing of events in computer algorithms and systems.
  4. Music: Understanding the patterns and harmonies in musical compositions.

By exploring different methods for calculating the LCM, such as prime factorization, listing multiples, and the division method, we gain a deeper appreciation for the versatility of mathematical tools. Each method offers a unique approach, and the choice of method often depends on the specific problem and individual preferences. This flexibility is crucial for effective problem-solving in various contexts.

In essence, this problem demonstrates that mathematics is not just a theoretical discipline but a practical toolkit that empowers us to understand and optimize the world around us. The ability to identify and apply concepts like the LCM is essential for making informed decisions, improving efficiency, and solving complex problems in a wide range of fields. As we continue to encounter similar challenges in our daily lives, the principles learned from this exercise will prove invaluable in finding effective and innovative solutions.