Two Trains Leave The Station At 8 A.m. One Train Completes Its Loop In 60 Minutes. The Other Train Completes Its Loop In 75 Minutes. In How Many Minutes Will The Two Trains Meet Again?

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Introduction

The problem of two trains leaving a station at the same time and meeting again at a later time is a classic example of a mathematical puzzle. In this article, we will explore the solution to this problem and provide a step-by-step guide on how to calculate the time it takes for the two trains to meet again.

The Problem

Two trains leave the station at 8 a.m. One train completes its loop in 60 minutes, while the other train completes its loop in 75 minutes. We need to find out in how many minutes the two trains will meet again.

Understanding the Problem

To solve this problem, we need to understand the concept of Least Common Multiple (LCM). The LCM of two numbers is the smallest number that is a multiple of both numbers. In this case, we need to find the LCM of 60 and 75.

Calculating the LCM

To calculate the LCM of 60 and 75, we need to find the prime factors of both numbers.

  • 60 = 2^2 × 3 × 5
  • 75 = 3 × 5^2

Finding the LCM

To find the LCM, we need to take the highest power of each prime factor that appears in either number.

  • LCM(60, 75) = 2^2 × 3 × 5^2 = 300

The Meeting Time

Since the LCM of 60 and 75 is 300, the two trains will meet again after 300 minutes. However, we need to consider the fact that the trains leave the station at 8 a.m. and meet again after a certain number of hours.

Converting Minutes to Hours

To convert 300 minutes to hours, we can divide by 60.

  • 300 minutes ÷ 60 = 5 hours

The Final Answer

Therefore, the two trains will meet again at 1 p.m. (8 a.m. + 5 hours).

Conclusion

In this article, we have explored the solution to the problem of two trains leaving a station at the same time and meeting again at a later time. We have used the concept of Least Common Multiple (LCM) to calculate the time it takes for the two trains to meet again. The final answer is 5 hours, which is equivalent to 300 minutes.

Real-World Applications

This problem has real-world applications in various fields such as transportation, logistics, and scheduling. For example, in a railway system, the LCM of the travel times of two trains can be used to determine the optimal time for them to meet at a station.

Mathematical Concepts

This problem involves several mathematical concepts, including:

  • Least Common Multiple (LCM): The LCM of two numbers is the smallest number that is a multiple of both numbers.
  • Prime Factors: The prime factors of a number are the prime numbers that multiply together to give the number.
  • Exponents: Exponents are used to represent the power of a number.

Solving Similar Problems

To solve similar problems, you can use the following steps:

  1. Understand the Problem: Read the problem carefully and understand what is being asked.
  2. Identify the Key Concepts: Identify the key mathematical concepts involved in the problem.
  3. Calculate the LCM: Calculate the LCM of the numbers involved in the problem.
  4. Convert Minutes to Hours: Convert the LCM from minutes to hours.
  5. Find the Meeting Time: Find the meeting time by adding the LCM to the initial time.

By following these steps, you can solve similar problems and find the meeting time of two trains or any other objects that leave a station at the same time.

Introduction

In our previous article, we explored the solution to the problem of two trains leaving a station at the same time and meeting again at a later time. In this article, we will provide a Q&A guide to help you understand the concept of Least Common Multiple (LCM) and how to calculate the meeting time of two trains.

Q: What is the Least Common Multiple (LCM)?

A: The LCM of two numbers is the smallest number that is a multiple of both numbers. It is used to find the meeting time of two trains or any other objects that leave a station at the same time.

Q: How do I calculate the LCM of two numbers?

A: To calculate the LCM of two numbers, you need to find the prime factors of both numbers and take the highest power of each prime factor that appears in either number.

Q: What are prime factors?

A: Prime factors are the prime numbers that multiply together to give the number. For example, the prime factors of 60 are 2, 2, 3, and 5.

Q: How do I find the prime factors of a number?

A: To find the prime factors of a number, you can use the following steps:

  1. Divide the number by the smallest prime number (2): If the number is divisible by 2, then 2 is a prime factor.
  2. Divide the number by the next prime number (3): If the number is divisible by 3, then 3 is a prime factor.
  3. Continue dividing the number by prime numbers: Continue dividing the number by prime numbers until you reach 1.

Q: How do I calculate the LCM of two numbers using prime factors?

A: To calculate the LCM of two numbers using prime factors, you need to take the highest power of each prime factor that appears in either number.

Q: What is the formula for calculating the LCM of two numbers?

A: The formula for calculating the LCM of two numbers is:

LCM(a, b) = (a × b) / GCD(a, b)

where GCD(a, b) is the Greatest Common Divisor of a and b.

Q: How do I calculate the GCD of two numbers?

A: To calculate the GCD of two numbers, you can use the following steps:

  1. List the factors of each number: List the factors of each number.
  2. Find the common factors: Find the common factors of both numbers.
  3. Take the highest common factor: Take the highest common factor of both numbers.

Q: How do I convert minutes to hours?

A: To convert minutes to hours, you can divide the number of minutes by 60.

Q: What is the meeting time of two trains?

A: The meeting time of two trains is the time at which they meet again after leaving the station at the same time.

Q: How do I find the meeting time of two trains?

A: To find the meeting time of two trains, you need to calculate the LCM of their travel times and convert it to hours.

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of Least Common Multiple (LCM) and how to calculate the meeting time of two trains. We hope this guide has been helpful in solving similar problems and finding the meeting time of two trains or any other objects that leave a station at the same time.

Real-World Applications

This problem has real-world applications in various fields such as transportation, logistics, and scheduling. For example, in a railway system, the LCM of the travel times of two trains can be used to determine the optimal time for them to meet at a station.

Mathematical Concepts

This problem involves several mathematical concepts, including:

  • Least Common Multiple (LCM): The LCM of two numbers is the smallest number that is a multiple of both numbers.
  • Prime Factors: The prime factors of a number are the prime numbers that multiply together to give the number.
  • Exponents: Exponents are used to represent the power of a number.
  • Greatest Common Divisor (GCD): The GCD of two numbers is the largest number that divides both numbers.

Solving Similar Problems

To solve similar problems, you can use the following steps:

  1. Understand the Problem: Read the problem carefully and understand what is being asked.
  2. Identify the Key Concepts: Identify the key mathematical concepts involved in the problem.
  3. Calculate the LCM: Calculate the LCM of the numbers involved in the problem.
  4. Convert Minutes to Hours: Convert the LCM from minutes to hours.
  5. Find the Meeting Time: Find the meeting time by adding the LCM to the initial time.

By following these steps, you can solve similar problems and find the meeting time of two trains or any other objects that leave a station at the same time.