Converting Seconds to Minutes and Hours

Oct 13, 2025 by wbrum.org 18 views

Introduction

Traffic lights are an essential part of modern transportation systems, ensuring the smooth flow of traffic and maintaining public safety. The timing of traffic lights is carefully planned to minimize congestion and reduce the risk of accidents. In this article, we will delve into a fascinating problem involving three traffic lights that change at different intervals. We will explore the concept of simultaneous changes and determine the time at which the traffic lights will change simultaneously again.

The Problem

Three traffic lights at different road crossings change at the following intervals:

Traffic light 1 changes every 48 seconds
Traffic light 2 changes every 72 seconds
Traffic light 3 changes every 108 seconds

If the traffic lights change simultaneously at 7:00 a.m., we need to find the time at which they will change simultaneously again.

Mathematical Approach

To solve this problem, we need to find the least common multiple (LCM) of the three time intervals. The LCM is the smallest number that is a multiple of all three numbers.

Step 1: Find the Prime Factors

First, we need to find the prime factors of each time interval.

48 = 2^4 × 3
72 = 2^3 × 3^2
108 = 2^2 × 3^3

Step 2: Identify the Common Factors

Next, we need to identify the common factors among the three time intervals.

The common factors are 2 and 3.

Step 3: Calculate the LCM

Now, we need to calculate the LCM by multiplying the highest power of each common factor.

LCM = 2^4 × 3^3 = 432

Conclusion

The least common multiple of 48, 72, and 108 is 432. Therefore, the traffic lights will change simultaneously again after 432 seconds.

Converting Seconds to Minutes and Hours

To make the answer more meaningful, we need to convert the time from seconds to minutes and hours.

432 seconds = 7.2 minutes
432 seconds = 0.72 hours

The Final Answer

The traffic lights will change simultaneously again at 7:00 a.m. + 0.72 hours = 7:04.48 a.m.

Discussion

This problem requires a deep understanding of mathematical concepts, particularly the least common multiple. The solution involves finding the prime factors of each time interval, identifying the common factors, and calculating the LCM. The result is a fascinating example of how mathematics can be applied to real-world problems.

Real-World Applications

The concept of simultaneous changes is crucial in various fields, including:

Traffic management: Understanding the timing of traffic lights is essential for optimizing traffic flow and reducing congestion.
Scheduling: The concept of simultaneous changes can be applied to scheduling tasks and events.
Engineering: The LCM is used in various engineering applications, such as designing electrical circuits and mechanical systems.

Conclusion

Q&A: The Traffic Lights Conundrum

Q: What is the least common multiple (LCM) and why is it important in this problem?

A: The least common multiple (LCM) is the smallest number that is a multiple of all three numbers. In this problem, the LCM is 432, which represents the time at which the traffic lights will change simultaneously again. The LCM is important because it helps us find the common time at which the traffic lights will change.

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

A: To find the prime factors of a number, we need to break it down into its smallest prime components. For example, the prime factors of 48 are 2^4 × 3, which means that 48 can be broken down into 2 × 2 × 2 × 2 × 3.

Q: What is the difference between the least common multiple (LCM) and the greatest common divisor (GCD)?

A: The least common multiple (LCM) is the smallest number that is a multiple of all three numbers, while the greatest common divisor (GCD) is the largest number that divides all three numbers. In this problem, the LCM is 432, while the GCD is 1.

Q: How can the concept of simultaneous changes be applied to real-world problems?

A: The concept of simultaneous changes can be applied to various real-world problems, including:

Traffic management: Understanding the timing of traffic lights is essential for optimizing traffic flow and reducing congestion.
Scheduling: The concept of simultaneous changes can be applied to scheduling tasks and events.
Engineering: The LCM is used in various engineering applications, such as designing electrical circuits and mechanical systems.

Q: What are some common applications of the least common multiple (LCM) in mathematics?

A: The least common multiple (LCM) has various applications in mathematics, including:

Number theory: The LCM is used to find the common factors of two or more numbers.
Algebra: The LCM is used to solve equations and find the common solutions.
Geometry: The LCM is used to find the common lengths of two or more lines.

Q: How can the traffic lights conundrum be used as a teaching tool in mathematics education?

A: The traffic lights conundrum can be used as a teaching tool in mathematics education to help students understand the concept of simultaneous changes and the least common multiple (LCM). This problem can be used to illustrate the importance of mathematical concepts in real-world applications.

Q: What are some common misconceptions about the least common multiple (LCM)?

A: Some common misconceptions about the least common multiple (LCM) include:

The LCM is always the largest number: This is not true, as the LCM can be smaller than the largest number.
The LCM is always the product of the numbers: This is not true, as the LCM can be a smaller number that is a multiple of all the numbers.

Conclusion

In conclusion, the traffic lights conundrum is a fascinating problem that requires a deep understanding of mathematical concepts. By finding the least common multiple (LCM) of the three time intervals, we can determine the time at which the traffic lights will change simultaneously again. This problem has real-world applications in traffic management, scheduling, and engineering.