Have you ever found yourself at a crossroad where two traffic lights turn red at the same time? One with a flash that turns red every 30 seconds and the other every 40 seconds? In this article, we will explore the mathematical and practical aspects of this fascinating phenomenon.

Understanding the Basics

When you encounter a solid red light, you wait for 30 seconds. This is because the first traffic light cycles every 30 seconds. For the flashing red light, you treat it as a stop sign and come to a complete stop, look around, and proceed if it is safe.

Mathematical Analysis

To determine when the two traffic lights will next turn red together, we need to understand the pattern of their timings. Let’s delve into the mathematical details.

Synthetic Approach to Periodicity

Let’s consider the sequence of times when each light turns red. We will number the seconds of a two-minute period from 0 to 119, since 2 minutes is equivalent to 120 seconds.

Both traffic signals turn red at the start, i.e., at time 0. Now, let’s list the subsequent times they turn red:

The first traffic light will turn red at 30, 60, 90, and again at 0 (next cycle). The second traffic light will turn red at 40, 80, and again at 0 (next cycle).

By looking at these timings, we can identify a pattern. The first traffic light turns red every 30 seconds, while the second traffic light turns red every 40 seconds. To find the next time they both turn red together, we need to find the least common multiple (LCM) of 30 and 40.

Least Common Multiple (LCM) Calculation

The LCM of two numbers is the smallest positive integer that is divisible by both numbers. To find the LCM of 30 and 40, we can use the prime factorization method:

Prime Factorization Method

Prime factorization of 30: 30 2 × 3 × 5 Prime factorization of 40: 40 23 × 5

The LCM of 30 and 40 is the product of the highest power of each prime number that appears in the factorizations:

LCM(30, 40) 23 × 3 × 5 120

Therefore, the two traffic lights will next turn red together after 120 seconds, which is exactly 2 minutes from the start.

Practical Implications

Understanding the synchronization of traffic lights can help drivers and pedestrians plan their movements more effectively. For instance, if the lights are synchronized every 2 minutes, you can plan your movements accordingly:

If you are driving, you can anticipate the signals turning red. If you are a pedestrian, you can synchronize your crossing with the green light sequences.

This knowledge can also be used for traffic management and planning, ensuring smoother and safer traffic flow.

Conclusion

In this article, we explored the synchronous pattern of two traffic lights turning red at specific intervals. We calculated the LCM to determine when they will next turn red together, which is every 2 minutes. Understanding this pattern can help enhance traffic flow and improve overall road safety.

For more information on traffic light patterns and synchronization, refer to additional resources on traffic engineering and urban planning.