Scheduling Buses: A Mathematical Discrepancy Resolved

Every day, bus stands operate with schedules that can sometimes be puzzling to passengers. For instance, two buses leave a bus stand at 10 AM for different destinations, with one leaving every 12 minutes and the other every 15 minutes. How can we determine when these two buses will leave together again after 10 AM? This article will walk you through the process of solving such a problem using mathematical concepts like the least common multiple (LCM).

Understanding the Problem

Let's break down the problem: The first bus leaves the stand every 12 minutes, and the second bus leaves every 15 minutes. Our goal is to find out when both buses will leave the stand simultaneously.

Step 1: Finding the Least Common Multiple (LCM)

To solve this, we need to determine the LCM of the departure intervals. LCM is a valuable tool in such scenarios as it provides us with the smallest interval that is a multiple of both numbers.

Step 1.1: Prime Factorization

First, we find the prime factorization of each number:

Next, we consider all the prime factors that appear in these factorizations, taking the highest power of each:

22 from 12, 31 common factor, and 51 from 15.

So the LCM is calculated as:

LCM 22 times; 31 times; 51

Step 1.2: Calculating the LCM

Now, let's do the multiplication:

LCM 4 times; 3 times; 5 60

Step 2: Determining the Time

The LCM tells us that the buses will leave together again after 60 minutes from the 10 AM departure. So, we simply add 60 minutes to 10 AM:

10 AM 60 minutes 11 AM

Therefore, the two buses will leave together again at 11 AM.

A Deeper Look into the Problem

It's worth noting that while our method involves some mathematical calculations, it can also be solved through logical reasoning. Observing that the first bus leaves 5 times in an hour (12-minute intervals) and the second bus leaves 4 times in an hour (15-minute intervals), it's clear that in one hour both buses will leave together.

While this can offer an easy solution, using the LCM provides a more systematic and universally applicable approach. This problem-solving technique can be applied to many real-world scenarios, enhancing efficiency and reliability in logistics and transportation planning.

Conclusion

The LCM of 12 and 15 is 60, meaning the two buses will leave together again after 60 minutes from 10 AM. Therefore, the next time you catch both these buses leaving from the bus stand will be at 11 AM. Understanding this method can help you anticipate and manage your travel schedules more effectively.