Determining When Three Runners Meet at the Starting Point

Introduction

Effective scheduling and race planning are essential in competitive running. For instance, when three runners with different lap completion times are in a race, understanding when they will all be at the starting point simultaneously is crucial. This article demonstrates how to find the least common multiple (LCM) to determine such meeting points.

Understanding the Problem

Suppose three runners are running around a circular track, with their lap completion times being 2 seconds, 4 seconds, and 5.5 seconds. The challenge is to find when they will all be at the starting point together.

Formulating the Problem

To solve this, we need to find the LCM of their lap completion times. The times can be expressed as follows:

Runner 1: 2 seconds Runner 2: 4 seconds Runner 3: 5.5 seconds, which can be converted to a fraction as 11/2 seconds

Converting Times to a Common Unit

First, let's convert all times to a common unit. Since 5.5 seconds can be expressed as a fraction, we convert it to 11/2 seconds.

Now, the times are:

Runner 1: 2 4/2 seconds Runner 2: 4 8/2 seconds Runner 3: 11/2 seconds

Calculating the LCM

The LCM of fractions can be found using the formula:

LCM(

Determining the numerators and denominators:

Numerators: 4, 8, 11 Denominators: 2, 2, 2

Step 1: Find the LCM of the numerators

LCM of 4 and 8 is 8 LCM of 8 and 11 is 88

So LCM(4, 8, 11) 88

Step 2: Find the GCD of the denominators

The GCD of 2, 2, 2 is 2

Step 3: Calculate the LCM of the fractions

Now, we can find the LCM:

LCM

This means the three runners will meet at the starting point after 44 seconds.

Conclusion

Understanding and calculating the LCM is essential for various applications, from sports to scheduling tasks. By following the steps above, you can determine when three runners, or any similar entities, will meet at the starting point.