Justin, Kris and Helen start at the same time walking around a circular track in the same direction. Justin takes 6 minutes to walk around the track, Kris takes 5 minutes, and Helen takes 4 minutes. How many times does each of them go around the track before all three meet again at their common starting point?

To solve this problem, we need to determine when Justin, Kris, and Helen will meet at their common starting point again. The key is finding the least common multiple (LCM) of their times taken to complete one lap.

- Justin takes 6 minutes to complete one lap.
- Kris takes 5 minutes to complete one lap.
- Helen takes 4 minutes to complete one lap.

### Step 1: Find the LCM of 6, 5, and 4.
We need to compute the LCM of the numbers 6, 5, and 4. 

1. Factorize the numbers:
   - 6 = \(2 \times 3\)
   - 5 = \(5\)
   - 4 = \(2^2\)

2. Take the highest powers of all prime factors:
   - The highest power of 2 is \(2^2\) from 4.
   - The highest power of 3 is \(3\) from 6.
   - The highest power of 5 is \(5\) from 5.

Thus, the LCM is:
\[
\text{LCM}(6, 5, 4) = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60 \text{ minutes}.
\]

### Step 2: Calculate how many laps each person completes in 60 minutes.
- Justin completes \( \frac{60}{6} = 10 \) laps.
- Kris completes \( \frac{60}{5} = 12 \) laps.
- Helen completes \( \frac{60}{4} = 15 \) laps.

### Conclusion:
Justin completes 10 laps, Kris completes 12 laps, and Helen completes 15 laps before they all meet again at the starting point.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the least common multiple of 12, 15, and 20, and how would it affect similar scenarios?
2. How does the prime factorization method help in finding the LCM of larger numbers?
3. Can you create a formula for determining when two or more people meet on a track, given their individual times?
4. What if one person changes their speed halfway through the track—how would that impact the LCM?
5. How does this method extend to non-integer times, like 4.5 minutes?

**Tip:** When finding the LCM, always focus on the highest powers of all prime factors involved to ensure accuracy.

Learn how to solve a time and distance problem where Justin, Kris, and Helen walk around a circular track and meet at the same starting point. This solution involves finding the least common multiple (LCM) of their times. Suitable for students in Grades 6-8.

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