Three runners Anisha, Bimala & Chetana run along a circular path of 12 km long with speeds of 5 km/hr, 7 km/hr, 13 km/hr. They start their race from the same point for the same destination. After how much time, they will meet again?

To find out after how much time the three runners Anisha, Bimala, and Chetana will meet again at the starting point, we need to calculate the least common multiple (LCM) of the times it takes each runner to complete one full lap around the circular path.

### Step-by-step Solution:

#### 1. Time taken by each runner to complete one lap:
The time taken by each runner to complete one lap of the circular path can be calculated using the formula:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]
Given:
- Distance = 12 km (since the circular path is 12 km long)
- Speeds: Anisha = 5 km/hr, Bimala = 7 km/hr, Chetana = 13 km/hr

Now, let's compute the time for each runner:

- Time taken by **Anisha** = \(\frac{12}{5}\) hours = 2.4 hours
- Time taken by **Bimala** = \(\frac{12}{7}\) hours ≈ 1.71 hours
- Time taken by **Chetana** = \(\frac{12}{13}\) hours ≈ 0.92 hours

#### 2. Find the LCM of the times:
We need to find the least common multiple (LCM) of the times taken by the runners.

To simplify this, let's first express these times as fractions:
- Anisha: \( \frac{12}{5} \)
- Bimala: \( \frac{12}{7} \)
- Chetana: \( \frac{12}{13} \)

Since they all have a common numerator (12), we can focus on finding the LCM of the denominators: 5, 7, and 13.

The LCM of 5, 7, and 13 (since they are all prime numbers) is simply their product:
\[
\text{LCM}(5, 7, 13) = 5 \times 7 \times 13 = 455
\]

Thus, the LCM of the times is:
\[
\text{LCM of times} = \frac{12 \times 455}{5 \times 7 \times 13} = 12 \times 91 = 1092 \text{ hours}.
\]

Therefore, the three runners will meet again after 1092 hours.

Let me know if you'd like further clarification or additional details.

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### Related Questions:
1. How would the result change if the speeds were doubled?
2. What if the circular track was 18 km long instead of 12 km?
3. How can we determine when only two of the runners will meet again?
4. How is the LCM of non-integer values calculated?
5. Can this problem be solved using relative velocities?

### Tip:
When solving circular track problems involving LCM, always focus on finding the times for each individual and then compute the LCM for accuracy.

This problem calculates when three runners, Anisha, Bimala, and Chetana, running at different speeds, will meet again on a 12 km circular track. Learn how to find the least common multiple (LCM) of their lap times and solve this real-world circular motion problem. Suitable for Grades 10-12.

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