Arjun and Bhishma run at the speed 30 m/s and 20 m/s respectively on the circular track of 600 m, as its circumference, when would the Arjun and Bhishma meet for the first time at the starting point if they start simultaneously from the same point?

Answer

To solve the problem of when Arjun and Bhishma will meet at the starting point on a circular track, we can follow these steps:

### Step 1: Determine the time taken by Arjun to complete one lap.
Arjun's speed is 30 m/s, and the circumference of the circular track is 600 m.

To find the time taken by Arjun to complete one lap, we use the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
So for Arjun:
\[ \text{Time}_{\text{Arjun}} = \frac{600 \text{ m}}{30 \text{ m/s}} = 20 \text{ seconds} \]

### Step 2: Determine the time taken by Bhishma to complete one lap.
Bhishma's speed is 20 m/s.

Using the same formula:
\[ \text{Time}_{\text{Bhishma}} = \frac{600 \text{ m}}{20 \text{ m/s}} = 30 \text{ seconds} \]

### Step 3: Find the least common multiple (LCM) of the times taken by Arjun and Bhishma.
We need to find the LCM of 20 seconds (Arjun's time) and 30 seconds (Bhishma's time) to determine when they will meet at the starting point.

The prime factorization of the times is:
- 20 = 2^2 × 5
- 30 = 2 × 3 × 5

To find the LCM, we take the highest power of each prime factor:
- LCM = 2^2 × 3^1 × 5^1 = 4 × 3 × 5 = 60 seconds

### Step 4: Conclusion
Arjun and Bhishma will meet at the starting point for the first time after 60 seconds.

### Summary of the Solution:
- Arjun takes 20 seconds to complete one lap.
- Bhishma takes 30 seconds to complete one lap.
- The LCM of 20 seconds and 30 seconds is 60 seconds.
- Therefore, they will meet at the starting point after 60 seconds.

---