A, B and C start at the same time in the same direction to run around a circular stadium. A complete a round in 10 minutes, B and C completes in 12 and 15 minutes respectively. In how many rounds, they will meet together according to A?

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem of how many rounds A, B, and C will meet together according to A, we can follow these steps:

### Step 1: Identify the time taken by each runner to complete one round.
- A completes a round in 10 minutes.
- B completes a round in 12 minutes.
- C completes a round in 15 minutes.

### Step 2: Find the Least Common Multiple (LCM) of the times taken by A, B, and C.
To find the LCM of 10, 12, and 15, we can use the prime factorization method.

- Prime factorization:
- 10 = 2 × 5
- 12 = 2² × 3
- 15 = 3 × 5

### Step 3: Determine the highest power of each prime factor.
- For 2: highest power is 2² (from 12)
- For 3: highest power is 3¹ (from 12 and 15)
- For 5: highest power is 5¹ (from 10 and 15)

### Step 4: Calculate the LCM using the highest powers.
LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60 minutes.

### Step 5: Calculate how many rounds A completes in 60 minutes.
Since A completes a round in 10 minutes:
Number of rounds completed by A in 60 minutes = 60 / 10 = 6 rounds.

### Conclusion:
A, B, and C will meet together after A has completed 6 rounds.

### Final Answer:
They will meet together according to A in **6 rounds**.

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