Three bells toll together at intervals of 9, 12, 15 minutes . If they start tolling together, after what time will they next toll together ?

Options

A. 1 hour
B. `1(1)/(2)` hours
C. `2(1)/(2)` hours
D. 3 hours

Answer

To find out when the three bells will toll together again after starting together, we need to calculate the Least Common Multiple (LCM) of the intervals at which they toll, which are 9, 12, and 15 minutes.

### Step-by-Step Solution:

1. **List the intervals**: The intervals for the three bells are 9 minutes, 12 minutes, and 15 minutes.

2. **Find the prime factorization of each number**:
- **9**: The prime factorization of 9 is \(3^2\).
- **12**: The prime factorization of 12 is \(2^2 \times 3^1\).
- **15**: The prime factorization of 15 is \(3^1 \times 5^1\).

3. **Identify the highest power of each prime factor**:
- For the prime number **2**: The highest power is \(2^2\) (from 12).
- For the prime number **3**: The highest power is \(3^2\) (from 9).
- For the prime number **5**: The highest power is \(5^1\) (from 15).

4. **Multiply these highest powers together to find the LCM**:
\[ LCM = 2^2 \times 3^2 \times 5^1 \]
\[ = 4 \times 9 \times 5 \]
\[ = 36 \times 5 = 180 \]

5. **Conclusion**: The three bells will toll together again after **180 minutes**.