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

Answer

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

### Step-by-Step Solution:

1. **Identify the Intervals**:
The intervals at which the bells toll are 9 minutes, 12 minutes, and 15 minutes.

2. **Find the Prime Factorization**:
- For 9: \( 9 = 3^2 \)
- For 12: \( 12 = 2^2 \times 3^1 \)
- For 15: \( 15 = 3^1 \times 5^1 \)

3. **List the Highest Powers of Each Prime**:
- The prime factors involved are 2, 3, and 5.
- The highest power of 2 is \( 2^2 \) (from 12).
- The highest power of 3 is \( 3^2 \) (from 9).
- The highest power of 5 is \( 5^1 \) (from 15).

4. **Calculate the LCM**:
\[ \text{LCM} = 2^2 \times 3^2 \times 5^1 \]
- Calculate \( 2^2 = 4 \)
- Calculate \( 3^2 = 9 \)
- Calculate \( 5^1 = 5 \)
- Now multiply these together:
\[ 4 \times 9 = 36 \]
\[ 36 \times 5 = 180 \]
So, the LCM of 9, 12, and 15 is 180 minutes.

5. **Convert Minutes to Hours**:
To convert 180 minutes into hours, we divide by 60:
\[ 180 \div 60 = 3 \]
Therefore, 180 minutes is equal to 3 hours.

### Final Answer:
The three bells will next toll together after **3 hours**.