Three bells ring at intervals of 15, 20 and 30 minutes respectively. If they all ring at 11.00 a.m. together, at what time will they next ring together ?
A. 11.30 a.m.
B. 12 noon
C. 12.30 p.m.
D. 1.00 p.m.

Options

A. C
B. A
C. B
D. D

Answer

To solve the problem of when the three bells will next ring together, we need to find the least common multiple (LCM) of the intervals at which they ring. The intervals are 15 minutes, 20 minutes, and 30 minutes.

### Step-by-Step Solution:

1. **Identify the Intervals**:
The three bells ring at intervals of 15, 20, and 30 minutes.

2. **Find the Prime Factorization**:
- For 15: \( 15 = 3 \times 5 \)
- For 20: \( 20 = 2^2 \times 5 \)
- For 30: \( 30 = 2 \times 3 \times 5 \)

3. **Determine the LCM**:
To find the LCM, we take the highest power of each prime factor present in the factorizations:
- The highest power of 2 is \( 2^2 \) (from 20).
- The highest power of 3 is \( 3^1 \) (from 15 and 30).
- The highest power of 5 is \( 5^1 \) (from all three).

Thus, the LCM is calculated as follows:
\[ LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 \]

4. **Calculate the LCM**:
\[ LCM = 4 \times 3 = 12 \]
\[ LCM = 12 \times 5 = 60 \]
So, the LCM of 15, 20, and 30 is 60 minutes.

5. **Determine the Next Ring Time**:
Since all three bells ring together at 11:00 a.m., we add the LCM (60 minutes) to this time:
\[ 11:00 \text{ a.m.} + 60 \text{ minutes} = 12:00 \text{ noon} \]

### Final Answer:
The three bells will next ring together at **12:00 noon**.