Three bells ring at intervals of 12, 15 and 18 min respectively. They started ringing simultaneously at 9:00 am. What will be the next time when they all ring simultaneously?

Options

A. 10:00 am
B. 11:00 am
C. 12:00 pm
D. 1:00 pm

Answer

To solve the problem of when the three bells will ring simultaneously after starting at 9:00 AM, we need to find the Least Common Multiple (LCM) of the intervals at which they ring: 12 minutes, 15 minutes, and 18 minutes.

### Step-by-Step Solution:

1. **Find the Prime Factorization of Each Interval:**
- For 12:
- 12 = 2 × 6
- 6 = 2 × 3
- So, 12 = 2² × 3¹
- For 15:
- 15 = 3 × 5
- So, 15 = 3¹ × 5¹
- For 18:
- 18 = 2 × 9
- 9 = 3 × 3
- So, 18 = 2¹ × 3²

2. **Identify the Highest Powers of Each Prime Factor:**
- The prime factors we have are 2, 3, and 5.
- The highest power of 2 from the factorizations is 2² (from 12).
- The highest power of 3 is 3² (from 18).
- The highest power of 5 is 5¹ (from 15).

3. **Calculate the LCM:**
- LCM = (Highest power of 2) × (Highest power of 3) × (Highest power of 5)
- LCM = 2² × 3² × 5¹
- LCM = 4 × 9 × 5

4. **Perform the Multiplication:**
- First, calculate 4 × 9 = 36.
- Then, calculate 36 × 5 = 180.

5. **Convert Minutes to Hours:**
- 180 minutes = 3 hours.

6. **Determine the Next Time They All Ring Simultaneously:**
- Since they started ringing at 9:00 AM, we add 3 hours to this time.
- 9:00 AM + 3 hours = 12:00 PM.

### Final Answer:
The next time when all three bells will ring simultaneously is **12:00 PM**.

---