Three bells ring simultaneously at 11 am. They ring at regular intervals of 20 minutes, 30 minutes, 40 minutes respectively. The time when all the three ring together next is :

Options

A. 2 p.m
B. 1 p.m.
C. 1.15 p.m.
D. 1.30 p.m

Answer

To solve the problem, we need to find the time when all three bells will ring together again after they first ring at 11 AM. The bells ring at intervals of 20 minutes, 30 minutes, and 40 minutes respectively.

### Step-by-Step Solution:

1. **Identify the intervals of the bells**:
- Bell 1 rings every 20 minutes.
- Bell 2 rings every 30 minutes.
- Bell 3 rings every 40 minutes.

2. **Find the Least Common Multiple (LCM)**:
- To determine when all three bells will ring together, we need to calculate the LCM of the intervals: 20, 30, and 40.

3. **Prime Factorization**:
- Break down each number into its prime factors:
- 20 = 2^2 × 5^1
- 30 = 2^1 × 3^1 × 5^1
- 40 = 2^3 × 5^1

4. **Determine the LCM**:
- For the LCM, take the highest power of each prime factor:
- For 2: the highest power is 2^3 (from 40)
- For 3: the highest power is 3^1 (from 30)
- For 5: the highest power is 5^1 (from all three)
- Therefore, LCM = 2^3 × 3^1 × 5^1 = 8 × 3 × 5.

5. **Calculate the LCM**:
- First, calculate 8 × 3 = 24.
- Then, calculate 24 × 5 = 120.
- Thus, LCM(20, 30, 40) = 120 minutes.

6. **Convert minutes to hours**:
- 120 minutes = 120 ÷ 60 = 2 hours.

7. **Determine the next time all bells ring together**:
- Since they all ring together at 11 AM, we add 2 hours to this time.
- 11 AM + 2 hours = 1 PM.

### Final Answer:
The next time all three bells will ring together is at **1 PM**.