Three bells ring simultaneously at 11a.m. 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 after they start ringing at 11 a.m. The bells ring at intervals of 20 minutes, 30 minutes, and 40 minutes.

### Step-by-Step Solution:

1. **Identify the intervals**: The three bells ring at intervals of 20 minutes, 30 minutes, and 40 minutes.

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

3. **Prime Factorization**:
- For 20: \(20 = 2^2 \times 5^1\)
- For 30: \(30 = 2^1 \times 3^1 \times 5^1\)
- For 40: \(40 = 2^3 \times 5^1\)

4. **Determine the LCM**: The LCM is found by taking the highest power of each prime factor present in the factorizations:
- For \(2\): highest power is \(2^3\) (from 40)
- For \(3\): highest power is \(3^1\) (from 30)
- For \(5\): highest power is \(5^1\) (from all three)

Therefore, the LCM is:
\[ LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]

5. **Calculate the LCM**:
- First, calculate \(8 \times 3 = 24\)
- Then, calculate \(24 \times 5 = 120\)

Thus, \(LCM(20, 30, 40) = 120\) minutes.

6. **Convert LCM to hours**: Since 120 minutes is equal to 2 hours, we add this to the initial time of 11 a.m.
- \(11:00 + 2 \text{ hours} = 1:00 \text{ p.m.}\)

### Final Answer:
The next time when all three bells will ring together is **1 p.m.**