Three bells, toll at interval of 36 sec, 40 sec and 48 sec respectively. They start ringing together at particular time. They will toll together next time after :

Options

A. 6 minutes
B. 12 minutes
C. 18 minutes
D. 24 minutes

Answer

To find out when the three bells will toll together next after starting at the same time, we need to calculate the Least Common Multiple (LCM) of their tolling intervals: 36 seconds, 40 seconds, and 48 seconds.

### Step-by-Step Solution:

1. **Identify the Tolling Intervals**:
- The tolling intervals of the bells are 36 seconds, 40 seconds, and 48 seconds.

2. **Prime Factorization**:
- We will find the prime factorization of each number:
- 36 = 2² × 3²
- 40 = 2³ × 5¹
- 48 = 2⁴ × 3¹

3. **Determine the LCM**:
- To find the LCM, we take the highest power of each prime number from the factorizations:
- For the prime number 2: the highest power is 2⁴ (from 48).
- For the prime number 3: the highest power is 3² (from 36).
- For the prime number 5: the highest power is 5¹ (from 40).
- Therefore, the LCM is calculated as:
\[ \text{LCM} = 2^4 × 3^2 × 5^1 \]

4. **Calculate the LCM**:
- Now we compute the LCM:
- \(2^4 = 16\)
- \(3^2 = 9\)
- \(5^1 = 5\)
- Now multiply these values together:
\[ \text{LCM} = 16 × 9 × 5 \]
- First, calculate \(16 × 9 = 144\).
- Then, calculate \(144 × 5 = 720\).

5. **Convert Seconds to Minutes**:
- The LCM is 720 seconds. To convert seconds into minutes:
\[ \text{Minutes} = \frac{720 \text{ seconds}}{60 \text{ seconds/minute}} = 12 \text{ minutes} \]

### Final Answer:
The three bells will toll together next time after **12 minutes**.

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