Traffic light at three different road crossings change after every 48 sec, 72 sec and 108 sec respectively. If they all change simultaneously at `7:00:00` hours, then at what time will they again change simultaneously ?

Options

A. `7:14 : 00` hours
B. `7:14:12` hours
C. `7: 07:12` hours
D. `7 : 09 :12` hours

Answer

To solve the problem of when the traffic lights will change simultaneously again, we need to find the least common multiple (LCM) of the three time intervals: 48 seconds, 72 seconds, and 108 seconds.

### Step-by-Step Solution:

1. **Identify the time intervals**:
The traffic lights change every 48 seconds, 72 seconds, and 108 seconds.

2. **Find the prime factorization of each time interval**:
- For 48:
\[ 48 = 2^4 \times 3^1 \]
- For 72:
\[ 72 = 2^3 \times 3^2 \]
- For 108:
\[ 108 = 2^2 \times 3^3 \]

3. **Determine the LCM**:
To find the LCM, take the highest power of each prime factor from the factorizations:
- For 2: the highest power is \(2^4\) (from 48).
- For 3: the highest power is \(3^3\) (from 108).

Therefore, the LCM is:
\[ LCM = 2^4 \times 3^3 = 16 \times 27 = 432 \text{ seconds} \]

4. **Convert seconds to minutes and seconds**:
To convert 432 seconds into minutes:
\[ 432 \div 60 = 7 \text{ minutes} \text{ with a remainder of } 12 \text{ seconds} \]
So, 432 seconds is equal to 7 minutes and 12 seconds.

5. **Calculate the time when they will change again**:
Since the traffic lights change simultaneously at 7:00:00, we add 7 minutes and 12 seconds to this time:
\[ 7:00:00 + 0:07:12 = 7:07:12 \]

### Final Answer:
The traffic lights will change simultaneously again at **7:07:12**.