The traffic lights at three different road crossings change after every 30 minutes, 45 minutes and 90 minutes respectively. If they all change simultaneously at 9:15 hours, then at what time will they again change simultaneously?

Options

A. `9:45` hrs
B. `10:15` hrs
C. `10 : 45` hrs
D. `10:35` hrs

Answer

To find out when the traffic lights will change simultaneously again, we need to calculate the least common multiple (LCM) of the time intervals at which the lights change: 30 minutes, 45 minutes, and 90 minutes.

### Step-by-Step Solution:

1. **Identify the time intervals**:
- Light 1 changes every 30 minutes.
- Light 2 changes every 45 minutes.
- Light 3 changes every 90 minutes.

2. **Find the LCM of the intervals**:
- First, we can find the prime factorization of each interval:
- \(30 = 2 \times 3 \times 5\)
- \(45 = 3^2 \times 5\)
- \(90 = 2 \times 3^2 \times 5\)

3. **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 \(2^1\) (from 30 and 90)
- For \(3\): highest power is \(3^2\) (from 45 and 90)
- For \(5\): highest power is \(5^1\) (from all three)
- Therefore, the LCM is:
\[ LCM = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90 \]

4. **Calculate the next simultaneous change time**:
- Since the traffic lights change simultaneously at 9:15 hours, we add the LCM (90 minutes) to this time.
- 90 minutes is equal to 1 hour and 30 minutes.
- Adding 1 hour and 30 minutes to 9:15 hours:
\[ 9:15 + 1:30 = 10:45 \]

5. **Final Answer**:
- The traffic lights will change simultaneously again at **10:45 hours**.