Question

The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8:20:00 hours, then at what time will they again change simultaneously?

• A. 8:25:12 hours
• B. 8:27:12 hours
• C. 9:25:12 hours
• D. 8:30:15 hours

Hint:

For problems involving events that repeat at different intervals (like bells ringing, lights flashing, or runners on a circular track), the key to finding when they coincide is always to calculate the LCM of their time intervals.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To find the time when the traffic lights will change simultaneously again, we need to find the least common multiple (LCM) of their individual time intervals. The LCM gives the smallest amount of time after which all events will coincide.
Step 2: Key Formula or Approach:
Calculate the LCM of 48, 72, and 108.
Step 3: Detailed Explanation:
First, find the prime factorization of each number:
\[ 48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3^1 \] \[ 72 = 2 \times 36 = 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 9 = 2^3 \times 3^2 \] \[ 108 = 2 \times 54 = 2 \times 2 \times 27 = 2^2 \times 3^3 \] The LCM is the product of the highest power of each prime factor present in the numbers.
\[ \text{LCM}(48, 72, 108) = 2^4 \times 3^3 \] \[ \text{LCM} = 16 \times 27 = 432 \text{ seconds} \] This means the lights will change together every 432 seconds.
Now, convert 432 seconds into minutes and seconds:
\[ \frac{432}{60} = 7 \text{ with a remainder of } 12 \] So, 432 seconds = 7 minutes and 12 seconds.
The initial simultaneous change was at 8:20:00 hours.
The next simultaneous change will be after 7 minutes and 12 seconds.
New time = 8 hours : 20 minutes : 00 seconds + 7 minutes : 12 seconds
New time = 8:27:12 hours.
Step 4: Final Answer:
The traffic lights will again change simultaneously at 8:27:12 hours.