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: Concept Identification: To determine the next simultaneous change of traffic lights, we must find the least common multiple (LCM) of their individual timing cycles. The LCM represents the shortest duration after which all cycles will align.
Step 2: Calculation Method: Compute the LCM for the intervals of 48, 72, and 108 seconds.
Step 3: Calculation Process:
First, determine the prime factorization of each number:
\[ 48 = 2^4 \times 3^1 \]
\[ 72 = 2^3 \times 3^2 \]
\[ 108 = 2^2 \times 3^3 \]
The LCM is derived by multiplying the highest power of each unique prime factor present:
\[ \text{LCM}(48, 72, 108) = 2^4 \times 3^3 \]
\[ \text{LCM} = 16 \times 27 = 432 \text{ seconds} \]
This indicates a simultaneous change occurs every 432 seconds.
Convert 432 seconds to minutes and seconds:
\[ 432 \div 60 = 7 \text{ with a remainder of } 12 \]
Thus, 432 seconds equals 7 minutes and 12 seconds.
The initial simultaneous change occurred at 8:20:00.
The subsequent simultaneous change will be 7 minutes and 12 seconds later.
New time = 8 hours : 20 minutes : 00 seconds + 7 minutes : 12 seconds
New time = 8:27:12 hours.
Step 4: Conclusion: The traffic lights will next change simultaneously at 8:27:12 hours.