Solving Traffic Light Simultaneous Change Time

This problem involves finding the time when three events, occurring at regular but different intervals, will happen simultaneously again. We are given the intervals at which three traffic lights change: 48 seconds, 72 seconds, and 108 seconds. They all changed together at 8:20:00 hours. We need to find the next time they will all change at the exact same moment.

Finding the Least Common Multiple (LCM)

To find when the traffic lights will change simultaneously again, we need to determine the smallest amount of time that is a multiple of all three intervals (48, 72, and 108 seconds). This is known as the Least Common Multiple (LCM).

Let's find the prime factorization of each number:

• 48 seconds: $48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3^1$
• 72 seconds: $72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$
• 108 seconds: $108 = 2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^3$

To calculate the LCM, we take the highest power of each prime factor present in any of the factorizations:

LCM$(48, 72, 108) = 2^{\max(4, 3, 2)} \times 3^{\max(1, 2, 3)}$

LCM$(48, 72, 108) = 2^4 \times 3^3

LCM$(48, 72, 108) = 16 \times 27

LCM$(48, 72, 108) = 432$ seconds

Calculating the Time Duration

The LCM is 432 seconds. This is the time interval after which the lights will change simultaneously again. We need to convert this duration into minutes and seconds to add it to the starting time.

To convert seconds to minutes, we divide by 60:

$432 \text{ seconds} \div 60 \text{ seconds/minute} = 7$ with a remainder of $12$.

So, 432 seconds is equal to 7 minutes and 12 seconds.

Determining the Next Simultaneous Change Time

The traffic lights last changed together at 8:20:00 hours. They will change together again after 7 minutes and 12 seconds.

Add the duration to the initial time:

Starting Time: 8:20:00 hours

Add Duration: + 0:07:12 hours

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Next Simultaneous Time: 8:27:12 hours

Therefore, the traffic lights will again change simultaneously at 8:27:12 hours.