There are three traffic signals. Each signal changes colour from green to red and then form red to green. The first signal takes 25 seconds, the second signal takes 39 seconds and the third signal takes 60 seconds to change the colour from green to red. The durations for green and red colour are time. At 2:00 p.m. they together turn green. At what time will they change to green next, simultaneously?

Correct Option (B)

To determine when the three traffic signals will next turn green simultaneously, it is necessary to calculate the Least Common Multiple (LCM) of their respective full cycle durations. The problem states that each signal takes a specific time to change from green to red, and the durations for both green and red colours are equal. Therefore, the full cycle duration for each signal (from green, through red, and back to green) is twice the given time:

• Signal 1 cycle duration: 25 seconds (green) + 25 seconds (red) = 50 seconds.
• Signal 2 cycle duration: 39 seconds (green) + 39 seconds (red) = 78 seconds.
• Signal 3 cycle duration: 60 seconds (green) + 60 seconds (red) = 120 seconds.

Next, calculate the LCM of these full cycle durations (50, 78, and 120):

• Prime factorization:
• $50=2\times 5^2$
• $78=2\times 3\times 13$
• $120=2^3\times 3\times 5$
• $LCM(50,78,120)=2^3\times 3\times 5^2\times 13=8\times 3\times 25\times 13=7800$ seconds.

Convert the total time from seconds to minutes and hours:

• $7800\text{ seconds}=\frac{7800}{60}\text{ minutes}=130\text{ minutes}$.
• $130\text{ minutes}=2\text{ hours and }10\text{ minutes}$.

The signals initially turn green together at 2:00 p.m. Adding the calculated duration:

• $2:00\text{ p.m.}+2\text{ hours }10\text{ minutes}=4:10\text{ p.m.}$.

Thus, the signals will next change to green simultaneously at 4:10 p.m.

Incorrect Options:

Options (A) 4:00 p.m., (C) 4:20 p.m., and (D) 4:30 p.m. do not correspond to the Least Common Multiple of the full cycle durations of the three traffic signals. These times would not result in all signals turning green simultaneously after the initial 2:00 p.m. instance, as they do not align with the calculated 2 hours and 10 minutes interval.