Question

# The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time they change simultaneously again?

Hint: In this question L.C.M of the time interval of different periods respectively of traffic lights will help us to get on the right track to reach the solution of the question.

If the Traffic light changes simultaneously at 7 a.m., then they will change again simultaneously by the L.C.M value of the respective times.
So, we have to take the L.C.M of the given times and add this value at 7 a.m., to get the required time at which they will change again simultaneously.
So first factorize the respective time,
Factors of 48 are
$48 = 2 \times 2 \times 2 \times 2 \times 3$.
Factors of 72 are
$72 = 2 \times 2 \times 2 \times 3 \times 3$
Factors of 108 are
$108 = 2 \times 2 \times 3 \times 3 \times 3$
So, the L.C.M of above numbers is
L.C.M $= 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$
L.C.M $= 432$seconds
Now convert these seconds into minutes.
As we know that in 1 minute there are 60 seconds.
So, divide 432 with 60.
$\Rightarrow 432{\text{seconds}} = \dfrac{{432}}{{60}}$ Minutes.
$\Rightarrow \dfrac{{432}}{{60}} = 7\dfrac{{12}}{{60}}$Minutes.
Or it can also be written as 7 minute 12 seconds.
Therefore the required time at which they will change again simultaneously,
$= 07:00:00 + 00:07:12$
$= 07:07:12$ a.m.

So, this is the required answer.

Note: In such types of questions the key concept we have to remember is that the traffic light will change simultaneously again by the L.C.M value of the respective times of traffic light so, first calculate the L.C.M of the numbers and then add this value in the previous time, so the new time is the required time at which they will again change again simultaneously.