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 $7a.m.,$ at what time will they change simultaneously again?

Answer 1

Let the traffic lights change in $1$ second,$2$ seconds,$3$ seconds.

At $6$ seconds, all of them change again.

Now, this $6$ is actually the LCM of $1,2,3$

So, time when they change again$=$LCM of $1,2,3$

traffic lights change every $48$ seconds,$72$ seconds and $108$ seconds

Now, the time, when they change again, will be more than $48,72,108$

So, we find LCM

Time when they change again$=$LCM of $48,72,108$

Factors of $48=1\times 2\times 2\times 2\times 2\times 3$

Factors of $72=1\times 2\times 2\times 2\times 3\times 3$

Factors of $108=1\times 2\times 2\times 3\times 3\times 3$

L.C.M$=2\times 2\times 2\times 2\times 3\times 3\times 3$

$=16\times 27=432$

$\therefore$ Time when they change again$=432$seconds

But we need to find time after $7$a.m

So, first we convert $432$ seconds in to minutes.

Time $=432$seconds$=\frac{432}{60}=7$minutes$12$ seconds

Thus, required times$=7a.m+7$ minutes $12$ seconds $=7:07:12$a.m