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 $7\mathrm{a}.\mathrm{m}.$ at what time will they change simultaneously again?

Answer 1

Determine the time when the traffic lights change simultaneously.

The traffic light at three different road crossings change after every $48$ seconds,$72$ seconds, and $108$ seconds, respectively and take the $\mathrm{LCM}$ of the given time.

$$\begin{gathered} 48=2\times 2\times 2\times 2\times 3 \\ 72=2\times 2\times 2\times 3\times 3 \\ 108=2\times 2\times 3\times 3\times 3 \end{gathered}$$

Hence, $\mathrm{LCM}$ of $48,72$ and $108$ $=(2\times 2\times 2\times 2\times 3\times 3\times 3)$

$\mathrm{LCM}(48,72,108)=432$

Therefore, after $432$ seconds, they will change simultaneously

We know that $60$ seconds $=1$ minute

After the division $\frac{432}{60}$, here $7$ is quotient and $12$ is reminder

Hence, $432$ seconds = $7$ minutes $12$ seconds

Therefore, time = $7\mathrm{a}.\mathrm{m}.+7$minutes $12$ seconds

Hence, the lights change simultaneously at $\mathbf{7}\mathbf{:}\mathbf{07}\mathbf{:}\mathbf{12}\mathbf{}\mathbf{a}\mathbf{.}\mathbf{m}\mathbf{.}$