Question

12. The traffic lights at three different road crossings change after every 48 seconds,
72 seconds, and 108 seconds respectively. If they all change simultaneously at 8
a.m. then at what time will they again change simultaneously?​

Answer 1

Answer:

in 80 seconds

Step-by-step explanation:

please follow me and Mark has bilant

Answer 2

Basic Concept Used :-

The traffic lights at three different road crossings change after every 48 seconds, 72 seconds, and 108 seconds respectively.

So, the time, they change together is equals to LCM (48, 72, 108).

Let's solve the problem now!!!

$\green{\bf :\longmapsto\:Prime \: factorization \: of \: 48}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:48\: \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:24\:\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:12\:\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:6\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:\:}} \\{\sf{}}&\underline{\sf{\:\:1\: \:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\green{\bf :\longmapsto\:Prime \: factorization \: of \: 48 = {2}^{4} \times 3}$

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 72}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:72\: \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:36\:\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:18\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:\:}} \\{\sf{}}&\underline{\sf{\:\:1\: \:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 72 = {2}^{3} \times {3}^{2}}$

$\blue{\bf :\longmapsto\:Prime \: factorization \: of \: 108}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:108\: \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:54\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:27\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:\:}} \\{\sf{}}&\underline{\sf{\:\:1\: \:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\blue{\bf :\longmapsto\:Prime \: factorization \: of \: 108 = {2}^{2} \times {3}^{3}}$

Hence,

$\pink{ \bf \: LCM (48,72,108)}$

$\rm \: = \: \: {2}^{4} \times {3}^{3}$

$\rm \: = \: \: 16 \times 27$

$\rm \: = \: \: 432$

$\bf\implies \:LCM (48,72,108) = 432$

Hence,

Three bells change together after 432 seconds

and

$\rm :\longmapsto\:432 \: seconds \: = 7 \: minutes \: 12 \: seconds$

It implies,

• They change simultaneously at 8 : 07 : 12 am