# How to tell when two advertising panels showing an ad will synchronize?

The problem is as follows:

A billboard turns on every $$5$$ minutes after turning off and displays a colored ad for $$45$$ seconds and then turns off. Another similar panel turns on every $$8$$ minutes after turning off and displays another colored advertisement for $$10$$ seconds and then turns off. If both panels are turned on at $$\textrm{3:00 a.m.}$$, what time will they turn on simultaneously for third time?

The alternatives given in my book are as follows:

$$\begin{array}{ll} 1.&\textrm{9:48 p.m}\\ 2.&\textrm{9:46 p.m}\\ 3.&\textrm{9:45 p.m}\\ 4.&\textrm{9:47 p.m}\\ 5.&\textrm{9:44 p.m}\\ \end{array}$$

How exactly should this problem be assessed?. I do remember there's a formula which related the number of single events to avoid falling in telephone post error

$$\textrm{number of individual events}=\frac{\textrm{total elapsed time}}{\textrm{time elapsed between each event}}+1$$

This is assuming that the time between each event is the same and by rearranging the above equation into:

$$(\textrm{number of individual events}-1)t=\textrm{total elapsed time}$$

From the looks of this situation to find when both panels will synchronize again will require to use the least common multiple. Is this part correct?.

But from reading at the problem it makes me feel confused the part of turns on every $$5$$ minutes after turning off. I think what it is intended to say is the panel will turn on every 5 minutes and display a 45 second ad and so on and on.

While the other will do the same but instead for $$8$$ minutes while its ad will take $$10$$ seconds.

But the part which makes me feel confused is what to find the lcm to?

Would it be to $$5$$ and $$10$$? Can someone guide me in the right path for this? Since I feel lost in this question, it would really help me a lot that an answer would include a detailed explanation for this problem and step by step using the approach mentioned. Is this the right adequate approach?.

The question is asking at what time will they simultaneously turn on for the third time. The key term to determine the inter-advertisement times is "after turning off". The first billboard turns on every 5 minutes and 45 seconds = 345 seconds. The second billboard turns on every 8 minutes and ten seconds = 490 seconds.

We need to find the least common multiple because we need to find when both billboards will be turned on. For example, if the first one turned on every 2 seconds and the second one every 3 seconds, we could enumerate the times they will turn on to find the common times. 0, 2, 4, 6, 8... 0, 3, 6, 9... We can see that 6, the least common multiple of 2 and 3, is the first time they turn on at the same time.

Knowing this, we can apply a similar principle to the given problem and find the least common multiple between 345 and 490 which is 33,810. We divide by 60 to get the number of minutes between each simultaneous turn-on event, which is 536.5 minutes, or 9 hours and 23.5 minutes.

Knowing this, we simply add to get the third time. The first time occurred at 3 am.

3 am -> first time

3:00 am + 9:23.5 = 12:23.5 pm -> second time

12:23.5 pm + 9:23.5 = 9:47 pm -> third time

Therefore we can see the correct answer is #4, 9:47 pm.

• correct approach (+1) Commented Nov 7, 2020 at 0:44
• @brightlySalty The formatting it doesn't help much but other than that I was intending to ask how to solve it using the approach mentioned in the question. Reading at your answer you mention that the key is to determine the inter-advertisement times, what you do is to add up $5\,min+45\,sec$ because after this time has elapsed the billboard will turn on. But it looks more like it is $45\,s+5\,m$ and so on and on. Then the thing is what's the justification for using the least common multiple?. Can you explain this part please?. Commented Nov 8, 2020 at 1:19
• @brightlySalty By the way the use of the term inter-advertisement time it is a bit misleading because if you look solely to that it could mean only the 5 minutes or 8 minutes which is the elapsed time by the two advertisement panels to turn on again. But this does not account for the time they display a message, in your approach you do count it. Hence I believe a more appropiate term would be the time which will take the billboard to reset after a cycle including the switching off. Don't you agree?. Again can you include the justification of least common multiple here please?. Commented Nov 8, 2020 at 1:26
• @ChrisSteinbeckBell I've edited it to add the justification. Commented Nov 9, 2020 at 23:21