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.