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# Old Textbook Question (Easy?!) watch

1. From a lowest common multiple standpoint (I assume):

A ship is at anchor between two lighthouses, L and H. The light from L shines on the ship every 30 seconds, the light from H shines on the ship every 40 seconds. How often do both lights shine on the ship at once, what assumption did you make?

2. Do you have any thoughts on this yourself? I'd prefer it if you did it rather than us giving you the answer...
3. LCM =120

So every 2 minutes both lights will shine once on the same object.
4. Oh I have the answer, 120 seconds, and I realise the method required, hence mentioning LCM, I'm just totally confuzzled as I think I'm over-thinking it. Sorry to sound like I just want an answer, it's the thinking that I need lol.
5. I just don't get why it should be the LCM - what quality of the LCM means that the light beams must meet? I realise this is a low-level question but I am just blocked...This happens a lot
6. (Original post by T-Dog)
Oh I have the answer, 120 seconds, and I realise the method required, hence mentioning LCM, I'm just totally confuzzled as I think I'm over-thinking it. Sorry to sound like I just want an answer, it's the thinking that I need lol.
Well, the lcm of 30 and 40 is 120. That is, 120 is the smallest number that's a multiple of 30 (so if L shines on the ship every 30 seconds, it will shine after 120 seconds) and a multiple of 40 (so if H shines every 40 seconds, it will shine after 120 seconds).

Do you know what assumptions you made?
7. (Original post by steve2005)
LCM =120

So every 2 minutes both lights will shine once on the same object.
Thanks Steve: I get that, but WHY is it that the LCM means that?
8. Assumption:

Spoiler:
Show
9. (Original post by generalebriety)
Well, the lcm of 30 and 40 is 120. That is, 120 is the smallest number that's a multiple of 30 (so if L shines on the ship every 30 seconds, it will shine after 120 seconds) and a multiple of 40 (so if H shines every 40 seconds, it will shine after 120 seconds).

Do you know what assumptions you made?
Well I can see why they shouldn't meet at all So I guess the opposite of that...Length of shine...Pathway...I just can't get my head around why things should change
10. (Original post by T-Dog)
Well I can see why they shouldn't meet at all So I guess the opposite of that...Length of shine...Pathway...I just can't get my head around why things should change
I don't quite understand what you're on about. Can you explain more clearly?
11. (Original post by generalebriety)
I don't quite understand what you're on about. Can you explain more clearly?
Sorry. Whilst I can take in abstract the LCM 'calculation' that leads from 30 and 40 to 120, I can't apply that to this specific example in my head and see why the light beams should suddenly meet at all...Quite possibly because I am very stuuuuuuuuuuuupid.
12. (Original post by T-Dog)
Sorry. Whilst I can take in abstract the LCM 'calculation' that leads from 30 and 40 to 120, I can't apply that to this specific example in my head and see why the light beams should suddenly meet at all...Quite possibly because I am very stuuuuuuuuuuuupid.
Ok. Let's focus on the first lighthouse for a while, and let's start our 'stopwatch' when it first shines on the ship, so at t = 0 seconds. Then it next shines on the ship after 30 seconds. Then again after 60 seconds. Then again after 90 seconds. Do you see that:

a) the light shines on the ship after every multiple of 30 seconds,
b) the light never shines on the ship if it hasn't been a multiple of 30 seconds?

That is, the first lighthouse shines on the ship if and only if it's been a multiple of 30 seconds. Similarly, the second lighthouse shines on the ship if and only if it's been a multiple of 40 seconds. Let's assume for a second that they both shine on the ship at t = 0; then when do they next shine on the ship? We need to look for a number that's both a multiple of 30 (for the first lighthouse to shine on it) and a multiple of 40 (for the second lighthouse to shine on it). There are loads of these, but 120 is the smallest one - the clue's in the name, "lowest common multiple".

Now, as for why they should shine on the ship at all: well, the first lighthouse shines on the ship after 0, 30, 60, 90, 120, 150 etc. seconds, and the second shines on the ship after 0, 40, 80, 120, 160 etc. seconds. Do you see that they both hit the ship after 120 seconds?

The assumption we made was, of course, that they both shone on the ship at the same time in the first place. If they were one second out, they'd never meet.
13. I guess I was over-thinking and under-thinking all at the same time. Your explanation is most appreciated, thank you. (And others)

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