Lighthouse problem - some kind of algrbraic approach?

Hey folks, Im looking at BMAT (admissions test for Oxford Medical School) and I can do most of the stuff in the preparatory book - or at least understand why and where I went wrong, however this has been bugging me for the last hour:

Consider two lighthouses that can be seen from the sea front at Shoreton, both lights switch on and off in regular repeating patterns. Lighthouse ONE is on for 3 seconds and off for 8 seconds, whilst Lighthouse 2 is on for 2 seconds and then off for 7 seconds.

15 seconds ago both lights became visable at precisely the same momment. How many seconds from now will both lights next disappear from view together?

My efforts (complete soln) are included below, I'm looking for a more efficent/elegant method though.

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I adopted a rather unwieldy and brute force approach......

Solution

Algebraic fumblings:

Hidden text

I bothered to post this because the next part is to consider the following:

If lighthouse 1 was on for 7 seconds and off for 16, and lighthouse 2 was on for 8 seconds and off for 23 with the same conditions and when they next turn off.

I got an answer for this part using the same method - but the end number is close to 600 and the equations become cumbersome and rather more mistake prone in exam conditions sans calculator. So I was wondering if there are more intuitive methods/other approaches? I'm always happy to look at new techniques :smile:

Thank you folks :smile:

Reply 1

DFranklin

18

To be honest, I don't see any other approach that I'd prefer to the brute force one. There are "clever" approaches but all the ones I can think of are going to be around the same amount of work, and you'd be more likely to make a mistake of some kind.

Someone else may have a better suggestion - I always hated these kind of questions when they came up in IQ tests.

Reply 2

OCC++

i am afraid that your algebraic way is the way to go.
even though there probably isnt any elegant way of doing it, algebraicmanipulations can shorten it(i duno whether u know because you never show ur workings)

23a + 7 = 31b + 8
23a = 31b + 1
23a = 23b + 8b + 1
now we can consider
23(a-b) = 8b + 1
23(a-b) - 1 = 8b
16(a-b) + 7(a-b) - 1=8b
now its just down to considering equation of the form
7x - 1 = 8 y

edited: then u can simplify even further:
7x=7y+y+1
7(x-y) = y+1

is this helpful?:P
bytheway such equation are typical of modular arithmetic and involve the concept of congruences.

Reply 3

Wangers

OP

16

OCC++

i am afraid that your algebraic way is the way to go.
even though there probably isnt any elegant way of doing it, algebraicmanipulations can shorten it(i duno whether u know because you never show ur workings)

I beg to differ - they're in the hidden text boxes - incase anyone wanted to do the Qs themselves. But thank you anyway, its interesting to see other peoples approaches.

Reply 4

Wangers

OP

16

DFranklin

To be honest, I don't see any other approach that I'd prefer to the brute force one. There are "clever" approaches but all the ones I can think of are going to be around the same amount of work, and you'd be more likely to make a mistake of some kind.

Someone else may have a better suggestion - I always hated these kind of questions when they came up in IQ tests.

So do I, personally this is meant to be biologically orientated so its not designed to test for elegant solutions, just to make sure the applicants are generally numerate - its just that working through the second half of the problem the answer comes to 587 - and takes a few minutes if done carefully. Granted its more testing speed of thought and improvisation, but its always more interesting (and indeed potentially lifesaving) to look and ask for other approaches.

Being a relaxed gapper I'm only slowly working my way through some old "pure" textbooks (the green ones) that I've managed to get my mitts on so I do apologise if I've missed something glaringly obvious, I take it there are standard methods to deal with convergence?

Thank you people :smile:

Lighthouse problem - some kind of algrbraic approach?

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