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Could someone good at maths explain this?

 
#1 Could someone good at maths explain this?
17/11/2009 15:51

Hmmm

I saw this example question on OW's website. The question is asking you to guess the number of trains on the London Underground.

Part of which is: What is the average number of stations on a route?

Again, this is the type of question where a sensible guess should be sanity checked, with the interviewer if needed. We estimate there are 25 stations for the average line. 25 stations on 12 lines gives 300 stations in total. This sounds reasonable.

However, to determine average number of stations on a route, we must be careful not to under-weight stations shared by multiple routes. One way to solve this problem is to consider each route to be isolated, and that every station that is shared between say 2 routes, counts as 2 stations. Now we must estimate how many stations are ‘doubles’ – say half for simplicity, given that there are on average two routes per line.

This means the effective number of stations is (300 + (1/2 x 300)) = 450 effective stations.

The average number of stations on a route is then 450/25 = 18 stations per route.

MY QUESTION: Wouldn't the number of stations be calculated when calculating the number of stations per line. If you assume some lines have branches and then factor this in for the total, surely it doesn't matter if lines share stations as you've already accounted for them when thinking about the total number of stations for separate lines? If two stations have 20 stops each and share half of them, the total number of stations will still be 40, not 60 as this question would imply. Am I being really rtarded here?

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#2 RE: Could someone good at maths explain this?
17/11/2009 16:28

Anon to Hmmm (#1)

Eh?

Why would anyone bother guessing this? You can get the stats off the TFL website.

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#3 RE: Could someone good at maths explain this?
17/11/2009 16:45

Mars A Day to Anon (#2)

You would bother guessing it because you wouldn't have the answer to hand randomly in an interview.

What OW would be looking for in this - or similar questions - is not an accurate answer but a well constructed equation to arrive at any answer where you can show what assumptions you have made and that they are reasonable.

OP your answer is actually perfectly adequate - you show your reasoning and that's what counts.

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#4 RE: Could someone good at maths explain this?
17/11/2009 22:02

Bob to Mars A Day (#3)

I think the original poster is asking which of two specific paths of logic is correct (its the simple one). The whole question of 'doubles' depends on how you define it:

- 'shared' stations where a single platform where both Circle and District line trains stop

- 'crossover' stations where train lines cross, but the trains stop at different platforms.

For the purpose of estimating the trains, crossover stations are completely irrelevant.

You could try to be a bit clever with shared stations (i.e. estimate number of these, and adjust this to the frequency of trains), or you could take a more top-down approach and either assume the trains on the lines still run at similar frequencies, or apply a 'reduced frequency' factor to the whole of any line that shares platforms with other lines.

But I don't see any purpose of having an 'effective station' concept for the purposes of answering this question.

As a further sanity check, you could use geographical parameters (e.g. the distance across london 20 miles, divide by 25 stations leaves 0.8 miles between each station).

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#5 RE: Could someone good at maths explain this?
18/11/2009 09:17

dutchconsultant to Hmmm (#1)

Hmmm, you are not being retarded ;-)

The answer, approach, logic and question you posted here is exactly what OW (and others) would be looking for.

Don't worry about getting the exact number (based on all kinds of estimations), it's the approach and logic that counts (as long as the outcome and estimations are not completely ridiculous).

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#6 RE: Could someone good at maths explain this?
18/11/2009 11:20

j to dutchconsultant (#5)

Easy answer: At 3am there are np service trains.

At other times look for assumptions and constraints:

At rush hour the network will run at near maximum capacity, though this will be less than maximum capacity.

Stations are a constraint on the underground because but not on other types of railway - the actual constraint is the number of signalling track sections in order to allow a safe separation between trains, but because of the nature of the underground stations roughly equate to line capacity. Note though each station on each line can hold at least two trains, one in each direction. At least one station on each line needs to be empty to allow for the trains to move (if every station was full you would have signalling gridlock) and you need to allow for joint stations.

Partners like to think in round numbers and accept so lets see.

10 lines (The Drain hardly counts) 30 stations each = 300. Quite a few share stations. Lets say we allow 10% for that gives us 270 plus we need to allow for our empty station on each line = 240. Multiply by 2 and you get 480. Lets say a ballpark 450 to 500.

Anyone know what the actual answer is?

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#7 RE: Could someone good at maths explain this?
18/11/2009 11:37

j to j (#6)

But don't try and type your answer on a phone's touch screen keyboard or you will make as many typos as I did, and fail whether you got the sums right or not.

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#8 RE: Could someone good at maths explain this?
18/11/2009 16:00

someguy to j (#7)

To your original question Hmmm: the only stations where the sharing would be relevant are those that share the same rail & platform. District and Circle, but also some of the Hammersmith&City stops as well. You need to make an assumption as to % of total stations these are and factor in accordingly to your trains calculation.

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