Fewest Number of StopsDate: 09/12/2001 at 11:42:48 From: John Subject: Algebra At some stops, the SLU Express bus picks up 5 people. At other stops, it picks up 2 and, at the same time, lets off 5. There are no other stops but these. It starts its run empty and picks up 5 people, and at the end it has 11 people aboard. If the number of stops is greater than 17, what is the fewest number of stops the bus makes? Date: 09/12/2001 at 14:34:56 From: Doctor Ian Subject: Re: Algebra Hi John, Basically, there are two things that can happen at a stop. The number of passengers can go down by 3, or it can go up by 5. So we can make a diagram in which each of these paths is represented as N-3 / N \ N+5 Starting with 5 passengers, it looks like this: back to 5 / 1 3 / / \ / \ / 2 4 6 8 / \ / \ / \ / \ 5 7 9 11 13 \ / \ / \ / \ / 10 12 14 16 \ / \ / \ / \ 15 17 19 21 \ / \ / \ / 20 22 24 Note that there are lots of 6-step paths from 5 to 11, e.g., 5 -> 2 -> 7 -> 12 -> 9 -> 14 -> 11 5 -> 10 -> 15 -> 12 -> 17 -> 14 -> 11 Note that we can also loop around, to increase the number of stops: [5 -> 2 -> 7 -> 4 -> 9 -> 6 -> 3 -> 8 ->] 5 -> 2 -> 7 -> 12 -> 9 -> 14 -> 11 \_______________________________________/ Repeat as often as desired So the problem now becomes one of finding a loop of the right size (k) such that nk + 6 > 17 Can you take it from here? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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