The following is a
generalization of the passing trains problem, with several solutions
actually presented by participants in the Algebraic Thinking Institute.
Assume that trains between City A and City B leave each city every
hour on the hour, with no stops. Due to a difference in elevation,
the trip from A to B takes 5 hours, but the trip from B to A takes
6 hours. We are on a train going from A to B. How many trains will
we pass on our trip?
Just as our train leaves A, one from B is pulling into the station.
We will agree to count this as a “passing”. Similarly,
when we arrive at B there is a train just pulling out of that station,
and we will count that one too.
Solution 1. When our train leaves A, there are already 7 trains
on the tracks approaching ours, spaced evenly along the tracks
as shown and labeled by the number of hours since they left B.
During the 5 hours we are traveling, 5 more trains will leave B.
We pass all 12 = 7 + 5 of them.
It is no harder to solve this problem if the given numbers 5 and
6 are replaced by any whole numbers m and n. The answer in that
case is m + n + 1. In particular, if we reverse the numbers, so
that our trip from A to B takes 6 hours and the trip back takes
5, the number of trains we pass is the same.
Solution 2. Same as solution 1, but picture the 5 extra trains
which have not yet left the station at B as being lined up behind
the station, traveling toward it. In the above diagram these
would be to the right of B, labeled –1, -2, -3, -4, and –5.
During our trip we pass the entire line of 12 trains. The distance
on a number line from –5 to 6 is 11; there are 11 equal
gaps between the 12 trains.
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