Teaser 3090: Main line
From The Sunday Times, 12th December 2021 [link] [link]
Anton and Boris live next to a railway line. One morning a goods train passed Anton’s house travelling south just as a slower train passed Boris’s house travelling north. The goods train passed Boris’s house at the same time as a passenger train, heading north at a speed that was half as fast again as the goods train. Similarly, as the slower train passed Anton’s house it passed a passenger train; this was heading south at a speed that was three times as great as that of the slower train.
The passenger trains then passed each other at a point 25 kilometres from Anton’s house before simultaneously passing the two houses.
All four trains travelled along the same route and kept to their own constant speeds.
How far apart do Anton and Boris live?
[teaser3090]
S2T2 
Jim Randell 1:38 pm on 10 December 2021 Permalink |
This is an exercise is generating and solving simultaneous equations. No programming necessary.
If we suppose B lives a distance d from A.
Initially (at time 0) if the goods train passes A travelling south at speed 2v, then it reaches B at a time of (d / 2v).
At this time, the passenger train, with a speed of 3v passes B, heading north.
And the slow train, travelling at speed 2fv (i.e. some fraction of v), reaches A at a time of (d / 2fv).
And at this time a train travelling at 6fv passes A heading south.
These trains pass at time t1 at a point 25 km south of A:
And then at time (t1 + t2) the two trains pass A and B:
Equating these:
So A and B are 65 km apart, and f = 4/5.
We are not given any times, so we cannot determine the actual speeds of the trains.
Solution: Anton and Boris live 65 km apart.
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