Trains Passing Each Other

At noon, a train leaves New York for Toronto while another leaves Toronto for New York. It takes one train 8 hours and the other 22 hours to make the trip. If both maintain constant speeds and travel along parallel tracks, at what time do they pass each other?
Source: NCTM Mathematics Teacher, October 2006

We use the following variables
t = time (duration) in hours when the trains pass each other
d = New York – Toronto distance in miles
d_1 = distance in miles from Toronto to the passing point
d_2 = distance in miles from New York to the passing point
We calculate speeds and distances
v_1=d/22  speed of slower train
v_2=d/8  speed of faster train
Divide both sides by d
t=5\,\dfrac{13}{15} = 5 hours 52 minutes

Answer: 5\!:\!52 p.m.

Alternative solution 1
Suppose train 1 is slower than train 2. Since distance is proportional to time
d_2 is to d_1 as 22 is to 8
t=(11/15)8=88/15=5\,\dfrac{13}{15} hours


About mvtrinh

Retired high school math teacher.
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