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

Solution
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
d_1=v_1t
d_2=v_2t
d_1+d_2=(v_1+v_2)t
d=(d/22+d/8)t
Divide both sides by d
1=(1/22+1/8)t
1=(30/176)t
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
\dfrac{d_2}{d_1}=\dfrac{22}{8}=\dfrac{11}{4}
\dfrac{d_2}{11}=\dfrac{d_1}{4}=\dfrac{d_1+d_2}{11+4}=\dfrac{d}{15}
d_2=(11/15)d
v_2t=(11/15)v_28
t=(11/15)8=88/15=5\,\dfrac{13}{15} hours

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About mvtrinh

Retired high school math teacher.
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