## 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