Question 117126: Problem: A train leaves Chicago for Detroit going 60 mph. At the same time, on an adjacent track, a train leaves Detroit heading for Chicago going 45 mph. Detroit is 280 miles from Chicago. How far are the trains from Chicago when they pass? Help, have been working on for 2 days and I think the answer to this problem is that the trains will be 160 miles from chicago when they meet, but i'm supposed to solve with a system on equations and solve eithe by subst. of combination. I solved this intuitively, thinking combined speeds (60x+45x=280) x=2.66 hrs when they meet, then plug back in...60*2.66=~160miles from chi.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Nothing wrong with what you did. You have the correct answer ... 160 miles from Chicago.
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I suppose that you could state that the distance each train travels is given by the equation:
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Distance = rate * time
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which in abbreviated form is D = R * T
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Since the two trains both start at the same time, each travels the same amount of time until
they meet. And when they meet the sum of their distances is 280 miles. Therefore, you can
set up the exact same equation as you did:
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(60*T + 45*T) = 280
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Factor T to get:
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T(60 + 45) = 280
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Add the two values in the parentheses to get:
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T(105) = 280
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Solve for T by dividing both sides by 105 to get:
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T = 280/105 = 2.666666 hrs
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So both trains have been traveling for 2.666 hours when they meet. The train that left
Chicago at 60 mph has traveled a distance equal to that rate times the amount of time it
has traveled ... or:
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D = 60*2.6666666 = 120 + 40 = 160 miles. It is 160 miles from Chicago when it meets the train
from Detroit that is headed to Chicago.
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This is essentially the way that you did the problem and this is an acceptable way to do the
problem. I suppose that you could expand the method a little by saying that the combine
distance that the two trains meet is 280 mile, the distance between the two cities, and
this distance equation can be written in equation form as:
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Note: the subscript c means that the distance is from Chicago, and subscript d means the
distance from Detroit.
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Next you could write the distance equations, using the rate and the time traveled.
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 and 
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Then substitute these two equations into the distance equation to get:
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and the rest you know how to do to solve for T. When you have T, then return to the equation:
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Substitute 2.666666 for T and get that 
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I inferred that you said "intuitive" problem solving is not the way you are supposed to
do this problem. I believe that it is exactly the way it should be done. You understood what
you were doing, did it correctly, and got the answer. Too many students just look for
an equation they can plug numbers into and get an answer without recognizing what they
are doing. Give me a student who thinks about the problem, analyzes it, and uses the tools
he/she has available to solve it any day! Good job ... keep up the good work, intuitively
and otherwise.
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