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Two trains leave a town at the same time heading in opposite directions. One train is traveling 12 mph faster than the
other. After two hours, they are 232 miles apart. What is the average speed of each train?

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Answer:

I'm going to paint you a picture in words of what this looks like on paper.  We have a train leaving from a point on your paper heading straight west.  We have another train leaving from the same point on your paper heading straight east.  This is the "opposite directions" that your problem gives you.  

Now let's make a table:

                distance       =        rate      *       time

Train 1

Train 2

We will fill in this table from the info in the problem then refer back to our drawing.  It says that one train is traveling 12 mph faster than the other train.  We don't know how fast "the other train" is going, so let's call that rate r.  If the first train is travelin 12 mph faster, that rate is r + 12.  Let's put that into the table

              distance        =        rate        *        time

Train 1                                        r

Train 2                                    (r + 12)

Then it says "after 2 hours", so the time for both trains is 2 hours:

         

              distance        =        rate        *        time

Train 1                                        r           *          2

Train 2                                  (r + 12)       *          2

Since distance = rate * time, the distance (or length of the arrow pointing straight west) for Train 1 is 2r.  The distance (or length of the arrow pointing straight east) for Train 2 is 2(r + 12) which is 2r + 24.  The distance between them (which is also the length of the whole entire arrow) is 232.  Thus:

2r + 2r + 24 = 232 and

4r = 208 so

r = 52

This means that Train 1 is traveling 52 mph and Train 2 is traveling 12 miles per hour faster than that at 64 mph

Step-by-step explanation:

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