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Distance = Rate x Time is the key to the solving of train problems.


Problem: One train with my luggage leaves the station going at a rate of 80 mph. I am finished with my meeting one hour later and take the commuter train that travels on a parallel track in the same direction as the freight train that has my luggage and my train travels at a rate of 120 mph. How far does my train travel to catch up with the train that has my luggage so I can get my bags?


Sloution: Let us agree that the letter f will represent the freight train my luggage is on and the letter p represent the passenger train I am on.


The distance the fright train will travel is: Df = rf tf

The distance the passenger train travels is: Dp = rp tp


The freight train is slower than the passenger train AND leaves the train station one hour sooner than I leave taking the passenger train so we have ti take that time difference into consideration when looking at the freight train’s time in motion.


The freight train’s time overall as the passenger train is catching up to it is t + 1 because it left one hour sooner than the passenger train.


The passenger train’s time in motion is just t.


The final consideration is the overall distance BOTH trains traveled. Because the faster passenger train will eventually catch up to the freight train, we set the distances BOTH trains traveled EQUAL to each other.


Df = Dp Because the passenger train catches up to the freight train so they travel the SAME distance.

This gives us by substitution: 80(t + 1) = 120t


Distributive property: Multiply the parenthesis contents by 80. 80t + 80 = 120t


Subtract 80t from BOTH sides of the equation:


  80t + 80 = 120t

-80t              -80t

              80  =  40t


Divide BOTH sides of the equation by 40:


80 = 40t

40    40


2 = t This means it took 2 hours for the passenger train to catch up to the freight train so now I can get my luggage.


The problem wants to know how far the distance was in order for the passenger train to catch up with the freight train. Because Dp = rp tp we now can answer the question.


Dp = rp tp


Dp = 120(2)


Dp = 240 miles.


The passenger train AND the freight train traveled 240 miles.