Algebra Help: How to Solve the Dreaded "Two Trains" Word Problem

Over two years ago I wrote the articles "Solving Equal-Distance Uniform Motion Word Problems in Algebra" and "Algebra Help: How to Solve Uniform Motion Word Problems of the Form D1 + D2 = k." As I was reviewing my library of articles I saw that these were fairly popular and regularly clicked on, but that they were probably too generic in description for the average Algebra student (unless he was specifically using the Saxon math textbooks) searching for homework help to find. To correct this, I decided to write a series of articles that were more specific to the kinds of uniform motion word problems students would be searching on: the boat-in-the-river word problem, the airplane-with-a-tail-wind word problem, and lastly here, the dreaded "two trains" word problem.

Like with all uniform motion word problems, we assume that the two trains are moving at a constant rate and are traveling in straight lines. For this article, we will assume that they always start at the same point and either go in the same direction or one goes in the opposite direction. (So if one train goes south, the other train will also go south. Or if one train goes east, the second train will go west.) The first kind of problem, where both trains go in the same direction and one overtakes the other is what we would call the "equal-distance" uniform motion word problem, while the second kind of problem, where the trains go in opposite directions, would be the "Form D1 + D2 = k" uniform motion word problem.

The most important basic algebraic equation to always remember when working with any uniform motion word problem is the equation:

Distance = Rate * Time

I recommend that the student uses the following variables:

D₁ = distance Train #1 travels
D₂ = distance Train #2 travels
R₁ = speed (or rate) of Train #1
R₂ = speed (or rate) of Train #2
T₁ = Train #1's travel time
T₂ = Train #2's travel time

The best way to learn how to solve these kinds of problems is to study solutions from actual problems.

Two Trains Word Problem #1: Both Trains Going in the Same Direction
Two trains start from the same point and travel in the same direction. One leaves 48 minutes later, travels 10 miles per hour faster than the other, and overtakes the first train in 4 hours. Find the rate of each train.

I always recommend immediately writing down the two essential distance equations to start.

D₁ = R₁*T₁
D₂ = R₂*T₂

Now, dissect the word problem to see what information we can write down in the form of Algebra equations.

We must be careful when we read statements like "one leaves 48 minutes later." This tells us that the first train actually travels 48 minutes longer than the first train. (The stopping point is when the second train overtakes the first.) Since we want to work in terms of hours and not minutes, we have to convert our 48 minutes to a fraction. 48 minutes becomes (48/60) hours.

T₁ = T₂ + (48/60) = T₂ + (4/5)

"travels 10 miles per hour faster" is easier to translate:

R₂ = R₁ + 10

"overtakes the first train in 4 hours" gives us two pieces of information:

D₁ = D₂ = D { to simplify matters, we use D for our distance. }
T₂ = 4

We can now solve for T₁.

T₁ = T₂ + (4/5) = 4 + (4/5) = (24/5)

Plug in what we've learned into our two distance equations.

D = R₁*(24/5)
D = (R₁ + 10)*4

Make the two equations equal to each other, simplify, and solve for R₁.

(24/5)*R₁ = 4*R₁ + 40
(24/5)*R₁ - (20/5)*R₁ = 40
(4/5)*R₁ = 40
R₁ = 50

We can now solve for R₂.

R₂ = 50 + 10 = 60

The speed of the first train is 50 mph, while the speed of the second train is 60 mph.

This was an example of an equal-distance uniform motion word problem in Algebra, as the problem, when defined, set the two distances to be the same.

Two Trains Word Problem #2: Trains Going in Opposite Directions
Two trains start from the same place and travel in opposite directions. One train travels 7 miles per hour faster than the other. In 5 hours they are 385 miles apart. Find the rate of each train.

As always, I recommend immediately writing down the two important distance equations.

D₁ = R₁*T₁
D₂ = R₂*T₂

Each distance given tells us how far each train travels from the starting point, one in one direction, the other in the opposite direction.

Dissect what information we are given from the problem. No mention is given of one train leaving before or after the other, so we can assume:

T₁ = T₂ = 5

"One train travels 7 miles per hour faster than the other" give us:

R₁ = R₂ + 7

"they are 385 miles apart" translates to:

D₁ + D₂ = 385

Let's solve D₁ in terms of D₂:

D₁ = 385 - D₂

Plug in the information we have into the two distance equations.

385 - D₂ = (R₂+7)*5
D₂ = R₂*5 = 5*R₂

Fortunately we can easily plug in the second distance equation into the first, simplify, and solve for R₂.

385 - 5*R₂ = 5*R₂ + 35
350 = 10*R₂
R₂ = 35

R₁ = 35 + 7 = 42

The first train travels at a rate of 42 mph, while the second train travels at a rate of 35 mph.

This was an example of a "D1 + D2 = k" form uniform motion word problem in Algebra, as the two unknown distances, when added together, were equal to some known constant.

Blessings!

Source
Virgil S. Mallory. A First Course in Algebra (1943)