How to Solve Motion Problems Part 2

This is the second part of the How to Solve Motion Word Problem series. I suggest that you read first the introduction to this series as well as the first part in order to understand better.

In the first part of this series, we discusses about a faster object overtaking a slower one who left first.  In this post, we continue to solve motion problems involving two objects traveling in the same direction and one object faster than the other; this time, they left at the same time. We want to know that given a particular time when they are a number of kilometers apart.

Problem 4

A freight train and a passenger train left the same station and traveled to the same direction. The passenger train travels at an average speed of 80 kilometers per hour while the freight train travels at an average speed of 65 kilometers per hour. In how many hours will they be 75 kilometers apart?

Solution 1

Like in the first part of this series, the problem can be solved using a table. The table below shows the distance traveled by the train after each hour. In the 5th hour, the passenger train traveled 400 km while the freight train traveled 325 kilometers. This means that after 5 hours, the distance between them is 75 kilometers.

Screen Shot 2014-03-29 at 3.53.17 PM


Solution 2

An algebraic solution can also be done to solve the problem above. Let us create a table like what we have done in the first part of this series. As shown in the table below, the rate of the passenger train is 80 kilometers per hour and the rate of the freight train is 65 kilometers per hour. The time traveled is the same since they left the same location at the same time. The distance is the product of the rate and time so we multiply them as shown below.

Screen Shot 2014-03-29 at 4.00.27 PM

Now, that we have all the given in place, let us analyze the problem. We are asked for the number of hours when the trains are 75 kilometers part.  The phrase “75 kilometers apart” means the

d traveled by the passenger train –  d traveled by the freight train = 75


80x - 65x = 75

15x = 75

Dividing both sides by 15, we have

x = 5.

Note that x is the number of hours in the table, therefore, in 5 hours, the trains will be 75 kilometers apart.

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