Distance/Rate/Time Word Problems
It's the most stereotypical math problem of all time. Train 1 leaves the station going west at 80 miles per hour. 600 miles to the west, train 2 leaves the station going east at 40 miles per hour. If both trains leave at the same time, how long before they crash?
This is a classic example of "why everybody hates word problems" because, while the math itself isn't so bad, there's a lot of information that needs to be evaluated beforehand to set up the equation. So one thing that almost always helps with these problems is to organize the information carefully before trying to solve it. This will help you to know what info you have, what info you need, and the relationships between them, which will allow you to set up an equation to solve it.
Solving these problems will almost always involve using two relationships: (1) the relationship between distance, speed, and time, and (2) a relationship between the items being measured.
So let's give this problem a try!
Step 1: start by writing the information from the problem in a table, so we know what we have and what we're missing. First, make columns for distance, rate, and time, and rows for each of the moving objects (such as the two trains in our example problem).
Then, fill in all the information that was provided by the problem. In our example problem, we know that the rates are 80 mph for train 1 and 40 mph for train 2, so let's fill those in.
Next, we'll fill in the time column. We know two things here: (1) the two trains leave their stations at the same time, and (2) the problem is asking us to find a time, specifically "how long until they crash?" Since the time is the unknown amount that we're trying to find, we can just call it "x." And since both trains leave at the same time, x can go in the time column for both rows. Since our speeds were in miles per hour, let's measure the time in hours as well, so we don't have to deal with any messy conversions.
Last, we need to fill in the distance column. We know that the two trains start out 600 miles apart, but end up in the same location. So how are we going to represent that mathematically?
Before we try to deal with that aspect of it, let's look at the distance column another way. The distance, rate, and time for a particular journey always have the same relationship: distance = rate × time, or d= r×t. (You probably use this relationship all the time in day-to-day life — for example, if you drive at 60 miles per hour for 2 hours, then you've driven 60×2 = 120 miles.) If we apply that relationship to our trains, then the distance for train 1 is 80 times x, and the distance for train 2 is 40 times x.
Now let's get back to that tricky bit from earlier. The two trains started out 600 miles apart, and after traveling for 80x and 40x miles respectively, they ended up in the same place. If we draw a picture of the scenario, it looks something like this:
The picture makes the situation much clearer: the 80x travelled by train 1, and the 40x travelled by train 2, together add up to the full 600 miles between the two stations. And with that, we have an equation! 80x + 40x = 600. From here, we can use algebra to solve it. By simplifying the left side, we get 120x=600, and then we can divide both sides by 120 to get x = 5 hours.
Thankfully, most word problems won't be as tricky as this one, but the principles to solve them will be the same for any distance/rate/time problem. Set up a table, fill in the given quantities, use d=rt to fill in missing boxes, and find some relationship between the two rows (typically either a connection between the two distances or a connection between the two times) to make an equation.
Credit: Train image by Erich Westendarp from Pixabay
