Distance Word Problems

In algebra distance word problems are a struggle for some. Use this free lesson to help you learn how to solve distance word problems.

Two-part and round-trip problems

Do you know how to solve this problem?

Bill took a trip to see a friend. His friend lives 225 miles away. He drove in town at an average of 30 mph, then he drove on the interstate at an average of 70 mph. The trip took three-and-a-half hours total. How far did Bill drive on the interstate?

This problem is a classic two-part trip problem because it's asking you to find information about one part of a two-part trip. This problem might seem complicated, but don't be intimidated!

You can solve it using the same tools we used to solve the simpler problems on the first page:

Let's start with the table. Take another look at the problem. This time, the information relating to distance, rate, and time has been underlined.

Bill took a trip to see a friend. His friend lives 225 miles away. He drove in town at an average of 30 mph, then he drove on the interstate at an average of 70 mph. The trip took three-and-a-half hours total. How far did Bill drive on the interstate?

If you tried to fill in the table the way we did on the last page, you might have noticed a problem: There's too much information. For instance, the problem contains two rates—30 mph and 70 mph. To include all of this information, let's create a table with an extra row. The top row of numbers and variables will be labeled in town, and the bottom row will be labeled interstate.

distanceratetime
in town30
interstate70

We filled in the rates, but what about the distance and time? If you look back at the problem, you'll see that these are the total figures, meaning they include both the time in town and on the interstate. So the total distance is 225. This means this is true:

Interstate distance + in-town distance = Total distance

Together, the interstate distance and in-town distance are equal to the total distance. See?

In any case, we're trying to find out how far Bill drove on the interstate, so let's represent this number with d. If the interstate distance is d, it means the in-town distance is a number that equals the total, 225, when added to d. In other words, it's equal to 225 - d.

We can fill in our chart like this:

distanceratetime
in town225 - d30
interstated70

We can use the same technique to fill in the time column. The total time is 3.5 hours. If we say the time on the interstate is t, then the remaining time in town is equal to 3.5 - t. We can fill in the rest of our chart.

distanceratetime
in town225 - d303.5 - t
interstated70t

Now we can work on solving the problem. The main difference between the problems on the first page and this problem is that this problem involves two equations. Here's the one for in-town travel:

225 - d = 30 ⋅ (3.5 - t)

And here's the one for interstate travel:

d = 70t

If you tried to solve either of these on its own, you might have found it impossible: since each equation contains two unknown variables, they can't be solved on their own. Try for yourself. If you work either equation on its own, you won't be able to find a numerical value for d. In order to find the value of d, we'll also have to know the value of t.

We can find the value of t in both problems by combining them. Let's take another look at our travel equation for interstate travel.

While we don't know the numerical value of d, this equation does tell us that d is equal to 70t.

d = 70t

Since 70t and d are equal, we can replace d with 70t. Substituting 70t for d in our equation for interstate travel won't help us find the value of t—all it tells us is that 70t is equal to itself, which we already knew.

70t = 70t

But what about our other equation, the one for in-town travel?

225 - d = 30 ⋅ (3.5 - t)

When we replace the d in that equation with 70t, the equation suddenly gets much easier to solve.

225 - 70t = 30 ⋅ (3.5 - t)

Our new equation might look more complicated, but it's actually something we can solve. This is because it only has one variable: t. Once we find t, we can use it to calculate the value of d—and find the answer to our problem.

To simplify this equation and find the value of t, we'll have to get the t alone on one side of the equals sign. We'll also have to simplify the right side as much as possible.

225 - 70t = 30 ⋅ (3.5 - t)

Let's start with the right side: 30 times (3.5 - t) is 105 - 30t.

225 - 70t = 105 - 30t

Next, let's cancel out the 225 next to 70t. To do this, we'll subtract 225 from both sides. On the right side, it means subtracting 225 from 105. 105 - 225 is -120.

- 70t = -120 - 30t

Our next step is to group like terms—remember, our eventual goal is to have t on the left side of the equals sign and a number on the right. We'll cancel out the -30t on the right side by adding 30t to both sides. On the right side, we'll add it to -70t. -70t + 30t is -40t.

- 40t = -120

Finally, to get t on its own, we'll divide each side by its coefficient: -40. -120 / - 40 is 3.

t = 3

So t is equal to 3. In other words, the time Bill traveled on the interstate is equal to 3 hours. Remember, we're ultimately trying to find the distance Bill traveled on the interstate. Let's look at the interstate row of our chart again and see if we have enough information to find out.

distanceratetime
interstated703

It looks like we do. Now that we're only missing one variable, we should be able to find its value pretty quickly.

To find the distance, we'll use the travel formula distance = rate ⋅ time.

d = rt

We now know that Bill traveled on the interstate for 3 hours at 70 mph, so we can fill in this information.

d = 3 ⋅ 70

Finally, we finished simplifying the right side of the equation. 3 ⋅ 70 is 210.

d = 210

So d = 210. We have the answer to our problem! The distance is 210. In other words, Bill drove 210 miles on the interstate.

Solving a round-trip problem

It might have seemed like it took a long time to solve the first problem. The more practice you get with these problems, the quicker they'll go. Let's try a similar problem. This one is called a round-trip problem because it describes a round trip—a trip that includes a return journey. Even though the trip described in this problem is slightly different from the one in our first problem, you should be able to solve it the same way. Let's take a look:

Eva drove to work at an average speed of 36 mph. On the way home, she hit traffic and only drove an average of 27 mph. Her total time in the car was 1 hour and 45 minutes, or 1.75 hours. How far does Eva live from work?

If you're having trouble understanding this problem, you might want to visualize Eva's commute like this:

As always, let's start by filling in a table with the important information. We'll make a row with information about her trip to work and from work.

1.75 - t to describe the trip from work. (Remember, the total travel time is 1.75 hours, so the time to work and from work should equal 1.75.)

From our table, we can write two equations:

In both equations, d represents the total distance. From the diagram, you can see that these two equations are equal to each other—after all, Eva drives the same distance to and from work.

Just like with the last problem we solved, we can solve this one by combining the two equations.

We'll start with our equation for the trip from work.

d = 27 (1.75 - t)

Next, we'll substitute in the value of d from our to work equation, d = 36t. Since the value of d is 36t, we can replace any occurrence of d with 36t.

36t = 27 (1.75 - t)

Now, let's simplify the right side. 27 ⋅(1.75 - t) is 47.25.

36t = 47.25 - 27t

Next, we'll cancel out -27t by adding 27t to both sides of the equation. 36t + 27t is 63t.

63t = 47.25

Finally, we can get t on its own by dividing both sides by its coefficient: 63. 47.25 / 63 is .75.

t = .75

t is equal to .75. In other words, the time it took Eva to drive to work is .75 hours. Now that we know the value of t, we'll be able to can find the distance to Eva's work.

If you guessed that we were going to use the travel equation again, you were right. We now know the value of two out of the three variables, which means we know enough to solve our problem.

d = rt

First, let's fill in the values we know. We'll work with the numbers for the trip to work. We already knew the rate: 36. And we just learned the time: .75.

d = 36 ⋅ .75

Now all we have to do is simplify the equation: 36 ⋅ .75 = 27.

d = 27

d is equal to 27. In other words, the distance to Eva's work is 27 miles. Our problem is solved.