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Distance, Rate, and Time

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- Relationship between Distance Rate and Time
- Solving an Equation for Distance, Rate, or Time

**Relationship between Distance, Rate, and Time**

An application of linear equations can be found in distance problems. When solving distance problems we will use the relationship rt = d or rate (speed) times time equals distance. There are a couple of ways we can represent this relationship:

If a person were to travel 30 mph for 4 hours, we can find the total distance by multiplying rate times time or (30)(4) = 120. This person travels a distance of 120 miles. The problems we will be solving here will be a few more steps than described above. So to keep the information in the problem organized we will use a table. An example of the basic structure of the table is below:

The third column, distance, will always be filled in by multiplying the rate and time columns together. If we are given a total distance of both persons or trips we will put this information below the distance column. We will now use this table to set up and solve the following example:

**Solving for a Distance, Rate, or Time**

Two joggers start from opposite ends of an 8 mile course running towards each other. One jogger is running at a rate of 4 mph, and the other is running at a rate of 6 mph. After how long will the joggers meet?

As the example illustrates, once the table is filled in, the equation to solve is very easy to find. This same process can be seen in the following example:

Bob and Fred start from the same point and walk in opposite directions. Bob walks 2 miles per hour faster than Fred. After 3 hours they are 30 miles apart. How fast did each walk?

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html