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# Distance Problems Help (page 3)

based on 5 ratings
By — McGraw-Hill Professional
Updated on Sep 26, 2011

## Three Unknowns

There are some distance/rate problems for which there are three unknowns. You must reduce the number of unknowns to one. The clues on how to do so are given in the problem.

#### Example 1

A semi-truck traveled from City A to City B at 50 mph. On the return trip, it averaged only 45 mph and took 15 minutes longer. How far is it from City A to City B?

There are three unknowns—the distance between City A and City B, the time spent traveling from City A to City B, and the time spent traveling from City B to City A. We must eliminate two of these unknowns. Let t represent the number of hours spent on the trip from City A to City B. We know that it took 15 minutes longer traveling from City B to City A (the return trip), so represents the number of hours traveling from City B to City A. We also know that the distance from City A to City B is the same as from City B to City A. Let d represent the distance between the two cities. We now have the following two equations.

But if the distance between them is the same, then 50 t = Distance from City A to City B is equal to the distance from City B to City A = 45 . Therefore,

We now know the time, but the problem asked for the distance. The distance from City A to City B is given by d = 50 t , so . The cities are miles apart.

Another approach to this problem would be to let t represent the number of hours the semi spent traveling from City B to City A. Then would represent the number of hours the semi spent traveling from City A to City B. The equation to solve would be .

#### Example 2

Kaye rode her bike to the library. The return trip took 5 minutes less. If she rode to the library at the rate of 10 mph and home from the library at the rate of 12 mph, how far is her house from the library?

Again there are three unknowns—the distance between Kaye’s house and the library, the time spent riding to the library and the time spent riding home. Let t represent the number of hours spent riding to the library. She spent 5 minutes less riding home, so represents the number of hours spent riding home. Let d represent the distance between Kaye’s house and the library.

The trip to the library is given by d = 10 t , and the trip home is given by . As these distances are equal, we have that .

The distance from home to the library is

Find practice problems and solutions at Distance Problems Practice Problems - Set 5.

In the above examples and practice problems, the number for t was substituted in the first distance equation to get d. It does not matter which equation you used to find d, you should get the same value. If you do not, then you have made an error somewhere.

More practice problems for this concept can be found at: Algebra Word Problems Practice Test.

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