# Applications of Solving Equations

### Rate Problems

#### Rate Equations

If a person travels for 3 hours at 50 miles per hour, we can find the total distance traveled by multiplying:

/times = 150miles

This is similar to ##converting between units#{math/measurments}#); here, miles per hour is treated as a conversion factor, since 50 miles is equivalent to 1 hour, in a sense.

In general, to find the total distance traveled, multiply the total time spent traveling by the rate of travel. This can be expressed as an equation:

d = rt where d = total distance, r = rate of travel, and t = time
Using this equation, we can write r and t in terms of the other variables:
r = and t =
Make sure units are consistent. If rate is given in miles per hour, time must be in hours and distance must be in miles.

Example 1: If Sue drives 45 minutes at 80 miles per hour, how far does she travel?

45 minutes = hour
d = rt = ×hour = 60miles

Example 2: If Sue drives 200 miles at 80 miles per hour, how long does it take her?

t = = = 200miles× = 2.5hours

Example 3: If Sue drives 200 miles in 4 hours at a steady pace, how fast is she traveling?

r = = = 50miles per hour

Hint: Make sure the units in your calculation cancel out to yield the units your answer should be in. If they don't, you've probably used the wrong units in your calculation or the wrong equation. Be careful, however, since units are not variables or numerical quantities, and multiplying or canceling them is only a way to check your work.

#### Solving Rate Problems with Two Travelers

Occasionally, we will run into a problem involving two "travelers."

For example:

Bonnie and Barbara are driving from Point A to Point B. Barbara arrives at Point B in 2 hours. Bonnie travels 15 miles per hour faster than Barbara, and arrives at Point B in 1.5 hours. How fast does each woman travel?

To solve a problem involving two travelers, follow these steps:

1. Figure out which quantity related to the travelers is equal (a time, distance, or rate).
2. Write two expressions for that quantity, one using each "traveler."
3. Set the two expressions equal to each other and solve the equation.

In our example, the steps are:

1. Distance is the same for both women.
2. Choose x = Barbara's rate (in mph). Bonnie's rate (in mph) = x + 15
3. Barbara's distance: d = rt = x(2) = 2x
4. Bonnie's distance: d = rt = (x + 15)(1.5) = 1.5x + 22.5
5. 2x = 1.5x + 22.5
6. x = 45

Barbara travels at 45 miles per hour. Bonnie travels at 60 miles per hour.

Here is another example:

Jeff and Jed are twins who run at exactly the same speed. 45 minutes after they begin running, Jeff pulls a muscle and has to stop. Jed finishes the run -- 4 more miles. If Jed runs for 75 minutes total, how many miles did Jeff run before he stopped?

1. Rate is the same for both men.
2. Choose x = Jeff's distance. Jed's distance = x + 4
3. Jeff's rate: r = =
4. Jed's rate: r = =
5. =
6. x = 6

Jeff ran 6 miles.

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