Learning Objectives
- Use the distance, rate, and time formula
- Solve a formula for a specific variable
Before you get started, take this readiness quiz.
- Write miles per gallon as a unit rate.
If you missed this problem, review Example 5.65. - Solve
If you missed this problem, review Example 8.20. - Find the simple interest earned after years on at an interest rate of
If you missed this problem, review Example 6.33.
Use the Distance, Rate, and Time Formula
One formula you’ll use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant speed. The basic idea is probably already familiar to you. Do you know what distance you travel if you drove at a steady rate of miles per hour for hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said miles, you already know how to use this formula!
The math to calculate the distance might look like this:
In general, the formula relating distance, rate, and time is
Distance, Rate and Time
For an object moving in at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula
where distance, rate, and time.
Notice that the units we used above for the rate were miles per hour, which we can write as a ratio Then when we multiplied by the time, in hours, the common units ‘hour’ divided out. The answer was in miles.
Example 9.57
Jamal rides his bike at a uniform rate of miles per hour for hours. How much distance has he traveled?
Lindsay drove for hours at miles per hour. How much distance did she travel?
Trinh walked for hours at miles per hour. How far did she walk?
Example 9.58
Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of miles. If he can drive at a steady rate of miles per hour, how many hours will the trip take?
Lee wants to drive from Phoenix to his brother’s apartment in San Francisco, a distance of miles. If he drives at a steady rate of miles per hour, how many hours will the trip take?
Yesenia is miles from Chicago. If she needs to be in Chicago in hours, at what rate does she need to drive?
Solve a Formula for a Specific Variable
In this chapter, you became familiar with some formulas used in geometry. Formulas are also very useful in the sciences and social sciences—fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft ExcelTM relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.
In Example 9.57 and Example 9.58, we used the formula This formula gives the value of when you substitute in the values of and But in Example 9.58, we had to find the value of We substituted in values of and and then used algebra to solve to If you had to do this often, you might wonder why there isn’t a formula that gives the value of when you substitute in the values of and We can get a formula like this by solving the formula for
To solve a formula for a specific variable means to get that variable by itself with a coefficient of on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable in general. This process is also called solving a literal equation. The result is another formula, made up only of variables. The formula contains letters, or literals.
Let’s try a few examples, starting with the distance, rate, and time formula we used above.
Example 9.59
Solve the formula for
- ⓐ when and
- ⓑ in general.
Solve the formula for
- ⓐ when and
- ⓐ in general
Solve the formula for
- ⓐ when and
- ⓑ in general
We used the formula in Use Properties of Rectangles, Triangles, and Trapezoids to find the area of a triangle when we were given the base and height. In the next example, we will solve this formula for the height.
Example 9.60
The formula for area of a triangle is Solve this formula for
- ⓐ when and
- ⓑ in general
Use the formula to solve for
- ⓐ when and
- ⓑ in general
Use the formula to solve for
- ⓐ when and
- ⓑ in general
In Solve Simple Interest Applications, we used the formula to calculate simple interest, where is interest, is principal, is rate as a decimal, and is time in years.
Example 9.61
Solve the formula to find the principal,
- ⓐ when
- ⓑ in general
Use the formula
Find ⓐ when ⓑ in general
Use the formula
Find ⓐ when ⓑ in general
Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually and You might be given an equation that is solved for and need to solve it for or vice versa. In the following example, we’re given an equation with both and on the same side and we’ll solve it for To do this, we will follow the same steps that we used to solve a formula for a specific variable.
Example 9.62
Solve the formula for
- ⓐ when
- ⓑ in general
Solve the formula for
- ⓐ when
- ⓑ in general
Solve the formula for
- ⓐ when
- ⓑ in general
In the previous examples, we used the numbers in part (a) as a guide to solving in general in part (b). Do you think you’re ready to solve a formula in general without using numbers as a guide?
Example 9.63
Solve the formula for
Solve the formula for
Solve the formula for
Example 9.64
Solve the equation for
Solve the formula for
Solve the formula for
Example 9.65
Solve the equation for
Solve the formula for
Solve the formula for
Links To Literacy
Media Access Additional Online Resources
Section 9.7 Exercises
Practice Makes Perfect
Use the Distance, Rate, and Time Formula
In the following exercises, solve.
Socorro drove for hours at miles per hour. How much distance did she travel?
Francie rode her bike for hours at miles per hour. How far did she ride?
Connor wants to drive from Tucson to the Grand Canyon, a distance of miles. If he drives at a steady rate of miles per hour, how many hours will the trip take?
Megan is taking the bus from New York City to Montreal. The distance is miles and the bus travels at a steady rate of miles per hour. How long will the bus ride be?
Aurelia is driving from Miami to Orlando at a rate of miles per hour. The distance is miles. To the nearest tenth of an hour, how long will the trip take?
Kareem wants to ride his bike from St. Louis, Missouri to Champaign, Illinois. The distance is miles. If he rides at a steady rate of miles per hour, how many hours will the trip take?
Javier is driving to Bangor, Maine, which is miles away from his current location. If he needs to be in Bangor in hours, at what rate does he need to drive?
Alejandra is driving to Cincinnati, Ohio, miles away. If she wants to be there in hours, at what rate does she need to drive?
Aisha took the train from Spokane to Seattle. The distance is miles, and the trip took hours. What was the speed of the train?
Philip got a ride with a friend from Denver to Las Vegas, a distance of miles. If the trip took hours, how fast was the friend driving?
Solve a Formula for a Specific Variable
In the following exercises, use the formula.
Solve for
- ⓐ when and
- ⓑ in general
Solve for
- ⓐ when and
- ⓑ in general
Solve for
- ⓐ when and
- ⓑ in general
Solve for
- ⓐ when and
- ⓑ in general.
In the following exercises, use the formula
Solve for
- ⓐ when and
- ⓑ in general
Solve for
- ⓐ when and
- ⓑ in general
In the following exercises, use the formula
Solve for the principal, for:
- ⓐ, ,
- ⓑ in general
Solve for the time, for:
- ⓐ, ,
- ⓑ in general
In the following exercises, solve.
Solve the formula for
- ⓐ when
- ⓑ in general
Solve the formula for
- ⓐ when
- ⓑ in general
Solve for
Solve for
Solve the formula for
Solve the formula for
Solve the formula for
Solve the formula for
Solve the formula for
Solve the formula for
Solve the formula for
Everyday Math
Converting temperature While on a tour in Greece, Tatyana saw that the temperature was Celsius. Solve for in the formula to find the temperature in Fahrenheit.
Converting temperature Yon was visiting the United States and he saw that the temperature in Seattle was Fahrenheit. Solve for in the formula to find the temperature in Celsius.
Writing Exercises
Solve the equation for
- ⓐ when
- ⓑ in general
- ⓒ Which solution is easier for you? Explain why.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?