# 1.14: Use the Formula for Distance to Find Distance, Rate and Time

**At Grade**Created by: CK-12

**Practice**Formulas for Problem Solving

Kevin and his family are going camping in Yellowstone National Park. Kevin tracked the family’s journey from Denver on his map and saw that they drove about 540 miles in nine hours. The family only made brief stops along the way, and Kevin wants to know how fast the car must have been traveling. How can Kevin calculate the car’s speed based on the information he already has?

In this concept, you will learn to use the formula for distance to calculate distances, rates and times.

### Distance, Rate and Time

Formulas can be used to calculate distance, rate and time.

**Distance** is the measure of how far you travel. The **rate** is the measure of the speed that you travel. How long you travel is the measure of **time**.

When using the formula involving distance, rate, and time, it is essential that you pay attention to the units used in the problem. If the rate in the problem gives miles per hour (mph), then time must be in hours. If the time is given in minutes, divide by sixty to determine the number of hours prior to solving the equation.

To find distance, use the formula:

\begin{align*}\text{Distance} = \text{rate} \times \text{time}\end{align*}

To find rate, divide both sides of the equation by time.

\begin{align*}\text{Rate} = \frac{\text{Distance}}{\text{Time}}\end{align*}

To find time, divide both sides of the equation by rate.

\begin{align*}\text{Time} = \frac{\text{Distance}}{\text{Rate}}\end{align*}

Let’s look at an example.

The Murphy Family drove for three and a half hours from Manhattan, New York to Providence, Rhode Island at a rate of fifty-three miles per hour. Determine the distance the Murphy’s traveled.

First, you need to think about what you are asked to solve for. In this problem, you need to figure out the distance that the Murphy family traveled.

\begin{align*}\text{Distance} = \text{rate} \times \text{time}\end{align*}

Next, take the given information from the problem and substitute that information into the formula.

\begin{align*}\begin{array}{rcl} \text{Distance} & = & \text{rate} \times \text{time}\\ \text{Distance} & = & 53 \ mph \times 3.5 \ h \end{array}\end{align*}

Then, calculate the distance by multiplying the rate by the time.

\begin{align*}\begin{array}{rcl} \text{Distance} & = & 53 \times 3.5\\ \text{Distance} & = & 185.5 \end{array}\end{align*}

The answer is 185.5.

The distance that the Murphy family traveled was 185.5 miles.

### Examples

#### Example 1

Earlier, you were given a problem about Kevin’s family vacation.

Kevin knows that they drove 540 miles in nine hours and wants to determine how fast they were driving, that is, the rate.

First, consider the formula to solve for the rate.

\begin{align*}\text{Rate} = \frac{\text{Distance}}{\text{Time}}\end{align*}

Next, fill in what you know.

\begin{align*}x=\frac{540}{9}\end{align*}

Then, solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} x & = & \frac{540}{9}\\ x & = & 60 \end{array}\end{align*}

The answer is 60.

The rate (speed) is 60 mph.

#### Example 2

A train traveled 255 miles in 300 minutes. Determine the rate at which the train was traveling.

First, convert the 300 minutes into hours since rate (or speed) is normally measured in miles per hour.

\begin{align*}\begin{array}{rcl} \# hours & = & 300 \ min \times \frac{1 \ hour}{60 \ min}\\ \# hours & = & 5 \ hr \end{array}\end{align*}

Next, use the rate formula to find the rate.

\begin{align*}\begin{array}{rcl} \text{Rate} & = & \frac{\text{Distance}}{\text{Time}}\\ \text{Rate} & = & \frac{255}{5}\\ \text{Rate} & = & 51 \end{array}\end{align*}

The answer is 50.

The train was travelling at a rate (or speed) of 51 miles per hour.

#### Example 3

Use the distance formula to solve for rate.

\begin{align*}\text{Distance}=285 \ miles\end{align*}

\begin{align*}\text{Time} = 9.5 \ hours\end{align*}

\begin{align*}\text{Rate} = x\end{align*}

First, write down the formula to solve for the rate.

\begin{align*}\text{Rate} = \frac{\text{Distance}}{\text{Time}}\end{align*}

Next, fill in what you know.

\begin{align*}x=\frac{285}{9.5}\end{align*}

Then, solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} x & = & \frac{285}{9.5}\\ x & = & 30 \end{array}\end{align*}

The answer is 30.

The rate (speed) is 30 mph.

#### Example 4

Use the distance formula to solve for time.

\begin{align*}\text{Distance} = 550 \ miles\end{align*}

\begin{align*}\text{Rate} = 55 \ mph\end{align*}

\begin{align*}\text{Time} = x\end{align*}

First, write down the formula to solve for the time.

\begin{align*}\text{Time} = \frac{\text{Distance}}{\text{Rate}}\end{align*}

Next, fill in what you know.

\begin{align*}x=\frac{550}{55}\end{align*}

Then, solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} x & = & \frac{550}{55}\\ x & = & 10 \end{array}\end{align*}

The answer is 10.

The time is 10 hours.

#### Example 5

A car traveled 45 mph for 6 hours. How many miles did it travel?

First, write down the formula to solve for the distance.

\begin{align*}\text{Distance} = \text{Rate} \times \text{Time}\end{align*}

Next, fill in what you know.

\begin{align*}x=45 \times 6\end{align*}

Then, solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} x & = & 45 \times 6\\ x & = & 270 \end{array}\end{align*}

The answer is 270.

The distance traveled was 270 miles.

### Review

Find the number of miles traveled given the rate and time.

1. 4 hours at a rate of 33 mph.

2. 6 hours at a rate of 55 mph.

3. 8 hours at a rate of 65 mph.

4. 12 hours at a rate of 50 mph.

5. 14 hours at a rate of 60 mph.

6. 19 hours at a rate of 50 mph.

7. 11 hours at a rate of 55 mph.

8. 18 hours at a rate of 35 mph.

9. 12 hours at a rate of 70 mph.

10. 10 hours at a rate of 58 mph.

11. 15 hours at a rate of 57 mph.

12. 21 hours at a rate of 66 mph.

Use the given information to figure out each rate or time.

13. A car traveled 450 miles at a speed of 30 mph. How many hours did it take?

14. A car traveled 600 miles in 12 hours. What was the speed of the car?

15. A runner traveled 6 miles in 30 minutes. How fast was the runner going?

16. A car traveled 520 miles at a speed of 65 mph. How many hours did it take?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.14.

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In this concept, you will learn to use the formula for distance to calculate distances, rates and times.

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