0.8 Working with Rates
Learning Objectives
By the end of this section, you will be able to:
- Combine conversion factors
- Use the distance, rate, and time formula
- Carry out rate conversions for computations involving two or more properties
Combining Conversion Factors
In Section 0.3, Unit Conversions, we saw that to convert between two units, it is sometimes easier to use more than one conversion factor in your calculations. Each successive conversion factor will cancel the units introduced by the previous conversion factor, and the final conversion factor will convert your result into the desired units.
Sometimes, a measurement may be given in square units (area) or cubic units (volume). In these cases, each instance of the unit needs to be converted separately. For example, one square foot contains 144 square inches - the length in feet is converted to a length in inches, and the width in feet is also converted to inches, as seen in the following example.
Converting Between Units of Area
A sheet of cardboard has an area of 6 square feet. How many square inches is this?
Solution
| Steps | Explanation |
|---|---|
| Step 1. Identify the starting unit and the target unit. | The measurement is in [latex]\text{ft}\cdot\text{ft}[/latex] and we will convert to [latex]\text{in}\cdot\text{in}[/latex]. |
| Step 2. Find a unit equivalence between the two units. | 1 foot = 12 inches |
| Step 3. Determine the correct conversion factor. | [latex]\frac{12\text{ in}}{1\text{ ft}}[/latex] cancels feet and leaves inches. We will need to multiply by this conversion factor twice. |
| Step 4. Multiply the measurement by the conversion factor. | [latex]6 \text{ ft}\cdot\text{ft} \times\frac{12 \text{ in}}{1 \text{ ft}}\times \frac{12 \text{ in}}{1 \text{ ft}} = 864 \text{ in}^2[/latex] |
| The cardboard has an area of 864 square inches. |
Try It
A 12-ounce can of soda contains 0.355 L of liquid. If 1 L = 1 dm3, how many cubic centimeters of liquid are in the can?
Solution
Try It
The area of a wall to be painted is 142,350 square inches. How many square feet is this?
Solution
Use the Distance, Rate, and Time Formula
One formula you will 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 60 miles per hour for 2 hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said 120 miles, you already know how to use this formula!
The math to calculate the distance might look like this:
[latex]\text{distance}=\left(\frac{60\text{ miles}}{1\text{ hour}}\right)\left(2\text{ hours}\right)[/latex]
[latex]\text{distance}=120\text{ miles}[/latex]
In general, the formula relating distance, rate, and time is
[latex]\text{distance}=\text{rate}\cdot\text{time}[/latex]
For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula [latex]d=rt[/latex] where [latex]d[/latex] is distance, [latex]r[/latex] is rate, and [latex]t[/latex] is time.
Notice that the units we used above for the rate were miles per hour, which we can write as a ratio: [latex]\frac{\text{miles}}{\text{hour}}[/latex]. Then when we multiplied by the time, in hours, the common unit ‘hour’ divided out. The answer was in miles.
Calculating Distance From Speed and Time
Jamal rides his bike at a uniform rate of 12 miles per hour for 312 hours. How much distance has he traveled?
Show Solution
| Steps | Explanation |
|---|---|
| Step 1. Read the problem. You may want to create a mini-chart to summarize the information in the problem. |
[latex]\begin{array}{cc}\hfill d&=?\hfill \\ \hfill r&= 12\text{ mph}\hfill \\ \hfill t&= 3\frac{1}{2}\text{ hours}\hfill \end{array}[/latex] |
| Step 2. Identify what you are looking for. | distance traveled |
| Step 3. Name. Choose a variable to represent it. | let [latex]d[/latex] = distance |
| Step 4. Translate. Write the appropriate formula for the situation. Substitute in the given information. |
[latex]\begin{array}{cc}\hfill d&=rt\hfill \\ \hfill d&=12\cdot 3\frac{1}{2}\hfill \end{array}[/latex] |
| Step 5. Solve the equation. | [latex]d=42\text{ miles}[/latex] |
| Step 6. Check: Does 42 miles make sense? |
 |
| Step 7. Answer the question with a complete sentence. | Jamal rode 42 miles. |
Try It
Lindsay drove for 5 hours at 60 miles per hour. How much distance did she travel?
Show Solution
300 miles
Try It
Trina walked for 2 hours at 3 miles per hour. How far did she walk?
Show Solution
6 miles
Calculating Time From Speed and Distance
Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of 520 miles. If he can drive at a steady rate of 65 miles per hour, how man hours will the trip take?
| Steps | Explanation |
|---|---|
| Step 1. Read the problem. Summarize the information in the problem. |
[latex]\begin{array}{cc}\hfill d&= 520 \text{ miles}\hfill \\ \hfill r&= 65\text{ mph}\hfill \\ \hfill t&= ?\hfill\end{array}[/latex] |
| Step 2. Identify what you are looking for. | how many hours (time) |
| Step 3. Name: Choose a variable to represent it. |
let [latex]t[/latex] = time |
| Step 4. Translate. Write the appropriate formula. Substitute in the given information. |
[latex]\begin{array}{cc}\hfill d&=rt\hfill \\ \hfill 520 &= 65t\hfill \end{array}[/latex] |
| Step 5. Solve the equation. | [latex]t=8[/latex] |
| Step 6. Check: Substitute the numbers into the formula and make sure the result is a true statement. |
[latex]\begin{array}{cc} \hfill d&=rt \hfill \\ \hfill 520 &\overset{?}{=}65\cdot 8\hfill \\ \hfill 520 &=520\,\checkmark\hfill \end{array}[/latex] |
| Step 7. Answer the question with a complete sentence. We know the units of time will be hours because we divided miles by miles per hour. |
Rey's trip will take 8 hours. |
Try It
Lee wants to drive from Phoenix to his brother’s apartment in San Francisco, a distance of 770 miles. If he drives a steady rate of 70 miles per hour, how many hours will the trip take?
Show Solution
11 hours
Try It
Yesenia is 168 miles from Chicago. If she needs to be in Chicago in 3 hours, at what rate does she need to drive?
Show Solution
56 miles per hour
Rate Conversions
It is often the case that a quantity of interest may not be easy (or even possible) to measure directly but instead must be calculated from other directly measured properties and appropriate mathematical relationships. A rate is a quantity which is computed from measurements in two or more different units.
The units of rates give us a hint as to how they are calculated. Table 4 gives some examples of rates and the calculations to determine them from direct measurements.
Some Common Rates
| Rate | Calculation |
|---|---|
| Speed in miles per hour | Divide the length (in miles) by the time (in hours) needed to travel that distance. |
| Density in g/mL | Divide the mass (in grams) of the substance by its volume (in mL). |
| Fuel efficiency in miles per gallon | Divide the number of miles driven by the number of gallons of gasolines used. |
Table 4 Some Common Rates
As you have seen, the method of multiplying quantities by conversion factors can be applied to computations ranging from simple unit conversions to more complex, multi-step conversions. Rate conversions are unit conversions involving several different quantities. When units are to be converted in a case like this, careful unit conversions are required for each unit involved.
How To
Carry Out Rate Conversions
Step 1. Calculate the quantity from the given measurements if necessary.
Step 2. Identify all of the units involved in the quantity.
Step 3. Identify the units that need to be converted and the target units.
Step 4. Determine appropriate conversion factors.
Step 5. Multiply the quantity by conversion factors to convert the starting combination of units to a combination of the desired units.
Calculating Speed From Distance and Time
Suppose that you drive the 10 km from your university to home in 20 min. Calculate your average speed (a) in kilometers per hour (km/h) and (b) in meters per second (m/s).
Solution for (a)
| Steps | Explanation |
|---|---|
| Step 1. Calculate the quantity from the given measurements. | Average speed is distance traveled divided by time of travel.
[latex]\text{average speed}=\frac{10\text{ km}}{20\text{ min}}=0.5\frac{\text{km}}{\text{min}}[/latex] |
| Step 2. Identify the units involved in the quantity. | The units present in the average speed are kilometers and minutes. |
| Step 3. Identify the units that need to be converted and the target units. | We want to give the speed in kilometers per hour, so we need to change minutes to hours. |
| Step 4. Determine appropriate conversion factors. | [latex]\frac{60\text{ min}}{\text{hr}}[/latex] cancels minutes from the denominator and leaves hours. |
| Step 5. Multiply the quantity by the conversion factor. | [latex]\frac{0.5\text{ km}}{\text{min}}\times\frac{60\text{ min}}{\text{hr}}=30\frac{\text{km}}{\text{ hr}}[/latex] |
| The average speed is 30 kilometers per hour. |
Solution for (b)
There are several ways to convert the average speed into meters per second.
1. Start with the answer to (a) and convert km/h to m/s.
| Steps | Explanation |
|---|---|
| Step 1. Calculate the quantity from the given measurements. | The quantity is 30 kilometers per hour. |
| Step 2. Identify the units involved in the quantity. | The units are kilometers and hours. |
| Step 3. Identify the units that need to be converted and the target units. | We need to change kilometers to meters and hours to seconds. |
| Step 4. Determine appropriate conversion factors. | [latex]\frac{1000\text{ m}}{\text{km}}[/latex] cancels kilometers from the numerator and leaves meters.
[latex]\frac{1\text{ h}}{3600\text{ s}}[/latex] cancels hours from the denominator and leaves seconds. |
| Step 5. Multiply the quantity by the conversion factors. | [latex]\frac{30\text{ km}}{\text{h}}\times\frac{1000\text{ m}}{\text{km}}\times\frac{1\text{ h}}{3600\text{ s}}=8.33 \text{ m/s}[/latex] |
| The average speed is 8.33 meters per second. |
2. If we had started with 0.5 km/min instead, we would have needed different conversion factors, but the answer would have been the same: 8.33 m/s.
Try It
If it takes a ball 8 seconds to fall 32 meters, what is its average speed in miles per hour?
Solution
Try It
Sha'Carri Richardson can run 100 meters in 10.65 seconds. Can she outrun a car traveling at 20 miles per hour?
Solution
Converting Between Units of Speed
An American is driving 103 km/h in a Canadian vehicle. How many miles per hour is this?
Solution
| Steps | Explanation |
|---|---|
| Step 1. Calculate the quantity from the given measurements. | The quantity is 103 km/h. |
| Step 2. Identify the units involved in the quantity. | The units are kilometers and hours. |
| Step 3. Identify the units to be converted and the target units. | We need to change kilometers to miles. |
| Step 4. Determine an appropriate conversion factor. | [latex]\frac{1\text{ mi}}{1.609\text{ km}}[/latex] cancels kilometers and leaves miles. |
| Step 5. Multiply the quantity by the conversion factor. | [latex]\frac{103\text{ km}}{\text{h}}\times\frac{1\text{ mi}}{1.609\text{ km}}=64\text{ mi/h}[/latex] |
| 103 km/h is equivalent to 64 mph. |
Try It
An object must travel at least 11.2 km/s in order to escape Earth's gravity and travel to outer space. How many miles per hour is this?
Solution
Try It
The speed limit on some interstate highways is roughly 100 km/h. How many miles per hour is this?
Solution
Calculating Density From Volume and Mass
What is the density of common antifreeze in units of g/mL? A 4-qt sample of the antifreeze weighs 9.26 lb.
Solution
| Steps | Explanation |
|---|---|
| Step 1. Calculate the quantity from the given measurements. | The density can be given in g/mL, so we need to divide the mass by the volume.
[latex]\frac{9.26\text{ lb}}{4\text{ qt}}=2.315\frac{\text{lb}}{\text{qt}}[/latex] |
| Step 2. Identify the units involved in the quantity. | The units are pounds and quarts. |
| Step 3. Identify the units to be converted and the target units. | We need to convert pounds to grams and quarts to milliliters. |
| Step 4. Determine the appropriate conversion factors. | [latex]\frac{453.59\text{ g}}{1\text{ lb}}[/latex] cancels pounds in the numerator and leaves grams.
[latex]\frac{1.0567\text{ qt}}{1\text{ L}}[/latex] cancels quarts in the denominator and leaves liters, and [latex]\frac{1\text{ L}}{1000\text{ mL}}[/latex] cancels liters to leave milliliters. |
| Step 5. Multiply the quantity by the conversion factors. | [latex]\frac{2.315\text{ lb}}{1\text{ qt}}\times\frac{453.59\text{ g}}{1\text{ lb}}\times\frac{1.0567\text{ qt}}{1\text{ L}}\times\frac{1000\text{ mL}}{1\text{ L}}=1.11 \text{g/mL}[/latex] |
| The density of common antifreeze is 1.11 grams per milliliter. |
Key Concepts
Carry Out Rate Conversions
- Step 1. Identify all of the units involved in the measurement.
- Step 2. Identify the units that need to be converted.
- Step 3. Determine appropriate conversion factors.
- Step 4. Multiply the measurement by conversion factors to convert the starting combination of units to a combination of the desired units.
Key Formulas
[latex]\text{distance} = \text{rate} \times \text{time}[/latex]
Section Exercises
Converting Area and Volume
Convert the following measurements to the indicated unit.
1. The area of the state of Oregon, 96,981 mi2, to square kilometers
Solution
2. The estimated volume of the oceans, 330,000,000 mi3, to cubic kilometers.
3. The area of an 8.5- × 11-inch sheet of paper in cm2.
Solution
4. The displacement volume of an automobile engine, 161 in.3, to liters.
Use the Distance, Rate, and Time Formula
In the following exercises, solve.
5. Steve drove for [latex]8\frac{1}{2}[/latex] hours at [latex]72[/latex] miles per hour. How much distance did he travel?
Show Solution
612 miles
6. Socorro drove for [latex]4\frac{5}{6}[/latex] hours at [latex]60[/latex] miles per hour. How much distance did she travel?
7. Yuki walked for [latex]1\frac{3}{4}[/latex] hours at [latex]4[/latex] miles per hour. How far did she walk?
Show Solution
7 miles
8. Francie rode her bike for [latex]2\frac{1}{2}[/latex] hours at [latex]12[/latex] miles per hour. How far did she travel?
9. Connor wants to drive from Tucson to the Grand Canyon, a distance of 338 miles. If he drives at a steady rate of 52 miles per hour, how long will the trip take?
Show Solution
6.5 hours
10. Megan is taking the bus from New York City to Montreal. The distance is 384 miles and the bus travels at a steady rate of 64 miles per hour. How long will the bus ride be?
11. Aurelia is driving from Miami to Orlando at a rate of 65 miles per hour. The distance is 235 miles. To the nearest tenth of an hour, how long will the trip take?
Show Solution
About 3.6 hours
12. Kareem wants to ride his bike from St. Louis, Missouri, to Champaign, Illinois. The distance is 180 miles. If he rides at a steady rate of 16 miles per hour, how many hours will the trip take?
13. Javier is driving to Bangor, Maine, which is 240 miles away from his current location. If he needs to be in Bangor in 4 hours, at what rate does he need to drive?
Show Solution
60 miles per hour
14. Alejandra is driving to Cincinnati, 450 miles away. If she wants to be there in 6 hours, at what rate does she need to drive?
15. Aisha took the train from Spokane to Seattle. The distance is 280 miles, and the trip took 3.5 hours. What was the speed of the train?
Show Solution
80 miles per hour
16. Philip got a ride with a friend from Denver to Las Vegas, a distance of 750 miles. If the trip took 10 hours, how fast was the friend driving?
Rate Conversions
17. In a recent Grand Prix, the winner completed the race with an average speed of 229.8 km/h. What was his speed in miles per hour, meters per second, and feet per second?
Solution
18. A car is traveling at a speed of 33 m/s. (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km/h speed limit?
19. If a marathon runner averages 9.5 mi/h, how long does it take him or her to run a 26.22-mi marathon?
Solution
20. A marathon runner completes a 42.188 km course in 2 hours, 30 minutes, and 12 seconds. What is her average speed in meters per second?
21. The speed of sound is measured to be 342 m/s on a certain day. What is this in km/h?
Solution
22. Tectonic plates are large segments of the Earth’s crust that move slowly. Suppose that one such plate has an average speed of 4.0 cm/year. (a) What distance does it move in 1 s at this speed? (b) What is its speed in kilometers per million years?
23. Amanda wants to take a 513 mile trip to see a friend. According to her car's owner's manual, the gas tank 22.3 gallons and the fuel efficiency is 38 miles per gallon.
(a) How many tanks of gas will Amanda use on her trip?
(b) If gasoline costs $2.87 per gallon, how much will she spend on gas?
Solution
Glossary
conversion factor
A conversion factor (or unit conversion factor) is a ratio of two equivalent quantities expressed with different measurement units.
metric system
The metric system is a decimal system of measurement which is based on a fundamental set of units including the meter, the second, and the kilogram.
rate
A rate is a quantity which is computed from measurements in two or more different units.
rate conversion
Rate conversions are unit conversions involving several different quantities.
unit
Units are standardized values used to express measurements of physical quantities.
unit equivalence
A unit equivalence tells you how many of one unit are equivalent to exactly one of another unit.
