Learning Objectives
- Convert mathematical relationships into conversion factors.
- Use conversion factors to perform unit conversions.
- Use scientific prefixes to create mathematical relationships.
Chapter 35 Unit Conversions
Section 35.1 Say Goodbye to the Mars Climate Orbiter
The $300,000,000 NASA project called the Mars Climate Orbiter launched on December 11, 1998. Approximately 9 months later, it crashed into the surface of Mars and was declared a total loss. The culprit was a piece of hardware that gave distance in meters to a piece of software that was working in feet.
This incident stands out among the many unit conversion errors (or simply the failure to convert units at all) because of a combination of the magnitude of the project and the high capacity minds that were working on it. Most errors of this type are not nearly as catastrophic, but that does not negate the importance of understanding how to perform unit conversions.
Units are names of measurements of quantities. Here are some examples:
- Time is measured in seconds, minutes, hours, days, and years.
- Lengths are measured in inches, feet, miles, centimeters, meters, and kilometers.
- Volumes are measured in tablespoons, cups, quarts, gallons, liters, and cubic meters.
- Quantities are measured in dozens, hundreds, thousands, millions, and moles.
Activity 35.1. Conversion Factors.
The most important feature about these units is that they represent specific quantities that can be related to each other. For example, there are 12 inches in a foot. And there are 60 seconds in a minute. These create mathematical relationships by simply translating the words into a formula:
\begin{equation*}
\begin{array}{ccccc}
\text{12 inches in a foot} \amp \longrightarrow \amp 1 \text{ foot} = 12 \text{ inches} \\
\text{60 seconds in a minute} \amp \longrightarrow \amp 1 \text{ minute} = 60 \text{ seconds}
\end{array}
\end{equation*}
These equations can be turned into fractions that we call conversion factors. Conversion factors have the property that they are equal to the number 1. Notice that every equality of this type creates two conversion factors.
\begin{equation*}
\begin{array}{ccccc}
1 \text{ foot} = 12 \text{ inches} \amp \longrightarrow
\amp \frac{1 \text{ foot}}{12 \text{ inches}} \amp \text{and} \amp \frac{12 \text{ inches}}{1 \text{ foot}} \\
1 \text{ minute} = 60 \text{ seconds} \amp \longrightarrow
\amp \frac{1 \text{ minute}}{60 \text{ seconds}} \amp \text{and} \amp \frac{60 \text{ seconds}}{1 \text{ minute}}
\end{array}
\end{equation*}
Try it!
Write an equation that relates feet to yards, then use it to determine two conversion factors.
Solution.
\begin{equation*}
\begin{array}{rl}
3 \text{ feet} = 1 \text{ yard}
\amp \hspace{1cm} \longrightarrow \hspace{1cm} \frac{3 \text{ feet}}{1 \text{ yard}} = 1 \hspace{0.5cm} \text{ and } \hspace{0.5cm} 1 = \frac{1 \text{ yard}}{3 \text{ feet}}
\end{array}
\end{equation*}
Activity 35.2. Using Conversion Factors.
The importance of conversion factors being equal to the number is that multiplying by does not change the value of a number. With an appropriate choice of conversion factors, it is possible to make original unit cancel out, leaving you with the value in the new unit. Here is an example:
\begin{equation*}
\begin{aligned}
8 \text{ minutes} \amp = 8 \text{ minutes} \cdot 1 \amp \eqnspacer \amp \text{Multiplying by $1$} \\
\amp = 8 \text{ minutes} \cdot \frac{60 \text{ seconds}}{1 \text{ minute}} \amp \amp \text{Conversion factor} \\
\amp = 8 \cancel{\text{ minutes}} \cdot \frac{60 \text{ seconds}}{1 \cancel{\text{ minute}}} \amp \amp \text{Cancel the units} \\
\amp = 480 \text{ seconds}
\end{aligned}
\end{equation*}
This example was drawn out with details for emphasis. Your presentation can be shortened.
\begin{equation*}
\begin{aligned}
8 \text{ minutes} \amp = 8 \cancel{\text{ minutes}} \cdot \frac{60 \text{ seconds}}{1 \cancel{\text{ minute}}}
\amp \eqnspacer \amp \text{Cancel the units} \\
\amp = 480 \text{ seconds}
\end{aligned}
\end{equation*}
Try it!
Convert 36 feet to yards using a conversion factor.
Solution.
\begin{equation*}
\begin{aligned}
36 \text{ feet} \amp = 36 \text{ }\cancel{feet} \cdot \frac{1 \text{ yard}}{3 \text{ }\cancel{feet}} \amp \eqnspacer \amp \text{Cancel the units} \\
\amp = 12 \text{ yards}
\end{aligned}
\end{equation*}
Activity 35.3. Scientific Prefixes.
You may be aware that there are certain prefixes that apply the the unit. These prefixes are another way that scientists avoid having to write really long numbers. Here are some of the common prefixes and their meaning:
\begin{equation*}
\begin{array} {|c|c|c|c|}
\hline
\text{Prefix} \amp \text{Symbol} \amp \text{Value as Power of 10} \amp \text{Value in Standard Form} \\ \hline
\text{nano} \amp n \amp 10^{-9} \amp 0.000000001 \\ \hline
\text{micro} \amp \mu \amp 10^{-6} \amp 0.000001 \\ \hline
\text{milli} \amp m \amp 10^{-3} \amp 0.001 \\ \hline
\text{centi} \amp c \amp 10^{-2} \amp 0.01 \\ \hline
\text{deci} \amp d \amp 10^{-1} \amp 0.1 \\ \hline
\text{deka} \amp da \amp 10^{1} \amp 10 \\ \hline
\text{hecto} \amp h \amp 10^{2} \amp 100 \\ \hline
\text{kilo} \amp k \amp 10^{3} \amp 1000 \\ \hline
\text{mega} \amp M \amp 10^{6} \amp 1000000 \\ \hline
\text{giga} \amp G \amp 10^{9} \amp 1000000000 \\ \hline
\text{tera} \amp T \amp 10^{12} \amp 1000000000000 \\ \hline
\end{array}
\end{equation*}
These prefixes are placed in front of a unit to change its value. For example, a kilometer is 1000 meters and a millimeter is 0.001 meters. This gives us mathematical relationships that we can use to write conversion factors. While it is not wrong to use decimals in the conversion factors with these prefixes, it’s generally considered better to use integers. So instead of using \(\frac{0.001 \text{ meters}}{1 \text{ millimeter}}\text{,}\) we would usually use \(\frac{1 \text{ meter}}{1000 \text{ millimeters}}\text{.}\)
Try it!
Write an equation that relates kilograms to grams, then use it to determine two conversion factors.
Solution.
\begin{equation*}
\begin{array}{rl}
1000 \text{ grams} = 1 \text{ kilogram} \amp \hspace{0.5cm} \longrightarrow \hspace{0.5cm} \frac{1000 \text{ grams}}{1 \text{ kilogram}} = 1 \hspace{0.5cm} \text{ and } \hspace{0.5cm} 1 = \frac{1 \text{ kilogram}}{1000 \text{ grams}}
\end{array}
\end{equation*}
Section 35.2 Worksheets
PDF Version of these Worksheets
1
external/worksheets/35-Worksheets.pdfWorksheet Worksheet 1
1.
Write an equation that relates inches to centimeters, then use it to determine two conversion factors.
2.
Convert 12 inches into centimeters.
3.
Convert 100 centimeters into inches.
4.
Write an equation that relates cups to quarts, then use it to determine two conversion factors.
5.
Convert 32 cups into quarts.
6.
Convert 4 quarts into cups.
Worksheet Worksheet 2
1.
Write an equation that relates kilowatts to watts, then use it to determine two conversion factors.
2.
Write an equation that relates deciliters to liters, then use it to determine two conversion factors.
3.
Write an equation that relates nanometers to meters, then use it to determine two conversion factors.
4.
Write an equation that relates hectares to acres, then use it to determine two conversion factors.
5.
Write an equation that relates centimeters to meters and another equation that relates meters to kilometers. Use these two equations together to determine two conversion factors that relate kilometers to centimeters.
Worksheet Worksheet 3
In science courses, you will encounter some interesting units that are built to describe specific situations. Although you may not have any intuition with these, as long as you have a formula and you understand the method, you can be begin to work with the problems.
1.
A mole of objects is \(6.022 \times 10^{23}\) of those objects. For example, a mole of carbon atoms is \(6.022 \times 10^{23}\) carbon atoms. How many molecules of oxygen are in 5.7 moles of oxygen?
2.
An astronomical unit is approximately \(1.496 \times 10^{11}\) meters. This is the approximate distance from the earth to the sun. Mars is approximately 1.52 astronomical units from the sun. About how far is it from the sun to Mars in meters?
3.
Words such as millions and billions can also be used as a unit conversion. This is often used when talking about finances at a state or national level. Earlier, we talked about the $300000000 Mars Climate Orbiter. This could have been written as $300 million.
Write the quantity 2753.78 billion in standard form and using scientific notation.
4.
A light-year is the distance that light can travels in one year in a vacuum. This distance is approximately \(9.46 \times 10^{12}\) kilometers. The distance between the Milky Way galaxy and the Andromeda galaxy is approximately 2.5 million light-years. Approximately how many kilometers is it between the two galaxies?
Worksheet Worksheet 4
1.
A stone is a measure of weight that is commonly used in the UK and Ireland. One stone is equal to 14 pounds. If an object weighs 37 stone, how many pounds does it weigh?
2.
A smoot is a unit of measurement devised as a prank by a fraternity at MIT. It is considered to be 67 inches, which corresponds to the height of Oliver Smoot, who was used to measure Harvard bridge. The bridge was determined to have a length of approximately 364.4 smoots. Convert this distance to feet.
3.
When thinking about very large or very small quantities, it is sometimes useful to relate them to quantities that you are more familiar with. For example, we can think of \(1 \text{ home} = \$400000\) as a relationship for converting dollars into homes. What is the equivalent of $1 billion in homes?
4.
Transistors are an important component for modern electronics. The smallest transistors are about 7 nanometers in size. A human hair is approximately 100 micrometers in diameter. How many transistors would need to be lined up to equate to the thickness of a human hair?
Worksheet Worksheet 5
A compound unit is a unit that mixes multiple other units together. Some common compound units are speeds like miles per hour (\(\frac{\text{miles}}{\text{hours}}\)) and pressures like pounds per square inch (\(\frac{\text{pounds}}{\text{inch}^2} = \frac{\text{pounds}}{\text{inch} \cdot \text{inch}}\)). When converting these, every unit much be converted individually. Here is an example of converting feet per day into inches per week:
\begin{equation*}
\begin{aligned}
10 \frac{\text{feet}}{\text{day}}
\amp = 10 \frac{\cancel{\text{feet}}}{\cancel{\text{day}}} \cdot \frac{12 \text{ inches}}{1 \cancel{\text{ foot}}} \cdot \frac{7 \cancel{\text{ days}}}{1 \text{ week}} \\
\amp = 840 \frac{\text{inches}}{\text{week}}
\end{aligned}
\end{equation*}
1.
One mile is equal to about 1.6 kilometers. If a car is traveling 75 miles per hour, how many kilometers per minute is it traveling?
2.
Water has a density of about 8.34 pounds per gallon. Convert this to ounces (weight) per ounce (liquid).
3.
A hectare is the area of a square that is 100 meters on each side. Determine the conversion factor for hectares to square meters.
Section 35.3 Deliberate Practice: Unit Conversions
Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.
Focus on these skills:
- You may need to use the internet to determine the appropriate conversion factor.
- Explicitly write out the conversion factor and show the cancellation step.
- Present your work legibly.
Worksheet Worksheet
Instructions: Convert the quantity to the indicated units. You may need to look up the conversion factor.
1.
Convert 39 deciliters to milliliters.
2.
Convert 4 kilometers to centimeters.
3.
Convert 13 micrograms to milligrams.
4.
Convert 3 astronomical units to meters.
5.
Convert 8 nautical miles to feet.
6.
How many ounces are in 12 cups?
7.
How many pints are in 3 gallons?
8.
How many teaspoons are in 9 tablespoons?
9.
How many ounces are in 3 pounds?
10.
How many liters are in 2 gallons?
Section 35.4 Closing Ideas
Understanding how to manipulate units is an important skill for science courses, but these types of conversions happen all the time in ways that you might not expect. For example, if you have $10 and a plate of 3 street tacos costs $2.50, you can convert $10 to 4 plates of tacos, and 4 plates of tacos to 12 individual street tacos. You probably won’t need to write out the conversion factors, and that’s okay. Remember that the goal is to understand what unit conversions do and how they work. The goal is not that you would learn that math is a set of rules you must follow.
This example may seem intuitive, but that intuition comes with familiarity and experience. Even though you may not have found a 3 street tacos for $2.50 deal, you understand how it works. If you have an intuitive sense for how these conversions work and you understand how unit manipulations work, then it’s not as difficult to solve problems in situations where you have less familiarity.
Every now and then, there’s an internet math meme that shows up that is an example of unit conversions gone wrong. Here is one example:
"The winner of the next lottery will win $1300 million. There are 300 million people in the US.
\begin{equation*}
\frac{\$1300 \text{ million}}{300 \text{ million}} = \$4.33 \text{ million}
\end{equation*}
Why not just give everyone $4.33 million?"
Can you spot the error? (Hint: Think about we have manipulated units in fractions throughout this entire section.)
