Section 1.6: Dimensional Analysis
- Page ID
- 182878
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- Multiply \(4.29(1000)\)
- Simplify \(\frac{30}{54}\)
- Multiply \(\frac{7}{15}\cdot\frac{25}{28}\)
Motivating Problem
A recipe from France calls for 500 milliliters of milk, but your measuring cup is in ounces. How can you figure out how much milk to use without guessing—or making the cake too soggy?
Fun Fact
The metric system was first introduced in France during the French Revolution as a way to unify the country’s many measurement systems. Today, it's used by every country in the world except three: the U.S., Myanmar, and Liberia!
The Goal
In this section, we learn how to convert between different units of measurement using multiplication, logic, and a clear visual structure. We’ll explore both the U.S. and metric systems and practice solving real-world problems, such as converting miles to kilometers, minutes to seconds, or gallons to ounces. Finally, we'll see how dimensional analysis can help us in science, travel, and even baking!
Make Unit Conversions in the U.S. System
There are two systems of measurement commonly used around the world. Most countries use the metric system. The U.S. uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.
The U.S. measurement system uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart, and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, and hours.
The table below shows the equivalencies of measurements. It also shows, in parentheses, the common abbreviations for each measurement.
| U.S. System of Measurement | |
|---|---|
| \(\begin{array} {llll} {} &{\text{1 foot (ft.)}} &{=} &{\text{12 inches (in.)}} \\ {\textbf{Length}} &{\text{1 yard (yd.)}} &{=} &{\text{3 feet (ft.)}} \\ {} &{\text{1 mile (mi.)}} &{=} &{\text{5280 feet (ft.)}} \end{array}\) | \(\begin{array} {llll} {} &{\text{3 teaspoons (t)}} &{=} &{\text{1 tablespoon (T)}} \\ {} &{\text{16 tablespoons (T)}} &{=} &{\text{1 cup (C)}} \\ {} &{\text{1 cup (C)}} &{=} &{\text{8 fluid ounces (fl.oz.)}} \\ {\textbf{Volume}} &{\text{1 pint (pt.)}} &{=} &{\text{2 cups (C)}} \\ {} &{\text{1 quart (qt.)}} &{=} &{\text{2 pints (pt.)}} \\ {} &{\text{1 gallon (gal)}} &{=} &{\text{4 quarts (qt.)}} \end{array}\) |
|
\(\begin{array} {llll} {\textbf{Weight}} &{\text{1 pound (lb.)}} &{=} &{\text{16 ounces (oz.)}} \\ {} &{\text{1 ton}} &{=} &{\text{2000 pounds (lb.)}} \end{array}\) |
\(\begin{array} {llll} {} &{\text{1 minute (min)}} &{=} &{\text{60 seconds (sec)}} \\ {} &{\text{1 hour (hr)}} &{=} &{\text{60 minutes (min)}} \\ {\textbf{Time}} &{\text{1 day}} &{=} &{\text{24 hours (hr)}} \\ {} &{\text{1 week (wk)}} &{=} &{\text{7 days}} \\ {} &{\text{1 year (yr)}} &{=} &{\text{365 days}} \end{array}\) |
In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, quarts and gallons, etc. We will use the identity property of multiplication to make these conversions. We’ll restate the identity property of multiplication here for easy reference.
Because for any real number \(a\) it is true that \(a \cdot 1 = a\) and that \(1 \cdot a = a\), we call 1 the multiplicative identity.
To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to change inches to feet. We know that 1 foot is equal to 12 inches, so we will write 1 as the fraction \(\frac{\text{1 foot}}{\text{12 inches}}\). When we multiply by this fraction we do not change the value, but just change the units.
But \(\frac{\text{12 inches}}{\text{1 foot}}\) also equals 1. How do we decide whether to multiply by \(\frac{\text{1 foot}}{\text{12 inches}}\) or \(\frac{\text{12 inches}}{\text{1 foot}}\)? We choose the fraction that will make the units we want to convert from divide out. Treat the unit words like factors and “divide out” common units like we do common factors. If we want to convert 66 inches to feet, which multiplication will eliminate the inches?
The inches divide out and leave only feet. The second form has no units that will divide out and so will not help us.
Mary Anne is 66 inches tall. Convert her height into feet.
Solution
Rene bought a hose that is 18 yards long. Convert the length to feet.
- Answer
-
54 feet
Ndula, an elephant at the San Diego Safari Park, weighs almost 3.2 tons. Convert her weight to pounds.
Solution
We will convert 3.2 tons into pounds. We will use the identity property of multiplication, writing 1 as the fraction \(\frac{\text{2000 pounds}}{\text{1 ton}}\).
| \(\text{3.2 tons}\) | |
| Multiply the measurement to be converted, by 1. | \(\text{3.2 tons} \cdot 1\) |
| Write 1 as a fraction relating tons and pounds. | \(\text{3.2 tons} \cdot \frac{\text{2000 pounds}}{\text{1 ton}}\) |
| Simplify. |  |
| Multiply. | 6400 pounds |
| Ndula weighs almost 6400 pounds. |
Arnold’s SUV weighs about 4.3 tons. Convert the weight to pounds.
- Answer
-
8600 pounds
Sometimes, to convert from one unit to another, we may need to use several other units in between, so we must multiply several fractions.
Juliet is going with her family to their summer home. She will be away from her boyfriend for 9 weeks. Convert the time to minutes.
Solution
To convert weeks into minutes, we will convert weeks into days, days into hours, and then hours into minutes. To do this, we will multiply by the conversion factors of 1.
| 9 weeks | |
| Write 1 as \(\frac{\text{7 days}}{\text{1 week}}\), and \(\frac{\text{60 minutes}}{\text{1 hour}}\). | \(\frac{\text{9 wk}}{\text{1}}\cdot\frac{\text{7 days}}{\text{1 wk}}\cdot\frac{\text{24 hr}}{\text{1 day}}\cdot\frac{\text{60 min}}{\text{1 hr}}\) |
| Divide out the common units. |  |
| Multiply. | \(\frac{9\cdot7\cdot24\cdot60\text{ min}}{1\cdot1\cdot1\cdot1}\) |
| Multiply. | 90,720 min |
Juliet and her boyfriend will be apart for 90,720 minutes (although it may seem like an eternity!).
The astronauts of Expedition 28 on the International Space Station spend 15 weeks in space. Convert the time to minutes.
- Answer
-
151,200 minutes
How many ounces are in 1 gallon?
Solution
We will convert gallons to ounces by multiplying by several conversion factors.
| 1 gallon | |
| Multiply the measurement to be converted by 1. | \(\frac{\text{1 gallon}}{\text{1}} \cdot \frac{\text{4 quarts}}{\text{1 gallon}} \cdot \frac{\text{2 pints}}{\text{1 quart}} \cdot \frac{\text{2 cups}}{\text{1 pint}} \cdot \frac{\text{8 ounces}}{\text{1 cup}}\) |
| Use conversion factors to get to the right unit. Simplify. |
 |
| Multiply. | \(\frac{1\cdot 4\cdot 2\cdot 2\cdot 8\text{ ounces}}{1\cdot 1\cdot 1\cdot 1\cdot 1 }\) |
| Simplify. | 128 ounces |
How many teaspoons are in 1 cup?
- Answer
-
48 teaspoons
Use Mixed Units of Measurement in the U.S. System
We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1-pound 2-ounce steak for dinner—all these measurements have mixed units.
Performing arithmetic operations on measurements with mixed units of measure requires care. Be sure to add or subtract like units!
Seymour bought three steaks for a barbecue. Their weights were 14 ounces, 1 pound 2 ounces, and 1 pound 6 ounces. How many total pounds of steak did he buy?
Solution
We will add the weights of the steaks to find the total weight of the steaks.
| Add the ounces. Then add the pounds. |  |
| Convert 22 ounces to pounds and ounces. | 2 pounds + 1 pound, 6 ounces |
| Add the pounds. | 3 pounds, 6 ounces |
| Seymour bought 3 pounds 6 ounces of steak. |
Stan cut two pieces of crown molding for his family room that were 8 feet 7 inches and 12 feet 11 inches. What was the total length of the molding?
- Answer
-
21 ft. 6 in.
Anthony bought four planks of wood that were each 6 feet 4 inches long. What is the total length of the wood he purchased?
Solution
We will multiply the length of one plank to find the total length.
| Multiply the inches and then the feet. |  |
| Convert the 16 inches to feet. Add the feet. |
 |
| Anthony bought 25 feet and 4 inches of wood. |
Joellen wants to double a solution of 5 gallons 3 quarts. How many gallons of solution will she have in all?
- Answer
-
11 gallons 2 qt.
Make Unit Conversions in the Metric System
In the metric system, units are related by powers of 10. The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is 1,000 meters; the prefix kilo means thousand. One centimeter is \(\frac{1}{100}\) of a meter, just like one cent is \(\frac{1}{100}\) of one dollar.
The table below shows the equivalencies of measurements in the metric system. The common abbreviations for each measurement are given in parentheses.
| Metric System of Measurement | ||
|---|---|---|
| Length | Mass | Capacity |
| 1 kilometer (km) = 1,000 m 1 hectometer (hm) = 100 m 1 dekameter (dam) = 10 m 1 meter (m) = 1 m 1 decimeter (dm) = 0.1 m 1 centimeter (cm) = 0.01 m 1 millimeter (mm) = 0.001 m |
1 kilogram (kg) = 1,000 g 1 hectogram (hg) = 100 g 1 dekagram (dag) = 10 g 1 gram (g) = 1 g 1 decigram (dg) = 0.1 g 1 centigram (cg) = 0.01 g 1 milligram (mg) = 0.001 g |
1 kiloliter (kL) = 1,000 L 1 hectoliter (hL) = 100 L 1 dekaliter (daL) = 10 L 1 liter (L) = 1 L 1 deciliter (dL) = 0.1 L 1 centiliter (cL) = 0.01 L 1 milliliter (mL) = 0.001 L |
| 1 meter = 100 centimeters 1 meter = 1,000 millimeters |
1 gram = 100 centigrams 1 gram = 1,000 milligrams |
1 liter = 100 centiliters 1 liter = 1,000 milliliters |
To convert in the metric system, we will use the same technique we used in the US system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.
Have you ever run a 5K or 10K race? The length of those races is measured in kilometers. The metric system is commonly used in the United States when discussing race length.
Nick ran a 10K race. How many meters did he run?
Solution
We will convert kilometers to meters using the identity property of multiplication.
| 10 kilometers | |
| Multiply the measurement to be converted by 1. |  |
| Write 1 as a fraction relating kilometers and meters. |  |
| Simplify. |  |
| Multiply. | 10,000 meters |
| Nick ran 10,000 meters. |
Herman bought a rug 2.5 meters in length. How many centimeters is the length?
- Answer
-
250 centimeters
Eleanor’s newborn baby weighed 3,200 grams. How many kilograms did the baby weigh?
Solution
We will convert grams into kilograms.
 |
|
| Multiply the measurement to be converted by 1. |  |
| Write 1 as a function relating kilograms and grams. |  |
| Simplify. |  |
| Multiply. | \(\frac{3,200 \text{ kilograms}}{1,000}\) |
| Divide. | 3.2 kilograms The baby weighed 3.2 kilograms. |
Anderson received a package that was marked 4,500 grams. How many kilograms did this package weigh?
- Answer
-
4.5 kilograms
As you become familiar with the metric system, you may see a pattern. Since the system is based on multiples of ten, the calculations involve multiplying by multiples of ten. We have learned how to simplify these calculations by just moving the decimal.
To multiply by 10, 100, or 1,000, we move the decimal to the right one, two, or three places, respectively. To multiply by 0.1, 0.01, or 0.001, we move the decimal to the left one, two, or three places, respectively.
We can apply this pattern when we make measurement conversions in the metric system. In the previous example, we changed 3,200 grams to kilograms by multiplying by \(\frac{1}{1000}\) (or 0.001). This is the same as moving the decimal three places to the left.
Convert
- 350 L to kiloliters
- 4.1 L to milliliters.
Solution
1. We will convert liters to kiloliters using the fact that 1 kiloliter = 1,000 liters and 1 kiloliter = 1,000 liters.
| 350 L | |
| Multiply by 1, writing 1 as a fraction relating liters to kiloliters. | \(350 \text{ L}\frac{\text{1 kL}}{\text{1000L}}\) |
| Simplify. | \(350 \not{\text{ L}}\frac{\text{1 kL}}{1000 \not\text{ L}}\) |
 |
0.35 kL |
2. We will convert liters to milliliters using the fact that 1 liter = 1,000 milliliters and 1 liter = 1,000 milliliters.
 |
|
| Multiply by 1, writing 1 as a fraction relating liters to milliliters. |  |
| Simplify. |  |
| Move the decimal 3 units to the right. |  |
 |
Convert:
- 350 hL to liters
- 4.1 L to centiliters
- Answer
-
- 35,000 liters
- 410 centiliters
Use Mixed Units of Measurement in the Metric System
Performing arithmetic operations on measurements with mixed units of measure in the metric system requires the same care we used in the US system. But it may be easier because of the relation of the units to the powers of 10. Make sure to add or subtract like units.
Ryland is 1.6 meters tall. His younger brother is 85 centimeters tall. How much taller is Ryland than his younger brother?
Solution
We can convert both measurements to either centimeters or meters. Since meters is the larger unit, we will subtract the lengths in meters. We convert 85 centimeters to meters by moving the decimal 2 places to the left.
\[\begin{array} { cc } { \text {Write the } 85 \text { centimeters as meters. } } & { 1.60 \mathrm { m } } \\ {} &{ \dfrac { - 0.85 \mathrm { m } } { 0.75 \mathrm { m } } } \end{array}\nonumber\]
Ryland is 0.75 m taller than his brother.
The fence around Hank’s yard is 2 meters high. Hank is 96 centimeters tall. How much shorter is Hank than the fence? Write the answer in meters.
- Answer
-
1.04 meters
Dena’s lentil soup recipe calls for 150 milliliters of olive oil. If she wants to triple the recipe, how many liters of olive oil will she need?
Solution
We will find the amount of olive oil in milliliters then convert to liters.
\(\begin{array} { ll } {} & { \text { Triple } 150 \text{ mL}} \\ { \text { Translate to algebra. } } &{3\cdot 150 \text{ mL}} \\ { \text { Multiply. } } &{450\text{ mL}}\\ { \text { Convert to liters. } } &{450\cdot \frac{0.001\text{ L}}{1 \text{ ml}}} \\ { \text { Simplify. } } &{0.45 \text{ L}}\\ {} &{ \text { Dena needs 0.45 liters of olive oil. } } \end{array}\)
To make one pan of baklava, Dorothea needs 400 grams of filo pastry. If Dorothea plans to make 6 pans of baklava, how many kilograms of filo pastry will she need?
- Answer
-
2.4 kilograms
Convert Between the U.S. and the Metric Systems of Measurement
Many measurements in the United States are made in metric units. Our soda may come in 2-liter bottles, our calcium may come in 500-mg capsules, and we may run a 5K race. To work easily in both systems, we need to be able to convert between the two systems.
The table below shows some of the most common conversions.
| Conversion Factors Between U.S. and Metric Systems | ||
|---|---|---|
| Length | Mass | Capacity |
| \(\begin{array} { l l l } {1 \text{ in.}} & {=} &{2.54 \text{ cm}} \\ {1\text{ ft.}} &{=} &{0.305 \text{ m}} \\ {1 \text{ yd.}} & {=} &{0.914 \text{ m}} \\ {1\text{ mi.}} &{=} &{1.61 \text{ km}} \\ {1 \text{ m}} & {=} &{3.28 \text{ ft}} \end{array}\) | \(\begin{array} { l l l } {1 \text{ lb.}} & {=} &{0.45 \text{ kg}} \\ {1\text{ oz.}} &{=} &{28 \text{ g}} \\ {1 \text{ kg}} & {=} &{2.2 \text{ lb}} \end{array}\) | \(\begin{array} { l l l } {1 \text{ qt.}} & {=} &{0.95 \text{ L}} \\ {1\text{ fl. oz.}} &{=} &{30 \text{ ml}} \\ {1 \text{ L}} & {=} &{1.06 \text{ lb}} \end{array}\) |
The figure below shows how inches and centimeters are related on a ruler.
The figure below shows the ounce and milliliter markings on a measuring cup.
The figure below shows how pounds and kilograms are marked on a bathroom scale.
We convert between the systems just as we do within them—by multiplying by unit conversion factors.
Lee’s water bottle holds 500 mL of water. How many ounces are in the bottle? Round to the nearest tenth of an ounce.
Solution
\(\begin{array} { l l } {} & {500 \text{ mL}} \\ {\text{Multiplying by a unit conversion factor relating}} &{500\text{ milliliters}\cdot\frac{1\text{ ounce}}{30\text{ milliliters}}} \\ {\text{mL and ounces}} &{} \\ {\text{Simplify.}} &{\frac{50\text{ ounce}}{30}} \\ {\text{Divide.}} &{16.7\text{ ounces}} \\ {} &{\text{The water bottle has 16.7 ounces}} \end{array}\)
How many liters are in 4 quarts of milk?
- Answer
-
3.8 liters
Soleil was on a road trip and saw a sign that said the next rest stop was in 100 kilometers. How many miles until the next rest stop?
Solution
\(\begin{array} { l l } {} & {100 \text{ kilometers}} \\ {\text{Multiplying by a unit conversion factor relating}} &{100\text{ kilometers}\cdot\frac{1\text{ mile}}{1.61\text{ kilometers}}} \\ {\text{km and mi.}} &{} \\ {\text{Simplify.}} &{\frac{100\text{ miles}}{1.61}} \\ {\text{Divide.}} &{62\text{ miles}} \\ {} &{\text{Soleil will travel 62 miles.}} \end{array}\)
The height of Mount Kilimanjaro is 5,895 meters. Convert the height to feet.
- Answer
-
19,335.6 feet