Elementary Algebra 2e · free preview

1.10 Systems of Measurement

Learning objectives: Make unit conversions in the US system; Use mixed units of measurement in the US system; Make unit conversions in the metric system; Use mixed units of measurement in the metric system; Convert between the US and the metric systems of measurement; Convert between Fahrenheit and Celsius temperatures

1. Learning objectives

Learning objectives

Make unit conversions in the US system

Use mixed units of measurement in the US system

Make unit conversions in the metric system

Use mixed units of measurement in the metric system

Convert between the US and the metric systems of measurement

Convert between Fahrenheit and Celsius temperatures

2. Make Unit Conversions in the U.S. System

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. system of measurement 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 equivalencies of measurements are shown in Table 1.58. The table also shows, in parentheses, the common abbreviations for each measurement.

U.S. System of Measurement

\begin{array}{lclcccccc} \mathrm{Length} & & \begin{array}{lll} 1\,\mathrm{foot}\,\mathrm{(ft.)} & = & 12\,\mathrm{inches}\,\mathrm{(in.)} \\ 1\,\mathrm{yard}\,\mathrm{(yd.)} & = & 3\,\mathrm{feet}\,\mathrm{(ft.)} \\ 1\,\mathrm{mile}\,\mathrm{(mi.)} & = & 5,280\,\mathrm{feet}\,\mathrm{(ft.)} \\ \end{array} \\ \end{array}

\begin{array}{lclcccccccccccc} \mathrm{Volume} & & \begin{array}{lll} 3\,\mathrm{teaspoons}\,\mathrm{(t)} & = & 1\,\mathrm{tablespoon}\,\mathrm{(T)} \\ \mathrm{16 tablespoons}\,\mathrm{(T)} & = & \mathrm{1 cup}\,\mathrm{(C)} \\ \mathrm{1 cup}\,\mathrm{(C)} & = & \mathrm{8 fluid ounces}\,\mathrm{(fl. oz.)} \\ \mathrm{1 pint}\,\mathrm{(pt.)} & = & \mathrm{2 cups}\,\mathrm{(C)} \\ \mathrm{1 quart}\,\mathrm{(qt.)} & = & \mathrm{2 pints}\,\mathrm{(pt.)} \\ \mathrm{1 gallon}\,\mathrm{(gal)} & = & \mathrm{4 quarts}\,\mathrm{(qt.)} \\ \end{array} \\ \end{array}

\begin{array}{lclcccc} \mathrm{Weight} & & \begin{array}{lll} \mathrm{1 pound}\,\mathrm{(lb.)} & = & \mathrm{16 ounces}\,\mathrm{(oz.)} \\ \mathrm{1 ton} & = & \mathrm{2000 pounds}\,\mathrm{(lb.)} \\ \end{array} \\ \end{array}

\begin{array}{lclcccccccccc} \mathrm{Time} & & \,\begin{array}{lll} \mathrm{1 minute}\,\mathrm{(min)} & = & \mathrm{60 seconds}\,\mathrm{(sec)} \\ \mathrm{1 hour}\,\mathrm{(hr)} & = & \mathrm{60 minutes}\,\mathrm{(min)} \\ \mathrm{1 day} & = & \mathrm{24 hours}\,\mathrm{(hr)} \\ \mathrm{1 week}\,\mathrm{(wk)} & = & \mathrm{7 days} \\ \mathrm{1 year}\,\mathrm{(yr)} & = & \mathrm{365 days} \\ \end{array} \\ \end{array}

Table

1.58

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 do these conversions. We’ll restate the identity property of multiplication here for easy reference.

$\begin{array}{lccccl} \mathrm{For any real number}\,a: & & & & & a\cdot{}1=a\,1\cdot{}a=a \\ 1\,\mathrm{is the}\,\mathrm{multiplicative identity} & & & & & \\ \end{array}$

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 $\frac{1\,\mathrm{foot}}{12\,\mathrm{inches}}.$ When we multiply by this fraction we do not change the value, but just change the units.

But $\frac{12\,\mathrm{inches}}{1\,\mathrm{foot}}$ also equals 1. How do we decide whether to multiply by $\frac{1\,\mathrm{foot}}{12\,\mathrm{inches}}$ or $\frac{12\,\mathrm{inches}}{1\,\mathrm{foot}}?$ 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 does not have any units that will divide out and so will not help us.

3. Example 1.140

Example 1.140

How to Make Unit Conversions

MaryAnne is 66 inches tall. Convert her height into feet.

Make Unit Conversions.

Step 1. Multiply the measurement to be converted by 1; write 1 as a fraction relating the units given and the units needed.
Step 2. Multiply.
Step 3. Simplify the fraction.
Step 4. Simplify.

When we use the identity property of multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.

4. Example 1.141

Example 1.141

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 $\frac{2000\,\mathrm{pounds}}{1\,\mathrm{ton}}.$

Sometimes, to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

5. Example 1.142

Example 1.142

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 conversion factors of 1.

Juliet and her boyfriend will be apart for 90,720 minutes (although it may seem like an eternity!).

6. Example 1.143

Example 1.143

How many ounces are in 1 gallon?

Solution

We will convert gallons to ounces by multiplying by several conversion factors. Refer to Table 1.58.

7. Use Mixed Units of Measurement in the U.S. System

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 measures requires care. Be sure to add or subtract like units!

8. Example 1.144

Example 1.144

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.

9. Example 1.145

Example 1.145

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.

10. Make Unit Conversions in the Metric System

Make Unit Conversions in the Metric System

In the metric system, units are related by powers of 10. The roots 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 equivalencies of measurements in the metric system are shown in Table 1.59. The common abbreviations for each measurement are given in parentheses.

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

Table

1.59

To make conversions in the metric system, we will use the same technique we did 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 are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.

11. Example 1.146

Example 1.146

Nick ran a 10K race. How many meters did he run?

Solution

We will convert kilometers to meters using the identity property of multiplication.

12. Example 1.147

Example 1.147

Eleanor’s newborn baby weighed 3,200 grams. How many kilograms did the baby weigh?

Solution

We will convert grams into 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 Example 1.147, 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.

13. Example 1.148

Example 1.148

Convert ⓐ 350 L to kiloliters ⓑ 4.1 L to milliliters.

Solution

ⓐ We will convert liters to kiloliters. In Table 1.59, we see that $1\,\mathrm{kiloliter}=\mathrm{1,000 liters.}$ $\mathrm{350 L}$ Multiply by 1, writing 1 as a fraction relating liters to kiloliters. $\mathrm{350 L}\cdot\frac{\mathrm{1 kL}}{\mathrm{1,000 L}}$ Simplify. $350\,L\cdot\frac{\mathrm{1 kL}}{1,000\,L}$ $\mathrm{0.35 kL}$

ⓑ We will convert liters to milliliters. From Table 1.59 we see that $\mathrm{1 liter}\,=\,\mathrm{1,000 milliliters.}$ Multiply by 1, writing 1 as a fraction relating liters to milliliters. Simplify. Move the decimal 3 units to the right.

14. Use Mixed Units of Measurement in the Metric System

Use Mixed Units of Measurement in the Metric System

Performing arithmetic operations on measurements with mixed units of measures 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.

15. Example 1.149

Example 1.149

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.


Write the 85 centimeters as meters.	$\begin{array}{l} \,1.60\,m \\ \underset{\mathrm{\_\_\_\_\_\_\_}}{-0.85\,m} \\ \,0.75\,m \\ \end{array}$

Table

1.60

Ryland is $\mathrm{0.75 m}$ taller than his brother.

16. Example 1.150

Example 1.150

Dena’s recipe for lentil soup calls for 150 milliliters of olive oil. Dena wants to triple the recipe. How many liters of olive oil will she need?

Solution

We will find the amount of olive oil in millileters then convert to liters.


	Triple $\mathrm{150 mL}$
Translate to algebra.	$3\cdot{}150\,\mathrm{mL}$
Multiply.	$\mathrm{450 mL}$
Convert to liters.	$450\cdot\frac{0.001\,L}{1\,\mathrm{mL}}$
Simplify.	$\mathrm{0.45 L}$
	Dena needs 0.45 liters of olive oil.

Table

1.61

17. Convert Between the U.S. and the Metric Systems of Measurement

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.

Table 1.62 shows some of the most common conversions.

Length	Mass	Capacity
$\begin{array}{lcccccccccc} \begin{array}{lll} \mathrm{1 in.} & = & \mathrm{2.54 cm} \\ \mathrm{1 ft.} & = & \mathrm{0.305 m} \\ \mathrm{1 yd.} & = & \mathrm{0.914 m} \\ \mathrm{1 mi.} & = & \mathrm{1.61 km} \\ \mathrm{1 m} & = & \mathrm{3.28 ft.} \\ \end{array} \\ \end{array}$	$\begin{array}{lcccccc} \begin{array}{lll} \mathrm{1 lb.} & = & \mathrm{0.45 kg} \\ \mathrm{1 oz.} & = & \mathrm{28 g} \\ \mathrm{1 kg} & = & \mathrm{2.2 lb.} \\ \end{array} \\ \end{array}$	$\begin{array}{lcccccc} \begin{array}{lll} \mathrm{1 qt.} & = & \mathrm{0.95 L} \\ \mathrm{1 fl. oz.} & = & \mathrm{30 mL} \\ \mathrm{1 L} & = & \mathrm{1.06 qt.} \\ \end{array} \\ \end{array}$

Table

1.62

Figure 1.22 shows how inches and centimeters are related on a ruler.

Figure 1.23 shows the ounce and milliliter markings on a measuring cup.

Figure 1.24 shows how pounds and kilograms marked on a bathroom scale.

We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.

18. Example 1.151

Example 1.151

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


	$\mathrm{500 mL}$
Multiply by a unit conversion factor relating mL and ounces.	$500\,\mathrm{milliliters}\cdot\frac{1\,\mathrm{ounce}}{30\,\mathrm{milliliters}}$
Simplify.	$\frac{\mathrm{500 ounce}}{30}$
Divide.	$\mathrm{16.7 ounces.}$
	The water bottle has 16.7 ounces.

Table

1.63

19. Example 1.152

Example 1.152

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


	$\mathrm{100 kilometers}$
Multiply by a unit conversion factor relating km and mi.	$100\,\mathrm{kilometers}\cdot\frac{1\,\mathrm{mile}}{1.61\,\mathrm{kilometer}}$
Simplify.	$\frac{\mathrm{100 miles}}{1.61}$
Divide.	$\mathrm{62 miles.}$
	Soleil will travel 62 miles.

Table

1.64

20. Convert between Fahrenheit and Celsius Temperatures

Convert between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for $22^{\circ}C,$ what does that mean?

The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written $^{\circ}F.$ The metric system uses degrees Celsius, written $^{\circ}C.$ Figure 1.25 shows the relationship between the two systems.

To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula

C=\frac{5}{9}(F-32).

To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula

F=\frac{9}{5}C+32.

21. Example 1.153

Example 1.153

Convert $50^{\circ}$ Fahrenheit into degrees Celsius.

Solution

We will substitute $50^{\circ}F$ into the formula to find C.

22. Example 1.154

Example 1.154

While visiting Paris, Woody saw the temperature was $20^{\circ}$ Celsius. Convert the temperature into degrees Fahrenheit.

Solution

We will substitute $20^{\circ}C$ into the formula to find F.

23. Section 1.10 Exercises

Section 1.10 Exercises

Make Unit Conversions in the U.S. System

In the following exercises, convert the units.

Use Mixed Units of Measurement in the U.S. System

In the following exercises, solve.

Make Unit Conversions in the Metric System

In the following exercises, convert the units.

Use Mixed Units of Measurement in the Metric System

In the following exercises, solve.

Convert Between the U.S. and the Metric Systems of Measurement

In the following exercises, make the unit conversions. Round to the nearest tenth.

Convert between Fahrenheit and Celsius Temperatures

In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.

In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

Everyday Math

Writing Exercises

ⓐ 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?

24. Practice 1

Lexie is 30 inches tall. Convert her height to feet.

Rene bought a hose that is 18 yards long. Convert the length to feet.

Arnold’s SUV weighs about 4.3 tons. Convert the weight to pounds.

The Carnival Destiny cruise ship weighs 51,000 tons. Convert the weight to pounds.

The distance between the earth and the moon is about 250,000 miles. Convert this length to yards.

25. Practice 2

The astronauts of Expedition 28 on the International Space Station spend 15 weeks in space. Convert the time to minutes.

How many cups are in 1 gallon?

How many teaspoons are in 1 cup?

Laura gave birth to triplets weighing 3 pounds 3 ounces, 3 pounds 3 ounces, and 2 pounds 9 ounces. What was the total birth weight of the three babies?

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?

26. Practice 3

Henri wants to triple his spaghetti sauce recipe that uses 1 pound 8 ounces of ground turkey. How many pounds of ground turkey will he need?

Joellen wants to double a solution of 5 gallons 3 quarts. How many gallons of solution will she have in all?

Sandy completed her first 5K race! How many meters did she run?

Herman bought a rug 2.5 meters in length. How many centimeters is the length?

Kari’s newborn baby weighed 2,800 grams. How many kilograms did the baby weigh?

27. Practice 4

Anderson received a package that was marked 4,500 grams. How many kilograms did this package weigh?

Convert: ⓐ 725 L to kiloliters ⓑ 6.3 L to milliliters

Convert: ⓐ 350 hL to liters ⓑ 4.1 L to centiliters

Mariella is 1.58 meters tall. Her daughter is 75 centimeters tall. How much taller is Mariella than her daughter? Write the answer in centimeters.

The fence around Hank’s yard is 2 meters high. Hank is 96 centimeters tall. How much shorter than the fence is Hank? Write the answer in meters.

28. Practice 5

A recipe for Alfredo sauce calls for 250 milliliters of milk. Renata is making pasta with Alfredo sauce for a big party and needs to multiply the recipe amounts by 8. How many liters of milk will she need?

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?

How many quarts of soda are in a 2-L bottle?

How many liters are in 4 quarts of milk?

The height of Mount Kilimanjaro is 5,895 meters. Convert the height to feet.

29. Practice 6

The flight distance from New York City to London is 5,586 kilometers. Convert the distance to miles.

Convert the Fahrenheit temperature to degrees Celsius: $59^{\circ}$ Fahrenheit.

Convert the Fahrenheit temperature to degrees Celsius: $41^{\circ}$ Fahrenheit.

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Helsinki, Finland, was $15^{\circ}$ Celsius.

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Sydney, Australia, was $10^{\circ}$ Celsius.

30. Practice 7

A park bench is 6 feet long. Convert the length to inches.

A ribbon is 18 inches long. Convert the length to feet.

A football field is 160 feet wide. Convert the width to yards.

Ulises lives 1.5 miles from school. Convert the distance to feet.

A killer whale weighs 4.6 tons. Convert the weight to pounds.

31. Practice 8

An empty bus weighs 35,000 pounds. Convert the weight to tons.

Rocco waited $1\frac{1}{2}$ hours for his appointment. Convert the time to seconds.

How many teaspoons are in a pint?

JJ’s cat, Posy, weighs 14 pounds. Convert her weight to ounces.

Crista will serve 20 cups of juice at her son’s party. Convert the volume to gallons.

32. Practice 9

Jon is 6 feet 4 inches tall. Convert his height to inches.

The voyage of the Mayflower took 2 months and 5 days. Convert the time to days.

Baby Preston weighed 7 pounds 3 ounces at birth. Convert his weight to ounces.

Eli caught three fish. The weights of the fish were 2 pounds 4 ounces, 1 pound 11 ounces, and 4 pounds 14 ounces. What was the total weight of the three fish?

One day Anya kept track of the number of minutes she spent driving. She recorded 45, 10, 8, 65, 20, and 35. How many hours did Anya spend driving?

33. Practice 10

Renee attached a 6 feet 6 inch extension cord to her computer’s 3 feet 8 inch power cord. What was the total length of the cords?

Leilani wants to make 8 placemats. For each placemat she needs 18 inches of fabric. How many yards of fabric will she need for the 8 placemats?

Ghalib ran 5 kilometers. Convert the length to meters.

Estrella is 1.55 meters tall. Convert her height to centimeters.

Mount Whitney is 3,072 meters tall. Convert the height to kilometers.

34. Practice 11

June’s multivitamin contains 1,500 milligrams of calcium. Convert this to grams.

One stick of butter contains 91.6 grams of fat. Convert this to milligrams.

The maximum mass of an airmail letter is 2 kilograms. Convert this to grams.

A bottle of wine contained 750 milliliters. Convert this to liters.

Matthias is 1.8 meters tall. His son is 89 centimeters tall. How much taller is Matthias than his son?

35. Practice 12

A typical dove weighs 345 grams. A typical duck weighs 1.2 kilograms. What is the difference, in grams, of the weights of a duck and a dove?

Harry mailed 5 packages that weighed 420 grams each. What was the total weight of the packages in kilograms?

Jonas drinks 200 milliliters of water 8 times a day. How many liters of water does Jonas drink in a day?

Bill is 75 inches tall. Convert his height to centimeters.

Marcus passed a football 24 yards. Convert the pass length to meters

36. Practice 13

Each American throws out an average of 1,650 pounds of garbage per year. Convert this weight to kilograms.

A 5K run is 5 kilometers long. Convert this length to miles.

Dawn’s suitcase weighed 20 kilograms. Convert the weight to pounds.

Ozzie put 14 gallons of gas in his truck. Convert the volume to liters.

$86^{\circ}$ Fahrenheit

37. Practice 14

$104^{\circ}$ Fahrenheit

$72^{\circ}$ Fahrenheit

$0^{\circ}$ Fahrenheit

$5^{\circ}$ Celsius

$-10^{\circ}$ Celsius

38. Practice 15

$22^{\circ}$ Celsius

$43^{\circ}$ Celsius

Nutrition Julian drinks one can of soda every day. Each can of soda contains 40 grams of sugar. How many kilograms of sugar does Julian get from soda in 1 year?

Some people think that $65^{\circ}\,\mathrm{to}\,75^{\circ}$ Fahrenheit is the ideal temperature range.

ⓐ What is your ideal temperature range? Why do you think so?

ⓑ Convert your ideal temperatures from Fahrenheit to Celsius.

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