Elementary Algebra

# 1.10Systems of Measurement

Elementary Algebra1.10 Systems of Measurement

### Learning Objectives

By the end of this section, you will be able to:

• 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

### Be Prepared 1.10

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, The Properties of Real Numbers.

### 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
$Length1foot(ft.)=12inches(in.)1yard(yd.)=3feet(ft.)1mile(mi.)=5,280feet(ft.)Length1foot(ft.)=12inches(in.)1yard(yd.)=3feet(ft.)1mile(mi.)=5,280feet(ft.)$ $Volume3teaspoons(t)=1tablespoon(T)16 tablespoons(T)=1 cup(C)1 cup(C)=8 fluid ounces(fl. oz.)1 pint(pt.)=2 cups(C)1 quart(qt.)=2 pints(pt.)1 gallon(gal)=4 quarts(qt.)Volume3teaspoons(t)=1tablespoon(T)16 tablespoons(T)=1 cup(C)1 cup(C)=8 fluid ounces(fl. oz.)1 pint(pt.)=2 cups(C)1 quart(qt.)=2 pints(pt.)1 gallon(gal)=4 quarts(qt.)$
$Weight1 pound(lb.)=16 ounces(oz.)1 ton=2000 pounds(lb.)Weight1 pound(lb.)=16 ounces(oz.)1 ton=2000 pounds(lb.)$ $Time1 minute(min)=60 seconds(sec)1 hour(hr)=60 minutes(min)1 day=24 hours(hr)1 week(wk)=7 days1 year(yr)=365 daysTime1 minute(min)=60 seconds(sec)1 hour(hr)=60 minutes(min)1 day=24 hours(hr)1 week(wk)=7 days1 year(yr)=365 days$
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.

### Identity Property of Multiplication

$For any real numbera:a·1=a1·a=a1is themultiplicative identityFor any real numbera:a·1=a1·a=a1is themultiplicative identity$

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 $1foot12inches.1foot12inches.$ When we multiply by this fraction we do not change the value, but just change the units.

But $12inches1foot12inches1foot$ also equals 1. How do we decide whether to multiply by $1foot12inches1foot12inches$ or $12inches1foot?12inches1foot?$ 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 $6666$ 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.

### Example 1.140

#### How to Make Unit Conversions

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

### Try It 1.279

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

### Try It 1.280

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

### How To

#### Make Unit Conversions.

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

### Example 1.141

Ndula, an elephant at the San Diego Safari Park, weighs almost 3.2 tons. Convert her weight to pounds.

### Try It 1.281

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

### Try It 1.282

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

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.

### 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.

### Try It 1.283

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

### Try It 1.284

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

### Example 1.143

How many ounces are in 1 gallon?

### Try It 1.285

How many cups are in 1 gallon?

### Try It 1.286

How many teaspoons are in 1 cup?

### 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!

### 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?

### Try It 1.287

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?

### Try It 1.288

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?

### 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?

### Try It 1.289

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?

### Try It 1.290

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

### 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 $11001100$ of a meter, just like one cent is $11001100$ 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.

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
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.

### Example 1.146

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

### Try It 1.291

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

### Try It 1.292

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

### Example 1.147

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

### Try It 1.293

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

### Try It 1.294

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

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 $1100011000$ (or 0.001). This is the same as moving the decimal three places to the left.

### Example 1.148

Convert 350 L to kiloliters 4.1 L to milliliters.

### Try It 1.295

Convert: 725 L to kiloliters 6.3 L to milliliters

### Try It 1.296

Convert: 350 hL to liters 4.1 L to centiliters

### 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.

### 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?

### Try It 1.297

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.

### Try It 1.298

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.

### 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?

### Try It 1.299

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?

### Try It 1.300

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?

### 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.

Conversion Factors Between U.S. and Metric Systems
Length Mass Capacity
$1 in.=2.54 cm1 ft.=0.305 m1 yd.=0.914 m1 mi.=1.61 km1 m=3.28 ft.1 in.=2.54 cm1 ft.=0.305 m1 yd.=0.914 m1 mi.=1.61 km1 m=3.28 ft.$ $1 lb.=0.45 kg1 oz.=28 g1 kg=2.2 lb.1 lb.=0.45 kg1 oz.=28 g1 kg=2.2 lb.$ $1 qt.=0.95 L1 fl. oz.=30 mL1 L=1.06 qt.1 qt.=0.95 L1 fl. oz.=30 mL1 L=1.06 qt.$
Table 1.62

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

Figure 1.22 This ruler shows inches and centimeters.

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

Figure 1.23 This measuring cup shows ounces and milliliters.

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

Figure 1.24 This scale shows pounds and kilograms.

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

### 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.

### Try It 1.301

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

### Try It 1.302

How many liters are in 4 quarts of milk?

### 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?

### Try It 1.303

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

### Try It 1.304

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

### Convert between Fahrenheit and Celsius Temperatures

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

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

Figure 1.25 The diagram shows normal body temperature, along with the freezing and boiling temperatures of water in degrees Fahrenheit and degrees Celsius.

### Temperature Conversion

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

$C=59(F−32).C=59(F−32).$

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

$F=95C+32.F=95C+32.$

### Example 1.153

Convert $50°50°$ Fahrenheit into degrees Celsius.

### Try It 1.305

Convert the Fahrenheit temperature to degrees Celsius: $59°59°$ Fahrenheit.

### Try It 1.306

Convert the Fahrenheit temperature to degrees Celsius: $41°41°$ Fahrenheit.

### Example 1.154

While visiting Paris, Woody saw the temperature was $20°20°$ Celsius. Convert the temperature into degrees Fahrenheit.

### Try It 1.307

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Helsinki, Finland, was $15°15°$ Celsius.

### Try It 1.308

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Sydney, Australia, was $10°10°$ Celsius.

### Section 1.10 Exercises

#### Practice Makes Perfect

Make Unit Conversions in the U.S. System

In the following exercises, convert the units.

825.

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

826.

A floor tile is 2 feet wide. Convert the width to inches.

827.

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

828.

Carson is 45 inches tall. Convert his height to feet.

829.

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

830.

On a baseball diamond, the distance from home plate to first base is 30 yards. Convert the distance to feet.

831.

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

832.

Denver, Colorado, is 5,183 feet above sea level. Convert the height to miles.

833.

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

834.

Blue whales can weigh as much as 150 tons. Convert the weight to pounds.

835.

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

836.

At take-off, an airplane weighs 220,000 pounds. Convert the weight to tons.

837.

Rocco waited $112112$ hours for his appointment. Convert the time to seconds.

838.

Misty’s surgery lasted $214214$ hours. Convert the time to seconds.

839.

How many teaspoons are in a pint?

840.

How many tablespoons are in a gallon?

841.

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

842.

April’s dog, Beans, weighs 8 pounds. Convert his weight to ounces.

843.

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

844.

Lance needs 50 cups of water for the runners in a race. Convert the volume to gallons.

845.

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

846.

Faye is 4 feet 10 inches tall. Convert her height to inches.

847.

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

848.

Lynn’s cruise lasted 6 days and 18 hours. Convert the time to hours.

849.

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

850.

Baby Audrey weighted 6 pounds 15 ounces at birth. Convert her weight to ounces.

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

In the following exercises, solve.

851.

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?

852.

Judy bought 1 pound 6 ounces of almonds, 2 pounds 3 ounces of walnuts, and 8 ounces of cashews. How many pounds of nuts did Judy buy?

853.

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?

854.

Last year Eric went on 6 business trips. The number of days of each was 5, 2, 8, 12, 6, and 3. How many weeks did Eric spend on business trips last year?

855.

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?

856.

Fawzi’s SUV is 6 feet 4 inches tall. If he puts a 2 feet 10 inch box on top of his SUV, what is the total height of the SUV and the box?

857.

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?

858.

Mireille needs to cut 24 inches of ribbon for each of the 12 girls in her dance class. How many yards of ribbon will she need altogether?

Make Unit Conversions in the Metric System

In the following exercises, convert the units.

859.

Ghalib ran 5 kilometers. Convert the length to meters.

860.

Kitaka hiked 8 kilometers. Convert the length to meters.

861.

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

862.

The width of the wading pool is 2.45 meters. Convert the width to centimeters.

863.

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

864.

The depth of the Mariana Trench is 10,911 meters. Convert the depth to kilometers.

865.

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

866.

A typical ruby-throated hummingbird weights 3 grams. Convert this to milligrams.

867.

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

868.

One serving of gourmet ice cream has 25 grams of fat. Convert this to milligrams.

869.

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

870.

Dimitri’s daughter weighed 3.8 kilograms at birth. Convert this to grams.

871.

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

872.

A bottle of medicine contained 300 milliliters. Convert this to liters.

Use Mixed Units of Measurement in the Metric System

In the following exercises, solve.

873.

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

874.

Stavros is 1.6 meters tall. His sister is 95 centimeters tall. How much taller is Stavros than his sister?

875.

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?

876.

Concetta had a 2-kilogram bag of flour. She used 180 grams of flour to make biscotti. How many kilograms of flour are left in the bag?

877.

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

878.

One glass of orange juice provides 560 milligrams of potassium. Linda drinks one glass of orange juice every morning. How many grams of potassium does Linda get from her orange juice in 30 days?

879.

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

880.

One serving of whole grain sandwich bread provides 6 grams of protein. How many milligrams of protein are provided by 7 servings of whole grain sandwich bread?

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

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

881.

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

882.

Frankie is 42 inches tall. Convert his height to centimeters.

883.

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

884.

Connie bought 9 yards of fabric to make drapes. Convert the fabric length to meters.

885.

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

886.

An average American will throw away 90,000 pounds of trash over his or her lifetime. Convert this weight to kilograms.

887.

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

888.

Kathryn is 1.6 meters tall. Convert her height to feet.

889.

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

890.

Jackson’s backpack weighed 15 kilograms. Convert the weight to pounds.

891.

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

892.

Bernard bought 8 gallons of paint. Convert the volume to liters.

Convert between Fahrenheit and Celsius Temperatures

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

893.

$86°86°$ Fahrenheit

894.

$77°77°$ Fahrenheit

895.

$104°104°$ Fahrenheit

896.

$14°14°$ Fahrenheit

897.

$72°72°$ Fahrenheit

898.

$4°4°$ Fahrenheit

899.

$0°0°$ Fahrenheit

900.

$120°120°$ Fahrenheit

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

901.

$5°5°$ Celsius

902.

$25°25°$ Celsius

903.

$−10°−10°$ Celsius

904.

$−15°−15°$ Celsius

905.

$22°22°$ Celsius

906.

$8°8°$ Celsius

907.

$43°43°$ Celsius

908.

$16°16°$ Celsius

#### Everyday Math

909.

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?

910.

Reflectors The reflectors in each lane-marking stripe on a highway are spaced 16 yards apart. How many reflectors are needed for a one mile long lane-marking stripe?

#### Writing Exercises

911.

Some people think that $65°to75°65°to75°$ Fahrenheit is the ideal temperature range.

1. What is your ideal temperature range? Why do you think so?
2. Convert your ideal temperatures from Fahrenheit to Celsius.
912.
1. Did you grow up using the U.S. or the metric system of measurement?
2. Describe two examples in your life when you had to convert between the two systems of measurement.

#### Self Check

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?

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