Prealgebra 2e

7.5Systems of Measurement

Prealgebra 2e7.5 Systems of Measurement

Learning Objectives

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

• Make unit conversions in the U.S. system
• Use mixed units of measurement in the U.S. system
• Make unit conversions in the metric system
• Use mixed units of measurement in the metric system
• Convert between the U.S. and the metric systems of measurement
• Convert between Fahrenheit and Celsius temperatures
Be Prepared 7.13

Before you get started, take this readiness quiz.

Multiply: $4.29(1000).4.29(1000).$
If you missed this problem, review Example 5.18.

Be Prepared 7.14

Simplify: $3054.3054.$
If you missed this problem, review Example 4.20.

Be Prepared 7.15

Multiply: $715·2528.715·2528.$
If you missed this problem, review Example 4.27.

In this section we will see how to convert among different types of units, such as feet to miles or kilograms to pounds. The basic idea in all of the unit conversions will be to use a form of $1,1,$ the multiplicative identity, to change the units but not the value of a quantity.

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 United States 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, or hours.

The equivalencies among the basic units of the U.S. system of measurement are listed in Table 7.2. The table also shows, in parentheses, the common abbreviations for each measurement.

U.S. System Units
Length Volume
$11$ foot (ft) = $1212$ inches (in)
$11$ yard (yd) = $33$ feet (ft)
$11$ mile (mi) = $52805280$ feet (ft)
$33$ teaspoons (t) = $11$ tablespoon (T)
$1616$ Tablespoons (T) = $11$ cup (C)
$11$ cup (C) = $88$ fluid ounces (fl oz)
$11$ pint (pt) = $22$ cups (C)
$11$ quart (qt) = $22$ pints (pt)
$11$ gallon (gal) = $44$ quarts (qt)
Weight Time
$11$ pound (lb) = $1616$ ounces (oz)
$11$ ton = $20002000$ pounds (lb)
$11$ minute (min) = $6060$ seconds (s)
$11$ hour (h) = $6060$ minutes (min)
$11$ day = $2424$ hours (h)
$11$ week (wk) = $77$ days
$11$ year (yr) = $365365$ days
Table 7.2

In many real-life applications, we need to convert between units of measurement. We will use the identity property of multiplication to do these conversions. We’ll restate the Identity Property of Multiplication here for easy reference.

$For any real numbera,a·1=a1·a=aFor any real numbera,a·1=a1·a=a$

To use the identity property of multiplication, we write $11$ in a form that will help us convert the units. For example, suppose we want to convert inches to feet. We know that $11$ foot is equal to $1212$ inches, so we can write $11$ as the fraction $1 ft12 in.1 ft12 in.$ When we multiply by this fraction, we do not change the value but just change the units.

But $12 in1 ft12 in1 ft$ also equals $1.1.$ How do we decide whether to multiply by $1 ft12 in1 ft12 in$ or $12 in1 ft?12 in1 ft?$ We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert $6060$ inches to feet. If we choose the fraction that has inches in the denominator, we can eliminate the inches.

$60in·1 ft12in=5 ft60in·1 ft12in=5 ft$

On the other hand, if we wanted to convert $55$ feet to inches, we would choose the fraction that has feet in the denominator.

$5 ft·12 in1ft=60 in5 ft·12 in1ft=60 in$

We treat the unit words like factors and ‘divide out’ common units like we do common factors.

How To

Make unit conversions.

1. Step 1. Multiply the measurement to be converted by $1;1;$ write $11$ as a fraction relating the units given and the units needed.
2. Step 2. Multiply.
3. Step 3. Simplify the fraction, performing the indicated operations and removing the common units.

Example 7.44

Mary Anne is $6666$ inches tall. What is her height in feet?

Try It 7.87

Lexie is $3030$ inches tall. Convert her height to feet.

Try It 7.88

Rene bought a hose that is $1818$ yards long. Convert the length to feet.

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 7.45

Ndula, an elephant at the San Diego Safari Park, weighs almost $3.23.2$ tons. Convert her weight to pounds.

Figure 7.5 (credit: Guldo Da Rozze, Flickr)
Try It 7.89

Arnold’s SUV weighs about $4.34.3$ tons. Convert the weight to pounds.

Try It 7.90

A cruise ship weighs $51,00051,000$ 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 7.46

Juliet is going with her family to their summer home. She will be away for $99$ weeks. Convert the time to minutes.

Try It 7.91

The distance between Earth and the moon is about $250,000250,000$ miles. Convert this length to yards.

Try It 7.92

A team of astronauts spends $1515$ weeks in space. Convert the time to minutes.

Example 7.47

How many fluid ounces are in $11$ gallon of milk?

Figure 7.6 (credit: www.bluewaikiki.com, Flickr)
Try It 7.93

How many cups are in $11$ gallon?

Try It 7.94

How many teaspoons are in $11$ cup?

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

Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units.

Example 7.48

Charlie bought three steaks for a barbecue. Their weights were $1414$ ounces, $11$ pound $22$ ounces, and $11$ pound $66$ ounces. How many total pounds of steak did he buy?

Figure 7.7 (credit: Helen Penjam, Flickr)
Try It 7.95

Laura gave birth to triplets weighing $33$ pounds $1212$ ounces, $33$ pounds $33$ ounces, and $22$ pounds $99$ ounces. What was the total birth weight of the three babies?

Try It 7.96

Seymour cut two pieces of crown molding for his family room that were $88$ feet $77$ inches and $1212$ feet $1111$ inches. What was the total length of the molding?

Example 7.49

Anthony bought four planks of wood that were each $66$ feet $44$ inches long. If the four planks are placed end-to-end, what is the total length of the wood?

Try It 7.97

Henri wants to triple his spaghetti sauce recipe, which calls for $11$ pound $88$ ounces of ground turkey. How many pounds of ground turkey will he need?

Try It 7.98

Joellen wants to double a solution of $55$ gallons $33$ 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 $10.10.$ The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is $10001000$ meters; the prefix kilo- means thousand. One centimeter is $11001100$ of a meter, because the prefix centi- means one one-hundredth (just like one cent is $11001100$ of one dollar).

The equivalencies of measurements in the metric system are shown in Table 7.3. The common abbreviations for each measurement are given in parentheses.

Metric Measurements
Length Mass Volume/Capacity
$11$ kilometer (km) = $10001000$ m
$11$ hectometer (hm) = $100100$ m
$11$ dekameter (dam) = $1010$ m
$11$ meter (m) = $11$ m
$11$ decimeter (dm) = $0.10.1$ m
$11$ centimeter (cm) = $0.010.01$ m
$11$ millimeter (mm) = $0.0010.001$ m
$11$ kilogram (kg) = $10001000$ g
$11$ hectogram (hg) = $100100$ g
$11$ dekagram (dag) = $1010$ g
$11$ gram (g) = $11$ g
$11$ decigram (dg) = $0.10.1$ g
$11$ centigram (cg) = $0.010.01$ g
$11$ milligram (mg) = $0.0010.001$ g
$11$ kiloliter (kL) = $10001000$ L
$11$ hectoliter (hL) = $100100$ L
$11$ dekaliter (daL) = $1010$ L
$11$ liter (L) = $11$ L
$11$ deciliter (dL) = $0.10.1$ L
$11$ centiliter (cL) = $0.010.01$ L
$11$ milliliter (mL) = $0.0010.001$ L
$11$ meter = $100100$ centimeters
$11$ meter = $10001000$ millimeters
$11$ gram = $100100$ centigrams
$11$ gram = $10001000$ milligrams
$11$ liter = $100100$ centiliters
$11$ liter = $10001000$ milliliters
Table 7.3

To make conversions in the metric system, we will use the same technique we did in the U.S. 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 $5 k5 k$ or $10 k10 k$ race? The lengths 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 7.50

Nick ran a $10-kilometer10-kilometer$ race. How many meters did he run?

Figure 7.8 (credit: William Warby, Flickr)
Try It 7.99

Sandy completed her first $5-km5-km$ race. How many meters did she run?

Try It 7.100

Herman bought a rug $2.52.5$ meters in length. How many centimeters is the length?

Example 7.51

Eleanor’s newborn baby weighed $32003200$ grams. How many kilograms did the baby weigh?

Try It 7.101

Kari’s newborn baby weighed $28002800$ grams. How many kilograms did the baby weigh?

Try It 7.102

Anderson received a package that was marked $45004500$ grams. How many kilograms did this package weigh?

Since the metric system is based on multiples of ten, conversions involve multiplying by multiples of ten. In Decimal Operations, we learned how to simplify these calculations by just moving the decimal.

To multiply by $10,100,or1000,10,100,or1000,$ we move the decimal to the right $1,2,or31,2,or3$ places, respectively. To multiply by $0.1,0.01,or0.0010.1,0.01,or0.001$ we move the decimal to the left $1,2,or31,2,or3$ places respectively.

We can apply this pattern when we make measurement conversions in the metric system.

In Example 7.51, we changed $32003200$ grams to kilograms by multiplying by $11000(or0.001).11000(or0.001).$ This is the same as moving the decimal $33$ places to the left.

Example 7.52

Convert:

1. $350350$ liters to kiloliters
2. $4.14.1$ liters to milliliters.
Try It 7.103

Convert: $7.257.25$ L to kL $6.36.3$ L to mL.

Try It 7.104

Convert: $350350$ hL to L $4.14.1$ L to cL.

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 U.S. system. But it may be easier because of the relation of the units to the powers of $10.10.$ We still must make sure to add or subtract like units.

Example 7.53

Ryland is $1.61.6$ meters tall. His younger brother is $8585$ centimeters tall. How much taller is Ryland than his younger brother?

Try It 7.105

Mariella is $1.581.58$ meters tall. Her daughter is $7575$ centimeters tall. How much taller is Mariella than her daughter? Write the answer in centimeters.

Try It 7.106

The fence around Hank’s yard is $22$ meters high. Hank is $9696$ centimeters tall. How much shorter than the fence is Hank? Write the answer in meters.

Example 7.54

Dena’s recipe for lentil soup calls for $150150$ milliliters of olive oil. Dena wants to triple the recipe. How many liters of olive oil will she need?

Try It 7.107

A recipe for Alfredo sauce calls for $250250$ milliliters of milk. Renata is making pasta with Alfredo sauce for a big party and needs to multiply the recipe amounts by $8.8.$ How many liters of milk will she need?

Try It 7.108

To make one pan of baklava, Dorothea needs $400400$ grams of filo pastry. If Dorothea plans to make $66$ pans of baklava, how many kilograms of filo pastry will she need?

Convert Between U.S. and Metric Systems of Measurement

Many measurements in the United States are made in metric units. A drink may come in $2-liter2-liter$ bottles, calcium may come in $500-mg500-mg$ capsules, and we may run a $5-K5-K$ race. To work easily in both systems, we need to be able to convert between the two systems.

Table 7.4 shows some of the most common conversions.

Conversion Factors Between U.S. and Metric Systems
Length Weight Volume
$11$ in = $2.542.54$ cm
$11$ ft = $0.3050.305$ m
$11$ yd = $0.9140.914$ m
$11$ mi = $1.611.61$ km

$11$ m = $3.283.28$ ft
$11$ lb = $0.450.45$ kg
$11$ oz = $2828$ g

$11$ kg = $2.22.2$ lb
$11$ qt = $0.950.95$ L
$11$ fl oz = $3030$ mL

$11$ L = $1.061.06$ qt
Table 7.4

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

Example 7.55

Lee’s water bottle holds $500500$ mL of water. How many fluid ounces are in the bottle? Round to the nearest tenth of an ounce.

Try It 7.109

How many quarts of soda are in a $2-liter2-liter$ bottle?

Try It 7.110

How many liters are in $44$ quarts of milk?

The conversion factors in Table 7.4 are not exact, but the approximations they give are close enough for everyday purposes. In Example 7.55, we rounded the number of fluid ounces to the nearest tenth.

Example 7.56

Soleil lives in Minnesota but often travels in Canada for work. While driving on a Canadian highway, she passes a sign that says the next rest stop is in $100100$ kilometers. How many miles until the next rest stop? Round your answer to the nearest mile.

Try It 7.111

The height of Mount Kilimanjaro is $5,8955,895$ meters. Convert the height to feet. Round to the nearest foot.

Try It 7.112

The flight distance from New York City to London is $5,5865,586$ kilometers. Convert the distance to miles. Round to the nearest mile.

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 7.9 shows the relationship between the two systems.

Figure 7.9 A temperature of $37°C37°C$ is equivalent to $98.6°F.98.6°F.$

If we know the temperature in one system, we can use a formula to convert it to the other system.

Temperature Conversion

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

$C=59(F−32)C=59(F−32)$

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

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

Example 7.57

Convert $50°F50°F$ into degrees Celsius.

Try It 7.113

Convert the Fahrenheit temperatures to degrees Celsius: $59°F.59°F.$

Try It 7.114

Convert the Fahrenheit temperatures to degrees Celsius: $41°F.41°F.$

Example 7.58

The weather forecast for Paris predicts a high of $20°C.20°C.$ Convert the temperature into degrees Fahrenheit.

Try It 7.115

Convert the Celsius temperatures to degrees Fahrenheit:

The temperature in Helsinki, Finland was $15°C.15°C.$

Try It 7.116

Convert the Celsius temperatures to degrees Fahrenheit:

The temperature in Sydney, Australia was $10°C.10°C.$

Section 7.5 Exercises

Practice Makes Perfect

Make Unit Conversions in the U.S. System

In the following exercises, convert the units.

214.

A park bench is $66$ feet long. Convert the length to inches.

215.

A floor tile is $22$ feet wide. Convert the width to inches.

216.

A ribbon is $1818$ inches long. Convert the length to feet.

217.

Carson is $4545$ inches tall. Convert his height to feet.

218.

Jon is $66$ feet $44$ inches tall. Convert his height to inches.

219.

Faye is $44$ feet $1010$ inches tall. Convert her height to inches.

220.

A football field is $160160$ feet wide. Convert the width to yards.

221.

On a baseball diamond, the distance from home plate to first base is $3030$ yards. Convert the distance to feet.

222.

Ulises lives $1.51.5$ miles from school. Convert the distance to feet.

223.

Denver, Colorado, is $5,1835,183$ feet above sea level. Convert the height to miles.

224.

A killer whale weighs $4.64.6$ tons. Convert the weight to pounds.

225.

Blue whales can weigh as much as $150150$ tons. Convert the weight to pounds.

226.

An empty bus weighs $35,00035,000$ pounds. Convert the weight to tons.

227.

At take-off, an airplane weighs $220,000220,000$ pounds. Convert the weight to tons.

228.

The voyage of the Mayflower took $22$ months and $55$ days. Convert the time to days (30 days = 1 month).

229.

Lynn’s cruise lasted $66$ days and $1818$ hours. Convert the time to hours.

230.

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

231.

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

232.

How many teaspoons are in a pint?

233.

How many tablespoons are in a gallon?

234.

JJ’s cat, Posy, weighs $1414$ pounds. Convert her weight to ounces.

235.

April’s dog, Beans, weighs $88$ pounds. Convert his weight to ounces.

236.

Baby Preston weighed $77$ pounds $33$ ounces at birth. Convert his weight to ounces.

237.

Baby Audrey weighed $66$ pounds $1515$ ounces at birth. Convert her weight to ounces.

238.

Crista will serve $2020$ cups of juice at her son’s party. Convert the volume to gallons.

239.

Lance needs $500500$ cups of water for the runners in a race. Convert the volume to gallons.

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

In the following exercises, solve and write your answer in mixed units.

240.

Eli caught three fish. The weights of the fish were $22$ pounds $44$ ounces, $11$ pound $1111$ ounces, and $44$ pounds $1414$ ounces. What was the total weight of the three fish?

241.

Judy bought $11$ pound $66$ ounces of almonds, $22$ pounds $33$ ounces of walnuts, and $88$ ounces of cashews. What was the total weight of the nuts?

242.

One day Anya kept track of the number of minutes she spent driving. She recorded trips of $45,10,8,65,20,and 35 minutes.45,10,8,65,20,and 35 minutes.$ How much time (in hours and minutes) did Anya spend driving?

243.

Last year Eric went on $66$ business trips. The number of days of each was $5,2,8,12,6,and 3.5,2,8,12,6,and 3.$ How much time (in weeks and days) did Eric spend on business trips last year?

244.

Renee attached a $6-foot-6-inch6-foot-6-inch$ extension cord to her computer’s $3-foot-8-inch3-foot-8-inch$ power cord. What was the total length of the cords?

245.

Fawzi’s SUV is $66$ feet $44$ inches tall. If he puts a $2-foot-10-inch2-foot-10-inch$ box on top of his SUV, what is the total height of the SUV and the box?

246.

Leilani wants to make $88$ placemats. For each placemat she needs $1818$ inches of fabric. How many yards of fabric will she need for the $88$ placemats?

247.

Mireille needs to cut $2424$ inches of ribbon for each of the $1212$ 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.

248.

Ghalib ran $55$ kilometers. Convert the length to meters.

249.

Kitaka hiked $88$ kilometers. Convert the length to meters.

250.

Estrella is $1.551.55$ meters tall. Convert her height to centimeters.

251.

The width of the wading pool is $2.452.45$ meters. Convert the width to centimeters.

252.

Mount Whitney is $3,0723,072$ meters tall. Convert the height to kilometers.

253.

The depth of the Mariana Trench is $10,91110,911$ meters. Convert the depth to kilometers.

254.

June’s multivitamin contains $1,5001,500$ milligrams of calcium. Convert this to grams.

255.

A typical ruby-throated hummingbird weights $33$ grams. Convert this to milligrams.

256.

One stick of butter contains $91.691.6$ grams of fat. Convert this to milligrams.

257.

One serving of gourmet ice cream has $2525$ grams of fat. Convert this to milligrams.

258.

The maximum mass of an airmail letter is $22$ kilograms. Convert this to grams.

259.

Dimitri’s daughter weighed $3.83.8$ kilograms at birth. Convert this to grams.

260.

A bottle of wine contained $750750$ milliliters. Convert this to liters.

261.

A bottle of medicine contained $300300$ milliliters. Convert this to liters.

Use Mixed Units of Measurement in the Metric System

In the following exercises, solve and write your answer in mixed units.

262.

Matthias is $1.81.8$ meters tall. His son is $8989$ centimeters tall. How much taller, in centimeters, is Matthias than his son?

263.

Stavros is $1.61.6$ meters tall. His sister is $9595$ centimeters tall. How much taller, in centimeters, is Stavros than his sister?

264.

A typical dove weighs $345345$ grams. A typical duck weighs $1.21.2$ kilograms. What is the difference, in grams, of the weights of a duck and a dove?

265.

Concetta had a $2-kilogram2-kilogram$ bag of flour. She used $180180$ grams of flour to make biscotti. How many kilograms of flour are left in the bag?

266.

Harry mailed $55$ packages that weighed $420420$ grams each. What was the total weight of the packages in kilograms?

267.

One glass of orange juice provides $560560$ 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 $3030$ days?

268.

Jonas drinks $200200$ milliliters of water $88$ times a day. How many liters of water does Jonas drink in a day?

269.

One serving of whole grain sandwich bread provides $66$ grams of protein. How many milligrams of protein are provided by $77$ servings of whole grain sandwich bread?

Convert Between U.S. and Metric Systems

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

270.

Bill is $7575$ inches tall. Convert his height to centimeters.

271.

Frankie is $4242$ inches tall. Convert his height to centimeters.

272.

Marcus passed a football $2424$ yards. Convert the pass length to meters.

273.

Connie bought $99$ yards of fabric to make drapes. Convert the fabric length to meters.

274.

Each American throws out an average of $1,6501,650$ pounds of garbage per year. Convert this weight to kilograms (2.20 pounds = 1 kilogram).

275.

An average American will throw away $90,00090,000$ pounds of trash over his or her lifetime. Convert this weight to kilograms (2.20 pounds = 1 kilogram).

276.

A $5K5K$ run is $55$ kilometers long. Convert this length to miles.

277.

Kathryn is $1.61.6$ meters tall. Convert her height to feet.

278.

Dawn’s suitcase weighed $2020$ kilograms. Convert the weight to pounds.

279.

Jackson’s backpack weighs $1515$ kilograms. Convert the weight to pounds.

280.

Ozzie put $1414$ gallons of gas in his truck. Convert the volume to liters.

281.

Bernard bought $88$ gallons of paint. Convert the volume to liters.

Convert between Fahrenheit and Celsius

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

282.

$86°F86°F$

283.

$77°F77°F$

284.

$104°F104°F$

285.

$14°F14°F$

286.

$72°F72°F$

287.

$4°F4°F$

288.

$0°F0°F$

289.

$120°F120°F$

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

290.

$5°C5°C$

291.

$25°C25°C$

292.

$−10°C−10°C$

293.

$−15°C−15°C$

294.

$22°C22°C$

295.

$8°C8°C$

296.

$43°C43°C$

297.

$16°C16°C$

Everyday Math

298.

Nutrition Julian drinks one can of soda every day. Each can of soda contains $4040$ grams of sugar. How many kilograms of sugar does Julian get from soda in $11$ year?

299.

Reflectors The reflectors in each lane-marking stripe on a highway are spaced $1616$ yards apart. How many reflectors are needed for a one-mile-long stretch of highway?

Writing Exercises

300.

Some people think that $65°65°$ to $75°75°$ 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.

301.

Did you grow up using the U.S. customary or the metric system of measurement? Describe two examples in your life when you had to convert between systems of measurement. Which system do you think is easier to use? Explain.

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?