Systems of Measurement

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 $1,$ 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 1. The table also shows, in parentheses, the common abbreviations for each measurement.

U.S. System Units
Length	Volume
$1$ foot (ft) = $12$ inches (in) $1$ yard (yd) = $3$ feet (ft) $1$ mile (mi) = $5280$ feet (ft)	$3$ teaspoons (t) = $1$ tablespoon (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)
Weight	Time
$1$ pound (lb) = $16$ ounces (oz) $1$ ton = $2000$ pounds (lb)	$1$ minute (min) = $60$ seconds (s) $1$ hour (h) = $60$ minutes (min) $1$ day = $24$ hours (h) $1$ week (wk) = $7$ days $1$ year (yr) = $365$ days

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 number a, a \cdot 1 = a 1 \cdot a = a

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 convert inches to feet. We know that $1$ foot is equal to $12$ inches, so we can write $1$ as the fraction $\frac{1 ft}{12 in} .$ When we multiply by this fraction, we do not change the value but just change the units.

But $\frac{12 in}{1 ft}$ also equals $1 .$ How do we decide whether to multiply by $\frac{1 ft}{12 in}$ or $\frac{12 in}{1 ft} ?$ We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert $60$ inches to feet. If we choose the fraction that has inches in the denominator, we can eliminate the inches.

60 in \cdot \frac{1 ft}{12 in} = 5 ft

On the other hand, if we wanted to convert $5$ feet to inches, we would choose the fraction that has feet in the denominator.

5 ft \cdot \frac{12 in}{1 ft} = 60 in

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

Mary Anne is $66$ inches tall. What is her height in feet?

Solution

This table demonstrates the step-by-step process and calculations for converting 66 inches into feet.
Convert 66 inches into feet.
Multiply the measurement to be converted by 1.	$66$ inches $\cdot 1$
Write 1 as a fraction relating the units given and the units needed.	$66 inches \cdot \frac{1 foot}{12 inches}$
Multiply.	$\frac{66 inches \cdot 1 foot}{12 inches}$
Simplify the fraction.	$\frac{66 inches \cdot 1 foot}{12 inches}$
	$\frac{66 feet}{12}$
	$5.5 feet$

Notice that the when we simplified the fraction, we first divided out the inches.

Mary Anne is $5.5$ feet tall.

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.

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

A photograph of an adult elephant. — (credit: Guldo Da Rozze, Flickr)

Solution

We will convert $3.2$ tons into pounds, using the equivalencies in Table 1. We will use the Identity Property of Multiplication, writing $1$ as the fraction $\frac{2000 pounds}{1 ton} .$

This table demonstrates the step-by-step conversion of 3.2 tons to pounds, culminating in a real-world example.
	$3.2 tons$
Multiply the measurement to be converted by 1.	$3.2 tons \cdot 1$
Write 1 as a fraction relating tons and pounds.	$3.2 tons \cdot \frac{2000 lbs}{1 ton}$
Simplify.	$\frac{3.2 tons \cdot 2000 lbs}{1 ton}$
Multiply.	$6400 lbs$
	Ndula weighs almost 6,400 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.

Juliet is going with her family to their summer home. She will be away for $9$ weeks. Convert the time to minutes.

Solution

To convert weeks into minutes, we will convert weeks to days, days to hours, and then hours to minutes. To do this, we will multiply by conversion factors of

1 .

	$9 weeks$
Write 1 as $\frac{7 days}{1 week}, \frac{24 hours}{1 day}, \frac{60 minutes}{1 hour}$ .
Cancel common units.
Multiply.	$\frac{9 \cdot 7 \cdot 24 \cdot 60 min}{1 \cdot 1 \cdot 1 \cdot 1} = 90, 720 min$
	Juliet will be away for 90,720 minutes.

How many fluid ounces are in $1$ gallon of milk?

A photograph of a milk display in a grocery store. — (credit: www.bluewaikiki.com, Flickr)

Solution

Use conversion factors to get the right units: convert gallons to quarts, quarts to pints, pints to cups, and cups to fluid ounces.

	1 gallon
Multiply the measurement to be converted by 1.	$\frac{1 gal}{1} \cdot \frac{4 qt}{1 gal} \cdot \frac{2 pt}{1 qt} \cdot \frac{2 C}{1 pt} \cdot \frac{8 fl oz}{1 C}$
Simplify.	$\frac{1 gal}{1} \cdot \frac{4 qt}{1 gal} \cdot \frac{2 pt}{1 qt} \cdot \frac{2 C}{1 pt} \cdot \frac{8 fl oz}{1 C}$
Multiply.	$\frac{1 \cdot 4 \cdot 2 \cdot 2 \cdot 8 fl oz}{1 \cdot 1 \cdot 1 \cdot 1 \cdot 1}$
Simplify.	128 fluid ounces
	There are 128 fluid ounces in a gallon.

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.

Charlie 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?

A photograph of meat being cooked on a charcoal grill. — (credit: Helen Penjam, Flickr)

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.
Add the pounds.	2 pounds + 1 pound, 6 ounces 3 pounds, 6 ounces
	Charlie bought 3 pounds 6 ounces of steak.

Anthony bought four planks of wood that were each $6$ feet $4$ inches long. If the four planks are placed end-to-end, what is the total length of the wood?

The image shows 4 planks of wood placed end-to-end horizontally. Each plank is labeled 6 feet 4 inches. A line starts at the left of the first plank and runs horizontally to the right of the fourth plank. The line is labeled with the letter l to represent length.

Solution

We will multiply the length of one plank by

4

to find the total length.

Multiply the inches and then the feet.
Convert 16 inches to feet.	24 feet + 1 foot 4 inches
Add the feet.	25 feet 4 inches
	Anthony bought 25 feet 4 inches of wood.

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 $1000$ meters; the prefix kilo- means thousand. One centimeter is $\frac{1}{100}$ of a meter, because the prefix centi- means one one-hundredth (just like one cent is $\frac{1}{100}$ of one dollar).

The equivalencies of measurements in the metric system are shown in Table 8. The common abbreviations for each measurement are given in parentheses.

Metric Measurements
Length	Mass	Volume/Capacity
$1$ kilometer (km) = $1000$ 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) = $1000$ 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) = $1000$ 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 = $1000$ millimeters	$1$ gram = $100$ centigrams $1$ gram = $1000$ milligrams	$1$ liter = $100$ centiliters $1$ liter = $1000$ milliliters

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 $5 k$ or $10 k$ 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.

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

A photograph of 8 male runners in a track race. — (credit: William Warby, Flickr)

Solution

We will convert kilometers to meters using the Identity Property of Multiplication and the equivalencies in Table 8.

	10 kilometers
Multiply the measurement to be converted by 1.
Write 1 as a fraction relating kilometers and meters.
Simplify.
Multiply.	10,000 m
	Nick ran 10,000 meters.

Eleanor’s newborn baby weighed $3200$ grams. How many kilograms did the baby weigh?

Solution

We will convert grams to kilograms.


Multiply the measurement to be converted by 1.
Write 1 as a fraction relating kilograms and grams.
Simplify.
Multiply.	$A fraction shows 3200 kilograms divided by 1000, illustrating a mathematical expression or unit conversion.$
Divide.	3.2 kilograms
	The baby weighed $3.2$ kilograms.

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 $10, 100, or 1000,$ we move the decimal to the right $1, 2, or 3$ places, respectively. To multiply by $0.1, 0.01, or 0.001$ we move the decimal to the left $1, 2, or 3$ places respectively.

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

In Example 8, we changed $3200$ grams to kilograms by multiplying by $\frac{1}{1000} (or 0.001) .$ This is the same as moving the decimal $3$ places to the left.

Multiplying 3200 by 1 over 1000 gives 3.2. Notice that the answer, 3.2, is similar to the original value, 3200, just with the decimal moved three places to the left.

Convert:

ⓐ $350$ liters to kiloliters
ⓑ $4.1$ liters to milliliters.

Solution

ⓐ We will convert liters to kiloliters. In Table 8, we see that

1 kiloliter = 1000 liters .

	350 L
Multiply by 1, writing 1 as a fraction relating liters to kiloliters.	$A mathematical expression showing the conversion of 350 liters to kiloliters: 350 L multiplied by the fraction 1 kL over 1000 L.$
Simplify.
Move the decimal 3 units to the left.
	0.35 kL

ⓑ We will convert liters to milliliters. In Table 8, we see that

1 liter = 1000 milliliters.

	4.1 L
Multiply by 1, writing 1 as a fraction relating milliliters to liters.
Simplify.	$An image displays the conversion of 4.1 liters to milliliters. The expression shows 4.1 L multiplied by a fraction (1000 mL / 1 L), with the 'L' unit diagonally crossed out in both the numerator and denominator.$
Move the decimal 3 units to the left.
	4100 mL

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 $10 .$ We still must 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 will subtract the lengths in meters. Convert $85$ centimeters to meters by moving the decimal $2$ places to the left; $85$ cm is the same as $0.85$ m.

Now that both measurements are in meters, subtract to find out how much taller Ryland is than his brother.

\begin{matrix} 1.60 m \\ \underset{_______}{−0.85 m} \\ 0.75 m \end{matrix}

Ryland is $0.75$ meters taller than his brother.

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 milliliters then convert to liters.

This table demonstrates the step-by-step calculation of triple 150 mL, including unit conversion to liters, and presents the final solution.
	Triple 150 mL
Translate to algebra.	$3 \cdot 150 mL$
Multiply.	$450 mL$
Convert to liters.	$450 mL \cdot \frac{0.001 L}{1 mL}$
Simplify.	$0.45 L$
	Dena needs 0.45 liter of olive oil.

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 $2-liter$ bottles, calcium may come in $500-mg$ capsules, and we may run a $5-K$ race. To work easily in both systems, we need to be able to convert between the two systems.

Table 14 shows some of the most common conversions.

Conversion Factors Between U.S. and Metric Systems
Length	Weight	Volume
$1$ in = $2.54$ cm $1$ ft = $0.305$ m $1$ yd = $0.914$ m $1$ mi = $1.61$ km $1$ m = $3.28$ ft	$1$ lb = $0.45$ kg $1$ oz = $28$ g $1$ kg = $2.2$ lb	$1$ qt = $0.95$ L $1$ fl oz = $30$ mL $1$ L = $1.06$ qt

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

Lee’s water bottle holds $500$ mL of water. How many fluid ounces are in the bottle? Round to the nearest tenth of an ounce.

Solution

This table demonstrates the step-by-step conversion of 500 mL to fluid ounces, resulting in 16.7 fl. oz.
	500 mL
Multiply by a unit conversion factor relating mL and ounces.	$500 mL \cdot \frac{1 fl oz}{30 mL}$
Simplify.	$\frac{500 fl oz}{30}$
Divide.	$16.7 fl. oz.$
	The water bottle holds 16.7 fluid ounces.

The conversion factors in Table 14 are not exact, but the approximations they give are close enough for everyday purposes. In Example 12, we rounded the number of fluid ounces to the nearest tenth.

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 $100$ kilometers. How many miles until the next rest stop? Round your answer to the nearest mile.

Solution

This table outlines the step-by-step conversion of 100 kilometers to approximately 62 miles, detailing the calculation process.
	100 kilometers
Multiply by a unit conversion factor relating kilometers and miles.	$100 kilometers \cdot \frac{1 mile}{1.61 kilometers}$ $100 \cdot \frac{1 mi}{1.61 km}$
Simplify.	$\frac{100 mi}{1.61}$
Divide.	62 mi
	It is about 62 miles to the next rest stop.

Convert Between Fahrenheit and Celsius Temperatures

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

The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written $°F .$ The metric system uses degrees Celsius, written $°C .$ Figure 5 shows the relationship between the two systems.

On the left side of the figure is a thermometer marked in degrees Celsius. The bottom of the thermometer begins with negative 20 degrees Celsius and ranges up to 100 degrees Celsius. There are tick marks on the thermometer every 5 degrees with every 10 degrees labeled. On the right side is a thermometer marked in degrees Fahrenheit. The bottom of the thermometer begins with negative 10 degrees Fahrenheit and ranges up to 212 degrees Fahrenheit. There are tick marks on the thermometer every 2 degrees with every 10 degrees labeled. Between the thermometers there is an arrow pointing on the left to 0 degrees Celsius and on the right to 32 degrees Fahrenheit. This is the temperature at which water freezes. Another arrow points on the left to 37 degrees Celsius and on the right to 98.6 degrees Fahrenheit. This is normal body temperature. A third arrow points on the left to 100 degrees Celsius and on the right to 212 degrees Fahrenheit. This is the temperature at which water boils. — A temperature of $37 °C$ is equivalent to $98.6 °F .$

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

Convert $50 °F$ into degrees Celsius.

Solution

We will substitute

50 °F

into the formula to find

C .

Use the formula for converting °F to °C	$C = \frac{5}{9} (F - 32)$

Simplify in parentheses.	$C = \frac{5}{9} (18)$
Multiply.	$C = 10$
	A temperature of 50°F is equivalent to 10°C.

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

Solution

We will substitute

20 °C

into the formula to find

F .

Use the formula for converting °F to °C	$F = \frac{9}{5} C + 32$

Multiply.	$F = 36 + 32$
Add.	$F = 68$
	So 20°C is equivalent to 68°F.

Section Exercises

Practice Makes Perfect

Make Unit Conversions in the U.S. System

In the following exercises, convert the units.

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

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

Solution

24 inches

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

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

Solution

3.75 feet

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

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

Solution

58 inches

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

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

Solution

90 feet

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

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

Solution

0.98 miles

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

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

Solution

300,000 pounds

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

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

Solution

110 tons

The voyage of the Mayflower took $2$ months and $5$ days. Convert the time to days (30 days = 1 month).

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

Solution

162 hours

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

Misty’s surgery lasted $2 \frac{1}{4}$ hours. Convert the time to seconds.

Solution

8100 seconds

How many teaspoons are in a pint?

How many tablespoons are in a gallon?

Solution

256 tablespoons

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

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

Solution

128 ounces

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

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

Solution

111 ounces

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

Lance needs $500$ cups of water for the runners in a race. Convert the volume to gallons.

Solution

31.25 gallons

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

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

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?

Judy bought $1$ pound $6$ ounces of almonds, $2$ pounds $3$ ounces of walnuts, and $8$ ounces of cashews. What was the total weight of the nuts?

Solution

4 lbs. 1 oz.

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

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

Solution

5 weeks and 1 day

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

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

Solution

9 ft 2 in

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?

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?

Solution

8 yards

Make Unit Conversions in the Metric System

In the following exercises, convert the units.

Ghalib ran $5$ kilometers. Convert the length to meters.

Kitaka hiked $8$ kilometers. Convert the length to meters.

Solution

8000 meters

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

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

Solution

245 centimeters

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

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

Solution

10.911 kilometers

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

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

Solution

3000 milligrams

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

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

Solution

25,000 milligrams

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

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

Solution

3800 grams

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

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

Solution

0.3 liters

Use Mixed Units of Measurement in the Metric System

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

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

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

Solution

65 centimeters

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?

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?

Solution

1.82 kilograms

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

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?

Solution

16.8 grams

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

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?

Solution

42,000 milligrams

Convert Between U.S. and Metric Systems

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

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

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

Solution

106.7 centimeters

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

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

Solution

8.2 meters

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

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

Solution

40,900 kilograms

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

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

Solution

5.2 feet

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

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

Solution

33 pounds

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

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

Solution

30.2 liters

Convert between Fahrenheit and Celsius

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

$86 °F$

$77 °F$

Solution

25°C

$104 °F$

$14 °F$

Solution

−10°C

$72 °F$

$4 °F$

Solution

−15.6°C

$0 °F$

$120 °F$

Solution

48.9°C

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

$5 °C$

$25 °C$

Solution

77°F

$−10 °C$

$−15 °C$

Solution

5°F

$22 °C$

$8 °C$

Solution

46.4°F

$43 °C$

$16 °C$

Solution

60.8°F

Everyday Math

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?

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 stretch of highway?

Solution

110 reflectors

Writing Exercises

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

ⓐ 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.

A self-assessment checklist for students to evaluate their proficiency in unit conversions and measurements across U.S. and metric systems, including temperature conversions.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?

Chapter Review Exercises

Rational and Irrational Numbers

In the following exercises, write as the ratio of two integers.

$6$

$−5$

Solution

$\frac{- 5}{1}$

$2.9$

$1.8$

Solution

$\frac{18}{10}$

In the following exercises, determine which of the numbers is rational.

$0.42, 0. \overset{–}{3}, 2.56813…$

$0.75319…, 0. \overset{—}{16}, 1.95$

Solution

$0. \overset{—}{16}, 1.95$

In the following exercises, identify whether each given number is rational or irrational.

ⓐ $\sqrt{49}$ ⓑ $\sqrt{55}$

ⓐ $\sqrt{72}$ ⓑ $\sqrt{64}$

Solution

ⓐ irrational
ⓑ rational

In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

$−9, 0, 0.361...., \frac{8}{9}, \sqrt{16}, 9$

$−5, - 2 \frac{1}{4}, - \sqrt{4}, 0. \overset{—}{25}, \frac{13}{5}, 4$

Solution

ⓐ $4$
ⓑ $- 5, - \sqrt{4}, 4$
ⓒ $- 5, - 2 \frac{1}{4}, - \sqrt{4}, 0. \overset{—}{25}, \frac{13}{5}, 4$
ⓓ none
ⓔ $- 5, - 2 \frac{1}{4}, - \sqrt{4}, 0. \overset{—}{25}, \frac{13}{5}, 4$

Commutative and Associative Properties

In the following exercises, use the commutative property to rewrite the given expression.

$6 + 4 =____$

$−14 \cdot 5 =____$

Solution

−14·5 = 5(−14)

$3 n =____$

$a + 8 =____$

Solution

a + 8 = 8 + a

In the following exercises, use the associative property to rewrite the given expression.

$(13 \cdot 5) \cdot 2 =_____$

$(22 + 7) + 3 =_____$

Solution

(22 + 7) + 3 = 22 + (7 + 3)

$(4 + 9 x) + x =_____$

$\frac{1}{2} (22 y) =_____$

Solution

$\frac{1}{2} (22 y) = (\frac{1}{2} \cdot 22) y$

In the following exercises, evaluate each expression for the given value.

If $y = \frac{11}{12},$ evaluate:
ⓐ $y + 0.7 + (- y)$
ⓑ $y + (- y) + 0.7$

If $z = - \frac{5}{3},$ evaluate:
ⓐ $z + 5.39 + (- z)$
ⓑ $z + (- z) + 5.39$

Solution

ⓐ 5.39
ⓑ 5.39

If $k = 65,$ evaluate:
ⓐ $\frac{4}{9} (\frac{9}{4} k)$
ⓑ $(\frac{4}{9} \cdot \frac{9}{4}) k$

If $m = −13,$ evaluate:
ⓐ $- \frac{2}{5} (\frac{5}{2} m)$
ⓑ $(- \frac{2}{5} \cdot \frac{5}{2}) m$

Solution

ⓐ 13
ⓑ 13

In the following exercises, simplify using the commutative and associative properties.

$6 y + 37 + (−6 y)$

$\frac{1}{4} + \frac{11}{15} + (- \frac{1}{4})$

Solution

$\frac{11}{15}$

$\frac{14}{11} \cdot \frac{35}{9} \cdot \frac{11}{14}$

$−18 \cdot 15 \cdot \frac{2}{9}$

Solution

−60

$(\frac{7}{12} + \frac{4}{5}) + \frac{1}{5}$

$(3.98 d + 0.75 d) + 1.25 d$

Solution

5.98 d

$−12 (4 m)$

$30 (\frac{5}{6} q)$

Solution

25 q

$11 x + 8 y + 16 x + 15 y$

$52 m + (−20 n) + (−18 m) + (−5 n)$

Solution

34 m + (−25 n)

Distributive Property

In the following exercises, simplify using the distributive property.

$7 (x + 9)$

$9 (u - 4)$

Solution

9y − 36

$−3 (6 m - 1)$

$−8 (−7 a - 12)$

Solution

56a + 96

$\frac{1}{3} (15 n - 6)$

$(y + 10) \cdot p$

Solution

yp + 10p

$(a - 4) - (6 a + 9)$

$4 (x + 3) - 8 (x - 7)$

Solution

−4x + 68

In the following exercises, evaluate using the distributive property.

If $u = 2,$ evaluate
ⓐ $3 (8 u + 9) and$
ⓑ $3 \cdot 8 u + 3 \cdot 9$ to show that $3 (8 u + 9) = 3 \cdot 8 u + 3 \cdot 9$

If $n = \frac{7}{8},$ evaluate
ⓐ $8 (n + \frac{1}{4})$ and
ⓑ $8 \cdot n + 8 \cdot \frac{1}{4}$ to show that $8 (n + \frac{1}{4}) = 8 \cdot n + 8 \cdot \frac{1}{4}$

Solution

ⓐ 9
ⓑ 9

If $d = 14,$ evaluate
ⓐ $−100 (0.1 d + 0.35)$ and
ⓑ $−100 \cdot (0.1 d) + (−100) (0.35)$ to show that $−100 (0.1 d + 0.35) = −100 \cdot (0.1 d) + (−100) (0.35)$

If $y = −18,$ evaluate
ⓐ $- (y - 18)$ and
ⓑ $- y + 18$ to show that $- (y - 18) = - y + 18$

Solution

ⓐ 36
ⓑ 36

Properties of Identities, Inverses, and Zero

In the following exercises, identify whether each example is using the identity property of addition or multiplication.

$−35 (1) = −35$

$29 + 0 = 29$

Solution

identity property of addition

$(6 x + 0) + 4 x = 6 x + 4 x$

$9 \cdot 1 + (- 3) = 9 + (- 3)$

Solution

identity property of multiplication

In the following exercises, find the additive inverse.

$−32$

$19.4$

Solution

−19.4

$\frac{3}{5}$

$- \frac{7}{15}$

Solution

$\frac{7}{15}$

In the following exercises, find the multiplicative inverse.

$\frac{9}{2}$

$−5$

Solution

$- \frac{1}{5}$

$\frac{1}{10}$

$- \frac{4}{9}$

Solution

$- \frac{9}{4}$

In the following exercises, simplify.

$83 \cdot 0$

$\frac{0}{9}$

Solution

$\frac{5}{0}$

$0 \div \frac{2}{3}$

Solution

$43 + 39 + (−43)$

$(n + 6.75) + 0.25$

Solution

n + 7

$\frac{5}{13} \cdot 57 \cdot \frac{13}{5}$

$\frac{1}{6} \cdot 17 \cdot 12$

Solution

$\frac{2}{3} \cdot 28 \cdot \frac{3}{7}$

$9 (6 x - 11) + 15$

Solution

54x − 84

Systems of Measurement

In the following exercises, convert between U.S. units. Round to the nearest tenth.

A floral arbor is $7$ feet tall. Convert the height to inches.

A picture frame is $42$ inches wide. Convert the width to feet.

Solution

3.5 feet

Kelly is $5$ feet $4$ inches tall. Convert her height to inches.

A playground is $45$ feet wide. Convert the width to yards.

Solution

15 yards

The height of Mount Shasta is $14,179$ feet. Convert the height to miles.

Shamu weighs $4.5$ tons. Convert the weight to pounds.

Solution

9000 pounds

The play lasted $1 \frac{3}{4}$ hours. Convert the time to minutes.

How many tablespoons are in a quart?

Solution

64 tablespoons

Naomi’s baby weighed $5$ pounds $14$ ounces at birth. Convert the weight to ounces.

Trinh needs $30$ cups of paint for her class art project. Convert the volume to gallons.

Solution

1.9 gallons

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

John caught $4$ lobsters. The weights of the lobsters were $1$ pound $9$ ounces, $1$ pound $12$ ounces, $4$ pounds $2$ ounces, and $2$ pounds $15$ ounces. What was the total weight of the lobsters?

Every day last week, Pedro recorded the amount of time he spent reading. He read for $50, 25, 83, 45, 32, 60, and 135$ minutes. How much time, in hours and minutes, did Pedro spend reading?

Solution

7 hours 10 minutes

Fouad is $6$ feet $2$ inches tall. If he stands on a rung of a ladder $8$ feet $10$ inches high, how high off the ground is the top of Fouad’s head?

Dalila wants to make pillow covers. Each cover takes $30$ inches of fabric. How many yards and inches of fabric does she need for $4$ pillow covers?

Solution

3 yards, 12 inches

In the following exercises, convert between metric units.

Donna is $1.7$ meters tall. Convert her height to centimeters.

Mount Everest is $8,850$ meters tall. Convert the height to kilometers.

Solution

8.85 kilometers

One cup of yogurt contains $488$ milligrams of calcium. Convert this to grams.

One cup of yogurt contains $13$ grams of protein. Convert this to milligrams.

Solution

13,000 milligrams

Sergio weighed $2.9$ kilograms at birth. Convert this to grams.

A bottle of water contained $650$ milliliters. Convert this to liters.

Solution

0.65 liters

In the following exercises, solve.

Minh is $2$ meters tall. His daughter is $88$ centimeters tall. How much taller, in meters, is Minh than his daughter?

Selma had a $1-liter$ bottle of water. If she drank $145$ milliliters, how much water, in milliliters, was left in the bottle?

Solution

855 milliliters

One serving of cranberry juice contains $30$ grams of sugar. How many kilograms of sugar are in $30$ servings of cranberry juice?

One ounce of tofu provides $2$ grams of protein. How many milligrams of protein are provided by $5$ ounces of tofu?

Solution

10,000 milligrams

In the following exercises, convert between U.S. and metric units. Round to the nearest tenth.

Majid is $69$ inches tall. Convert his height to centimeters.

A college basketball court is $84$ feet long. Convert this length to meters.

Solution

25.6 meters

Caroline walked $2.5$ kilometers. Convert this length to miles.

Lucas weighs $78$ kilograms. Convert his weight to pounds.

Solution

171.6 pounds

Steve’s car holds $55$ liters of gas. Convert this to gallons.

A box of books weighs $25$ pounds. Convert this weight to kilograms.

Solution

11.4 kilograms

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

$95 °F$

$23 °F$

Solution

−5°C

$20 °F$

$64 °F$

Solution

17.8°C

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

$30 °C$

$−5 °C$

Solution

23°F

$−12 °C$

$24 °C$

Solution

75.2°F

Chapter Practice Test

For the numbers $0.18349…, 0. \overset{–}{2}, 1.67,$ list the ⓐ rational numbers and ⓑ irrational numbers.

Is $\sqrt{144}$ rational or irrational?

Solution

$\sqrt{144} = 12 therefore rational.$

From the numbers $−4, - 1 \frac{1}{2}, 0, \frac{5}{8}, \sqrt{2}, 7,$ which are ⓐ integers ⓑ rational ⓒ irrational ⓓ real numbers?

Rewrite using the commutative property: $x \cdot 14 =_________$

Solution

x·14 = 14·x

Rewrite the expression using the associative property: $(y + 6) + 3 =_______________$

Rewrite the expression using the associative property: $(8 \cdot 2) \cdot 5 =___________$

Solution

(8·2)·5 = 8·(2·5)

Evaluate $\frac{3}{16} (\frac{16}{3} n)$ when $n = 42 .$

For the number $\frac{2}{5}$ find the ⓐ additive inverse ⓑ multiplicative inverse.

Solution

ⓐ $- \frac{2}{5}$
ⓑ $\frac{5}{2}$

In the following exercises, simplify the given expression.

$\frac{3}{4} (−29) (\frac{4}{3})$

$−3 + 15 y + 3$

Solution

15y

$(1.27 q + 0.25 q) + 0.75 q$

$(\frac{8}{15} + \frac{2}{9}) + \frac{7}{9}$

Solution

$\frac{23}{15}$

$−18 (\frac{3}{2} n)$

$14 y + (−6 z) + 16 y + 2 z$

Solution

30y − 4z

$9 (q + 9)$

$6 (5 x - 4)$

Solution

30x − 24

$−10 (0.4 n + 0.7)$

$\frac{1}{4} (8 a + 12)$

Solution

2a + 3

$m (n + 2)$

$8 (6 p - 1) + 2 (9 p + 3)$

Solution

66p − 2

$(12 a + 4) - (9 a + 6)$

$\frac{0}{8}$

Solution

$\frac{4.5}{0}$

$0 \div (\frac{2}{3})$

Solution

In the following exercises, solve using the appropriate unit conversions.

Azize walked $4 \frac{1}{2}$ miles. Convert this distance to feet. $(1 mile = 5,280 feet).$

One cup of milk contains $276$ milligrams of calcium. Convert this to grams. $(1 milligram = 0.001 gram)$

Solution

.276 grams

Larry had $5$ phone customer phone calls yesterday. The calls lasted $28, 44, 9, 75, and 55$ minutes. How much time, in hours and minutes, did Larry spend on the phone? $(1 hour = 60 minutes)$

Janice ran $15$ kilometers. Convert this distance to miles. Round to the nearest hundredth of a mile. $(1 mile = 1.61 kilometers)$

Solution

9.317 miles

Yolie is $63$ inches tall. Convert her height to centimeters. Round to the nearest centimeter. $(1 inch = 2.54 centimeters)$

Use the formula $F = \frac{9}{5} C + 32$ to convert $35 °C$ to degrees $F$

Solution

95°F