Foundations

11 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

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 (Figure). The table also shows, in parentheses, the common abbreviations for each measurement.

U.S. System of Measurement
\begin{array}{ccc}\mathbf{\text{Length}}\hfill & & \begin{array}{ccc}1\phantom{\rule{0.2em}{0ex}}\text{foot}\phantom{\rule{0.2em}{0ex}}\text{(ft.)}\hfill & =\hfill & 12\phantom{\rule{0.2em}{0ex}}\text{inches}\phantom{\rule{0.2em}{0ex}}\text{(in.)}\hfill \\ 1\phantom{\rule{0.2em}{0ex}}\text{yard}\phantom{\rule{0.2em}{0ex}}\text{(yd.)}\hfill & =\hfill & 3\phantom{\rule{0.2em}{0ex}}\text{feet}\phantom{\rule{0.2em}{0ex}}\text{(ft.)}\hfill \\ 1\phantom{\rule{0.2em}{0ex}}\text{mile}\phantom{\rule{0.2em}{0ex}}\text{(mi.)}\hfill & =\hfill & 5,280\phantom{\rule{0.2em}{0ex}}\text{feet}\phantom{\rule{0.2em}{0ex}}\text{(ft.)}\hfill \end{array}\hfill \end{array} \begin{array}{ccc}\mathbf{\text{Volume}}\hfill & & \begin{array}{ccc}3\phantom{\rule{0.2em}{0ex}}\text{teaspoons}\phantom{\rule{0.2em}{0ex}}\text{(t)}\hfill & =\hfill & 1\phantom{\rule{0.2em}{0ex}}\text{tablespoon}\phantom{\rule{0.2em}{0ex}}\text{(T)}\hfill \\ \text{16 tablespoons}\phantom{\rule{0.2em}{0ex}}\text{(T)}\hfill & =\hfill & \text{1 cup}\phantom{\rule{0.2em}{0ex}}\text{(C)}\hfill \\ \text{1 cup}\phantom{\rule{0.2em}{0ex}}\text{(C)}\hfill & =\hfill & \text{8 fluid ounces}\phantom{\rule{0.2em}{0ex}}\text{(fl. oz.)}\hfill \\ \text{1 pint}\phantom{\rule{0.2em}{0ex}}\text{(pt.)}\hfill & =\hfill & \text{2 cups}\phantom{\rule{0.2em}{0ex}}\text{(C)}\hfill \\ \text{1 quart}\phantom{\rule{0.2em}{0ex}}\text{(qt.)}\hfill & =\hfill & \text{2 pints}\phantom{\rule{0.2em}{0ex}}\text{(pt.)}\hfill \\ \text{1 gallon}\phantom{\rule{0.2em}{0ex}}\text{(gal)}\hfill & =\hfill & \text{4 quarts}\phantom{\rule{0.2em}{0ex}}\text{(qt.)}\hfill \end{array}\hfill \end{array}
\begin{array}{ccc}\mathbf{\text{Weight}}\hfill & & \begin{array}{ccc}\text{1 pound}\phantom{\rule{0.2em}{0ex}}\text{(lb.)}\hfill & =\hfill & \text{16 ounces}\phantom{\rule{0.2em}{0ex}}\text{(oz.)}\hfill \\ \text{1 ton}\hfill & =\hfill & \text{2000 pounds}\phantom{\rule{0.2em}{0ex}}\text{(lb.)}\hfill \end{array}\hfill \end{array} \begin{array}{ccc}\mathbf{\text{Time}}\hfill & & \phantom{\rule{1em}{0ex}}\begin{array}{ccc}\text{1 minute}\phantom{\rule{0.2em}{0ex}}\text{(min)}\hfill & =\hfill & \text{60 seconds}\phantom{\rule{0.2em}{0ex}}\text{(sec)}\hfill \\ \text{1 hour}\phantom{\rule{0.2em}{0ex}}\text{(hr)}\hfill & =\hfill & \text{60 minutes}\phantom{\rule{0.2em}{0ex}}\text{(min)}\hfill \\ \text{1 day}\hfill & =\hfill & \text{24 hours}\phantom{\rule{0.2em}{0ex}}\text{(hr)}\hfill \\ \text{1 week}\phantom{\rule{0.2em}{0ex}}\text{(wk)}\hfill & =\hfill & \text{7 days}\hfill \\ \text{1 year}\phantom{\rule{0.2em}{0ex}}\text{(yr)}\hfill & =\hfill & \text{365 days}\hfill \end{array}\hfill \end{array}

In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, quarts and gallons, etc. We will use the identity property of multiplication to do these conversions. We’ll restate the identity property of multiplication here for easy reference.

Identity Property of Multiplication

\begin{array}{cccccc}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a:\hfill & & & & & a·1=a\phantom{\rule{3em}{0ex}}1·a=a\hfill \\ 1\phantom{\rule{0.2em}{0ex}}\text{is the}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{multiplicative identity}}\hfill & & & & & \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\phantom{\rule{0.2em}{0ex}}\text{foot}}{12\phantom{\rule{0.2em}{0ex}}\text{inches}}. When we multiply by this fraction we do not change the value, but just change the units.

But \frac{12\phantom{\rule{0.2em}{0ex}}\text{inches}}{1\phantom{\rule{0.2em}{0ex}}\text{foot}} also equals 1. How do we decide whether to multiply by \frac{1\phantom{\rule{0.2em}{0ex}}\text{foot}}{12\phantom{\rule{0.2em}{0ex}}\text{inches}} or \frac{12\phantom{\rule{0.2em}{0ex}}\text{inches}}{1\phantom{\rule{0.2em}{0ex}}\text{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?

Two expressions are given: 66 inches times the fraction (1 foot) over (12 inches), and 66 inches times the fraction (12 inches) over (1 foot). This second expression is crossed out. Below this, it is stated that “The first form works since 66 inches times the fraction (1 foot) over (12 inches), with inches crossed off in both instances.

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.

How to Make Unit Conversions

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

Solution

A table is given with three columns. In the first column are directions. The second column has exposition, and the third column has the mathematical steps. In the first row, the direction is “Step 1. Multiply the measurement to be converted by; write as a fraction relating the units given and the units needed.” The exposition is “Multiply inches by, writing as a fraction relating inches and feet. We need inches in the denominator so that the inches will divide out!” The mathematical step is 66 inches times the fraction (1 foot) over (12 inches).In the following row, we have “Step 2. Multiply.” The hint is “Think of 66 inches as the quantity 66 inches divided by 1.” The math portion is the fraction (66 inches times 1 foot) over 12 inches.In the following row, we have “Step 3. Simplify the fraction.” The hint is that “Notice: inches divide out.” We obtain 66 feet divided by 12.Then the last step is “Step 4. Simplify.” The hint is “Divide 66 by 12.” Hence, our final mathematical statement is 5.5 feet.

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

2.5 feet

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

54 feet

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

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\phantom{\rule{0.2em}{0ex}}\text{pounds}}{1\phantom{\rule{0.2em}{0ex}}\text{ton}}.

\text{3.2 tons}
Multiply the measurement to be converted, by 1. \text{3.2 tons}\cdot 1
Write 1 as a fraction relating tons and pounds. \text{3.2 tons}\cdot \frac{\text{2,000 pounds}}{\text{1 ton}}
Simplify. .
Multiply. \text{6,400 pounds}
Ndula weighs almost 6,400 pounds.

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

8,600 pounds

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

102,000,000 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 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.

9 weeks
Write 1 as \frac{\text{7 days}}{\text{1 week}}, \frac{\text{24 hours}}{\text{1 day}}, and \frac{\text{60 minutes}}{\text{1 hour}}. \frac{\text{9 wk}}{1}\cdot \frac{\text{7 days}}{\text{1 wk}}\cdot \frac{\text{24 hr}}{\text{1 day}}\cdot \frac{\text{60 min}}{\text{1 hr}}
Divide out the common units. .
Multiply. \frac{9\cdot 7\cdot 24\cdot \text{60 min}}{1\cdot 1\cdot 1\cdot 1}
Multiply. \text{90,720 min}

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

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

440,000,000 yards

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

151,200 minutes

How many ounces are in 1 gallon?

Solution

We will convert gallons to ounces by multiplying by several conversion factors. Refer to (Figure).

1 gallon
Multiply the measurement to be converted by 1. \frac{\text{1 gallon}}{1}\cdot \frac{\text{4 quarts}}{\text{1 gallon}}\cdot \frac{\text{2 pints}}{\text{1 quart}}\cdot \frac{\text{2 cups}}{\text{1 pint}}\cdot \frac{\text{8 ounces}}{\text{1 cup}}
Use conversion factors to get to the right unit.
Simplify.
.
Multiply. \frac{1\cdot 4\cdot 2\cdot 2\cdot \text{8 ounces}}{1\cdot 1\cdot 1\cdot 1\cdot 1}
Simplify. \text{128 ounces}
There are 128 ounces in a gallon.

How many cups are in 1 gallon?

16 cups

How many teaspoons are in 1 cup?

48 teaspoons

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

We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1 pound 2 ounce steak for dinner—all these measurements have mixed units.

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

Seymour bought three steaks for a barbecue. Their weights were 14 ounces, 1 pound 2 ounces and 1 pound 6 ounces. How many total pounds of steak did he buy?

Solution

We will add the weights of the steaks to find the total weight of the steaks.

Add the ounces. Then add the pounds. .
Convert 22 ounces to pounds and ounces. \phantom{\rule{0.35em}{0ex}}2 pounds + 1 pound, 6 ounces
Add the pounds. \phantom{\rule{0.35em}{0ex}}3 pounds, 6 ounces
Seymour bought 3 pounds 6 ounces of steak.

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?

\text{9 lbs}.\phantom{\rule{0.2em}{0ex}}\text{8 oz}

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?

21 ft. 6 in.

Anthony bought four planks of wood that were each 6 feet 4 inches long. What is the total length of the wood he purchased?

Solution

We will multiply the length of one plank to find the total length.

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

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?

4 lbs. 8 oz.

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

11 gallons 2 qt.

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 (Figure). The common abbreviations for each measurement are given in parentheses.

Metric System of Measurement
Length Mass Capacity
1 kilometer (km) = 1,000 m

1 hectometer (hm) = 100 m

1 dekameter (dam) = 10 m

1 meter (m) = 1 m

1 decimeter (dm) = 0.1 m

1 centimeter (cm) = 0.01 m

1 millimeter (mm) = 0.001 m
1 kilogram (kg) = 1,000 g

1 hectogram (hg) = 100 g

1 dekagram (dag) = 10 g

1 gram (g) = 1 g

1 decigram (dg) = 0.1 g

1 centigram (cg) = 0.01 g

1 milligram (mg) = 0.001 g
1 kiloliter (kL) = 1,000 L

1 hectoliter (hL) = 100 L

1 dekaliter (daL) = 10 L

1 liter (L) = 1 L

1 deciliter (dL) = 0.1 L

1 centiliter (cL) = 0.01 L

1 milliliter (mL) = 0.001 L
1 meter = 100 centimeters

1 meter = 1,000 millimeters
1 gram = 100 centigrams

1 gram = 1,000 milligrams
1 liter = 100 centiliters

1 liter = 1,000 milliliters

To 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.

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

Solution

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

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

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

5,000 meters

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

250 centimeters

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

Solution

We will convert grams into kilograms.

.
Multiply the measurement to be converted by 1. .
Write 1 as a function relating kilograms and grams. .
Simplify. .
Multiply. \frac{\text{3,200 kilograms}}{1,000}
Divide. 3.2 kilograms
The baby weighed 3.2 kilograms.

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

2.8 kilograms

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

4.5 kilograms

As you become familiar with the metric system you may see a pattern. Since the system is based on multiples of ten, the calculations involve multiplying by multiples of ten. We have learned how to simplify these calculations by just moving the decimal.

To multiply by 10, 100, or 1,000, we move the decimal to the right one, two, or three places, respectively. To multiply by 0.1, 0.01, or 0.001, we move the decimal to the left one, two, or three places, respectively.

We can apply this pattern when we make measurement conversions in the metric system. In (Figure), 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.

We have the statement 3200 g times the fraction 1 kg over 1000 g, with the g’s crossed out. Below this, we have 3.2. We also have the statement 3200 times 1/1000, with an arrow drawn from the right of the final 0 in 3200 to the space between the 0’s, to the space between the 2 and the 0, and then to the space between the 3 and the 2. Below this, we have 3.2.

Convert 350 L to kiloliters 4.1 L to milliliters.

Solution
  1. We will convert liters to kiloliters. In (Figure), we see that 1\phantom{\rule{0.2em}{0ex}}\text{kiloliter}=\text{1,000 liters.}

    \text{350 L}
    Multiply by 1, writing 1 as a fraction relating liters to kiloliters. \text{350 L}\cdot \frac{\text{1 kL}}{\text{1,000 L}}
    Simplify. 350\phantom{\rule{.2em}{0ex}}\overline{)\text{L}}\cdot \frac{\text{1 kL}}{1,000\phantom{\rule{.2em}{0ex}}\overline{)\text{L}}}
    . \text{0.35 kL}



  2. We will convert liters to milliliters. From (Figure) we see that \text{1 liter}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\text{1,000 milliliters.}

    .
    Multiply by 1, writing 1 as a fraction relating liters to milliliters. .
    Simplify. .
    Move the decimal 3 units to the right. .
    .

Convert: 725 L to kiloliters 6.3 L to milliliters

7,250 kiloliters 6,300 milliliters

Convert: 350 hL to liters 4.1 L to centiliters

35,000 liters 410 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.

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}{c}\phantom{\rule{0.7em}{0ex}}1.60\phantom{\rule{0.2em}{0ex}}\text{m}\hfill \\ \underset{\text{_______}}{-0.85\phantom{\rule{0.2em}{0ex}}\text{m}}\hfill \\ \phantom{\rule{0.7em}{0ex}}0.75\phantom{\rule{0.2em}{0ex}}\text{m}\hfill \end{array}

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

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.

83 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.

1.04 meters

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 \text{150 mL}
Translate to algebra. 3·150\phantom{\rule{0.2em}{0ex}}\text{mL}
Multiply. \text{450 mL}
Convert to liters. 450·\frac{0.001\phantom{\rule{0.2em}{0ex}}\text{L}}{1\phantom{\rule{0.2em}{0ex}}\text{mL}}
Simplify. \text{0.45 L}
Dena needs 0.45 liters of olive oil.

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?

2 liters

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?

2.4 kilograms

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

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

(Figure) shows some of the most common conversions.

Conversion Factors Between U.S. and Metric Systems
Length Mass Capacity
\begin{array}{c}\begin{array}{ccc}\text{1 in.}\hfill & =\hfill & \text{2.54 cm}\hfill \\ \text{1 ft.}\hfill & =\hfill & \text{0.305 m}\hfill \\ \text{1 yd.}\hfill & =\hfill & \text{0.914 m}\hfill \\ \text{1 mi.}\hfill & =\hfill & \text{1.61 km}\hfill \\ \text{1 m}\hfill & =\hfill & \text{3.28 ft.}\hfill \end{array}\hfill \end{array} \begin{array}{c}\begin{array}{ccc}\text{1 lb.}\hfill & =\hfill & \text{0.45 kg}\hfill \\ \text{1 oz.}\hfill & =\hfill & \text{28 g}\hfill \\ \text{1 kg}\hfill & =\hfill & \text{2.2 lb.}\hfill \end{array}\hfill \end{array} \begin{array}{c}\begin{array}{ccc}\text{1 qt.}\hfill & =\hfill & \text{0.95 L}\hfill \\ \text{1 fl. oz.}\hfill & =\hfill & \text{30 mL}\hfill \\ \text{1 L}\hfill & =\hfill & \text{1.06 qt.}\hfill \end{array}\hfill \end{array}

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

This ruler shows inches and centimeters.

A ruler with inches and centimeters.

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

This measuring cup shows ounces and milliliters.

A measuring cup showing milliliters and ounces.

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

This scale shows pounds and kilograms.

We are given an image of a bathroom scale showing pounds.

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 ounces are in the bottle? Round to the nearest tenth of an ounce.

Solution
\text{500 mL}
Multiply by a unit conversion factor relating mL and ounces. 500\phantom{\rule{0.2em}{0ex}}\text{milliliters}·\frac{1\phantom{\rule{0.2em}{0ex}}\text{ounce}}{30\phantom{\rule{0.2em}{0ex}}\text{milliliters}}
Simplify. \frac{\text{50 ounce}}{30}
Divide. \text{16.7 ounces.}
The water bottle has 16.7 ounces.

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

2.12 quarts

How many liters are in 4 quarts of milk?

3.8 liters

Soleil was on a road trip and saw a sign that said the next rest stop was in 100 kilometers. How many miles until the next rest stop?

Solution
\text{100 kilometers}
Multiply by a unit conversion factor relating km and mi. 100\phantom{\rule{0.2em}{0ex}}\text{kilometers}·\frac{1\phantom{\rule{0.2em}{0ex}}\text{mile}}{1.61\phantom{\rule{0.2em}{0ex}}\text{kilometer}}
Simplify. \frac{\text{100 miles}}{1.61}
Divide. \text{62 ounces.}
Soleil will travel 62 miles.

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

19,335.6 feet

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

8,993.46 km

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\text{°}\text{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 \text{°}\text{F}\text{.} The metric system uses degrees Celsius, written \text{°}\text{C}\text{.} (Figure) shows the relationship between the two systems.

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

Two thermometers are shown, one in Celsius (°C) and another in Fahrenheit (°F). They are marked “Water boils” at 100°C and 212°F. They are marked “Normal body temperature” at 37°C and 98.6°F. They are marked “Water freezes” at 0°C and 32°F.

Temperature Conversion

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

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

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

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

Convert 50\text{°} Fahrenheit into degrees Celsius.

Solution

We will substitute 50\text{°}\text{F} into the formula to find C.

.
. .
Simplify in parentheses. .
Multiply. .
So we found that 50°F is equivalent to 10°C.

Convert the Fahrenheit temperature to degrees Celsius: 59\text{°} Fahrenheit.

15\text{°}\text{C}

Convert the Fahrenheit temperature to degrees Celsius: 41\text{°} Fahrenheit.

5\text{°}\text{C}

While visiting Paris, Woody saw the temperature was 20\text{°} Celsius. Convert the temperature into degrees Fahrenheit.

Solution

We will substitute 20\text{°}\text{C} into the formula to find F.

.
. .
Multiply. .
Add. .
So we found that 20°C is equivalent to 68°F.

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Helsinki, Finland, was 15\text{°} Celsius.

59\text{°}\text{F}

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Sydney, Australia, was 10\text{°} Celsius.

50\text{°}\text{F}

Key Concepts

  • Metric System of Measurement
    • Length
      \begin{array}{ccc}\text{1 kilometer (km)}\hfill & =\hfill & \text{1,000 m}\hfill \\ \text{1 hectometer (hm)}\hfill & =\hfill & \text{100 m}\hfill \\ \text{1 dekameter (dam)}\hfill & =\hfill & \text{10 m}\hfill \\ \text{1 meter (m)}\hfill & =\hfill & \text{1 m}\hfill \\ \text{1 decimeter (dm)}\hfill & =\hfill & \text{0.1 m}\hfill \\ \text{1 centimeter (cm)}\hfill & =\hfill & \text{0.01 m}\hfill \\ \text{1 millimeter (mm)}\hfill & =\hfill & \text{0.001 m}\hfill \\ \text{1 meter}\hfill & =\hfill & \text{100 centimeters}\hfill \\ \text{1 meter}\hfill & =\hfill & \text{1,000 millimeters}\hfill \end{array}
    • Mass
      \begin{array}{ccc}\text{1 kilogram (kg)}\hfill & =\hfill & \text{1,000 g}\hfill \\ \text{1 hectogram (hg)}\hfill & =\hfill & \text{100 g}\hfill \\ \text{1 dekagram (dag)}\hfill & =\hfill & \text{10 g}\hfill \\ \text{1 gram (g)}\hfill & =\hfill & \text{1 g}\hfill \\ \text{1 decigram (dg)}\hfill & =\hfill & \text{0.1 g}\hfill \\ \text{1 centigram (cg)}\hfill & =\hfill & \text{0.01 g}\hfill \\ \text{1 milligram (mg)}\hfill & =\hfill & \text{0.001 g}\hfill \\ \text{1 gram}\hfill & =\hfill & \text{100 centigrams}\hfill \\ \text{1 gram}\hfill & =\hfill & \text{1,000 milligrams}\hfill \end{array}
    • Capacity
      \begin{array}{ccc}\text{1 kiloliter (kL)}\hfill & =\hfill & \text{1,000 L}\hfill \\ \text{1 hectoliter (hL)}\hfill & =\hfill & \text{100 L}\hfill \\ \text{1 dekaliter (daL)}\hfill & =\hfill & \text{10 L}\hfill \\ \text{1 liter (L)}\hfill & =\hfill & \text{1 L}\hfill \\ \text{1 deciliter (dL)}\hfill & =\hfill & \text{0.1 L}\hfill \\ \text{1 centiliter (cL)}\hfill & =\hfill & \text{0.01 L}\hfill \\ \text{1 milliliter (mL)}\hfill & =\hfill & \text{0.001 L}\hfill \\ \text{1 liter}\hfill & =\hfill & \text{100 centiliters}\hfill \\ \text{1 liter}\hfill & =\hfill & \text{1,000 milliliters}\hfill \end{array}
  • Temperature Conversion
    • To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula \text{C}=\frac{5}{9}\left(\text{F}-32\right)
    • To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula \text{F}=\frac{9}{5}\text{C}+32

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.

72 inches

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

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

1.5 feet

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

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

53\frac{1}{3} yards

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

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

7,920 feet

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

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

9,200 pounds

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

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

17\frac{1}{2} tons

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

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

5,400 s

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

How many teaspoons are in a pint?

96 teaspoons

How many tablespoons are in a gallon?

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

224 ounces

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

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

1\frac{1}{4} gallons

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

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

26 in.

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

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

65 days

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

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

115 ounces

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.

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?

8 lbs. 13 oz.

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?

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?

3.05 hours

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?

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?

10 ft. 2 in.

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?

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?

4 yards

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.

Ghalib ran 5 kilometers. Convert the length to meters.

5,000 meters

Kitaka hiked 8 kilometers. Convert the length to meters.

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

155 centimeters

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

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

3.072 kilometers

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

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

1.5 grams

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

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

91,600 milligrams

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

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

2,000 grams

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

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

0.75 liters

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.

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

91 centimeters

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

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?

855 grams

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?

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

2.1 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?

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

1.6 liters

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.

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

190.5 centimeters

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

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

21.9 meters

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

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

742.5 kilograms

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

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

3.1 miles

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

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

44 pounds

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

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

53.2 liters

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.

86\text{°} Fahrenheit

30\text{°}\text{C}

77\text{°} Fahrenheit

104\text{°} Fahrenheit

40\text{°}\text{C}

14\text{°} Fahrenheit

72\text{°} Fahrenheit

22.2\text{°}\text{C}

4\text{°} Fahrenheit

0\text{°} Fahrenheit

\text{−}17.8\text{°}\text{C}

120\text{°} Fahrenheit

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

5\text{°} Celsius

41\text{°}\text{F}

25\text{°} Celsius

\text{−}10\text{°} Celsius

14\text{°}\text{F}

\text{−}15\text{°} Celsius

22\text{°} Celsius

71.6\text{°}\text{F}

8\text{°} Celsius

43\text{°} Celsius

109.4\text{°}\text{F}

16\text{°} Celsius

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?

14.6 kilograms

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

Some people think that 65\text{°}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}75\text{°} 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.

Answers may vary.

  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.

This is a table that has seven rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “define US units of measurement and convert from one unit to another,” “use US units of measurement,” “define metric units of measurement and convert from one unit to another,” “use metric units of measurement,” “convert between the US and the metric system of measurement,” and “convert between Fahrenheit and Celsius temperatures.” The rest of the cells are blank.

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

Chapter Review Exercises

Introduction to Whole Numbers

Use Place Value with Whole Number

In the following exercises find the place value of each digit.

26,915

1
2
9
5
6

tens ten thousands hundreds ones thousands

359,417

9
3
4
7
1

58,129,304

5
0
1
8
2

ten millions tens hundred thousands millions ten thousands

9,430,286,157

6
4
9
0
5

In the following exercises, name each number.

6,104

six thousand, one hundred four

493,068

3,975,284

three million, nine hundred seventy-five thousand, two hundred eighty-four

85,620,435

In the following exercises, write each number as a whole number using digits.

three hundred fifteen

315

sixty-five thousand, nine hundred twelve

ninety million, four hundred twenty-five thousand, sixteen

90,425,016

one billion, forty-three million, nine hundred twenty-two thousand, three hundred eleven

In the following exercises, round to the indicated place value.

Round to the nearest ten.

407 8,564

4108,560

Round to the nearest hundred.

25,846 25,864

In the following exercises, round each number to the nearest hundred thousand ten thousand.

864,951

865,000865,000860,000

3,972,849

Identify Multiples and Factors

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.

168

\text{by}\phantom{\rule{0.2em}{0ex}}2,3,6

264

375

\text{by}\phantom{\rule{0.2em}{0ex}}3,5

750

1430

\text{by}\phantom{\rule{0.2em}{0ex}}2,5,10

1080

Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

420

2·2·3·5·7

115

225

3·3·5·5

2475

1560

2·2·2·3·5·13

56

72

2·2·2·3·3

168

252

2·2·3·3·7

391

In the following exercises, find the least common multiple of the following numbers using the multiples method.

6,15

30

60, 75

In the following exercises, find the least common multiple of the following numbers using the prime factors method.

24, 30

120

70, 84

Use the Language of Algebra

Use Variables and Algebraic Symbols

In the following exercises, translate the following from algebra to English.

25-7

25 minus 7, the difference of twenty-five and seven

5·6

45÷5

45 divided by 5, the quotient of forty-five and five

x+8

42\ge 27

forty-two is greater than or equal to twenty-seven

3n=24

3\le 20÷4

3 is less than or equal to 20 divided by 4, three is less than or equal to the quotient of twenty and four

a\ne 7·4

In the following exercises, determine if each is an expression or an equation.

6·3+5

expression

y-8=32

Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

{3}^{5}

243

{10}^{8}

In the following exercises, simplify

6+10\text{/}2+2

13

9+12\text{/}3+4

20÷\left(4+6\right)·5

10

33÷\left(3+8\right)·2

{4}^{2}+{5}^{2}

41

{\left(4+5\right)}^{2}

Evaluate an Expression

In the following exercises, evaluate the following expressions.

9x+7 when x=3

34

5x-4 when x=6

{x}^{4} when x=3

81

{3}^{x} when x=3

{x}^{2}+5x-8 when x=6

58

2x+4y-5 when
x=7,y=8

Simplify Expressions by Combining Like Terms

In the following exercises, identify the coefficient of each term.

12n

12

9{x}^{2}

In the following exercises, identify the like terms.

3n,{n}^{2},12,12{p}^{2},3,3{n}^{2}

12\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3,{n}^{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3{n}^{2}

5,18{r}^{2},9s,9r,5{r}^{2},5s

In the following exercises, identify the terms in each expression.

11{x}^{2}+3x+6

11{x}^{2},3x,6

22{y}^{3}+y+15

In the following exercises, simplify the following expressions by combining like terms.

17a+9a

26a

18z+9z

9x+3x+8

12x+8

8a+5a+9

7p+6+5p-4

12p+2

8x+7+4x-5

Translate an English Phrase to an Algebraic Expression

In the following exercises, translate the following phrases into algebraic expressions.

the sum of 8 and 12

8+12

the sum of 9 and 1

the difference of x and 4

x-4

the difference of x and 3

the product of 6 and y

6y

the product of 9 and y

Adele bought a skirt and a blouse. The skirt cost ?15 more than the blouse. Let b represent the cost of the blouse. Write an expression for the cost of the skirt.

b+15

Marcella has 6 fewer boy cousins than girl cousins. Let g represent the number of girl cousins. Write an expression for the number of boy cousins.

Add and Subtract Integers

Use Negatives and Opposites of Integers

In the following exercises, order each of the following pairs of numbers, using < or >.


6___2
-7___4
-9___-1
9___-3

> < < >


-5___1
-4___-9
6___10
3___-8

In the following exercises,, find the opposite of each number.

-8 1

8 -1

-26

In the following exercises, simplify.

\text{−}\left(-19\right)

19

\text{−}\left(-53\right)

In the following exercises, simplify.

\text{−}m when
m=3
m=-3

-3 3

\text{−}p when
p=6
p=-6

Simplify Expressions with Absolute Value

In the following exercises,, simplify.

|7||-25||0|

7 25 0

|5||0||-19|

In the following exercises, fill in <, >, or = for each of the following pairs of numbers.


-8___|-8|
\text{−}|-2|___-2

< =


|-3|___-|-3|
4___-|-4|

In the following exercises, simplify.

|8-4|

4

|9-6|

8\left(14-2|-2|\right)

80

6\left(13-4|-2|\right)

In the following exercises, evaluate.

|x| when x=-28

28 15


|y| when y=-37
|\text{−}z| when z=-24

Add Integers

In the following exercises, simplify each expression.

-200+65

-135

-150+45

2+\left(-8\right)+6

0

4+\left(-9\right)+7

140+\left(-75\right)+67

132

-32+24+\left(-6\right)+10

Subtract Integers

In the following exercises, simplify.

9-3

6

-5-\left(-1\right)

15-615+\left(-6\right)

9 9

12-912+\left(-9\right)

8-\left(-9\right)8+9

17 17

4-\left(-4\right)4+4

In the following exercises, simplify each expression.

10-\left(-19\right)

29

11-\left(-18\right)

31-79

-48

39-81

-31-11

-42

-32-18

-15-\left(-28\right)+5

18

71+\left(-10\right)-8

-16-\left(-4+1\right)-7

-20

-15-\left(-6+4\right)-3

Multiply Integers

In the following exercises, multiply.

-5\left(7\right)

-35

-8\left(6\right)

-18\left(-2\right)

36

-10\left(-6\right)

Divide Integers

In the following exercises, divide.

-28÷7

-4

56÷\left(-7\right)

-120÷\left(-20\right)

6

-200÷25

Simplify Expressions with Integers

In the following exercises, simplify each expression.

-8\left(-2\right)-3\left(-9\right)

43

-7\left(-4\right)-5\left(-3\right)

{\left(-5\right)}^{3}

-125

{\left(-4\right)}^{3}

-4·2·11

-88

-5·3·10

-10\left(-4\right)÷\left(-8\right)

-5

-8\left(-6\right)÷\left(-4\right)

31-4\left(3-9\right)

55

24-3\left(2-10\right)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

x+8 when
x=-26
x=-95

-18-87

y+9 when
y=-29
y=-84

When b=-11, evaluate:
b+6
\text{−}b+6

-5 17

When c=-9, evaluate:
c+\left(-4\right)
\text{−}c+\left(-4\right)

{p}^{2}-5p+2 when
p=-1

8

{q}^{2}-2q+9 when q=-2

6x-5y+15 when x=3 and y=-1

38

3p-2q+9 when p=8 and q=-2

Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

the sum of -4 and -17, increased by 32

\left(-4+\left(-17\right)\right)+32;11

the difference of 15 and -7 subtract 15 from -7

the quotient of -45 and -9

\frac{-45}{-9};5

the product of -12 and the difference of c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d

Use Integers in Applications

In the following exercises, solve.

Temperature The high temperature one day in Miami Beach, Florida, was 76\text{°}. That same day, the high temperature in Buffalo, New York was \text{−}8\text{°}. What was the difference between the temperature in Miami Beach and the temperature in Buffalo?

84 degrees

CheckingAccount Adrianne has a balance of \text{−}?22 in her checking account. She deposits ?301 to the account. What is the new balance?

Visualize Fractions

Find Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

\frac{1}{4}

\frac{2}{8},\frac{3}{12},\frac{4}{16} answers may vary

\frac{1}{3}

\frac{5}{6}

\frac{10}{12},\frac{15}{18},\frac{20}{24} answers may vary

\frac{2}{7}

Simplify Fractions

In the following exercises, simplify.

\frac{7}{21}

\frac{1}{3}

\frac{8}{24}

\frac{15}{20}

\frac{3}{4}

\frac{12}{18}

-\phantom{\rule{0.2em}{0ex}}\frac{168}{192}

-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}

-\phantom{\rule{0.2em}{0ex}}\frac{140}{224}

\frac{11x}{11y}

\frac{x}{y}

\frac{15a}{15b}

Multiply Fractions

In the following exercises, multiply.

\frac{2}{5}·\frac{1}{3}

\frac{2}{15}

\frac{1}{2}·\frac{3}{8}

\frac{7}{12}\left(-\phantom{\rule{0.2em}{0ex}}\frac{8}{21}\right)

-\phantom{\rule{0.2em}{0ex}}\frac{2}{9}

\frac{5}{12}\left(-\phantom{\rule{0.2em}{0ex}}\frac{8}{15}\right)

-28p\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}\right)

7p

-51q\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}\right)

\frac{14}{5}\left(-15\right)

-42

-1\left(-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}\right)

Divide Fractions

In the following exercises, divide.

\frac{1}{2}÷\frac{1}{4}

2

\frac{1}{2}÷\frac{1}{8}

-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}÷\frac{4}{7}

-\phantom{\rule{0.2em}{0ex}}\frac{7}{5}

-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}÷\frac{3}{5}

\frac{5}{8}÷\frac{a}{10}

\frac{25}{4a}

\frac{5}{6}÷\frac{c}{15}

\frac{7p}{12}÷\frac{21p}{8}

\frac{2}{9}

\frac{5q}{12}÷\frac{15q}{8}

\frac{2}{5}÷\left(-10\right)

-\phantom{\rule{0.2em}{0ex}}\frac{1}{25}

-18÷-\left(\frac{9}{2}\right)

In the following exercises, simplify.

\frac{\frac{2}{3}}{\frac{8}{9}}

\frac{3}{4}

\frac{\frac{4}{5}}{\frac{8}{15}}

\frac{-\phantom{\rule{0.2em}{0ex}}\frac{9}{10}}{3}

-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}

\frac{2}{\frac{5}{8}}

\frac{\frac{r}{5}}{\frac{s}{3}}

\frac{3r}{5s}

\frac{-\phantom{\rule{0.2em}{0ex}}\frac{x}{6}}{-\phantom{\rule{0.2em}{0ex}}\frac{8}{9}}

Simplify Expressions Written with a Fraction Bar

In the following exercises, simplify.

\frac{4+11}{8}

\frac{15}{8}

\frac{9+3}{7}

\frac{30}{7-12}

-6

\frac{15}{4-9}

\frac{22-14}{19-13}

\frac{4}{3}

\frac{15+9}{18+12}

\frac{5·8}{-10}

-4

\frac{3·4}{-24}

\frac{15·5-{5}^{2}}{2·10}

\frac{5}{2}

\frac{12·9-{3}^{2}}{3·18}

\frac{2+4\left(3\right)}{-3-{2}^{2}}

-2

\frac{7+3\left(5\right)}{-2-{3}^{2}}

Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

the quotient of c and the sum of d and 9.

\frac{c}{d+9}

the quotient of the difference of h and k, and -5.

Add and Subtract Fractions

Add and Subtract Fractions with a Common Denominator

In the following exercises, add.

\frac{4}{9}+\frac{1}{9}

\frac{5}{9}

\frac{2}{9}+\frac{5}{9}

\frac{y}{3}+\frac{2}{3}

\frac{y+2}{3}

\frac{7}{p}+\frac{9}{p}

-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}\right)

-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}

-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\right)

In the following exercises, subtract.

\frac{4}{5}-\phantom{\rule{0.2em}{0ex}}\frac{1}{5}

\frac{3}{5}

\frac{4}{5}-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}

\frac{y}{17}-\phantom{\rule{0.2em}{0ex}}\frac{9}{17}

\frac{y-9}{17}

\frac{x}{19}-\phantom{\rule{0.2em}{0ex}}\frac{8}{19}

-\phantom{\rule{0.2em}{0ex}}\frac{8}{d}-\phantom{\rule{0.2em}{0ex}}\frac{3}{d}

-\phantom{\rule{0.2em}{0ex}}\frac{11}{d}

-\phantom{\rule{0.2em}{0ex}}\frac{7}{c}-\phantom{\rule{0.2em}{0ex}}\frac{7}{c}

Add or Subtract Fractions with Different Denominators

In the following exercises, add or subtract.

\frac{1}{3}+\frac{1}{5}

\frac{8}{15}

\frac{1}{4}+\frac{1}{5}

\frac{1}{5}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{10}\right)

\frac{3}{10}

\frac{1}{2}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}\right)

\frac{2}{3}+\frac{3}{4}

\frac{17}{12}

\frac{3}{4}+\frac{2}{5}

\frac{11}{12}-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}

\frac{13}{24}

\frac{5}{8}-\phantom{\rule{0.2em}{0ex}}\frac{7}{12}

-\phantom{\rule{0.2em}{0ex}}\frac{9}{16}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}\right)

\frac{19}{80}

-\phantom{\rule{0.2em}{0ex}}\frac{7}{20}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\right)

1+\frac{5}{6}

\frac{11}{6}

1-\phantom{\rule{0.2em}{0ex}}\frac{5}{9}

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

\frac{{\left(\frac{1}{5}\right)}^{2}}{2+{3}^{2}}

\frac{1}{275}

\frac{{\left(\frac{1}{3}\right)}^{2}}{5+{2}^{2}}

\frac{\frac{2}{3}+\frac{1}{2}}{\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}}

14

\frac{\frac{3}{4}+\frac{1}{2}}{\frac{5}{6}-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}}

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

x+\frac{1}{2} when
x=-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}
x=-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}

\frac{3}{8} 0

x+\frac{2}{3} when
x=-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}
x=-\phantom{\rule{0.2em}{0ex}}\frac{5}{3}

4{p}^{2}q when p=-\phantom{\rule{0.2em}{0ex}}\frac{1}{2} and q=\frac{5}{9}

\frac{5}{9}

5{m}^{2}n when m=-\phantom{\rule{0.2em}{0ex}}\frac{2}{5} and n=\frac{1}{3}

\frac{u+v}{w} when
u=-4,v=-8,w=2

-6

\frac{m+n}{p} when
m=-6,n=-2,p=4

Decimals

Name and Write Decimals

In the following exercises, write as a decimal.

Eight and three hundredths

8.03

Nine and seven hundredths

One thousandth

0.001

Nine thousandths

In the following exercises, name each decimal.

7.8

seven and eight tenths

5.01

0.005

five thousandths

0.381

Round Decimals

In the following exercises, round each number to the nearest hundredth tenth whole number.

5.7932

5.79 5.8 6

3.6284

12.4768

12.48 12.5 12

25.8449

Add and Subtract Decimals

In the following exercises, add or subtract.

18.37+9.36

27.73

256.37-85.49

15.35-20.88

−5.53

37.5+12.23

-4.2+\left(-9.3\right)

−13.5

-8.6+\left(-8.6\right)

100-64.2

35.8

100-65.83

2.51+40

42.51

9.38+60

Multiply and Divide Decimals

In the following exercises, multiply.

\left(0.3\right)\left(0.4\right)

0.12

\left(0.6\right)\left(0.7\right)

\left(8.52\right)\left(3.14\right)

26.7528

\left(5.32\right)\left(4.86\right)

\left(0.09\right)\left(24.78\right)

2.2302

\left(0.04\right)\left(36.89\right)

In the following exercises, divide.

0.15÷5

0.03

0.27÷3

?8.49÷12

?0.71

?16.99÷9

12÷0.08

150

5÷0.04

Convert Decimals, Fractions, and Percents

In the following exercises, write each decimal as a fraction.

0.08

\frac{2}{25}

0.17

0.425

\frac{17}{40}

0.184

1.75

\frac{7}{4}

0.035

In the following exercises, convert each fraction to a decimal.

\frac{2}{5}

0.4

\frac{4}{5}

-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}

-0.375

-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}

\frac{5}{9}

0.\stackrel{\text{-}}{5}

\frac{2}{9}

\frac{1}{2}+6.5

7

\frac{1}{4}+10.75

In the following exercises, convert each percent to a decimal.

5%

0.05

9%

40%

0.4

50%

115%

1.15

125%

In the following exercises, convert each decimal to a percent.

0.18

18%

0.15

0.009

0.9%

0.008

1.5

150%

2.2

The Real Numbers

Simplify Expressions with Square Roots

In the following exercises, simplify.

\sqrt{64}

8

\sqrt{144}

\text{−}\sqrt{25}

−5

\text{−}\sqrt{81}

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

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

9 8.47

\frac{9}{1}\frac{847}{100}

-15 3.591

In the following exercises, list the rational numbers, irrational numbers.

0.84,0.79132\text{…},1.\stackrel{\text{-}}{3}

0.84,1.\stackrel{\text{-}}{3}0.79132\text{…},

2.3\stackrel{\text{-}}{8},0.572,4.93814\text{…}

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

\sqrt{121}\sqrt{48}

rational irrational

\sqrt{56}\sqrt{16}

In the following exercises, identify whether each number is a real number or not a real number.

\sqrt{-9}\text{−}\sqrt{169}

not a real number real number

\sqrt{-64}\text{−}\sqrt{81}

In the following exercises, list the whole numbers, integers, rational numbers, irrational numbers, real numbers for each set of numbers.

-4,0,\frac{5}{6},\sqrt{16},\sqrt{18},5.2537\text{…}

0,\sqrt{16}-4,0,\sqrt{16}-4,0,\frac{5}{6},\sqrt{16}\sqrt{18},5.2537\text{…}-4,0,\frac{5}{6},\sqrt{16},\sqrt{18},5.2537\text{…}

\text{−}\sqrt{4},0.\stackrel{\text{—}}{36},\frac{13}{3},6.9152\text{…},\sqrt{48},10\frac{1}{2}

Locate Fractions on the Number Line

In the following exercises, locate the numbers on a number line.

\frac{2}{3},\frac{5}{4},\frac{12}{5}

This figure is a number line ranging from 0 to 6 with tick marks for each integer. 2 thirds, 5 fourths, and 12 fifths are plotted.

\frac{1}{3},\frac{7}{4},\frac{13}{5}

2\frac{1}{3},-2\frac{1}{3}

This figure is a number line ranging from negative 4 to 4 with tick marks for each integer. Negative 2 and 1 third, and 2 and 1 third are plotted.

1\frac{3}{5},-1\frac{3}{5}

In the following exercises, order each of the following pairs of numbers, using < or >.

-1___-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}

<

-3\frac{1}{4}___-4

-\phantom{\rule{0.2em}{0ex}}\frac{7}{9}___-\phantom{\rule{0.2em}{0ex}}\frac{4}{9}

>

-2___-\phantom{\rule{0.2em}{0ex}}\frac{19}{8}

Locate Decimals on the Number Line

In the following exercises, locate on the number line.

0.3

This figure is a number line ranging from 0 to 1 with tick marks for each tenth of an integer. 0.3 is plotted.

-0.2

-2.5

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. Negative 2.5 is plotted.

2.7

In the following exercises, order each of the following pairs of numbers, using < or >.

0.9___0.6

>

0.7___0.8

-0.6___-0.59

>

-0.27___-0.3

Properties of Real Numbers

Use the Commutative and Associative Properties

In the following exercises, use the Associative Property to simplify.

-12\left(4m\right)

-48m

30\left(\frac{5}{6}q\right)

\left(a+16\right)+31

a+47

\left(c+0.2\right)+0.7

In the following exercises, simplify.

6y+37+\left(-6y\right)

37

\frac{1}{4}+\frac{11}{15}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}\right)

\frac{14}{11}·\frac{35}{9}·\frac{14}{11}

\frac{35}{9}

-18·15·\frac{2}{9}

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

1\frac{7}{12}

\left(3.98d+0.75d\right)+1.25d

11x+8y+16x+15y

27x+23y

52m+\left(-20n\right)+\left(-18m\right)+\left(-5n\right)

Use the Identity and Inverse Properties of Addition and Multiplication

In the following exercises, find the additive inverse of each number.


\frac{1}{3}
5.1
-14
-\phantom{\rule{0.2em}{0ex}}\frac{8}{5}

-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}-5.1 14 \frac{8}{5}


-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}
-0.03
17
\frac{12}{5}

In the following exercises, find the multiplicative inverse of each number.

10 -\phantom{\rule{0.2em}{0ex}}\frac{4}{9} 0.6

\frac{1}{10}-\phantom{\rule{0.2em}{0ex}}\frac{9}{4}\frac{5}{3}

-\phantom{\rule{0.2em}{0ex}}\frac{9}{2}-7 2.1

Use the Properties of Zero

In the following exercises, simplify.

83·0

0

\frac{0}{9}

\frac{5}{0}

undefined

0÷\frac{2}{3}

In the following exercises, simplify.

43+39+\left(-43\right)

39

\left(n+6.75\right)+0.25

\frac{5}{13}·57·\frac{13}{5}

57

\frac{1}{6}·17·12

\frac{2}{3}·28·\frac{3}{7}

8

9\left(6x-11\right)+15

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the Distributive Property.

7\left(x+9\right)

7x+63

9\left(u-4\right)

-3\left(6m-1\right)

-18m+3

-8\left(-7a-12\right)

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

5n-2

\left(y+10\right)·p

\left(a-4\right)-\left(6a+9\right)

-5a-13

4\left(x+3\right)-8\left(x-7\right)

Systems of Measurement

1.1 Define U.S. Units of Measurement and Convert from One Unit to Another

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

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

84 inches

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

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

64 inches

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

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

2.7 miles

Shamu weights 4.5 tons. Convert the weight to pounds.

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

105 minutes

How many tablespoons are in a quart?

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

94 ounces

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

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

In the following exercises, solve.

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?

10 lbs. 6 oz.

Every day last week Pedro recorded the number of minutes he spent reading. The number of minutes were 50, 25, 83, 45, 32, 60, 135. How many hours did Pedro spend reading?

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?

15 feet

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

Make Unit Conversions in the Metric System

In the following exercises, convert the units.

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

170 centimeters

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

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

0.488 grams

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

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

2,900 grams

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

Use Mixed Units of Measurement in the Metric System

In the following exerices, solve.

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

1.12 meter

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

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

0.9 kilograms

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

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

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

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

175.3 centimeters

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

Caroline walked 2.5 kilometers. Convert this length to miles.

1.6 miles

Lucas weighs 78 kilograms. Convert his weight to pounds.

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

14.6 gallons

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

Convert between Fahrenheit and Celsius Temperatures

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

95° Fahrenheit

35° C

23° Fahrenheit

20° Fahrenheit

–6.7° C

64° Fahrenheit

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

30° Celsius

86° F

–5° Celsius

–12° Celsius

10.4° F

24° Celsius

Chapter Practice Test

Write as a whole number using digits: two hundred five thousand, six hundred seventeen.

205,617

Find the prime factorization of 504.

Find the Least Common Multiple of 18 and 24.

72

Combine like terms: 5n+8+2n-1.

In the following exercises, evaluate.

\text{−}|x| when x=-2

-2

11-a when a=-3

Translate to an algebraic expression and simplify: twenty less than negative 7.

-7-20;-27

Monique has a balance of \text{−}?18 in her checking account. She deposits ?152 to the account. What is the new balance?

Round 677.1348 to the nearest hundredth.

677.13

Convert \frac{4}{5} to a decimal.

Convert 1.85 to a percent.

185%

Locate \frac{2}{3},-1.5,\text{and}\phantom{\rule{0.2em}{0ex}}\frac{9}{4} on a number line.

In the following exercises, simplify each expression.

4+10\left(3+9\right)-{5}^{2}

99

-85+42

-19-25

-44

{\left(-2\right)}^{4}

-5\left(-9\right)÷15

3

\frac{3}{8}·\frac{11}{12}

\frac{4}{5}÷\frac{9}{20}

\frac{16}{9}

\frac{12+3·5}{15-6}

\frac{m}{7}+\frac{10}{7}

\frac{m+10}{7}

\frac{7}{12}-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}

-5.8+\left(-4.7\right)

-10.5

100-64.25

\left(0.07\right)\left(31.95\right)

2.2365

9÷0.05

-14\left(\frac{5}{7}p\right)

-10p

\left(u+8\right)-9

6x+\left(-4y\right)+9x+8y

15x+4y

\frac{0}{23}

\frac{75}{0}

undefined

-2\left(13q-5\right)

A movie lasted 1\frac{2}{3} hours. How many minutes did it last? (1 hour = 60 minutes)

100 minutes

Mike’s SUV is 5 feet 11 inches tall. He wants to put a rooftop cargo bag on the the SUV. The cargo bag is 1 foot 6 inches tall. What will the total height be of the SUV with the cargo bag on the roof? (1 foot = 12 inches)

Jennifer ran 2.8 miles. Convert this length to kilometers. (1 mile = 1.61 kilometers)

4.508 km

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Elementary Algebra by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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