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Mathematics LibreTexts

1.11: Systems of Measurement

  • Page ID
    30356
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    Skills to Develop

    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

    Note

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

    Make Unit Conversions in the U.S. System

    There are two systems of measurement commonly used around the world. Most countries use the metric system. The U.S. uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.

    The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart, and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, and hours.

    The equivalencies of measurements are shown in Table \(\PageIndex{1}\). The table also shows, in parentheses, the common abbreviations for each measurement.

    U.S. System of Measurement
    \(\begin{array} {llll} {} &{\text{1 foot (ft.)}} &{=} &{\text{12 inches (in.)}} \\ {\textbf{Length}} &{\text{1 yard (yd.)}} &{=} &{\text{3 feet (ft.)}} \\ {} &{\text{1 mile (mi.)}} &{=} &{\text{5280 feet (ft.)}} \end{array}\) \(\begin{array} {llll} {} &{\text{3 teaspoons (t)}} &{=} &{\text{1 tablespoon (T)}} \\ {} &{\text{16 tablespoons (T)}} &{=} &{\text{1 cup (C)}} \\ {} &{\text{1 cup (C)}} &{=} &{\text{8 fluid ounces (fl.oz.)}} \\ {\textbf{Volume}} &{\text{1 pint (pt.)}} &{=} &{\text{2 cups (C)}} \\ {} &{\text{1 quart (qt.)}} &{=} &{\text{2 pints (ft.)}} \\ {} &{\text{1 gallon (gal)}} &{=} &{\text{4 quarts (qt.)}} \end{array}\)

    \(\begin{array} {llll} {\textbf{Weight}} &{\text{1 pound (lb.)}} &{=} &{\text{16 ounces (oz.)}} \\ {} &{\text{1 ton}} &{=} &{\text{2000 pounds (lb.)}} \end{array}\)

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

    Table \(\PageIndex{1}\)

    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

    \(\begin{array} { l l } { \text {For any real number } a : } & { a \cdot 1 = a \quad 1 \cdot a = a } \\ { \textbf{1} \text { is the } \textbf{multiplicative identity } } \end{array}\)

    To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to change inches to feet. We know that 1 foot is equal to 12 inches, so we will write 1 as the fraction \(\frac{\text{1 foot}}{\text{12 inches}}\). When we multiply by this fraction we do not change the value, but just change the units.

    But \(\frac{\text{12 inches}}{\text{1 foot}}\) also equals 1. How do we decide whether to multiply by \(\frac{\text{1 foot}}{\text{12 inches}}\) or \(\frac{\text{12 inches}}{\text{1 foot}}\)? 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 6666 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.

    Figure \(\PageIndex{1}\)

    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.

    Exercise \(\PageIndex{1}\)

    Mary Anne is 66 inches tall. Convert her height into feet.

    Answer

    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.

    Exercise \(\PageIndex{2}\)

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

    Answer

    2.5 feet

    Exercise \(\PageIndex{3}\)

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

    Answer

    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.

    Exercise \(\PageIndex{4}\)

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

    Answer

    We will convert 3.2 tons into pounds. We will use the identity property of multiplication, writing 1 as the fraction \(\frac{\text{2000 pounds}}{\text{1 ton}}\).

    \(\text{3.2 tons}\)
    Multiply the measurement to be converted, by 1. \(\text{3.2 tons} \cdot 1\)
    Write 1 as a fraction relating tons and pounds. \(\text{3.2 tons} \cdot \frac{\text{2000 pounds}}{\text{1 ton}}\)
    Simplify. .
    Multiply. 6400 pounds
    Ndula weighs almost 6400 pounds.

    Exercise \(\PageIndex{5}\)

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

    Answer

    8600 pounds

    Exercise \(\PageIndex{6}\)

    The Carnival Destiny cruise ship weighs 51000 tons. Convert the weight to pounds.

    Answer

    102000000 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.

    Exercise \(\PageIndex{7}\)

    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.

    Answer

    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 \(\frac{\text{7 days}}{\text{1 week}}\), and \(\frac{\text{60 minutes}}{\text{1 hour}}\). \(\frac{\text{9 wk}}{\text{1}}\cdot\frac{\text{7 days}}{\text{1 wk}}\cdot\frac{\text{24 hr}}{\text{1 day}}\cdot\frac{\text{60 min}}{\text{1 hr}}\)
    Divide out the common units. .
    Multiply. \(\frac{9\cdot7\cdot24\cdot60\text{ min}}{1\cdot1\cdot1\cdot1}\)
    Multiply. 90,720 min

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

    Exercise \(\PageIndex{8}\)

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

    Answer

    440,000,000 yards

    Exercise \(\PageIndex{9}\)

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

    Answer

    151,200 minutes

    Exercise \(\PageIndex{10}\)

    How many ounces are in 1 gallon?

    Answer

    We will convert gallons to ounces by multiplying by several conversion factors. Refer to Table \(\PageIndex{1}\).

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

    Exercise \(\PageIndex{11}\)

    How many cups are in 1 gallon?

    Answer

    16 cups

    Exercise \(\PageIndex{12}\)

    How many teaspoons are in 1 cup?

    Answer

    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!

    Exercise \(\PageIndex{13}\)

    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?

    Answer

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

    Exercise \(\PageIndex{14}\)

    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?

    Answer

    9 lbs.8 oz

    Exercise \(\PageIndex{15}\)

    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?

    Answer

    21 ft. 6 in.

    Exercise \(\PageIndex{16}\)

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

    Answer

    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.

    Exercise \(\PageIndex{17}\)

    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?

    Answer

    4 lbs. 8 oz.

    Exercise \(\PageIndex{18}\)

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

    Answer

    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 Table \(\PageIndex{2}\). The common abbreviations for each measurement are given in parentheses.

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

    1 hectometer (hm) = 100 m

    1 dekameter (dam) = 10 m

    1 meter (m) = 1 m

    1 decimeter (dm) = 0.1 m

    1 centimeter (cm) = 0.01 m

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

    1 hectogram (hg) = 100 g

    1 dekagram (dag) = 10 g

    1 gram (g) = 1 g

    1 decigram (dg) = 0.1 g

    1 centigram (cg) = 0.01 g

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

    1 hectoliter (hL) = 100 L

    1 dekaliter (daL) = 10 L

    1 liter (L) = 1 L

    1 deciliter (dL) = 0.1 L

    1 centiliter (cL) = 0.01 L

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

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

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

    1 liter = 1,000 milliliters

    Table \(\PageIndex{2}\)

    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.

    Exercise \(\PageIndex{19}\)

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

    Answer

    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.

    Exercise \(\PageIndex{20}\)

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

    Answer

    5,000 meters

    Exercise \(\PageIndex{21}\)

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

    Answer

    250 centimeters

    Exercise \(\PageIndex{22}\)

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

    Answer

    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. \(\frac{3,200 \text{ kilograms}}{1,000}\)
    Divide. 3.2 kilograms
    The baby weighed 3.2 kilograms.

    Exercise \(\PageIndex{23}\)

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

    Answer

    2.8 kilograms

    Exercise \(\PageIndex{24}\)

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

    Answer

    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 Exercise \(\PageIndex{25}\), 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.

    Figure \(\PageIndex{2}\)

    Exercise \(\PageIndex{25}\)

    Convert

    1. 350 L to kiloliters
    2. 4.1 L to milliliters.
    Answer

    1. We will convert liters to kiloliters. In Table \(\PageIndex{2}\), we see that 1 kiloliter=1,000 liters.1kiloliter=1,000 liters.

    350 L
    Multiply by 1, writing 1 as a fraction relating liters to kiloliters. \(350 \text{ L}\frac{\text{1 kL}}{\text{1000L}}\)
    Simplify. \(350 \not{\text{ L}}\frac{\text{1 kL}}{1000 \not\text{ L}}\)
    . 0.35 kL


    2. We will convert liters to milliliters. From Table \(\PageIndex{2}\) we see that 1 liter=1,000 milliliters.1 liter=1,000 milliliters.
    .
    Multiply by 1, writing 1 as a fraction relating liters to milliliters. .
    Simplify. .
    Move the decimal 3 units to the right. .
    .

    Exercise \(\PageIndex{26}\)

    Convert:

    1. 725 L to kiloliters
    2. 6.3 L to milliliters
    Answer
    1. 7,250 kiloliters
    2. 6,300 milliliters

    Exercise \(\PageIndex{27}\)

    Convert:

    1. 350 hL to liters
    2. 4.1 L to centiliters
    Answer
    1. 35,000 liters
    2. 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.

    Exercise \(\PageIndex{28}\)

    Ryland is 1.6 meters tall. His younger brother is 85 centimeters tall. How much taller is Ryland than his younger brother?

    Answer

    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.

    \[\begin{array} { cc } { \text {Write the } 85 \text { centimeters as meters. } } & { 1.60 \mathrm { m } } \\ {} &{ \dfrac { - 0.85 \mathrm { m } } { 0.75 \mathrm { m } } } \end{array}\]

    Ryland is 0.75 m0.75 m taller than his brother.

    Exercise \(\PageIndex{29}\)

    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.

    Answer

    83 centimeters

    Exercise \(\PageIndex{30}\)

    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.

    Answer

    1.04 meters

    Exercise \(\PageIndex{31}\)

    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?

    Answer

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

    \(\begin{array} { ll } {} & { \text { Triple } 150 \text{ mL}} \\ { \text { Translate to algebra. } } &{3\cdot 150 \text{ mL}} \\ { \text { Multiply. } } &{450\text{ mL}}\\ { \text { Convert to liters. } } &{450\cdot \frac{0.001\text{ L}}{1 \text{ ml}}} \\ { \text { Simplify. } } &{0.45 \text{ L}}\\ {} &{ \text { Dena needs 0.45 liters of olive oil. } } \end{array}\)

    Exercise \(\PageIndex{32}\)

    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?

    Answer

    2 liters

    Exercise \(\PageIndex{33}\)

    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?

    Answer

    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.

    Table \(\PageIndex{3}\) shows some of the most common conversions.

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

    Table \(\PageIndex{3}\)

    Figure \(\PageIndex{3}\) shows how inches and centimeters are related on a ruler.

    A ruler with inches and centimeters.

    Figure \(\PageIndex{3}\): This ruler shows inches and centimeters.

    Figure \(\PageIndex{4}\) shows the ounce and milliliter markings on a measuring cup.

    A measuring cup showing milliliters and ounces.

    Figure \(\PageIndex{4}\): This measuring cup shows ounces and milliliters.

    Figure \(\PageIndex{5}\) shows how pounds and kilograms marked on a bathroom scale.

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

    Figure \(\PageIndex{5}\): This scale shows pounds and kilograms.

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

    Exercise \(\PageIndex{34}\)

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

    Answer

    \(\begin{array} { l l } {} & {500 \text{ mL}} \\ {\text{Multiplying by a unit conversion factor relating}} &{500\text{ milliliters}\cdot\frac{1\text{ ounce}}{30\text{ milliliters}}} \\ {\text{mL and ounces}} &{} \\ {\text{Simplify.}} &{\frac{50\text{ ounce}}{30}} \\ {\text{Divide.}} &{16.7\text{ ounces}} \\ {} &{\text{The water bottle has 16.7 ounces}} \end{array}\)

    Exercise \(\PageIndex{35}\)

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

    Answer

    2.12 quarts

    Exercise \(\PageIndex{36}\)

    How many liters are in 4 quarts of milk?

    Answer

    3.8 liters

    Exercise \(\PageIndex{37}\)

    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?

    Answer

    \(\begin{array} { l l } {} & {100 \text{ kilometers}} \\ {\text{Multiplying by a unit conversion factor relating}} &{100\text{ kilometers}\cdot\frac{1\text{ mile}}{1.61\text{ kilometers}}} \\ {\text{km and mi.}} &{} \\ {\text{Simplify.}} &{\frac{100\text{ miles}}{1.61}} \\ {\text{Divide.}} &{62\text{ miles}} \\ {} &{\text{Soleil will travel 62 miles.}} \end{array}\)

    Exercise \(\PageIndex{38}\)

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

    Answer

    19,335.6 feet

    Exercise \(\PageIndex{39}\)

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

    Answer

    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°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 \(\PageIndex{6}\) shows the relationship between the two systems.

    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.

    Figure \(\PageIndex{6}\): The diagram shows normal body temperature, along with the freezing and boiling temperatures of water in degrees Fahrenheit and degrees Celsius.

    TEMPERATURE CONVERSION

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

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

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

    \[F = \frac { 9 } { 5 } C + 32\]

    Exercise \(\PageIndex{40}\)

    Convert 50° Fahrenheit into degrees Celsius.

    Answer

    We will substitute 50°F into the formula to find C.

    .
    . .
    Simplify in parentheses. .
    Multiply. .

    So we found that 50°F is equivalent to 10°C.

    Exercise \(\PageIndex{41}\)

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

    Answer

    15°C

    Exercise \(\PageIndex{42}\)

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

    Answer

    5°C

    Exercise \(\PageIndex{43}\)

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

    Answer

    We will substitute 20°C into the formula to find F.

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

    Exercise \(\PageIndex{44}\)

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

    Answer

    59°F

    Exercise \(\PageIndex{45}\)

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

    Answer

    50°F

    Key Concepts

    • Metric System of Measurement
      • Length

    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 meter = 100 centimeters

    1 meter = 1,000 millimeters

    • Mass

    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 gram = 100 centigrams

    1 gram = 1,000 milligrams

    • Capacity

    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 liter = 100 centiliters

    1 liter = 1,000 milliliters

    • Temperature Conversion
      • To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula \(C=\frac{5}{9}(F−32)\)
      • To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula \(F=\frac{9}{5}C+32\)

    Section Exercises

    Practice Makes Perfect

    Make Unit Conversions in the U.S. System

    In the following exercises, convert the units.

    Exercise \(\PageIndex{1}\)

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

    Answer

    72 inches

    Exercise \(\PageIndex{2}\)

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

    Exercise \(\PageIndex{3}\)

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

    Answer

    1.5 feet

    Exercise \(\PageIndex{4}\)

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

    Exercise \(\PageIndex{5}\)

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

    Answer

    53\(\frac{1}{3}\) yards

    Exercise \(\PageIndex{6}\)

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

    Exercise \(\PageIndex{7}\)

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

    Answer

    7,920 feet

    Exercise \(\PageIndex{8}\)

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

    Exercise \(\PageIndex{9}\)

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

    Answer

    9,200 pounds

    Exercise \(\PageIndex{10}\)

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

    Exercise \(\PageIndex{11}\)

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

    Answer

    17\(\frac{1}{2}\) tons

    Exercise \(\PageIndex{12}\)

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

    Exercise \(\PageIndex{13}\)

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

    Answer

    5,400 s

    Exercise \(\PageIndex{14}\)

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

    Exercise \(\PageIndex{15}\)

    How many teaspoons are in a pint?

    Answer

    96 teaspoons

    Exercise \(\PageIndex{16}\)

    How many tablespoons are in a gallon?

    Exercise \(\PageIndex{17}\)

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

    Answer

    224 ounces

    Exercise \(\PageIndex{18}\)

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

    Exercise \(\PageIndex{19}\)

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

    Answer

    1\(\frac{1}{4}\) gallons

    Exercise \(\PageIndex{20}\)

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

    Exercise \(\PageIndex{21}\)

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

    Answer

    26 in.

    Exercise \(\PageIndex{22}\)

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

    Exercise \(\PageIndex{23}\)

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

    Answer

    65 days

    Exercise \(\PageIndex{24}\)

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

    Exercise \(\PageIndex{25}\)

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

    Answer

    115 ounces

    Exercise \(\PageIndex{26}\)

    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.

    Exercise \(\PageIndex{27}\)

    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?

    Answer

    8 lbs. 13 oz.

    Exercise \(\PageIndex{28}\)

    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?

    Exercise \(\PageIndex{29}\)

    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?

    Answer

    3.05 hours

    Exercise \(\PageIndex{30}\)

    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?

    Exercise \(\PageIndex{31}\)

    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?

    Answer

    10 ft. 2 in.

    Exercise \(\PageIndex{32}\)

    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?

    Exercise \(\PageIndex{33}\)

    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?

    Answer

    4 yards

    Exercise \(\PageIndex{34}\)

    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.

    Exercise \(\PageIndex{35}\)

    Ghalib ran 5 kilometers. Convert the length to meters.

    Answer

    5,000 meters

    Exercise \(\PageIndex{36}\)

    Kitaka hiked 8 kilometers. Convert the length to meters.

    Exercise \(\PageIndex{37}\)

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

    Answer

    155 centimeters

    Exercise \(\PageIndex{38}\)

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

    Exercise \(\PageIndex{39}\)

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

    Answer

    3.072 kilometers

    Exercise \(\PageIndex{40}\)

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

    Exercise \(\PageIndex{41}\)

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

    Answer

    1.5 grams

    Exercise \(\PageIndex{42}\)

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

    Exercise \(\PageIndex{43}\)

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

    Answer

    91,600 milligrams

    Exercise \(\PageIndex{44}\)

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

    Exercise \(\PageIndex{45}\)

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

    Answer

    2,000 grams

    Exercise \(\PageIndex{46}\)

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

    Exercise \(\PageIndex{47}\)

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

    Answer

    0.75 liters

    Exercise \(\PageIndex{48}\)

    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.

    Exercise \(\PageIndex{49}\)

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

    Answer

    91 centimeters

    Exercise \(\PageIndex{50}\)

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

    Exercise \(\PageIndex{51}\)

    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?

    Answer

    855 grams

    Exercise \(\PageIndex{52}\)

    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?

    Exercise \(\PageIndex{53}\)

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

    Answer

    2.1 kilograms

    Exercise \(\PageIndex{54}\)

    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?

    Exercise \(\PageIndex{55}\)

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

    Answer

    1.6 liters

    Exercise \(\PageIndex{56}\)

    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.

    Exercise \(\PageIndex{57}\)

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

    Answer

    190.5 centimeters

    Exercise \(\PageIndex{58}\)

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

    Exercise \(\PageIndex{59}\)

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

    Answer

    21.9 meters

    Exercise \(\PageIndex{60}\)

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

    Exercise \(\PageIndex{61}\)

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

    Answer

    742.5 kilograms

    Exercise \(\PageIndex{62}\)

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

    Exercise \(\PageIndex{63}\)

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

    Answer

    3.1 miles

    Exercise \(\PageIndex{64}\)

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

    Exercise \(\PageIndex{65}\)

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

    Answer

    44 pounds

    Exercise \(\PageIndex{66}\)

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

    Exercise \(\PageIndex{67}\)

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

    Answer

    53.2 liters

    Exercise \(\PageIndex{68}\)

    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.

    Exercise \(\PageIndex{69}\)

    86° Fahrenheit

    Answer

    30°C

    Exercise \(\PageIndex{70}\)

    77° Fahrenheit

    Exercise \(\PageIndex{71}\)

    104° Fahrenheit

    Answer

    40°C

    Exercise \(\PageIndex{72}\)

    14° Fahrenheit

    Exercise \(\PageIndex{73}\)

    72° Fahrenheit

    Answer

    22.2°C

    Exercise \(\PageIndex{74}\)

    4° Fahrenheit

    Exercise \(\PageIndex{75}\)

    0° Fahrenheit

    Answer

    −17.8°C

    Exercise \(\PageIndex{76}\)

    120° Fahrenheit

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

    Exercise \(\PageIndex{77}\)

    5° Celsius

    Answer

    41°F

    Exercise \(\PageIndex{78}\)

    25° Celsius

    Exercise \(\PageIndex{79}\)

    −10° Celsius

    Answer

    14°F

    Exercise \(\PageIndex{80}\)

    −15° Celsius

    Exercise \(\PageIndex{81}\)

    22° Celsius

    Answer

    71.6°F

    Exercise \(\PageIndex{82}\)

    8° Celsius

    Exercise \(\PageIndex{83}\)

    43° Celsius

    Answer

    109.4°F

    Exercise \(\PageIndex{84}\)

    16° Celsius

    Everyday Math

    Exercise \(\PageIndex{85}\)

    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?

    Answer

    14.6 kilograms

    Exercise \(\PageIndex{86}\)

    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

    Exercise \(\PageIndex{87}\)

    Some people think that 65° to 75° Fahrenheit is the ideal temperature range.

    1. What is your ideal temperature range? Why do you think so?
    2. Convert your ideal temperatures from Fahrenheit to Celsius.
    Answer

    Answers may vary.

    Exercise \(\PageIndex{88}\)

    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.

    Exercise \(\PageIndex{1}\)

    26,915

    1. 1
    2. 2
    3. 9
    4. 5
    5. 6
    Answer
    1. tens
    2. ten thousands
    3. hundreds
    4. ones
    5. thousands

    Exercise \(\PageIndex{2}\)

    359,417

    1. 9
    2. 3
    3. 4
    4. 7
    5. 1

    Exercise \(\PageIndex{3}\)

    58,129,304

    1. 5
    2. 0
    3. 1
    4. 8
    5. 2
    Answer
    1. ten millions
    2. tens
    3. hundred thousands
    4. millions
    5. ten thousands

    Exercise \(\PageIndex{4}\)

    9,430,286,157

    1. 6
    2. 4
    3. 9
    4. 0
    5. 5

    In the following exercises, name each number.

    Exercise \(\PageIndex{5}\)

    6,104

    Answer

    six thousand, one hundred four

    Exercise \(\PageIndex{6}\)

    493,068

    Exercise \(\PageIndex{7}\)

    3,975,284

    Answer

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

    Exercise \(\PageIndex{8}\)

    85,620,435

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

    Exercise \(\PageIndex{9}\)

    three hundred fifteen

    Answer

    315

    Exercise \(\PageIndex{10}\)

    sixty-five thousand, nine hundred twelve

    Exercise \(\PageIndex{11}\)

    ninety million, four hundred twenty-five thousand, sixteen

    Answer

    90,425,016

    Exercise \(\PageIndex{12}\)

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

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

    Exercise \(\PageIndex{13}\)

    Round to the nearest ten.

    1. 407
    2. 8,564
    Answer
    1. 410
    2. 8,560

    Exercise \(\PageIndex{14}\)

    Round to the nearest hundred.

    1. 25,846
    2. 25,864

    In the following exercises, round each number to the nearest 1. hundred 2. thousand 3. ten thousand.

    Exercise \(\PageIndex{15}\)

    864,951

    Answer
    1. 865,000865,000
    2. 865,000865,000
    3. 860,000

    Exercise \(\PageIndex{16}\)

    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.

    Exercise \(\PageIndex{17}\)

    168

    Answer

    by 2,3,6

    Exercise \(\PageIndex{18}\)

    264

    Exercise \(\PageIndex{19}\)

    375

    Answer

    by 3,5

    Exercise \(\PageIndex{20}\)

    750

    Exercise \(\PageIndex{21}\)

    1430

    Answer

    by 2,5,10

    Exercise \(\PageIndex{22}\)

    1080

    Find Prime Factorizations and Least Common Multiples

    In the following exercises, find the prime factorization.

    Exercise \(\PageIndex{23}\)

    420

    Answer

    2\(\cdot 2 \cdot 3 \cdot 5 \cdot 7\)

    Exercise \(\PageIndex{24}\)

    115

    Exercise \(\PageIndex{25}\)

    225

    Answer

    3\(\cdot 3 \cdot 5 \cdot 5\)

    Exercise \(\PageIndex{26}\)

    2475

    Exercise \(\PageIndex{27}\)

    1560

    Answer

    \(2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 13\)

    Exercise \(\PageIndex{28}\)

    56

    Exercise \(\PageIndex{29}\)

    72

    Answer

    \(2 \cdot 2 \cdot 2 \cdot 3 \cdot 3\)

    Exercise \(\PageIndex{30}\)

    168

    Exercise \(\PageIndex{31}\)

    252

    Answer

    \(2 \cdot 2 \cdot 3 \cdot 3 \cdot 7\)

    Exercise \(\PageIndex{32}\)

    391

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

    Exercise \(\PageIndex{33}\)

    6,15

    Answer

    30

    Exercise \(\PageIndex{34}\)

    60, 75

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

    Exercise \(\PageIndex{35}\)

    24, 30

    Answer

    120

    Exercise \(\PageIndex{36}\)

    70, 84

    Use the Language of Algebra

    Use Variables and Algebraic Symbols

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

    Exercise \(\PageIndex{37}\)

    25−7

    Answer

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

    Exercise \(\PageIndex{38}\)

    5\(\cdot 6\)

    Exercise \(\PageIndex{39}\)

    \(45 \div 5\)

    Answer

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

    Exercise \(\PageIndex{40}\)

    x+8

    Exercise \(\PageIndex{41}\)

    \(42 \geq 27\)

    Answer

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

    Exercise \(\PageIndex{42}\)

    3n=24

    Exercise \(\PageIndex{43}\)

    \(3 \leq 20 \div 4\)

    Answer

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

    Exercise \(\PageIndex{44}\)

    \(a \neq 7 \cdot 4\)

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

    Exercise \(\PageIndex{45}\)

    \(6 \cdot 3+5\)

    Answer

    expression

    Exercise \(\PageIndex{46}\)

    y−8=32

    Simplify Expressions Using the Order of Operations

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{47}\)

    \(3^{5}\)

    Answer

    243

    Exercise \(\PageIndex{48}\)

    \(10^{8}\)

    In the following exercises, simplify

    Exercise \(\PageIndex{49}\)

    6+10/2+2

    Answer

    13

    Exercise \(\PageIndex{50}\)

    9+12/3+4

    Exercise \(\PageIndex{51}\)

    \(20 \div(4+6) \cdot 5\)

    Answer

    10

    Exercise \(\PageIndex{52}\)

    \(33 \div(3+8) \cdot 2\)

    Exercise \(\PageIndex{53}\)

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

    Answer

    41

    Exercise \(\PageIndex{54}\)

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

    Evaluate an Expression

    In the following exercises, evaluate the following expressions.

    Exercise \(\PageIndex{55}\)

    9x+7 when x=3

    Answer

    34

    Exercise \(\PageIndex{56}\)

    5x−4 when x=6

    Exercise \(\PageIndex{57}\)

    \(x^{4}\) when \(x=3\)

    Answer

    81

    Exercise \(\PageIndex{58}\)

    \(3^{x}\) when \(x=3\)

    Exercise \(\PageIndex{59}\)

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

    Answer

    58

    Exercise \(\PageIndex{60}\)

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

    Simplify Expressions by Combining Like Terms

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

    Exercise \(\PageIndex{61}\)

    12n

    Answer

    12

    Exercise \(\PageIndex{62}\)

    9\(x^{2}\)

    In the following exercises, identify the like terms.

    Exercise \(\PageIndex{63}\)

    \(3 n, n^{2}, 12,12 p^{2}, 3,3 n^{2}\)

    Answer

    12 and \(3, n^{2}\) and 3\(n^{2}\)

    Exercise \(\PageIndex{64}\)

    \(5,18 r^{2}, 9 s, 9 r, 5 r^{2}, 5 s\)

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

    Exercise \(\PageIndex{65}\)

    \(11 x^{2}+3 x+6\)

    Answer

    \(11 x^{2}, 3 x, 6\)

    Exercise \(\PageIndex{66}\)

    \(22 y^{3}+y+15\)

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

    Exercise \(\PageIndex{67}\)

    17a+9a

    Answer

    26a

    Exercise \(\PageIndex{68}\)

    18z+9z

    Exercise \(\PageIndex{69}\)

    9x+3x+8

    Answer

    12x+8

    Exercise \(\PageIndex{70}\)

    8a+5a+9

    Exercise \(\PageIndex{71}\)

    7p+6+5p−4

    Answer

    12p+2

    Exercise \(\PageIndex{72}\)

    8x+7+4x−5

    Translate an English Phrase to an Algebraic Expression

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

    Exercise \(\PageIndex{73}\)

    the sum of 8 and 12

    Answer

    8+12

    Exercise \(\PageIndex{74}\)

    the sum of 9 and 1

    Exercise \(\PageIndex{75}\)

    the difference of x and 4

    Answer

    x−4

    Exercise \(\PageIndex{76}\)

    the difference of x and 3

    Exercise \(\PageIndex{77}\)

    the product of 6 and y

    Answer

    6y

    Exercise \(\PageIndex{78}\)

    the product of 9 and y

    Exercise \(\PageIndex{79}\)

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

    Answer

    b+15

    Exercise \(\PageIndex{80}\)

    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 >.

    Exercise \(\PageIndex{81}\)

    1. 6___2
    2. −7___4
    3. −9___−1
    4. 9___−3

    Answer
    1. >
    2. <
    3. <
    4. >

    Exercise \(\PageIndex{82}\)

    1. −5___1
    2. −4___−9
    3. 6___10
    4. 3___−8

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

    Exercise \(\PageIndex{83}\)

    1. −8
    2. 1
    Answer
    1. 8
    2. −1

    Exercise \(\PageIndex{84}\)

    1. −2
    2. 6

    In the following exercises, simplify.

    Exercise \(\PageIndex{85}\)

    −(−19)

    Answer

    19

    Exercise \(\PageIndex{86}\)

    −(−53)

    In the following exercises, simplify.

    Exercise \(\PageIndex{87}\)

    −m when

    1. m=3
    2. m=−3
    Answer
    1. −3
    2. 3

    Exercise \(\PageIndex{88}\)

    −p when

    1. p=6
    2. p=−6

    Simplify Expressions with Absolute Value

    In the following exercises,, simplify.

    Exercise \(\PageIndex{89}\)

    1. |7|
    2. |−25|
    3. |0|
    Answer
    1. 7
    2. 25
    3. 0

    Exercise \(\PageIndex{90}\)

    1. |5|
    2. |0|
    3. |−19|

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

    Exercise \(\PageIndex{91}\)

    1. −8___|−8|
    2. −|−2|___−2
    Answer
    1. <
    2. =

    Exercise \(\PageIndex{92}\)

    1. |−3|___−|−3|
    2. 4___−|−4|

    In the following exercises, simplify.

    Exercise \(\PageIndex{93}\)

    |8−4|

    Answer

    4

    Exercise \(\PageIndex{94}\)

    |9−6|

    Exercise \(\PageIndex{95}\)

    8(14−2|−2|)

    Answer

    80

    Exercise \(\PageIndex{96}\)

    6(13−4|−2|)

    In the following exercises, evaluate.

    Exercise \(\PageIndex{97}\)

    1. |x| when x=−28

    Answer
    1. 28
    2. 15

    Exercise \(\PageIndex{98}\)

    1. ∣y∣ when y=−37
    2. |−z| when z=−24

    Add Integers

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{99}\)

    −200+65

    Answer

    −135

    Exercise \(\PageIndex{100}\)

    −150+45

    Exercise \(\PageIndex{101}\)

    2+(−8)+6

    Answer

    0

    Exercise \(\PageIndex{102}\)

    4+(−9)+7

    Exercise \(\PageIndex{103}\)

    140+(−75)+67

    Answer

    132

    Exercise \(\PageIndex{104}\)

    −32+24+(−6)+10

    Subtract Integers

    In the following exercises, simplify.

    Exercise \(\PageIndex{105}\)

    9−3

    Answer

    6

    Exercise \(\PageIndex{106}\)

    −5−(−1)

    Exercise \(\PageIndex{107}\)

    1. 15−6
    2. 15+(−6)
    Answer
    1. 9
    2. 9

    Exercise \(\PageIndex{108}\)

    1. 12−9
    2. 12+(−9)

    Exercise \(\PageIndex{109}\)

    1. 8−(−9)
    2. 8+9
    Answer
    1. 17
    2. 17

    Exercise \(\PageIndex{110}\)

    1. 4−(−4)
    2. 4+4

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{111}\)

    10−(−19)

    Answer

    29

    Exercise \(\PageIndex{112}\)

    11−(−18)

    Exercise \(\PageIndex{113}\)

    31−79

    Answer

    −48

    Exercise \(\PageIndex{114}\)

    39−81

    Exercise \(\PageIndex{115}\)

    −31−11

    Answer

    −42

    Exercise \(\PageIndex{116}\)

    −32−18

    Exercise \(\PageIndex{117}\)

    −15−(−28)+5

    Answer

    18

    Exercise \(\PageIndex{118}\)

    71+(−10)−8

    Exercise \(\PageIndex{119}\)

    −16−(−4+1)−7

    Answer

    -20

    Exercise \(\PageIndex{120}\)

    −15−(−6+4)−3

    Multiply Integers

    In the following exercises, multiply.

    Exercise \(\PageIndex{121}\)

    −5(7)

    Answer

    −35

    Exercise \(\PageIndex{122}\)

    −8(6)

    Exercise \(\PageIndex{123}\)

    −18(−2)

    Answer

    36

    Exercise \(\PageIndex{124}\)

    −10(−6)

    Divide Integers

    In the following exercises, divide.

    Exercise \(\PageIndex{125}\)

    \(-28 \div 7\)

    Answer

    -4

    Exercise \(\PageIndex{126}\)

    \(56 \div(-7)\)

    Exercise \(\PageIndex{127}\)

    \(-120 \div(-20)\)

    Answer

    6

    Exercise \(\PageIndex{128}\)

    \(-200 \div 25\)

    Simplify Expressions with Integers

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{129}\)

    −8(−2)−3(−9)

    Answer

    43

    Exercise \(\PageIndex{130}\)

    −7(−4)−5(−3)

    Exercise \(\PageIndex{131}\)

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

    Answer

    −125

    Exercise \(\PageIndex{132}\)

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

    Exercise \(\PageIndex{133}\)

    \(-4 \cdot 2 \cdot 11\)

    Answer

    −88

    Exercise \(\PageIndex{134}\)

    \(-5 \cdot 3 \cdot 10\)

    Exercise \(\PageIndex{135}\)

    \(-10(-4) \div(-8)\)

    Answer

    -5

    Exercise \(\PageIndex{136}\)

    \(-8(-6) \div(-4)\)

    Exercise \(\PageIndex{137}\)

    31−4(3−9)

    Answer

    55

    Exercise \(\PageIndex{138}\)

    24−3(2−10)

    Evaluate Variable Expressions with Integers

    In the following exercises, evaluate each expression.

    Exercise \(\PageIndex{139}\)

    x+8 when

    1. x=−26
    2. x=−95
    Answer
    1. −18
    2. −87

    Exercise \(\PageIndex{140}\)

    y+9 when

    1. y=−29
    2. y=−84

    Exercise \(\PageIndex{141}\)

    When b=−11, evaluate:

    1. b+6
    2. −b+6
    Answer
    1. −5
    2. 17

    Exercise \(\PageIndex{142}\)

    When c=−9, evaluate:

    1. c+(−4)c+(−4)
    2. −c+(−4)

    Exercise \(\PageIndex{143}\)

    \(p^{2}-5 p+2\) when
    \(p=-1\)

    Answer

    8

    Exercise \(\PageIndex{144}\)

    \(q^{2}-2 q+9\) when \(q=-2\)

    Exercise \(\PageIndex{145}\)

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

    Answer

    38

    Exercise \(\PageIndex{146}\)

    \(3 p-2 q+9\) 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.

    Exercise \(\PageIndex{147}\)

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

    Answer

    (−4+(−17))+32;11

    Exercise \(\PageIndex{148}\)

    1. the difference of 15 and −7
    2. subtract 15 from −7

    Exercise \(\PageIndex{149}\)

    the quotient of −45 and −9

    Answer

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

    Exercise \(\PageIndex{150}\)

    the product of −12 and the difference of c and d

    Use Integers in Applications

    In the following exercises, solve.

    Exercise \(\PageIndex{151}\)

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

    Answer

    84 degrees

    Exercise \(\PageIndex{152}\)

    Checking Account Adrianne has a balance of −$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.

    Exercise \(\PageIndex{153}\)

    \(\frac{1}{4}\)

    Answer

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

    Exercise \(\PageIndex{154}\)

    \(\frac{1}{3}\)

    Exercise \(\PageIndex{155}\)

    \(\frac{5}{6}\)

    Answer

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

    Exercise \(\PageIndex{156}\)

    \(\frac{2}{7}\)

    Simplify Fractions

    In the following exercises, simplify.

    Exercise \(\PageIndex{157}\)

    \(\frac{7}{21}\)

    Answer

    \(\frac{1}{3}\)

    Exercise \(\PageIndex{158}\)

    \(\frac{8}{24}\)

    Exercise \(\PageIndex{159}\)

    \(\frac{15}{20}\)

    Answer

    \(\frac{3}{4}\)

    Exercise \(\PageIndex{160}\)

    \(\frac{12}{18}\)

    Exercise \(\PageIndex{161}\)

    \(-\frac{168}{192}\)

    Answer

    \(-\frac{7}{8}\)

    Exercise \(\PageIndex{162}\)

    \(-\frac{140}{224}\)

    Exercise \(\PageIndex{163}\)

    \(\frac{11 x}{11 y}\)

    Answer

    \(\frac{x}{y}\)

    Exercise \(\PageIndex{164}\)

    \(\frac{15 a}{15 b}\)

    Multiply Fractions

    In the following exercises, multiply.

    Exercise \(\PageIndex{165}\)

    \(\frac{2}{5} \cdot \frac{1}{3}\)

    Answer

    \(\frac{2}{15}\)

    Exercise \(\PageIndex{166}\)

    \(\frac{1}{2} \cdot \frac{3}{8}\)

    Exercise \(\PageIndex{167}\)

    \(\frac{7}{12}\left(-\frac{8}{21}\right)\)

    Answer

    \(-\frac{2}{9}\)

    Exercise \(\PageIndex{168}\)

    \(\frac{5}{12}\left(-\frac{8}{15}\right)\)

    Exercise \(\PageIndex{169}\)

    \(-28 p\left(-\frac{1}{4}\right)\)

    Answer

    7p

    Exercise \(\PageIndex{170}\)

    \(-51 q\left(-\frac{1}{3}\right)\)

    Exercise \(\PageIndex{172}\)

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

    Answer

    −42

    Exercise \(\PageIndex{173}\)

    \(-1\left(-\frac{3}{8}\right)\)

    Divide Fractions

    In the following exercises, divide

    Exercise \(\PageIndex{174}\)

    \(\frac{1}{2} \div \frac{1}{4}\)

    Answer

    2

    Exercise \(\PageIndex{175}\)

    \(\frac{1}{2} \div \frac{1}{8}\)

    Exercise \(\PageIndex{176}\)

    \(-\frac{4}{5} \div \frac{4}{7}\)

    Answer

    \(-\frac{7}{5}\)

    Exercise \(\PageIndex{177}\)

    \(-\frac{3}{4} \div \frac{3}{5}\)

    Exercise \(\PageIndex{178}\)

    \(\frac{5}{8} \div \frac{a}{10}\)

    Answer

    \(\frac{25}{4 a}\)

    Exercise \(\PageIndex{179}\)

    \(\frac{5}{6} \div \frac{c}{15}\)

    Exercise \(\PageIndex{180}\)

    \(\frac{7 p}{12} \div \frac{21 p}{8}\)

    Answer

    \(\frac{2}{9}\)

    Exercise \(\PageIndex{181}\)

    \(\frac{5 q}{12} \div \frac{15 q}{8}\)

    Exercise \(\PageIndex{182}\)

    \(\frac{2}{5} \div(-10)\)

    Answer

    \(-\frac{1}{25}\)

    Exercise \(\PageIndex{183}\)

    \(-18 \div-\left(\frac{9}{2}\right)\)

    In the following exercises, simplify.

    Exercise \(\PageIndex{184}\)

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

    Answer

    \(\frac{3}{4}\)

    Exercise \(\PageIndex{185}\)

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

    Exercise \(\PageIndex{186}\)

    \(\frac{-\frac{9}{10}}{3}\)

    Answer

    \(-\frac{3}{10}\)

    Exercise \(\PageIndex{187}\)

    \(\frac{2}{\frac{5}{8}}\)

    Exercise \(\PageIndex{188}\)

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

    Answer

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

    Exercise \(\PageIndex{189}\)

    \(\frac{-\frac{x}{6}}{-\frac{8}{9}}\)

    Simplify Expressions Written with a Fraction Bar

    In the following exercises, simplify.

    Exercise \(\PageIndex{190}\)

    \(\frac{4+11}{8}\)

    Answer

    \(\frac{15}{8}\)

    Exercise \(\PageIndex{191}\)

    \(\frac{9+3}{7}\)

    Exercise \(\PageIndex{192}\)

    \(\frac{30}{7-12}\)

    Answer

    -6

    Exercise \(\PageIndex{193}\)

    \(\frac{15}{4-9}\)

    Exercise \(\PageIndex{194}\)

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

    Answer

    \(\frac{4}{3}\)

    Exercise \(\PageIndex{195}\)

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

    Exercise \(\PageIndex{196}\)

    \(\frac{5 \cdot 8}{-10}\)

    Answer

    -4

    Exercise \(\PageIndex{197}\)

    \(\frac{3 \cdot 4}{-24}\)

    Exercise \(\PageIndex{198}\)

    \(\frac{15 \cdot 5-5^{2}}{2 \cdot 10}\)

    Answer

    \(\frac{5}{2}\)

    Exercise \(\PageIndex{199}\)

    \(\frac{12 \cdot 9-3^{2}}{3 \cdot 18}\)

    Exercise \(\PageIndex{200}\)

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

    Answer

    -2

    Exercise \(\PageIndex{201}\)

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

    Translate Phrases to Expressions with Fractions

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

    Exercise \(\PageIndex{202}\)

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

    Answer

    \(\frac{c}{d+9}\)

    Exercise \(\PageIndex{203}\)

    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.

    Exercise \(\PageIndex{204}\)

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

    Answer

    \(\frac{5}{9}\)

    Exercise \(\PageIndex{205}\)

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

    Exercise \(\PageIndex{206}\)

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

    Answer

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

    Exercise \(\PageIndex{207}\)

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

    Exercise \(\PageIndex{208}\)

    \(-\frac{1}{8}+\left(-\frac{3}{8}\right)\)

    Answer

    \(-\frac{1}{2}\)

    Exercise \(\PageIndex{209}\)

    \(-\frac{1}{8}+\left(-\frac{5}{8}\right)\)

    In the following exercises, subtract.

    Exercise \(\PageIndex{210}\)

    \(\frac{4}{5}-\frac{1}{5}\)

    Answer

    \(\frac{3}{5}\)

    Exercise \(\PageIndex{211}\)

    \(\frac{4}{5}-\frac{3}{5}\)

    Exercise \(\PageIndex{212}\)

    \(\frac{y}{17}-\frac{9}{17}\)

    Answer

    \(\frac{y-9}{17}\)

    Exercise \(\PageIndex{213}\)

    \(\frac{x}{19}-\frac{8}{19}\)

    Exercise \(\PageIndex{214}\)

    \(-\frac{8}{d}-\frac{3}{d}\)

    Answer

    \(-\frac{11}{d}\)

    Exercise \(\PageIndex{215}\)

    \(-\frac{7}{c}-\frac{7}{c}\)

    Add or Subtract Fractions with Different Denominators

    In the following exercises, add or subtract.

    Exercise \(\PageIndex{216}\)

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

    Answer

    \(\frac{8}{15}\)

    Exercise \(\PageIndex{217}\)

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

    Exercise \(\PageIndex{218}\)

    \(\frac{1}{5}-\left(-\frac{1}{10}\right)\)

    Answer

    \(\frac{3}{10}\)

    Exercise \(\PageIndex{219}\)

    \(\frac{1}{2}-\left(-\frac{1}{6}\right)\)

    Exercise \(\PageIndex{220}\)

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

    Answer

    \(\frac{17}{12}\)

    Exercise \(\PageIndex{221}\)

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

    Exercise \(\PageIndex{222}\)

    \(\frac{11}{12}-\frac{3}{8}\)

    Answer

    \(\frac{13}{24}\)

    Exercise \(\PageIndex{223}\)

    \(\frac{5}{8}-\frac{7}{12}\)

    Exercise \(\PageIndex{224}\)

    \(-\frac{9}{16}-\left(-\frac{4}{5}\right)\)

    Answer

    \(\frac{19}{80}\)

    Exercise \(\PageIndex{225}\)

    \(-\frac{7}{20}-\left(-\frac{5}{8}\right)\)

    Exercise \(\PageIndex{226}\)

    \(1+\frac{5}{6}\)

    Answer

    \(\frac{11}{6}\)

    Exercise \(\PageIndex{227}\)

    \(1-\frac{5}{9}\)

    Use the Order of Operations to Simplify Complex Fractions

    In the following exercises, simplify.

    Exercise \(\PageIndex{228}\)

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

    Answer

    \(\frac{1}{275}\)

    Exercise \(\PageIndex{229}\)

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

    Exercise \(\PageIndex{230}\)

    \(\frac{\frac{2}{3}+\frac{1}{2}}{\frac{3}{4}-\frac{2}{3}}\)

    Answer

    14

    Exercise \(\PageIndex{231}\)

    \(\frac{\frac{3}{4}+\frac{1}{2}}{\frac{5}{6}-\frac{2}{3}}\)

    Evaluate Variable Expressions with Fractions

    In the following exercises, evaluate.

    Exercise \(\PageIndex{232}\)

    \(x+\frac{1}{2}\) when

    1. \(x=-\frac{1}{8}\)
    2. \(x=-\frac{1}{2}\)
    Answer
    1. \(\frac{3}{8}\)
    2. \(0\)

    Exercise \(\PageIndex{233}\)

    \(x+\frac{2}{3}\) when

    1. \(x=-\frac{1}{6}\)
    2. \(x=-\frac{5}{3}\)

    Exercise \(\PageIndex{234}\)

    4\(p^{2} q\) when \(p=-\frac{1}{2}\) and \(q=\frac{5}{9}\)

    Answer

    \(\frac{5}{9}\)

    Exercise \(\PageIndex{235}\)

    5\(m^{2} n\) when \(m=-\frac{2}{5}\) and \(n=\frac{1}{3}\)

    Exercise \(\PageIndex{236}\)

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

    Answer

    -6

    Exercise \(\PageIndex{237}\)

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

    Decimals

    Name and Write Decimals

    In the following exercises, write as a decimal.

    Exercise \(\PageIndex{238}\)

    Eight and three hundredths

    Answer

    8.03

    Exercise \(\PageIndex{239}\)

    Nine and seven hundredths

    Exercise \(\PageIndex{240}\)

    One thousandth

    Answer

    0.001

    Exercise \(\PageIndex{241}\)

    Nine thousandths

    In the following exercises, name each decimal.

    Exercise \(\PageIndex{242}\)

    7.8

    Answer

    seven and eight tenths

    Exercise \(\PageIndex{243}\)

    5.01

    Exercise \(\PageIndex{244}\)

    0.005

    Answer

    five thousandths

    Exercise \(\PageIndex{245}\)

    0.381

    Round Decimals

    In the following exercises, round each number to the nearest

    1. hundredth
    2. tenth
    3. whole number.

    Exercise \(\PageIndex{246}\)

    5.7932

    Answer
    1. 5.79
    2. 5.8
    3. 6

    Exercise \(\PageIndex{247}\)

    3.6284

    Exercise \(\PageIndex{248}\)

    12.4768

    Answer
    1. 12.48
    2. 12.5
    3. 12

    Exercise \(\PageIndex{249}\)

    25.8449

    Add and Subtract Decimals

    In the following exercises, add or subtract.

    Exercise \(\PageIndex{250}\)

    18.37+9.36

    Answer

    27.73

    Exercise \(\PageIndex{251}\)

    256.37−85.49

    Exercise \(\PageIndex{252}\)

    15.35−20.88

    Answer

    −5.53

    Exercise \(\PageIndex{253}\)

    37.5+12.23

    Exercise \(\PageIndex{254}\)

    −4.2+(−9.3)

    Answer

    −13.5

    Exercise \(\PageIndex{255}\)

    −8.6+(−8.6)

    Exercise \(\PageIndex{256}\)

    100−64.2

    Answer

    35.8

    Exercise \(\PageIndex{257}\)

    100−65.83

    Exercise \(\PageIndex{258}\)

    2.51+40

    Answer

    42.51

    Exercise \(\PageIndex{259}\)

    9.38+60

    Multiply and Divide Decimals

    In the following exercises, multiply.

    Exercise \(\PageIndex{260}\)

    (0.3)(0.4)

    Answer

    0.12

    Exercise \(\PageIndex{261}\)

    (0.6)(0.7)

    Exercise \(\PageIndex{262}\)

    (8.52)(3.14)

    Answer

    26.7528

    Exercise \(\PageIndex{263}\)

    (5.32)(4.86)

    Exercise \(\PageIndex{264}\)

    (0.09)(24.78)

    Answer

    2.2302

    Exercise \(\PageIndex{265}\)

    (0.04)(36.89)

    In the following exercises, divide.

    Exercise \(\PageIndex{266}\)

    \(0.15 \div 5\)

    Answer

    0.03

    Exercise \(\PageIndex{267}\)

    \(0.27 \div 3\)

    Exercise \(\PageIndex{268}\)

    \(\$ 8.49 \div 12\)

    Answer

    $0.71

    Exercise \(\PageIndex{269}\)

    \(\$ 16.99 \div 9\)

    Exercise \(\PageIndex{270}\)

    \(12 \div 0.08\)

    Answer

    150

    Exercise \(\PageIndex{271}\)

    \(5 \div 0.04\)

    Convert Decimals, Fractions, and Percents

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

    Exercise \(\PageIndex{272}\)

    0.08

    Answer

    \(\frac{2}{25}\)

    Exercise \(\PageIndex{273}\)

    0.17

    Exercise \(\PageIndex{274}\)

    0.425

    Answer

    \(\frac{17}{40}\)

    Exercise \(\PageIndex{275}\)

    0.184

    Exercise \(\PageIndex{276}\)

    1.75

    Answer

    \(\frac{7}{4}\)

    Exercise \(\PageIndex{277}\)

    0.035

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

    Exercise \(\PageIndex{278}\)

    \(\frac{2}{5}\)

    Answer

    0.4

    Exercise \(\PageIndex{279}\)

    \(\frac{4}{5}\)

    Exercise \(\PageIndex{280}\)

    \(-\frac{3}{8}\)

    Answer

    −0.375

    Exercise \(\PageIndex{281}\)

    \(-\frac{5}{8}\)

    Exercise \(\PageIndex{282}\)

    \(\frac{5}{9}\)

    Answer

    \(0 . \overline{5}\)

    Exercise \(\PageIndex{283}\)

    \(\frac{2}{9}\)

    Exercise \(\PageIndex{284}\)

    \(\frac{1}{2}+6.5\)

    Answer

    7

    Exercise \(\PageIndex{285}\)

    \(\frac{1}{4}+10.75\)

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

    Exercise \(\PageIndex{286}\)

    5%

    Answer

    0.05

    Exercise \(\PageIndex{287}\)

    9%

    Exercise \(\PageIndex{288}\)

    40%

    Answer

    0.4

    Exercise \(\PageIndex{289}\)

    50%

    Exercise \(\PageIndex{290}\)

    115%

    Answer

    1.15

    Exercise \(\PageIndex{291}\)

    125%

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

    Exercise \(\PageIndex{292}\)

    0.18

    Answer

    18%

    Exercise \(\PageIndex{293}\)

    0.15

    Exercise \(\PageIndex{294}\)

    0.009

    Answer

    0.9%

    Exercise \(\PageIndex{295}\)

    0.008

    Exercise \(\PageIndex{296}\)

    1.5

    Answer

    150%

    Exercise \(\PageIndex{297}\)

    2.2

    The Real Numbers

    Simplify Expressions with Square Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{298}\)

    \(\sqrt{64}\)

    Answer

    8

    Exercise \(\PageIndex{299}\)

    \(\sqrt{144}\)

    Exercise \(\PageIndex{300}\)

    \(-\sqrt{25}\)

    Answer

    -5

    Exercise \(\PageIndex{301}\)

    \(-\sqrt{81}\)

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

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

    Exercise \(\PageIndex{302}\)

    1. 9
    2. 8.47
    Answer
    1. \(\frac{9}{1}\)
    2. \(\frac{847}{100}\)

    Exercise \(\PageIndex{303}\)

    1. −15
    2. 3.591

    In the following exercises, list the

    1. rational numbers,
    2. irrational numbers.

    Exercise \(\PageIndex{304}\)

    \(0.84,0.79132 \ldots, 1 . \overline{3}\)

    Answer
    1. \(0.84,1.3\)
    2. \(0.79132 \ldots\)

    Exercise \(\PageIndex{305}\)

    \(2.3 \overline{8}, 0.572,4.93814 \ldots\)

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

    Exercise \(\PageIndex{306}\)

    1. \(\sqrt{121}\)
    2. \(\sqrt{48}\)
    Answer
    1. rational
    2. irrational

    Exercise \(\PageIndex{307}\)

    1. \(\sqrt{56}\)
    2. \(\sqrt{16}\)

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

    Exercise \(\PageIndex{308}\)

    1. \(\sqrt{-9}\)
    2. \(-\sqrt{169}\)
    Answer
    1. not a real number
    2. real number

    Exercise \(\PageIndex{309}\)

    1. \(\sqrt{-64}\)
    2. \(-\sqrt{81}\)

    In the following exercises, list the

    1. whole numbers,
    2. integers,
    3. rational numbers,
    4. irrational numbers,
    5. real numbers for each set of numbers.

    Exercise \(\PageIndex{310}\)

    \(-4,0, \frac{5}{6}, \sqrt{16}, \sqrt{18}, 5.2537 \ldots\)

    Answer
    1. \(0, \sqrt{16}\)
    2. \(-4,0, \sqrt{16}\)
    3. \(-4,0, \frac{5}{6}, \sqrt{16}\)
    4. \(\sqrt{18}, 5.2537 \ldots\)
    5. \(-4,0, \frac{5}{6}, \sqrt{16}, \sqrt{18}, 5.2537 \ldots\)

    Exercise \(\PageIndex{311}\)

    \(-\sqrt{4}, 0 . \overline{36}, \frac{13}{3}, 6.9152 \ldots, \sqrt{48}, 10 \frac{1}{2}\)

    Locate Fractions on the Number Line

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

    Exercise \(\PageIndex{312}\)

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

    Answer

    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.

    Exercise \(\PageIndex{313}\)

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

    Exercise \(\PageIndex{314}\)

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

    Answer

    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.

    Exercise \(\PageIndex{315}\)

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

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

    Exercise \(\PageIndex{316}\)

    −1___\(-\frac{1}{8}\)

    Answer

    <

    Exercise \(\PageIndex{317}\)

    \(-3 \frac{1}{4}\)___−4

    Exercise \(\PageIndex{318}\)

    \(-\frac{7}{9}\) ___ \(\frac{4}{9}\)

    Answer

    >

    Exercise \(\PageIndex{319}\)

    \(-2\) ___ \(\frac{19}{8}\)

    Locate Decimals on the Number Line

    In the following exercises, locate on the number line.

    Exercise \(\PageIndex{320}\)

    0.3

    Answer

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

    Exercise \(\PageIndex{321}\)

    −0.2

    Exercise \(\PageIndex{322}\)

    −2.5

    Answer

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

    Exercise \(\PageIndex{323}\)

    2.7

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

    Exercise \(\PageIndex{324}\)

    0.9___0.6

    Answer

    >

    Exercise \(\PageIndex{325}\)

    0.7___0.8

    Exercise \(\PageIndex{326}\)

    −0.6___−0.59

    Answer

    >

    Exercise \(\PageIndex{327}\)

    −0.27___−0.3

    Properties of Real Numbers

    Use the Commutative and Associative Properties

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

    Exercise \(\PageIndex{328}\)

    −12(4m)

    Answer

    −48m

    Exercise \(\PageIndex{329}\)

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

    Exercise \(\PageIndex{330}\)

    (a+16)+31

    Answer

    a+47

    Exercise \(\PageIndex{331}\)

    (c+0.2)+0.7

    In the following exercises, simplify.

    Exercise \(\PageIndex{332}\)

    6y+37+(−6y)

    Answer

    37

    Exercise \(\PageIndex{333}\)

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

    Exercise \(\PageIndex{334}\)

    \(\frac{14}{11} \cdot \frac{35}{9} \cdot \frac{14}{11}\)

    Answer

    \(\frac{35}{9}\)

    Exercise \(\PageIndex{335}\)

    \(-18 \cdot 15 \cdot \frac{2}{9}\)

    Exercise \(\PageIndex{336}\)

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

    Answer

    1\(\frac{7}{12}\)

    Exercise \(\PageIndex{337}\)

    (3.98d+0.75d)+1.25d

    Exercise \(\PageIndex{338}\)

    11x+8y+16x+15y

    Answer

    27x+23y

    Exercise \(\PageIndex{339}\)

    52m+(−20n)+(−18m)+(−5n)

    Use the Identity and Inverse Properties of Addition and Multiplication

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

    Exercise \(\PageIndex{340}\)

    1. \(\frac{1}{3}\)
    2. 5.1
    3. \(-14\)
    4. \(-\frac{8}{5}\)
    Answer
    1. \(-\frac{1}{3}\)
    2. \(-5.1\)
    3. -14
    4. \(-\frac{8}{5}\)

    Exercise \(\PageIndex{341}\)

    1. \(-\frac{7}{8}\)
    2. \(-0.03\)
    3. 17
    4. \(\frac{12}{5}\)

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

    Exercise \(\PageIndex{342}\)

    1. \(10\)
    2. \(-\frac{4}{9}\)
    3. 0.6
    Answer
    1. \(\frac{1}{10}\)
    2. \(-\frac{9}{4}\)
    3. \(\frac{5}{3}\)

    Exercise \(\PageIndex{343}\)

    1. \(-\frac{9}{2}\)
    2. -7
    3. 2.1

    Use the Properties of Zero

    In the following exercises, simplify.

    Exercise \(\PageIndex{344}\)

    83\(\cdot 0\)

    Answer

    0

    Exercise \(\PageIndex{345}\)

    \(\frac{0}{9}\)

    Exercise \(\PageIndex{346}\)

    \(\frac{5}{0}\)

    Answer

    undefined

    Exercise \(\PageIndex{347}\)

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

    In the following exercises, simplify.

    Exercise \(\PageIndex{348}\)

    43+39+(−43)

    Answer

    39

    Exercise \(\PageIndex{349}\)

    (n+6.75)+0.25

    Exercise \(\PageIndex{350}\)

    \(\frac{5}{13} \cdot 57 \cdot \frac{13}{5}\)

    Answer

    57

    Exercise \(\PageIndex{351}\)

    \(\frac{1}{6} \cdot 17 \cdot 12\)

    Exercise \(\PageIndex{352}\)

    \(\frac{2}{3} \cdot 28 \cdot \frac{3}{7}\)

    Answer

    8

    Exercise \(\PageIndex{353}\)

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

    Simplify Expressions Using the Distributive Property

    In the following exercises, simplify using the Distributive Property.

    Exercise \(\PageIndex{354}\)

    7(x+9)

    Answer

    7x+63

    Exercise \(\PageIndex{355}\)

    9(u−4)

    Exercise \(\PageIndex{356}\)

    −3(6m−1)

    Answer

    −18m+3

    Exercise \(\PageIndex{357}\)

    −8(−7a−12)

    Exercise \(\PageIndex{358}\)

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

    Answer

    5n−2

    Exercise \(\PageIndex{359}\)

    \((y+10) \cdot p\)

    Exercise \(\PageIndex{360}\)

    (a−4)−(6a+9)

    Answer

    −5a−13

    Exercise \(\PageIndex{361}\)

    4(x+3)−8(x−7)

    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.

    Exercise \(\PageIndex{362}\)

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

    Answer

    84 inches

    Exercise \(\PageIndex{363}\)

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

    Exercise \(\PageIndex{364}\)

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

    Answer

    64 inches

    Exercise \(\PageIndex{365}\)

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

    Exercise \(\PageIndex{366}\)

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

    Answer

    2.7 miles

    Exercise \(\PageIndex{367}\)

    Shamu weights 4.5 tons. Convert the weight to pounds.

    Exercise \(\PageIndex{368}\)

    The play lasted

    \[1\(\frac{3}{4}\)\]

    hours. Convert the time to minutes.

    Answer

    105 minutes

    Exercise \(\PageIndex{369}\)

    How many tablespoons are in a quart?

    Exercise \(\PageIndex{370}\)

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

    Answer

    94 ounces

    Exercise \(\PageIndex{371}\)

    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.

    Exercise \(\PageIndex{372}\)

    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?

    Answer

    10 lbs. 6 oz.

    Exercise \(\PageIndex{373}\)

    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?

    Exercise \(\PageIndex{374}\)

    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?

    Answer

    15 feet

    Exercise \(\PageIndex{375}\)

    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.

    Exercise \(\PageIndex{376}\)

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

    Answer

    170 centimeters

    Exercise \(\PageIndex{377}\)

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

    Exercise \(\PageIndex{378}\)

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

    Answer

    0.488 grams

    Exercise \(\PageIndex{379}\)

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

    Exercise \(\PageIndex{380}\)

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

    Answer

    2,900 grams

    Exercise \(\PageIndex{381}\)

    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.

    Exercise \(\PageIndex{382}\)

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

    Answer

    1.12 meter

    Exercise \(\PageIndex{383}\)

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

    Exercise \(\PageIndex{384}\)

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

    Answer

    0.9 kilograms

    Exercise \(\PageIndex{385}\)

    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.

    Exercise \(\PageIndex{386}\)

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

    Answer

    175.3 centimeters

    Exercise \(\PageIndex{387}\)

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

    Exercise \(\PageIndex{388}\)

    Caroline walked 2.5 kilometers. Convert this length to miles.

    Answer

    1.6 miles

    Exercise \(\PageIndex{389}\)

    Lucas weighs 78 kilograms. Convert his weight to pounds.

    Exercise \(\PageIndex{390}\)

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

    Answer

    14.6 gallons

    Exercise \(\PageIndex{391}\)

    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.

    Exercise \(\PageIndex{392}\)

    95° Fahrenheit

    Answer

    35° C

    Exercise \(\PageIndex{393}\)

    23° Fahrenheit

    Exercise \(\PageIndex{394}\)

    20° Fahrenheit

    Answer

    –6.7° C

    Exercise \(\PageIndex{395}\)

    64° Fahrenheit

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

    Exercise \(\PageIndex{396}\)

    30° Celsius

    Answer

    86° F

    Exercise \(\PageIndex{397}\)

    –5° Celsius

    Exercise \(\PageIndex{398}\)

    –12° Celsius

    Answer

    10.4° F

    Exercise \(\PageIndex{399}\)

    24° Celsius

    Chapter Practice Test

    Exercise \(\PageIndex{400}\)

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

    Answer

    205,617

    Exercise \(\PageIndex{401}\)

    Find the prime factorization of 504.

    Exercise \(\PageIndex{402}\)

    Find the Least Common Multiple of 18 and 24.

    Answer

    72

    Exercise \(\PageIndex{403}\)

    Combine like terms: 5n+8+2n−1.

    In the following exercises, evaluate.

    Exercise \(\PageIndex{404}\)

    \(-|x|\) when \(x=-2\)

    Answer

    −2

    Exercise \(\PageIndex{405}\)

    11−a when a=−3

    Exercise \(\PageIndex{406}\)

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

    Answer

    −7−20;−27

    Exercise \(\PageIndex{407}\)

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

    Exercise \(\PageIndex{408}\)

    Round 677.1348 to the nearest hundredth.

    Answer

    677.13

    Exercise \(\PageIndex{409}\)

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

    Exercise \(\PageIndex{410}\)

    Convert 1.85 to a percent.

    Answer

    185%

    Exercise \(\PageIndex{411}\)

    Locate \(\frac{2}{3},-1.5,\) and \(\frac{9}{4}\) on a number line.

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{412}\)

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

    Answer

    99

    Exercise \(\PageIndex{413}\)

    −85+42

    Exercise \(\PageIndex{414}\)

    −19−25

    Answer

    −44

    Exercise \(\PageIndex{415}\)

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

    Exercise \(\PageIndex{416}\)

    \(-5(-9) \div 15\)

    Answer

    3

    Exercise \(\PageIndex{417}\)

    \(\frac{3}{8} \cdot \frac{11}{12}\)

    Exercise \(\PageIndex{418}\)

    \(\frac{4}{5} \div \frac{9}{20}\)

    Answer

    \(\frac{16}{9}\)

    Exercise \(\PageIndex{419}\)

    \(\frac{12+3 \cdot 5}{15-6}\)

    Exercise \(\PageIndex{420}\)

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

    Answer

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

    Exercise \(\PageIndex{421}\)

    \(\frac{7}{12}-\frac{3}{8}\)

    Exercise \(\PageIndex{422}\)

    \(-5.8+(-4.7)\)

    Answer

    −10.5

    Exercise \(\PageIndex{423}\)

    100−64.25

    Exercise \(\PageIndex{424}\)

    (0.07)(31.95)

    Answer

    2.2365

    Exercise \(\PageIndex{425}\)

    \(9 \div 0.05\)

    Exercise \(\PageIndex{426}\)

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

    Answer

    −10p

    Exercise \(\PageIndex{427}\)

    (u+8)−9

    Exercise \(\PageIndex{428}\)

    6x+(−4y)+9x+8y

    Answer

    15x+4y

    Exercise \(\PageIndex{429}\)

    \(\frac{0}{23}\)

    Exercise \(\PageIndex{430}\)

    \(\frac{75}{0}\)

    Answer

    undefined

    Exercise \(\PageIndex{431}\)

    −2(13q−5)

    Exercise \(\PageIndex{432}\)

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

    Answer

    100 minutes

    Exercise \(\PageIndex{433}\)

    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)

    Exercise \(\PageIndex{434}\)

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

    Answer

    4.508 km

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