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# 1.11: Systems of Measurement

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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?

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.

Exercise $$\PageIndex{2}$$

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

2.5 feet

Exercise $$\PageIndex{3}$$

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.

Exercise $$\PageIndex{4}$$

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

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.

8600 pounds

Exercise $$\PageIndex{6}$$

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

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.

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.

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.

151,200 minutes

Exercise $$\PageIndex{10}$$

How many ounces are in 1 gallon?

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?

16 cups

Exercise $$\PageIndex{12}$$

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!

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?

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?

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?

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?

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?

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?

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?

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?

5,000 meters

Exercise $$\PageIndex{21}$$

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

250 centimeters

Exercise $$\PageIndex{22}$$

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

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?

2.8 kilograms

Exercise $$\PageIndex{24}$$

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 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.

Figure $$\PageIndex{2}$$

Exercise $$\PageIndex{25}$$

Convert

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

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
1. 7,250 kiloliters
2. 6,300 milliliters

Exercise $$\PageIndex{27}$$

Convert:

1. 350 hL to liters
2. 4.1 L to centiliters
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?

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.

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.

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?

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?

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?

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.

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

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

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

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

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.

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

Exercise $$\PageIndex{35}$$

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

2.12 quarts

Exercise $$\PageIndex{36}$$

How many liters are in 4 quarts of milk?

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?

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

Exercise $$\PageIndex{38}$$

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

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.

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.

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.

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.

15°C

Exercise $$\PageIndex{42}$$

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

5°C

Exercise $$\PageIndex{43}$$

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

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.

59°F

Exercise $$\PageIndex{45}$$

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

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.

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.

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.

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.

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.

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.

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.

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?

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.

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.

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.

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.

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.

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?

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?

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?

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?

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.

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.

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.

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.

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.

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.

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.

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?

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?

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?

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?

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.

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

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.

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.

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.

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.

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

30°C

Exercise $$\PageIndex{70}$$

77° Fahrenheit

Exercise $$\PageIndex{71}$$

104° Fahrenheit

40°C

Exercise $$\PageIndex{72}$$

14° Fahrenheit

Exercise $$\PageIndex{73}$$

72° Fahrenheit

22.2°C

Exercise $$\PageIndex{74}$$

4° Fahrenheit

Exercise $$\PageIndex{75}$$

0° Fahrenheit

−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

41°F

Exercise $$\PageIndex{78}$$

25° Celsius

Exercise $$\PageIndex{79}$$

−10° Celsius

14°F

Exercise $$\PageIndex{80}$$

−15° Celsius

Exercise $$\PageIndex{81}$$

22° Celsius

71.6°F

Exercise $$\PageIndex{82}$$

8° Celsius

Exercise $$\PageIndex{83}$$

43° Celsius

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?

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.

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.

ⓑ 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
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
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

six thousand, one hundred four

Exercise $$\PageIndex{6}$$

493,068

Exercise $$\PageIndex{7}$$

3,975,284

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

315

Exercise $$\PageIndex{10}$$

sixty-five thousand, nine hundred twelve

Exercise $$\PageIndex{11}$$

ninety million, four hundred twenty-five thousand, sixteen

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
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

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

by 2,3,6

Exercise $$\PageIndex{18}$$

264

Exercise $$\PageIndex{19}$$

375

by 3,5

Exercise $$\PageIndex{20}$$

750

Exercise $$\PageIndex{21}$$

1430

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

2$$\cdot 2 \cdot 3 \cdot 5 \cdot 7$$

Exercise $$\PageIndex{24}$$

115

Exercise $$\PageIndex{25}$$

225

3$$\cdot 3 \cdot 5 \cdot 5$$

Exercise $$\PageIndex{26}$$

2475

Exercise $$\PageIndex{27}$$

1560

$$2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 13$$

Exercise $$\PageIndex{28}$$

56

Exercise $$\PageIndex{29}$$

72

$$2 \cdot 2 \cdot 2 \cdot 3 \cdot 3$$

Exercise $$\PageIndex{30}$$

168

Exercise $$\PageIndex{31}$$

252

$$2 \cdot 2 \cdot 3 \cdot 3 \cdot 7$$

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

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

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

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

Exercise $$\PageIndex{38}$$

5$$\cdot 6$$

Exercise $$\PageIndex{39}$$

$$45 \div 5$$

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

Exercise $$\PageIndex{40}$$

x+8

Exercise $$\PageIndex{41}$$

$$42 \geq 27$$

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

Exercise $$\PageIndex{42}$$

3n=24

Exercise $$\PageIndex{43}$$

$$3 \leq 20 \div 4$$

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

Exercise $$\PageIndex{44}$$

$$a \neq 7 \cdot 4$$

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

Exercise $$\PageIndex{45}$$

$$6 \cdot 3+5$$

expression

Exercise $$\PageIndex{46}$$

y−8=32

Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

Exercise $$\PageIndex{47}$$

$$3^{5}$$

243

Exercise $$\PageIndex{48}$$

$$10^{8}$$

In the following exercises, simplify

Exercise $$\PageIndex{49}$$

6+10/2+2

13

Exercise $$\PageIndex{50}$$

9+12/3+4

Exercise $$\PageIndex{51}$$

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

10

Exercise $$\PageIndex{52}$$

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

Exercise $$\PageIndex{53}$$

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

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

34

Exercise $$\PageIndex{56}$$

5x−4 when x=6

Exercise $$\PageIndex{57}$$

$$x^{4}$$ when $$x=3$$

81

Exercise $$\PageIndex{58}$$

$$3^{x}$$ when $$x=3$$

Exercise $$\PageIndex{59}$$

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

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

12

Exercise $$\PageIndex{62}$$

9$$x^{2}$$

In the following exercises, identify the like terms.

Exercise $$\PageIndex{63}$$

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

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 $$\PageIndex{65}$$

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

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

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

26a

Exercise $$\PageIndex{68}$$

18z+9z

Exercise $$\PageIndex{69}$$

9x+3x+8

12x+8

Exercise $$\PageIndex{70}$$

8a+5a+9

Exercise $$\PageIndex{71}$$

7p+6+5p−4

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

8+12

Exercise $$\PageIndex{74}$$

the sum of 9 and 1

Exercise $$\PageIndex{75}$$

the difference of x and 4

x−4

Exercise $$\PageIndex{76}$$

the difference of x and 3

Exercise $$\PageIndex{77}$$

the product of 6 and y

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 $$\PageIndex{270}$$

$$12 \div 0.08$$

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

$$\frac{2}{25}$$

Exercise $$\PageIndex{273}$$

0.17

Exercise $$\PageIndex{274}$$

0.425

$$\frac{17}{40}$$

Exercise $$\PageIndex{275}$$

0.184

Exercise $$\PageIndex{276}$$

1.75

$$\frac{7}{4}$$

Exercise $$\PageIndex{277}$$

0.035

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

Exercise $$\PageIndex{278}$$

$$\frac{2}{5}$$

0.4

Exercise $$\PageIndex{279}$$

$$\frac{4}{5}$$

Exercise $$\PageIndex{280}$$

$$-\frac{3}{8}$$

−0.375

Exercise $$\PageIndex{281}$$

$$-\frac{5}{8}$$

Exercise $$\PageIndex{282}$$

$$\frac{5}{9}$$

$$0 . \overline{5}$$

Exercise $$\PageIndex{283}$$

$$\frac{2}{9}$$

Exercise $$\PageIndex{284}$$

$$\frac{1}{2}+6.5$$

7

Exercise $$\PageIndex{285}$$

$$\frac{1}{4}+10.75$$

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

Exercise $$\PageIndex{286}$$

5%

0.05

Exercise $$\PageIndex{287}$$

9%

Exercise $$\PageIndex{288}$$

40%

0.4

Exercise $$\PageIndex{289}$$

50%

Exercise $$\PageIndex{290}$$

115%

1.15

Exercise $$\PageIndex{291}$$

125%

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

Exercise $$\PageIndex{292}$$

0.18

18%

Exercise $$\PageIndex{293}$$

0.15

Exercise $$\PageIndex{294}$$

0.009

0.9%

Exercise $$\PageIndex{295}$$

0.008

Exercise $$\PageIndex{296}$$

1.5

150%

Exercise $$\PageIndex{297}$$

2.2

## The Real Numbers

Simplify Expressions with Square Roots

In the following exercises, simplify.

Exercise $$\PageIndex{298}$$

$$\sqrt{64}$$

8

Exercise $$\PageIndex{299}$$

$$\sqrt{144}$$

Exercise $$\PageIndex{300}$$

$$-\sqrt{25}$$

-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
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 $$\PageIndex{304}$$

$$0.84,0.79132 \ldots, 1 . \overline{3}$$

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. $$\sqrt{48}$$
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. $$-\sqrt{169}$$
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 $$\PageIndex{310}$$

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

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 $$\PageIndex{312}$$

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

Exercise $$\PageIndex{313}$$

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

Exercise $$\PageIndex{314}$$

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

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___$$-\frac{1}{8}$$

<

Exercise $$\PageIndex{317}$$

$$-3 \frac{1}{4}$$___−4

Exercise $$\PageIndex{318}$$

$$-\frac{7}{9}$$ ___ $$\frac{4}{9}$$

>

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

Exercise $$\PageIndex{321}$$

−0.2

Exercise $$\PageIndex{322}$$

−2.5

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

>

Exercise $$\PageIndex{325}$$

0.7___0.8

Exercise $$\PageIndex{326}$$

−0.6___−0.59

>

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)

−48m

Exercise $$\PageIndex{329}$$

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

Exercise $$\PageIndex{330}$$

(a+16)+31

a+47

Exercise $$\PageIndex{331}$$

(c+0.2)+0.7

In the following exercises, simplify.

Exercise $$\PageIndex{332}$$

6y+37+(−6y)

37

Exercise $$\PageIndex{333}$$

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

Exercise $$\PageIndex{334}$$

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

$$\frac{35}{9}$$

Exercise $$\PageIndex{335}$$

$$-18 \cdot 15 \cdot \frac{2}{9}$$

Exercise $$\PageIndex{336}$$

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

1$$\frac{7}{12}$$

Exercise $$\PageIndex{337}$$

(3.98d+0.75d)+1.25d

Exercise $$\PageIndex{338}$$

11x+8y+16x+15y

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. $$-\frac{8}{5}$$
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
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$$\cdot 0$$

0

Exercise $$\PageIndex{345}$$

$$\frac{0}{9}$$

Exercise $$\PageIndex{346}$$

$$\frac{5}{0}$$

undefined

Exercise $$\PageIndex{347}$$

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

In the following exercises, simplify.

Exercise $$\PageIndex{348}$$

43+39+(−43)

39

Exercise $$\PageIndex{349}$$

(n+6.75)+0.25

Exercise $$\PageIndex{350}$$

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

57

Exercise $$\PageIndex{351}$$

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

Exercise $$\PageIndex{352}$$

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

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)

7x+63

Exercise $$\PageIndex{355}$$

9(u−4)

Exercise $$\PageIndex{356}$$

−3(6m−1)

−18m+3

Exercise $$\PageIndex{357}$$

−8(−7a−12)

Exercise $$\PageIndex{358}$$

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

5n−2

Exercise $$\PageIndex{359}$$

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

Exercise $$\PageIndex{360}$$

(a−4)−(6a+9)

−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.

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.

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.

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.

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.

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?

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?

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.

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.

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.

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?

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?

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.

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.

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.

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

35° C

Exercise $$\PageIndex{393}$$

23° Fahrenheit

Exercise $$\PageIndex{394}$$

20° Fahrenheit

–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

86° F

Exercise $$\PageIndex{397}$$

–5° Celsius

Exercise $$\PageIndex{398}$$

–12° Celsius

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.

205,617

Exercise $$\PageIndex{401}$$

Find the prime factorization of 504.

Exercise $$\PageIndex{402}$$

Find the Least Common Multiple of 18 and 24.

72

Exercise $$\PageIndex{403}$$

Combine like terms: 5n+8+2n−1.

In the following exercises, evaluate.

Exercise $$\PageIndex{404}$$

$$-|x|$$ when $$x=-2$$

−2

Exercise $$\PageIndex{405}$$

11−a when a=−3

Exercise $$\PageIndex{406}$$

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

−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.

677.13

Exercise $$\PageIndex{409}$$

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

Exercise $$\PageIndex{410}$$

Convert 1.85 to a percent.

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 $$\PageIndex{412}$$

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

99

Exercise $$\PageIndex{413}$$

−85+42

Exercise $$\PageIndex{414}$$

−19−25

−44

Exercise $$\PageIndex{415}$$

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

Exercise $$\PageIndex{416}$$

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

3

Exercise $$\PageIndex{417}$$

$$\frac{3}{8} \cdot \frac{11}{12}$$

Exercise $$\PageIndex{418}$$

$$\frac{4}{5} \div \frac{9}{20}$$

$$\frac{16}{9}$$

Exercise $$\PageIndex{419}$$

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

Exercise $$\PageIndex{420}$$

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

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

Exercise $$\PageIndex{421}$$

$$\frac{7}{12}-\frac{3}{8}$$

Exercise $$\PageIndex{422}$$

$$-5.8+(-4.7)$$

−10.5

Exercise $$\PageIndex{423}$$

100−64.25

Exercise $$\PageIndex{424}$$

(0.07)(31.95)

2.2365

Exercise $$\PageIndex{425}$$

$$9 \div 0.05$$

Exercise $$\PageIndex{426}$$

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

−10p

Exercise $$\PageIndex{427}$$

(u+8)−9

Exercise $$\PageIndex{428}$$

6x+(−4y)+9x+8y

15x+4y

Exercise $$\PageIndex{429}$$

$$\frac{0}{23}$$

Exercise $$\PageIndex{430}$$

$$\frac{75}{0}$$

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)

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)