© 2010 Pearson Prentice Hall. All rights reserved Chapter 1 Review of Real Numbers.

© 2010 Pearson Prentice Hall. All rights reserved

Chapter 1

Review of Real Numbers


1.1

Tips for Success in Mathematics

Martin-Gay, Prealgebra, 6ed 33

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Positive Attitude

Believe you can succeed.

Scheduling Make sure you have time for your classes.

Be Prepared

Have all the materials you need, like a lab manual, calculator, or other supplies.

Getting Ready for This Course



General Tips for Success

Tip Details

Get a contact person.Exchange names, phone numbers or e-mail addresses with at least one other person in class.

Attend all class periods.Sit near the front of the classroom to make hearing the presentation, and participating easier.

Do you homework.The more time you spend solving mathematics, the easier the process becomes.

Check your work.Review your steps, fix errors, and compare answers with the selected answers in the back of the book.

Learn from your mistakes.

Find and understand your errors. Use them to become a better math student.

Continued



General Tips for Success

Tip Details

Get help if you need it.

Ask for help when you don’t understand something. Know when your instructors office hours are, and whether tutoring services are available.

Organize class materials.Organize your assignments, quizzes, tests, and notes for use as reference material throughout your course.

Read your textbook.Review your section before class to help you understand its ideas more clearly.

Ask questions.Speak up when you have a question. Other students may have the same one.

Hand in assignments on time.

Don’t lose points for being late. Show every step of a problem on your assignment.



Using This Text

Resource Details

Practice Problems.Try each Practice Problem after you’ve finished its corresponding example.

Chapter Test Prep Video CD.Chapter Test exercises are worked out by the author, these are available off of the CD this book contains.

Lecture Video CDs.Exercises marked with a CD symbol are worked out by the author on a video CD. Check with your instructor to see if these are available.

Symbols before an exercise set.

Symbols listed at the beginning of each exercise set will remind you of the available supplements.

Objectives.The main section of exercises in an exercise set is referenced by an objective. Use these if you are having trouble with an assigned problem.

Continued



Using This Text

Resource Details

Icons (Symbols).

A CD symbol tells you the corresponding exercise may be viewed on a video segment. A pencil symbol means you should answer using complete sentences.

Integrated Reviews.Reviews found in the middle of each chapter can be used to practice the previously learned concepts.

End of Chapter Opportunities.Use Chapter Highlights, Chapter Reviews, Chapter Tests, and Cumulative Reviews to help you understand chapter concepts.

Study Skills Builder.Read and answer questions in the Study Skills Builder to increase your chance of success in this course.

The Bigger Picture.This can help you make the transition from thinking “section by section” to thinking about how everything corresponds in the bigger picture.



Getting Help

Tip Details

Get help as soon as you

need it.

Material presented in one section builds on your understanding of the previous section. If you don’t understand a concept covered during a class period, there is a good chance you won’t understand the concepts covered in the next period.For help try your instructor, a tutoring center, or a math lab. A study group can also help increase your understanding of covered materials.



Preparing for and Taking an Exam

Steps for Preparing for a Test1. Review previous homework assignments.2. Review notes from class and section-level quizzes you have

taken.3. Read the Highlights at the end of each chapter to review

concepts and definitions.4. Complete the Chapter Review at the end of each chapter to

practice the exercises.5. Take a sample test in conditions similar to your test

conditions.6. Set aside plenty of time to arrive where you will be taking

the exam.

Continued



Preparing for and Taking an Exam

Steps for Taking Your Test1. Read the directions on the test carefully.2. Read each problem carefully to make sure that you

answer the question asked.3. Pace yourself so that you have enough time to

attempt each problem on the test.4. Use extra time checking your work and answers.5. Don’t turn in your test early. Use extra time to

double check your work.



Managing Your Time

Tips for Making a Schedule1. Make a list of all of your weekly commitments for

the term.2. Estimate the time needed and how often it will be

performed, for each item.3. Block out a typical week on a schedule grid, start

with items with fixed time slots.4. Next, fill in items with flexible time slots.5. Remember to leave time for eating, sleeping, and

relaxing.6. Make changes to your workload, classload, or other

areas to fit your needs.


§ 1.2

Place Value, Names for Numbers, and

Reading Tables


The position of each digit in a number determines its place value.

3 5 6 8 9 4 0 2

One

s

Hun

dred

-tho

usan

ds

Hun

dred

-bil

lion

s

Ten-

bill

ions

Bil

lion

s

Hun

dred

-mil

lion

s

Ten-

mil

lion

s

Mil

lion

s

Ten-

thou

sand

s

Tho

usan

ds

Hun

dred

s

Tens

Place Value


A whole number such as 35,689,402 is written in standard form. The columns separate the digits into groups of threes. Each group of three digits is a period.

Millions ThousandsBillions Ones

3 5 6 8 9 4 0 2

One

s

Hun

dred

-tho

usan

ds

Hun

dred

-bil

lion

s

Ten-

bill

ions

Bil

lion

s

Hun

dred

-mil

lion

s

Ten-

mil

lion

s

Mil

lion

s

Ten-

thou

sand

s

Tho

usan

ds

Hun

dred

s

Tens

Writing a Number in Words


To write a whole number in words, write the number in each period followed by the name of the period.

thirty-five million, six hundred eighty-nine thousand, four hundred two

3 5 6 8 9 4 0 2

One

s

Hun

dred

-tho

usan

ds

Hun

dred

-bil

lion

s

Ten-

bill

ions

Bil

lion

s

Hun

dred

-mil

lion

s

Ten-

mil

lion

s

Mil

lion

s

Ten-

thou

sand

s

Tho

usan

ds

Hun

dred

s

Tens

Writing a Number in Words


The name of the ones period is not used when reading and writing whole numbers. Also, the word “and” is not used when reading and writing whole numbers. It is used when reading and writing mixed numbers and some decimal values as shown later.

Helpful Hint


The place value of a digit can be used to write a number in expanded form. The expanded form of a number shows each digit of the number with its place value.

4,786 = 4000 + 700 + 80 + 6

Standard Form Expanded Form

Expanded Form


Comparing Whole Numbers

We can picture whole numbers as equally spaced points on a line called the number line.

A whole number is graphed by placing a dot on the number line. The graph of 4 is shown.

0 541 2 3


Comparing Numbers

For any two numbers graphed on a number line, the number to the right is the greater number, and the number to the left is the smaller number.

2 is to the left of 5, so 2 is less than 5

5 is to the right of 2, so 5 is greater than 2

0 541 2 3


Comparing Numbers . . .

2 is less than 5can be written in symbols as

2 < 55 is greater than 2

is written as5 > 2


One way to remember the meaning of the inequality

symbols < and > is to think of them as arrowheads

“pointing” toward the smaller number.

For example,

2 < 5 and 5 > 2

are both true statements.

Helpful Hint


Reading Tables

Gold Silver Bronze Total

107 104 86 297

113 83 78 274

94 92 74 260

69 71 51 191

41 57 64 162

Source: The Sydney Morning Herald, Flags courtesy of www.theodora.com/flags used with permission

Germany

Russia

Norway

USA

Austria

Most Medals Olympic Winter (1924 – 2002) Games


1.3

Adding and Subtracting Whole

Numbers, and Perimeter


Addition Property of 0

The sum of 0 and any number is that number.

8 + 0 = 8 and 0 + 8 = 8


Changing the order of two addends does not change their sum.

4 + 2 = 6 and 2 + 4 = 6

Commutative Property of Addition


Changing the grouping of addends does not change their sum.

3 + (4 + 2) = 3 + 6 = 9

and

(3 + 4) + 2 = 7 + 2 = 9

Associative Property of Addition


Subtraction Properties of 0

The difference of any number and that same number is 0.

9 – 9 = 0

The difference of any number and 0 is the same number.

7 – 0 = 7


Polygons

A polygon is a flat figure formed by line segments connected at their ends.

rectanglesquare

triangle

Geometric figures such as triangles, squares, and rectangles are called polygons.


Perimeter

The perimeter of a polygon is the distance around the polygon.


Descriptions of problems solved through addition may include any of these key words or phrases:

Key Words Examples Symbols

added to 3 added to 9 3 + 9

plus 5 plus 22 5 + 22

more than 7 more than 8 7 + 8

total total of 6 and 5 6 + 5

increased by 16 increased by 7 16 + 7

sum sum of 50 and 11 50 + 11

Addition Problems


Descriptions of problems solved by subtraction may include any of these key words or phrases:


subtract subtract 3 from 9 9 – 3

difference difference of 8 and 2 8 – 2

less 12 less 8 12 – 8

take away 14 take away 9 14 – 9

decreased by 16 decreased by 7 16 – 7

subtracted from 5 subtracted from 9 9 – 5

Subtraction Problems


Be careful when solving applications that suggest subtraction. Although order does not matter when adding, order does matter when subtracting. For example, 10 – 3 and 3 – 10 do not simplify to the same number.

Helpful Hint


Since subtraction and addition are reverse operations, don’t forget that a subtraction problem can be checked by adding.

Helpful Hint


The graph shows the number of endangered species in each country.

Nu

mb

er o

f E

nd

ange

red

Sp

ecie

s

146

89 8373

Country

Reading a Bar Graph

Source: The Top 10 of Everything, Russell Ash.

7264


1.4

Rounding and Estimating



23 rounded to the nearest ten is 20.



10 2015

40 5048

20 3023

Rounding



Rounding Whole Numbers

Step 1: Locate the digit to the right of the given place value.

Step 2: If this digit is 5 or greater, add 1 to the digit in the given place value and replace each digit to its right by 0.

Step 3: If this digit is less than 5, replace it and each digit to its right by 0.



Making estimates is often the quickest way to solve real-life problems when their solutions do not need to be exact.

Estimates



Estimation is useful to check for incorrect answers when using a calculator. For example, pressing a key too hard may result in a double digit, while pressing a key too softly may result in the number not appearing in the display.

Helpful Hint


1.5

Multiplying Whole Numbers and Area



4 + 4 + 4 + 4 + 4 = 5 ∙ 4 = 20

5 fours factor product

Multiplication is repeated addition with a different notation.

Multiplication



Multiplication Property of 0

The product of 0 and any number is 0.

9 0 = 0

0 6 = 0



Multiplication Property of 1

The product of 1 and any number is that same number.

9 1 = 9

1 6 = 6



Commutative Property of Multiplication

Changing the order of two factors does not change their product.

6 3 = 18 and 3 6 = 18



Associative Property of Multiplication

Changing the grouping of factors does not change their product.

5 ( 2 3) = 5 6 = 30

and

(5 2) 3 = 10 3 = 30



Distributive Property

Multiplication distributes over addition.

5(3 + 4) = 5 3 + 5 4



Area

1 square inch1

15 inches

3 inches

Area of a rectangle = length width = (5 inches)(3 inches) = 15 square inches



Remember that perimeter (distance around a plane figure) is measured in units. Area (space enclosed by a plane figure) is measured in square units.

5 inches

4 inchesRectangle5 inches + 4 inches + 5 inches + 4 inches = 18 inches

Perimeter =

Area = (5 inches)(4 inches) = 20 square inches

Helpful Hint



There are several words or phrases that indicate the operation of multiplication. Some of these are as follows:


multiply multiply 4 by 3 4 3

product product of 2 and 5 2 5times 7 times 6 7 6

Multiplication Words


1.6

Dividing Whole Numbers



The process of separating a quantity into equal parts is called division.

204

5 3 186

14 2 7

quotient

divisordividend

Division



Division Properties of 1

The quotient of any number, except 0, and that same number is 1.

66

1 5 51

7 7 1 = ¸ =




61

6 1 55

7 1 7= ¸ =

The quotient of any number and 1 is that same number.




The quotient of 0 and any number (except 0) is 0.

06

0 00

0 7 0= ¸ =5




The quotient of any number and 0 is not a number. We say that

are undefined.

60

0 5 7 0¸



Since division and multiplication are reverse operations, don’t forget that a division problem can be checked by multiplying.

Helpful Hint



Here are some key words and phrases that indicate the operation of division.


divide divide 15 by 3 15 3

quotient quotient of 12 and 6

divided by 8 divided by 4

divided or shared equally

$20 divided equally among five people 20 5

Division Words

126

4 8



How do you find an average?

A student’s prealgebra grades at the end of the semester are:

90, 85, 95, 70, 80, 100, 98, 82, 90, 90.

How do you find his average?

Find the sum of the scores and then divide the sum by the number of scores.

Average = 880 ÷ 10 = 88

Sum = 880

Average


1.7

Exponents and Order of Operations



An exponent is a shorthand notation for repeated multiplication.

3 • 3 • 3 • 3 • 3

3 is a factor 5 times

Using an exponent, this product can be written as

35exponent

base

Exponents



This is called exponential notation. The exponent, 5, indicates how many times the base, 3, is a factor.

35exponent

base

Read as “three to the fifth power” or “the fifth power of three.”

3 • 3 • 3 • 3 • 3

3 is a factor 5 times

Exponential Notation



4 = 41

4 4 = 42

is read as “four to the first power.”

is read as “four to the second power” or “four squared.”

Reading Exponential Notation



4 4 4 = 43

4 4 4 4 = 44

is read as “four to the third power” or “four cubed.”

is read as “four to the fourth power.”

Reading Exponential Notation



Usually, an exponent of 1 is not written, so when no exponent appears, we assume that the exponent is 1. For example,

2 = 21 and 7 = 71.

Helpful Hint



To evaluate an exponential expression, we write the expression as a product and then find the value of the product.

35 = 3 • 3 • 3 • 3 • 3 = 243

Evaluating Exponential Expressions



An exponent applies only to its base. For example,

Don’t forget that 24 is not 2 • 4. 24 means repeated multiplication of the same factor.

4 • 23 means 4 • 2 • 2 • 2.

24 = 2 • 2 • 2 • 2 = 16, whereas 2 • 4 = 8

Helpful Hint



1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars, starting with the innermost set.

2. Evaluate any expressions with exponents.

3. Multiply or divide in order from left to right.

4. Add or subtract in order from left to right.

Order of Operations


1.8

Introduction to Variables, Algebraic

Expressions, and Equations



A combination of operations on letters (variables) and numbers is called an algebraic expression.

Algebraic Expressions5 + x 6 y 3 y – 4 + x

4x means 4 xand

xy means x y

Algebraic Expressions



Replacing a variable in an expression by a number and then finding the value of the expression is called evaluating the expression for the variable.

Algebraic Expressions



Evaluate x + y for x = 5 and y = 2.

x + y = ( ) + ( )

Replace x with 5 and y with 2 in x + y.

5 2

= 7

Evaluating Algebraic Expressions



Equation

Statements like 5 + 2 = 7 are called equations.

An equation is of the form expression = expression

An equation can be labeled as

Equal sign

left side right side

x + 5 = 9



Solutions

When an equation contains a variable, deciding which values of the variable make an equation a true statement is called solving an equation for the variable.

A solution of an equation is a value for the variable that makes an equation a true statement.



Solutions

Determine whether a number is a solution:

Is –2 a solution of the equation 2y + 1 = –3?

Replace y with –2 in the equation.

2y + 1 = –3

2(–2) + 1 = –3?

–4 + 1 = –3

–3 = –3

?

True

Since –3 = –3 is a true statement, –2 is a solution of the equation.



Solutions

Determine whether a number is a solution:

Is 6 a solution of the equation 5x – 1 = 30?

Replace x with 6 in the equation.5x – 1 = 30

5(6) – 1 = 30?

30 – 1 = 30

29 = 30

?

False

Since 29 = 30 is a false statement, 6 is not a solution of the equation.



Solutions

To solve an equation, we will use properties of equality to write simpler equations, all equivalent to the original equation, until the final equation has the form x = number or number = x

Equivalent equations have the same solution. The word “number” above represents the solution of the original equation.



Keywords and phrases suggesting addition, subtraction, multiplication, division or equals.

Addition Subtraction Multiplication Division Equal Sign

sum difference product quotient equals

plus minus times into gives

added to less than of per is/was/ will be

more than less twice divide yields

total decreased by multiply divided by amounts to

increased by

subtracted from

double is equal to

Keywords and Phrases



Translating Word Phrases

the product of 5 and a number5x

twice a number2x

a number decreased by 3n – 3

a number increased by 2z + 2

four times a number4w



Additional Word Phrases

x + 7

three times the sum of a number and 7

3(x + 7)

the quotient of 5 and a number

the sum of a number and 7

5

x



Remember that order is important when subtracting. Study the order of numbers and variables below.

Phrase Translation

a number decreased by 5 x – 5

a number subtracted from 5 5 – x

Helpful Hint


Chapter 2

Integers and Introduction to

Integers


2.1

Introduction to Integers



Numbers greater than 0 are called positive numbers. Numbers less than 0 are called negative numbers.

negative numbers zero

positive numbers

6543210-1-2-3-4-5-6

Positive and Negative Numbers



Some signed numbers are integers.


positive numbers

6543210–1–2–3–4–5–6

The integers are{ …, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, …}

Integers



–3 indicates “negative three.”3 and + 3 both indicate “positive three.”The number 0 is neither positive nor negative.


positive numbers

6543210–1–2–3–4–5–6

Negative and Positive Numbers



We compare integers just as we compare whole numbers. For any two numbers graphed on a number line, the number to the right is the greater number and the number to the left is the smaller number.

<means

“is less than”

>means

“is greater than”

Comparing Integers



The graph of –5 is to the left of –3, so –5 is less than –3, written as 5 < –3 . We can also write –3 > –5. Since –3 is to the right of –5, –3 is greater than –5.

6543210-1-2-3-4-5-66543210–1–2–3–4–5–6

Graphs of Integers



The absolute value of a number is the number’s distance from 0 on the number line. The symbol for absolute value is | |.

2 is 2 because 2 is 2 units from 0.

6543210–1–2–3–4–5–6

is 2 because –2 is 2 units from 0.2

6543210–1–2–3–4–5–6

Absolute Value



Since the absolute value of a number is that number’s distance from 0, the absolute value of a number is always 0 or positive. It is never negative.

0 = 0 6 = 6

zero a positive number

Helpful Hint



Two numbers that are the same distance from 0 on the number line but are on the opposite sides of 0 are called opposites.

5 units 5 units

5 and –5 are opposites.

6543210–1–2–3–4–5–6

Opposite Numbers



5 is the opposite of –5 and –5 is the opposite of 5.

The opposite of 4 is – 4 is written as

–(4) = –4

The opposite of – 4 is 4 is written as

–(– 4) = 4

–(–4) = 4

If a is a number, then –(– a) = a.

Opposite Numbers



Remember that 0 is neither positive nor negative. Therefore, the opposite of 0 is 0.

Helpful Hint


2.2

Adding Integers


6543210-1-2-3-4-5-6

Adding Two Numbers with the Same Sign

2 + 3 = 2Start End

– 2 + (– 3) =–2–3

StartEnd

3

6543210–1–2–3–4–5–6

6543210–1–2–3–4–5–6

5

– 5


Adding Two Numbers with the Same Sign

Step 1: Add their absolute values.

Step 2: Use their common sign as the sign of the sum.

Examples: – 3 + (–5) = – 8

5 + 2 = 7


Adding Two Numbers with Different Signs

2 + (–3) =

2

– 3

– 2 + 3 =– 2

3

Start

End

Start

End

6543210–1–2–3–4–5–6

6543210–1–2–3–4–5–6

–1

1


Step 1: Find the larger absolute value minus the smaller absolute value.

Step 2: Use the sign of the number with the larger absolute value as the sign of the sum.

Examples: –4 + 5 = 1 6 + (–8) = –2

Adding Two Numbers with Different Signs


If a is a number, then –a is its opposite.

a + (–a) = 0 –a + a = 0

The sum of a number and its opposite is 0.

Helpful Hint


Don’t forget that addition is commutative and associative. In other words, numbers may be added in any order.

Helpful Hint


Evaluate x + y for x = 5 and y = –9.

x + y = ( ) + ( )

Replace x with 5 and y with –9 in x + y.

5 –9

= –4



2.3

Subtracting Integers


To subtract integers, rewrite the subtraction problem as an addition problem. Study the examples below.

9 5 = 4

9 + (–5) = 4

equal 4, we can say

9 5 = 9 + (–5) = 4

Since both expressions

Subtracting Integers


Subtracting Two Numbers

If a and b are numbers,

then

a b = a + (–b).

To subtract two numbers, add the first number to the opposite (called additive inverse) of the second number.


subtraction

=first

number

+opposite of second number

7 – 4 = 7 + (– 4) = 3

– 5 – 3 = – 5 + (– 3) = – 8

3 – (–6) = 3 + 6 = 9

– 8 – (– 2) = – 8 + 2 = – 6

Subtracting Two Numbers


If a problem involves adding or subtracting more than two integers, rewrite differences as sums and add. By applying the associative and commutative properties, add the numbers in any order.

9 – 3 + (–5) – (–7) = 9 + (–3) + (–5) + 7

6 + (–5) + 7

1 + 7

8

Adding and Subtracting Integers


Evaluate x – y for x = –6 and y = 8.

x – y

Replace x with –6 and y with 8 in x – y.

= ( ) – ( )–6 8

= –14

= ( ) + ( )–6 –8



2.4

Multiplying and Dividing Integers


Consider the following pattern of products.

3 5 = 15

2 5 = 10

1 5 = 5

0 5 = 0This pattern continues as follows.

– 1 5 = - 5

– 2 5 = - 10

– 3 5 = - 15This suggests that the product of a negative number and a positive number is a negative number.

First factordecreases by 1each time.

Productdecreases by 5each time.

Multiplying Integers


2 (– 5) = –10

0 (– 5) = 0

This pattern continues as follows.

–1 (–5) = 5

–2 (–5) = 10

– 3 (–5) = 15This suggests that the product of two negative numbers is a positive number.

Product increases by 5 each time.

1 (– 5) = –5

Observe the following pattern.




The product of two numbers having the same sign is a positive number.

–2 (– 4) = 82 4 = 8

2 (– 4) = –8 – 2 4 = –8

The product of two numbers having different signs is a negative number.



Product of Like Signs

( + )( + ) = +

(–)( + ) = – ( + )(–) = –

Product of Different Signs

(–)(–) = +


If we let ( – ) represent a negative number and ( + ) represent a positive number, then

( – ) ( – ) = ( + )

( – ) ( – ) ( – ) = ( – )

( – ) ( – ) ( – ) ( – ) = ( + )

( – ) ( – ) ( – ) ( – ) ( – ) = ( – )

The product of an even number of negative numbers is a positive result.

The product of an odd number of negative numbers is a negative result.

Helpful Hint


Division of integers is related to multiplication of integers.

3 3 2 6= · =because62

= · =– 3 – 3 2 – 6because– 62

– 3 – 3 (– 2) 6= · =because6– 2

= 3 – 6because = 3 (– 2)·

– 6– 2

Dividing Integers


Dividing Integers

The quotient of two numbers having the same sign is a positive number.

–12 ÷ (–4 ) = 312 ÷ 4 = 3

– 12 ÷ 4 = –3 12 ÷ (– 4) = – 3

The quotient of two numbers having different signs is a negative number.


Dividing Numbers

Quotient of Like Signs

( )

( )

( )

( )

Quotient of Different Signs

( )

( )

( )

( )


2.5

Order of Operations



Order of Operations







Simplify 4(5 – 2) + 42.

Simplify inside parentheses.

4(5 – 2) + 42 = 4(3) + 42

= 4(3) + 16 Write 42 as 16.

= 12 + 16 Multiply.

= 28 Add.

Using the Order of Operations



When simplifying expressions with exponents, parentheses make an important difference.

(–5)2 and –52 do not mean the same thing.

(–5)2 means (–5)(–5) = 25.

–52 means the opposite of 5 ∙ 5, or –25.

Only with parentheses is the –5 squared.

Helpful Hint


Chapter 3

Solving Equations and Problem Solving


3.1

Simplifying Algebraic

Expressions


A term that is only a number is called a constant term, or simply a constant. A term that contains a variable is called a variable term.

x + 3

Constantterms

Variableterms

3y2 + (–4y) + 2

Constant and Variable Terms


The number factor of a variable term is called the numerical coefficient. A numerical coefficient of 1 is usually not written.

5x x or 1x –7y 3y

2

Numerical coefficient is 5.

Numerical coefficient is –7.

Numerical coefficient is 3.

Understood numerical coefficient is 1.

Coefficients


Terms that are exactly the same, except that they may have different numerical coefficients are called like terms.

Like Terms Unlike Terms

3x, 2x–6y, 2y, y–3, 4

7x, 7y5y, 56a, ab

The order of the variablesdoes not have to be the same.

2ab2, –5b

2a

5x, x 2

Like Terms


A sum or difference of like terms can be simplified using the distributive property.


If a, b, and c are numbers, then

ac + bc = (a + b)c

Also,

ac – bc = (a – b)c



By the distributive property,

7x + 5x = (7 + 5)x = 12x

This is an example of combining like terms.

An algebraic expression is simplified when all like terms have been combined.



The commutative and associative properties of addition and multiplication help simplify expressions.

Properties of Addition and Multiplication



a + b = b + a


a ∙ b = b ∙ a

The order of adding or multiplying two numbers can be changed without changing their sum or product.

Addition and Multiplication Properties


The grouping of numbers in addition or multiplication can be changed without changing their sum or product.


(a + b) + c = a + (b + c)


(a ∙ b) ∙ c = a ∙ (b ∙ c)

Associative Properties


Examples of Commutative and Associative Properties of Addition and Multiplication

4 + 3 = 3 + 4

6 ∙ 9 = 9 ∙ 6

(3 + 5) + 2 = 3 + (5 + 2)

(7 ∙ 1) ∙ 8 = 7 ∙ (1 ∙ 8)





Helpful Hint


We can also use the distributive property to multiply expressions.

2(5 + x) = 2 ∙ 5 + 2 ∙ x = 10 + 2xor

2(5 – x) = 2 ∙ 5 – 2 ∙ x = 10 – 2x

The distributive property says that multiplication distributes over addition and subtraction.

Multiplying Expressions


To simply expressions, use the distributive property first to multiply and then combine any like terms.

3(5 + x) – 17 =

Simplify: 3(5 + x) – 17

= 15 + 3x + (–17)

Apply the Distributive Property

Multiply

= 3x + (–2) or 3x – 2 Combine like terms

Note: 3 is not distributed to the –17 since –17 is not within the parentheses.

3 ∙ 5 + 3 ∙ x + (–17)

Simplifying Expressions


Finding Perimeter

3z feet

9z feet

7z feet

Perimeter = 3z + 7z + 9z = 19z feet

Don’t forget to insert proper units.

Perimeter is the distance around the figure.


Finding Area

A = length ∙ width = 3(2x – 5) = 6x – 15 square meters

Don’t forget to insert proper units.

3 meters

(2x – 5) meters


Don’t forget . . .

Area:• surface enclosed• measured in square units

Perimeter:• distance around• measured in units

Helpful Hint


3.2

Solving Equations: Review of the Addition

and Multiplication Properties


Statements like 5 + 2 = 7 are called equations.

An equation is of the form expression = expression.

An equation can be labeled as

Equal sign

left side right side

x + 5 = 9

Equation vs. Expression


Addition Property of Equality

Let a, b, and c represent numbers.

If a = b, then

a + c = b + c

and

a – c = b - c

In other words, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation.


Multiplication Property of Equality

Let a, b, and c represent numbers and let c 0. If a = b, then

a ∙ c = b ∙ c and

In other words, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation.

a b=

c c


Solve for x.

x - 4 = 3To solve the equation for x, we need to rewrite the equation in the form x = number. To do so, we add 4 to both sides of the equation. x - 4 = 3 x - 4 + 4 = 3 + 4 Add 4 to both sides. x = 7 Simplify.


Check

x - 4 = 3 Original equation

7 - 4 = 3 Replace x with 7.

3 = 3 True.

Since 3 = 3 is a true statement, 7 is the solution of the equation.

To check, replace x with 7 in the original equation.

?


Solve for x

4x = 8To solve the equation for x, notice that 4 is multiplied by x.

To get x alone, we divide both sides of the equation by 4 and then simplify.

4 8

4 4

x= 1∙x = 2 or x = 2


Check


4x = 8 Original equation

4 ∙ 2 = 8 Let x = 2.

8 = 8 True.

The solution is 2.

?


Using Both Properties to Solve Equations

2(2x – 3) = 10 Use the distributive property to simplify the left side.

4x – 6 = 10

Add 6 to both sides of the equation

x = 4

4x – 6 + 6 = 10 + 6

4x = 16

Divide both sides by 4.


Check


2(2x – 3) = 10 Original equation

2(2 · 4 – 3) = 10 Let x = 4.

2(8 – 3) = 10

(2)5 = 10 True.

The solution is 4.

?

?


Chapter 4

Fractions and Mixed Numbers


4.1

Introduction to Fractions and Mixed

Numbers


Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used.

A fraction is a number of the form ,

where a and b are integers and b is not 0.

The parts of a fraction are

ab

numerator abdenominator

fraction bar

Parts of a Fraction


Remember that the bar in a fraction means

47

division. Since division by 0 is undefined, a fraction with a denominator of 0 is undefined.

Helpful Hint


One way to visualize fractions is to picture them as shaded parts of a whole figure.

Visualizing Fractions


56

parts shaded

equal partsfive-sixths

Picture Fraction Read as

73

parts shaded

equal partsseven-thirds

14

part shaded

equal partsone-fourth

Visualizing Fractions


Types of Fractions

A proper fraction is a fraction whose numerator is less than its denominator.

Proper fractions have values that are less than 1.

An improper fraction is a fraction whose numerator is greater than or equal to its denominator.Improper fractions have values that are greater than or equal to 1.

A mixed number is a sum of a whole number and a proper fraction.

1 3 2

2 4 5, ,

8 5 4

3 5 1, ,

2 1 22

3 5 7,3 ,4


Another way to visualize fractions is to graph them on a number line.

0 1

5 equal parts

3

35

1

5

1

5

1

5

1

5

1

5

Fractions on Number Lines


If n is any integer other than 0, then

=1n

n

5

5=1

If n is any integer, then

1=

nn

3

1= 3

Fraction Properties of 1


If n is any integer other than 0, then 0

0n

=0

5= 0

If n is any integer, then

0= undefined

n 3

0= undefined

Fraction Properties of 0


Writing a Mixed Number as an Improper Fraction

Step 1: Multiply the denominator of the fraction by the whole number.

Step 2: Add the numerator of the fraction to the product from Step 1.

Step 3: Write the sum from Step 2 as the numerator of the improper fraction over the original denominator.

23

4

2 4 3

4

8 3

4

11

4=

∙ +=

+=


Writing an Improper Fraction as a Mixed Number or a Whole Number

Step 1: Divide the denominator into the numerator.

Step 2: The whole number part of the mixed number is the quotient. The fraction part of the mixed number is the remainder over the original denominator.

remainder

original denominatorquotient =


4.2

Factors and Simplest Form


Prime and Composite Numbers

A prime number is a natural number greater than 1 whose only factors are 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . .

A composite number is a natural number greater than 1 that is not prime.


The natural number 1 is neither prime nor composite.

Helpful Hint


Prime Factorization

A prime factorization of a number expresses the number as a product of its factors and the factors must be prime numbers.


Remember a factor is any number that divides a number evenly (with a remainder of 0).

Helpful Hints


Prime Factorization

Every whole number greater than 1 has exactly one prime factorization.

12 = 2 • 2 • 3

2 and 3 are prime factors of 12 because they are prime numbers and they divide evenly into 12.


Divisibility Tests

A whole number is divisible by

2 if its last digit is 0, 2, 4, 6, or 8.

3 if the sum of its digits is divisible by 3.

196 is divisible by 2

117 is divisible by 3 since 1 + 1 + 7 = 9 is divisible by 3.


Divisibility Tests

A whole number is divisible by

5 if the ones digit is 0 or 5.

10 if its last digit is 0.

2,345 is divisible by 5.

8,470 is divisible by 10.


Graph on the number line.34

34

Graph on the number line.68

68

0 1

1

418

Equivalent Fractions


Equivalent Fractions

Fractions that represent the same portion of a whole or the same point on the number line are called equivalent fractions.

6 ÷ 2 3

8 ÷ 2 4

6= =

8

3 2 6

4 2 8

3= =

4


Fundamental Property of Fractions


and also

×a a c

b b×c=

a a ÷ c

b b ÷ c=

as long as b and c are not 0. If the numerator and denominator are multiplied or divided by the same nonzero number, the result is an equivalent fraction.


Simplest Form

A fraction is in simplest form, or lowest terms, when the numerator and denominator have no common factors other than 1.

14 14 ÷ 7 2= =

21 21÷ 7 3

Using the fundamental principle of fractions, divide the numerator and denominator by the common factor of 7.

Using the prime factorization of the numerator and denominator, divide out common factors.

14 7 2= =

21 7 3

2

3=7 2

7 3


Writing a Fraction in Simplest Form

To write a fraction in simplest form, write the prime factorization of the numerator and the denominator and then divide both by all common factors.

The process of writing a fraction in simplest form is called simplifying the fraction.


When all factors of the numerator or denominator are divided out, don’t forget that 1 still remains in that numerator or denominator.

5 5 1= =

10 5 • 2 2

15 3 • 5 5= = = 5

3 3 1

Helpful Hints


4.3

Multiplying and Dividing Fractions



Multiplying Fractions

34

12 of is

38

0 134

38

68

The word “of” means multiplication and “is” means equal to.



34

12 of is

38

means

1 3 32 4 8

Multiplying Fractions



Multiplying Two Fractions

If a, b, c, and d are numbers and b and d are not 0, then

a c a c

b d b d

In other words, to multiply two fractions, multiply the numerators and multiply the denominators.


Examples

15

14

3 5 3 5

2 7 2 7

If the numerators have common factors with the denominators, divide out common factors before multiplying.

3 2 3 2

4 5 2 2 5

3

10 3 2

4 53

102

1

or


3 8

4 5

x

x

Examples

3 4 2

4 5

x

x 3 8

4 5

x

x

1

2

6

5

6

5or



Recall that when the denominator of a fraction contains a variable, such as

,8

5xwe assume that the variable is not 0.

Helpful Hint


Expressions with Fractional Bases

The base of an exponential expression can also be a fraction.

23

3FHIK 23

23

23

2 2 23 3 3

8

27



Reciprocal of a Fraction

a

b

b

a

because

1

a b a b ab

b a b a ab

Two numbers are reciprocals of each other if their product is 1. The reciprocal of the fraction

is



Dividing Two Fractions

If b, c, and d are not 0, then

a c a d a d

b d b c b c

In other words, to divide fractions, multiply the first fraction by the reciprocal of the second fraction.

3 2 3 7 21

5 7 5 2 10



Every number has a reciprocal except 0. The number 0 has no reciprocal. Why?

There is no number that when multiplied by 0 gives the result 1.

Helpful Hint



When dividing by a fraction, do not look for common factors to divide out until you rewrite the division as multiplication.

12

23

12

32

34

Do not try to divide out these two 2s.

Helpful Hint


Fractional Replacement Values

x y

56

25

Replace x with and y with .

If x = and y = , evaluate . 56

25

x y

56

25

56

52

2512


4.4

Adding and Subtracting Like Fractions, Least

Common Denominator, and Equivalent Fractions



Fractions that have the same or common denominator are called like fractions.

Fractions that have different denominators are called unlike fractions.

Like Fractions Unlike Fractions

2

5and

4

5

2

3and

3

4

5

6and

5

12

Like and Unlike Fractions

5

7and

3

7



Adding or Subtracting Like Fractions

If a, b, and c, are numbers and b is not 0, then

alsoa c a c a c a c

b b b b b b

To add or subtract fractions with the same denominator, add or subtract their numerators and write the sum or difference over the common denominator.



0 11

7

2

7

4

7

2 4

7 7

6

7=

To add like fractions, add the numerators and write the sum over the common denominator.

6

7

Start End

Adding or Subtracting Like Fractions



Do not forget to write the answer in simplest form. If it is not in simplest form, divide out all common factors larger than 1.

Helpful Hint



Equivalent Negative Fractions

2 2 2 2

3 3 3 3



To add or subtract fractions that have unlike, or different, denominators, we write the fractions as equivalent fractions with a common denominator. The smallest common denominator is called the least common denominator (LCD) or the least common multiple (LCM).

Least Common Denominator



The least common denominator (LCD) of a list of fractions is the smallest positive number divisible by all the denominators in the list. (The least common denominator is also the least common multiple (LCM) of the denominators.)

Least Common Multiple



To find the LCD of and5 5

12 18

First, write each denominator as a product of primes.

Then write each factor the greatest number of times it appears in any one prime factorization.

The greatest number of times that 2 appears is 2 times. The greatest number of times that 3 appears is 2 times.

12 = 2 • 2 • 3

LCD = 2 • 2 • 3 • 3 = 36

18 = 2 • 3 • 3

Least Common Denominator


4.5

Adding and Subtracting Unlike Fractions



Adding or Subtracting Unlike Fractions

Step 1: Find the LCD of the denominators of the fractions.

Step 2: Write each fraction as an equivalent fraction whose denominator is the LCD.

Step 3: Add or subtract the like fractions.

Step 4: Write the sum or difference in simplest form.



Add:

Step 1: Find the LCD of 9 and 12.

LCD = 2 ∙ 2 ∙ 3 ∙ 3 = 36

Step 2: Rewrite equivalent fractions with the LCD.


1 79 12

9 = 3 ∙ 3 and 12 = 2 ∙ 2 ∙ 3

1 1 4 49 9 4 36

7 7 3 21

12 12 3 36

Continued.



Continued:

Step 3: Add like fractions.

Step 4: Write the sum in simplest form.


1 4 7 3 4 21 259 4 12 3 36 36 36

2536



One important application of the least common denominator is to use the LCD to help order or compare fractions.

Writing Fractions in Order

Insert < or > to form a true sentence.

The LCD for these fractions is 35.

Write each fraction as an equivalent fraction with a denominator of 35.

3 5

? 4 7

3 3 7 215 5 7 35

4 4 5 207 7 5 35

Continued.



Compare the numerators of the equivalent fractions.

Writing Fractions in Order

Continued:

Since 21 > 20, then 21 20 > 35 35

Thus, 3 4 > 5 7



Evaluating Expressions

Evaluate x – y if x = and y = .

199Martin-Gay, Prealgebra, 5ed

23

34

Replacing x with and y with , 23

34

then, x – y2 33 4

2 4 3 3 8 9 13 4 4 3 12 12 12



Solving Equations Containing Fractions

Solve: 1 53 12

x

To get x by itself, add to both sides. 13

1 133 3

1 512

x

1 412 3 45

Continued.




Write fraction in simplest form.

Continued:

5 412 12

x

9 312 4


4.6

Complex Fractions and Review of Order of

Operations



Complex Fraction

A fraction whose numerator or denominator or both numerator and denominator contain fractions is called a complex fraction.

2 33 5

15 7y

2

45

78

23

4x



Method 1: Simplifying Complex Fractions

2 9

3 84

1

1

3

3

4

2389

This method makes use of the fact that a fraction bar means division.

When dividing fractions, multiply by the reciprocal of the divisor.



1 12 63 24 3

3 16 69 8

12 12

461

12

4 12

6 18

1

2

33

3

1 12 6

24

4333 4


Recall the order of operations. Since the fraction bar is a grouping symbol, simplify the numerator and denominator separately. Then divide.

When dividing fractions, multiply by the reciprocal of the divisor.




This method is to multiply the numerator and the denominator of the complex fraction by the LCD of all the fractions in its numerator and its denominator. Since this LCD is divisible by all denominators, this has the effect of leaving sums and differences of terms in the numerator and the denominator and thus a simple fraction.

Let’s use this method to simplify the complex fraction of the previous example.




12

16

34

23

1212

16

1234

23

FH IKFH IK

12

12

1216

1234

1223

FHIK FHIKFHIK FHIK

Step 1: The complex fraction contains fractions with denominators of 2, 6, 4, and 3. The LCD is 12. By the fundamental property of fractions, multiply the numerator and denominator of the complex fraction by 12.

Step 2: Apply the distributive propertyContinued.




12

16

34

23

1212

1216

1234

1223

FHIK FHIKFHIK FHIK

Step 3: Multiply.

Step 4: Simplify.

6 29 8

81

8

The result is the same nomatter which method is used.

Continued:



Reviewing the Order of Operations






4.7

Operations on Mixed Numbers



Recall that a mixed number is the sum of a whole number and a proper fraction.

19

53

4

5=

34

53

4

5= +

0 1 2 543

Mixed Numbers



Multiplying or Dividing with Mixed Numbers

To multiply or divide with mixed numbers or whole numbers, first write each mixed number as an improper fraction.

Write the solutionas a mixed numberif possible.

Remove common factorsand multiply.

Change mixed numbersto improper fractions.

Multiply: 1 1

3 25 4

1 1 16 9

3 25 4 5 4

4 4 9

5 4

36

5

17

5



We can add or subtract mixed numbers by first writing each mixed number as an improper fraction. But it is often easier to add or subtract the whole number parts and add or subtract the proper fraction parts vertically.

Adding or Subtracting Mixed Numbers



Add: 2 5514

67

The LCD of 14 and 7 is 14.

2 2

5 5

514

514

67

1214

71714

1714 1 3

14Since is , write the sum as

71714 7 1 3

14 8 3

14

Write equivalent fractions with the LCD of 14.

Notice that the fractional part is improper.

Add the fractions, then add the whole numbers.

Make sure the fractionalpart is always proper.




When subtracting mixed numbers, borrowing may be needed.

313

0 1 2 543

313= +2 1

13= + +2 1

13= + +2

33

13= 2

43

Borrow 1 from 3.




Subtract: 5 3314

67

The LCD of 14 and 7 is 14.

5 5

3 3

314

314

67

1214

Write equivalent fractions with the LCD of 14.

To subtract the fractions, we have to borrow.

Notice that the fractionalpart is proper.

4 1714

4 1714

Subtract the fractions, then subtract the whole numbers.

5 5 4

3 3 3

314

314

1714

67

1214

1214

1 514

5 4 1314

314



4.8




Addition Property of Equality

Let a, b, and c represent numbers.If a = b, then

a + c = b + c and

a – c = b - c

In other words, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation.


Multiplication Property of Equality

Let a, b, and c represent numbers and let c 0. If a = b, then

a ∙ c = b ∙ c and

In other words, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation.

a bc c



Solving an Equation in x

Step 1: If fractions are present, multiply both sides of the equation by the LCD of the fractions.

Step 2: If parentheses are present, use the distributive property.

Step 3: Combine any like terms on each side of the equation.



Solving an Equation in x

Step 4: Use the addition property of equality to rewrite the equation so that variable terms are on one side of the equation and constant terms are on the other side.

Step 5: Divide both sides of the equation by the numerical coefficient of x to solve.

Step 6: Check the answer in the original equation.


Solve for x

1 5=7 9x

Multiply both sides by 7. 7 17 75= 9x

Simplify both sides.35= 9x


Solve for x

3( +3) = 2 +65y y

Multiply both sides by 5. 3( +3) = 2 +6

5y y5 5

Simplify both sides.3 +9 = 10 +30y yAdd – 3y to both sides.9 = 7 +30yAdd – 30 to both sides. 21= 7yDivide both sides by 7. 3 = y


Chapter 5

Decimals


5.1

Introduction to Decimals



Whole number part

Decimal point

Decimal part

16.743

Like fractional notation, decimal notation is used to denote a part of a whole. Numbers written in decimal notation are called decimal numbers, or simply decimals. The decimal 16.734 has three parts.

Decimal Notation



The position of each digit in a number determines its place value.

1 6 7 3 4 on

es

thou

san

dth

s

hu

nd

red

s

ten

s

ten

ths

hu

nd

red

ths

ten

-th

ousa

nd

ths

hu

nd

red

-th

ousa

nd

ths

Place Value

decimal point

100 10 1

1100

110,000

110

1100,000

11000

Place Value



Notice that the value of each place is

of the value of the place to its left.

110

Place Value



16.734

The digit 3 is in the hundredths place, so

its value is 3 hundredths or .3

100

Place Value



Writing a Decimal in Words

Step 1: Write the whole number part in words.

Step 2: Write “and” for the decimal point.

Step 3: Write the decimal part in words as though it were a whole number, followed by the place value of the last digit.



Writing a Decimal in Words

Write the decimal 143.056 in words.

143.056

one hundred forty-three and fifty-six thousandths

whole number part decimal part



A decimal written in words can be written in standard form by reversing the procedure.

Writing Decimals in Standard Form

Write one hundred six and five hundredths in standard form.

one hundred six and five hundredths

106 . 05

decimal partwhole-number part decimal

5 must be in thehundredths place

5 must be in thehundredths place



When writing a decimal from words to decimal notation, make sure the last digit is in the correct place by inserting 0s after the decimal point if necessary.

For example,

three and fifty-four thousandths is 3.054

thousandths place

Helpful Hint



Once you master writing and reading decimals correctly, then you write a decimal as a fraction using the fractions associated with the words you use when you read it.

0.9is read “nine tenths” and written as a fraction as 9

10

Writing Decimals as Fractions



twenty-one hundredthsand written as a fraction as 21

100

0.011 is read as eleven thousandthsand written as a fraction as

111000

0.21 is read as

Writing Decimals as Fractions



Notice that the number of decimal places in a decimal number is the same as the number of zeros in the denominator of the equivalent fraction. We can use this fact to write decimals as fractions.

0 3737

100. =

2 decimal places

2 zeros

0 02929

1000. =

3 decimal places

3 zeros

Comparing Two Positive Decimals



Comparing Decimals

One way to compare decimals is to compare their graphs on a number line. Recall that for any two numbers on a number line, the number to the left is smaller and the number to the right is larger. To compare 0.3 and 0.7 look at their graphs.

0 10.3

3

10

7

10

0.70.3 < 0.7 or 0.7 > 0.3



Comparing decimals by comparing their graphs on a number line can be time consuming, so we compare the size of decimals by comparing digits in corresponding places.





Compare digits in the same places from left to right. When two digits are not equal, the number with the larger digit is the larger decimal. If necessary, insert 0s after the last digit to the right of the decimal point to continue comparing.Compare hundredths place digits.

3 5<

35.638 35.657<

35.638 35.657



For any decimal, writing 0s after the last digit to the right of the decimal point does not change the value of the number.

8.5 = 8.50 = 8.500, and so onWhen a whole number is written as a decimal, the decimal point is placed to the right of the ones digit. 15 = 15.0 = 15.00, and so on

Helpful Hint



We round the decimal part of a decimal number in nearly the same way as we round whole numbers. The only difference is that we drop digits to the right of the rounding place, instead of replacing these digits by 0s. For example,

63.782 rounded to the nearest hundredth is

63.78

Rounding Decimals



Rounding Decimals

Step 1: Locate the digit to the right of the given place value.

Step 2: If this digit is 5 or greater, add 1 to the digit in the given place value and drop all digits to the right. If this digit is less than 5, drop all digits to the right of the given place.



Rounding Decimals to a Place Value

Round 326.4386 to the nearest tenth.

Locate the digit to the right of the tenths place.

326.4386

tenths place

digit to the right

Since the digit to the right is less than 5, drop it and all digits to its right.

326.4386 rounded to the nearest tenths is 326.4


5.2

Adding and Subtracting Decimals



Adding or Subtracting Decimals

Step 1: Write the decimals so that the decimal points line up vertically.

Step 2: Add or subtract as with whole numbers.

Step 3: Place the decimal point in the sum or difference so that it lines up vertically with the decimal points in the problem.



Recall that 0s may be inserted to the right of the decimal point after the last digit without changing the value of the decimal. This may be used to help line up place values when adding or subtracting decimals.

85 - 13.26 becomes

85.00 - 13.26

71.74

two 0s inserted

Helpful Hint



Don’t forget that the decimal point in a whole number is after the last digit.

Helpful Hint



Estimating sums, differences, products, and quotients of decimal numbers is an important skill whether you use a calculator or perform decimal operations by hand.

Estimating Operations on Decimals



Add 23.8 + 32.1.

Estimating When Adding Decimals

Exact Estimate

23.8+32.1

55.9

rounds to 24rounds to 32

56

This is a reasonable answer.



When rounding to check a calculation, you may want to round the numbers to a place value of your choosing so that your estimates are easy to compute mentally.

Helpful Hint



Evaluate x + y for x = 5.5 and y = 2.8.

Evaluating with Decimals

x + y = ( ) + ( )

Replace x with 5.5 and y with 2.8 in x + y.

5.5 2.8

= 8.3


5.3

Multiplying Decimals and Circumference of a

Circle



Multiplying decimals is similar to multiplying whole numbers. The difference is that we place a decimal point in the product.

0.7 0.03 = 7

10

3

100

1 decimal place

2 decimal places

21

1000

= 0.021

=

3 decimal places

Multiplying Decimals



Step 1: Multiply the decimals as though they were whole numbers.

Step 2: The decimal point in the product is placed so the number of decimal places in the product is equal to the sum of the number of decimal places in the factors.

Multiplying Decimals



Multiply 32.3 1.9.

Estimating when Multiplying Decimals

Exact Estimate

32.3

1.9

290.7323.061.37

rounds to 32rounds to 2

64




Multiplying Decimals by Powers of 10

There are some patterns that occur when we multiply a number by a power of ten, such as 10, 100, 1000, 10,000, and so on.



76.543 10 = 765.43

76.543 100 = 7654.3

76.543 100,000 = 7,654,300

Decimal point moved 1 place to the right.

Decimal point moved 2 places to the right.

Decimal point moved 5 places to the right.

2 zeros

5 zeros

1 zero

The decimal point is moved the same number of places as there are zeros in the power of 10.




Move the decimal point to the right the same number of places as there are zeros in the power of 10.

Multiply: 3.4305 100

Since there are two zeros in 100, move the decimal place two places to the right.

3.4305 100 = 343.053.4305 =




Move the decimal point to the left the same number of places as there are decimal places in the power of 10.

Multiply: 8.57 0.01

Since there are two decimal places in 0.01, move the decimal place two places to the left.

8.57 0.01 = 0.0857

Notice that zeros had to be inserted.

008.57 =




The distance around a polygon is called its perimeter.

The distance around a circle is called the circumference.

This distance depends on the radius or the diameter of the circle.

The Circumference of a Circle



r

d

Circumference = 2·p ·radiusor

Circumference = p ·diameterC = 2 p r or C = p d




The symbol p is the Greek letter pi, pronounced “pie.” It is a constant between 3 and 4. A decimal approximation for p is 3.14.A fraction approximation for p is .

p

227



Find the circumference of a circle whose radius is 4 inches.

4 inches

C = 2pr = 2 p ·4 = 8p inches8 p inches is the exact circumference of this circle.

If we replace with the approximation 3.14, C = 8 8(3.14) = 25.12 inches.25.12 inches is the approximate circumference of the circle.



5.4

Dividing Decimals



The only difference is the placement of a decimal point in the quotient. If the divisor is a whole number, divide as for whole numbers; then place the decimal point in the quotient directly above the decimal point in the dividend.

8

- 5 0 4

2 52-2 52

0

divisor

quotient

dividend0 4

Division of decimal numbers is similar to division of whole numbers.

63 52.92

Dividing by a Decimal



863 52 9.2

- 504

25 2-252

0

4

If the divisor is not a whole number, we need to move the decimal point to the right until the divisor is a whole number before we divide.

divisor dividend6 3 52 92. .

63 529 2. .





Step 1: Move the decimal point in the divisor to the right until the divisor is a whole number.

Step 2: Move the decimal point in the dividend to the right the same number of places as the decimal point was moved in Step 1.

Step 3: Divide. Place the decimal point in the quotient directly over the moved decimal point in the dividend.



Divide 258.3 ÷ 2.8

Estimating When Dividing Decimals

Exact Estimate

28. 2583. - 252 63 - 56 70 - 56 140 -140 0

rounds to 3 300100


92.25



There are patterns that occur when dividing by powers of 10, such as 10, 100, 1000, and so on.

The decimal point moved 1 place to the left.

1 zero

3 zeros

The decimal point moved 3 places to the left.

The pattern suggests the following rule.

.45 6210

=456.2

1 0000 4562

,.=456.2

Dividing Decimals by Powers of 10



Move the decimal point of the dividend to the left the same number of places as there are zeros in the power of 10.

Dividing Decimals by Powers of 10

Notice that this is the same pattern as multiplying by powers of 10 such as 0.1, 0.01, or 0.001. Because dividing by a power of 10 such as 100 is the same as multiplying by its reciprocal , or 0.01.

1100

463 7100

463 71

100463 7 0 01 4 637

.. . . .

To divide by a number is the same as multiplying by its reciprocal.


5.5

Fractions, Decimals, and Order of Operations



To write a fraction as a decimal, divide the numerator by the denominator.

Writing Fractions as Decimals

3 = 3 4 = 0.754

2 = 2 5 = 0.405



Comparing Fractions and Decimals

To compare decimals and fractions, write the fraction as an equivalent decimal.

Compare 0.125 and .14

1 = 0.254

Therefore, 0.125 < 0.25



Order of Operations







Using the Order of Operations

Simplify ( –2.3)2 + 4.1(2.2 + 3.1)

Simplify inside parentheses.

( –2.3)2 + 4.1(2.2 + 3.1)= ( –2.3)2 + 4.1(5.3)

= 5.29 + 4.1(5.3) Write ( –2.3)2 as 5.29.

= 5.29 + 21.73 Multiply.

= 27.02 Add.



Finding the Area of a Triangle

base

height

A base • height=1

2

A bh=1

2


5.6

Equations Containing Decimals



Steps for Solving an Equation in x

Step 1: If fractions are present, multiply both sides of the equation by the LCD of the fractions.

Step 2: If parentheses are present, use the distributive property.

Step 3: Combine any like terms on each side of the equation.



Steps for Solving an Equation in x

Step 4: Use the addition property of equality to rewrite the equation so that variable terms are on one side of the equation and constant terms are on the other side.

Step 5: Divide both sides by the numerical coefficient of x to solve.

Step 6: Check the answer in the original equation.



–0.01(5a + 4) = 0.04 – 0.01(a + 4)

Solving Equations with Decimals

Multiply both sides by 100. –1(5a + 4) = 4 – 1(a + 4)

Apply the distributive property. –5a – 4 = 4 – a – 4

Add a to both sides. –4a – 4 = 4 – 4

Add 4 to both sides and simplify. –4a = 4

Divide both sides by 4. a = –1


5.7

Decimal Applications: Mean, Median, and

Mode



The mean, the median, and the mode are called measures of central tendency. They describe a set of data, or a set of numbers, by a single “middle” number.

Measures of Central Tendency



Mean (Average)

The most common measure of central tendency is the mean (sometimes called the “arithmetic mean” or the “average”).

The mean (average) of a set of number items is the sum of the items divided by the number of items.



Finding the Mean

Find the mean of the following list of numbers.

2.5

5.1

9.5

6.8

2.5

Continued.



The mean is the average of the numbers:

2.5

5.1

9.5

6.8

2.5

2.5 +5.1+9.5 +6.8 +2.55

= 5.28

Finding the Mean



Median

You may have noticed that a very low number or a very high number can affect the mean of a list of numbers. Because of this, you may sometimes want to use another measure of central tendency, called the median.

The median of an ordered set of numbers is the middle number. If the number of items is even, the median is the mean (average) of the two middle numbers.



Finding the Median

Find the median of the following list of numbers.

2.5

5.1

9.5

6.8

2.5Continued.



Finding the Median

List the numbers in numerical order:

2.5

2.5

5.1

6.8

9.5

Median



In order to compute the median, the numbers must first be placed in order.

Helpful Hint



Mode

The mode of a set of numbers is the number that occurs most often. (It is possible for a set of numbers to have more than one mode or to have no mode.)



Finding the Mode

Find the mode of the following list of numbers.

2.5

5.1

9.5

6.8

2.5

Continued.



Finding the Mode

The mode occurs the most often:

2.5

5.1

9.5

6.8

2.5

The mode is 2.5.



Don’t forget that it is possible for a list of numbers to have no mode. For example, the list

2, 4, 5, 6, 8, 9has no mode. There is no number or numbers that occur more often than the others.

Helpful Hint


Chapter 6

Ratio, Proportion, and

Triangle Applications


6.1

Ratio and Rates



A ratio is the quotient of two quantities.

Writing Ratios as Fractions

For example, a percent can be thought of as a ratio, since it is the quotient of a number and 100.

53% = 53

100 or the ratio of 53 to 100



The ratio of a number a to a number b is their quotient. Ways of writing ratios are

andab

a to b, a : b,

Ratio



A rate is a special kind of ratio. It is used to compare different kinds of quantities.

Writing Rates as Fractions

5 miles 1 mile55 minutes 11 minutes



To write a rate as a unit rate, divide the numerator of the rate by the denominator.

Finding Unit Rates

miles g

314.al15

7 lons 314.5 ÷ 17 = 18.5

The unit rate is .18.5 miles1 gallon



When a unit rate is “money per item,” it is also called a unit price.

Finding Unit Prices

price unit price = number of units

A store charges $2.76 for a 12-ounce jar of pickles. What is the unit price?

$2.76 $0.23 unit price = 12 ounces 1 ounce

($0.23 per ounce )


6.2

Proportions



A proportion is a statement that two ratios or rates are equal.

Solving Proportions

If and are two ratios, then

is a proportion.

ab

cd

ab

cd

=



Solving Proportions

A proportion contains four numbers. If any three numbers are known, the fourth number can be found by solving the proportion. To solve use cross products.

bdFHGIKJ bd

FHGIKJ

a

b

c

d

a

b

c

d=

Multiply both sides by the LCD, bd

Simplify ad = bc

cross product cross product

ad

bc

These are calledcross products.



Determining Whether Proportions are True

3 12Is = a true proportion?8 32

3 12=8 32

3 32 = 8 12?

96 = 96 True proportion



Finding Unknown Numbers in Proportions

26 28Solve = . 49x

26 49 = 28x Cross multiply.

1274 = 28x Simplify the left side.

45.5 = x Divide both sides by 28.

Check: 45.526 28=

490.57143 = 0.57143 (Rounded)


6.3

Proportions and Problem Solving



A 16-oz Cinnamon Mocha Iced Tea at a local coffee shop has 80 calories. How many calories are there in a 28-oz Cinnamon Mocha Iced Tea?

Solving Problems by Writing Proportions

16 ounces 28 ounces80 calories caloriesx

Solve the proportion.

16 80 28x Cross multiply.

16 2240x Simplify the right side.

140x Divide both side by 140.

A 28-oz Cinnamon Mocha Iced Tea has 140 calories.



When writing proportions to solve problems, write the proportions so that the numerators have the same unit measures and the denominators have the same unit measures.

For example, 2 7inches5 miles

inchesmiles

n

Helpful Hint


6.4

Square Roots and the Pythagorean

Theorem



The square of a number is the number times itself.

The square of 6 is 36 because 62 = 36.

The square of –6 is also 36 because

The Square of a Number

(–6)2 = (–6) (–6) = 36.



The reverse process of squaring is finding a square root.

A square root of 36 is 6 because 62 = 36.

A square root of 36 is also –6 because (–6)2 = 36.

Square Root of a Number

We use the symbol , called a radical sign, to

indicate the positive square root.

because 42 = 16 and 4 is positive. 16 4

25 5 because 52 = 25 and 5 is positive.



Square Root of a Number

The square root, , of a positive number a is the

positive number b whose square is a. In symbols,

2 if .a b b a

29 3 because 3 9.

Also, 0 0.



Remember that the radical sign is

used to indicate the positive square root

of a nonnegative number.

Helpful Hint



Numbers like are

called perfect squares because their square

root is a whole number or a fraction.

Perfect Squares

1 4, 36, , and 14 25



A square root such as cannot be

written as a whole number or a fraction

since 6 is not a perfect square. It can be

approximated by estimating by using a

table or by using a calculator.

Approximating Square Roots

6



One important application of square roots has to do with right triangles.

A right triangle is a triangle in which one of the angles is a right angle or measures 90º (degrees).

The hypotenuse of a right triangle is the side opposite the right angle.

hypotenuseleg

leg

The legs of a right triangle are the other two sides.

Right Triangles



Pythagorean Theorem

If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then

In other words,

c a

b

(leg)2 + (other leg)2 = (hypotenuse)2.

2 2 2a b c


6.5

Congruent and Similar Triangles



Two triangles are congruent when they have the same shape and the same size. Corresponding angles are equal, and corresponding sides are equal.

Congruent Triangles

a = 6 c = 11

b = 9

d = 6e = 11

f = 9

equal angles

equal anglesequal angles



Similar triangles are found in art, engineering, architecture, biology, and chemistry. Two triangles are similar when they have the same shape but not necessarily the same size.

Similar Triangles



In similar triangles, the measures of corresponding angles are equal and corresponding sides are in proportion.

a = 3

c = 8

b = 5 d = 6 e = 10

f = 16

Side a corresponds to side d, side b corresponds to side e, and side c corresponds to side f.

Similar Triangles

3 16 2

ad 5 1

10 2be 8 1

16 2cf


Chapter 7

Percents


7.1

Percents, Decimals, and Fractions



The word percent comes from the Latin phrase per centum, which means “per 100.”

Percent means per one hundred. The “%” symbol is used to denote percent.

Understanding Percent

1

1% 0.01100



0.65 = 0.65(100%) = 65.% or 65%

Writing a Decimal as a Percent

Multiply by 1 in the form of 100%.



Writing a Percent as a Decimal

43% = 43(0.01) = 0.43

100% = 100(0.01) = 1.00 or 1

Replace the percent symbol with its decimal equivalent, 0.01; then multiply.



Writing a Percent as a Fraction

43% 431 43

100 100

Replace the percent symbol with its fraction

equivalent, ; then multiply. Don’t forget to

simplify the fraction, if possible.

1

100



Writing a Fraction as a Percent

•3 3

5 5 100% •

3 100%

5 1

300%

560%

Multiply by 1 in the form of 100%.



We know that

100% = 1

Recall that when we multiply a number by 1, we are not changing the value of that number.

Therefore, when we multiply a number by 100%, we are not changing its value but rather writing the number as an equivalent percent.

Helpful Hint



• To write a percent as a fraction, replace the % symbol with its fraction equivalent, ; then multiply.

1

100

Summary

• To write a percent as a decimal, replace the % symbol with its decimal equivalent, 0.01; then multiply.

• To write a decimal or fraction as a percent, multiply by 100%.


7.2

Solving Percent Problems with

Equations



Key Words

of means multiplication (∙)

is means equals (=)

what (or some equivalent) means the unknown number

Let x stand for the unknown number.



Remember that an equation is simply a mathematical statement that contains an equal sign (=).

6 = 18x

equal sign

Helpful Hint



20% of 50 = 10

20% • 50 = 10percent base amount

Percent Equationpercent ∙ base = amount

Solving Percent Problems



When solving a percent equation, write the percent as a decimal or fraction.

If your unknown in the percent equation is a percent, don’t forget to convert your answer to a percent.

Helpful Hint



Use the following to see if your answers are reasonable.

a percent greater than 100%

a percent less than 100%

a number larger than the original number

a number less than the original number

=

=

100% of a number = the number

Helpful Hint


7.3

Solving Percent Problems with

Proportions



To understand the proportion method, recall that

30% means the ratio of 30 to 100, or .

Writing Percent Problems as Proportions

30

100

30 3

30%100 10



Since the ratio is equal to the ratio , we

have the proportion

called the percent proportion.

Writing Percent Problems as Proportions

,

30

1003

10

30 3

100 10



always 100

or

percent

base

amount

Percent Proportion

amount percent

base 100

100

a p

b



When we translate percent problems to proportions, the percent can be identified by looking for the symbol % or the word percent. The base usually follows the word of. The amount is the part compared to the whole.

Symbols and Key Words



Part of Proportion

How It’s Identified

Percent % or percent

Base

Amount

Appears after of

Part compared to whole

Helpful Hints



What number is 20% of 8?

amount percent base

amount

base

percent

Solving Percent Proportions for the Amount

20

8 100

a



20 is 40% of what number?

amount percent base

amount

base

percent

20 40

100b

Solving Percent Proportions for the Base



What percent of 40 is 8?

amountpercent base

amount

base

percent

8

40 100

p

Helpful HintRecall from our percent proportion that this number, p already is a percent. Just keep the number the same and attach a % symbol.

Solving Percent Proportions for the Percent



A ratio in a proportion may be simplified before

solving the proportion. The unknown number in

both

and

is 20.

6 30

4 b

3 30

2 b

Helpful Hint


7.4

Applications of Percent



The freshman class of 450 students is 36% of all students at State College. How many students go to State College?

State the problem in words, then translate to an equation.

Equation Method

In words: 450 is 36% of what number?

Solve: 450 = 0.36x

Translate: 450 = 36% • x

Equation Method



The freshman class of 450 students is 36% of all students at State College. How many students go to State College?

State the problem in words, then translate to a proportion.

Proportion Equation Method

In words: 450 is 36% of what number?

Translate and Solve:

amount basepercent

450 36=

100b

Proportion Method



Percent Increase Percent Decrease

percent increase =

percent decrease =

In each case write the quotient as a percent.

amount of increase

original amount

amount of decrease

original amount

Helpful HintMake sure that this number in the denominator is the original number and not the new number.


7.5

Percent and Problem Solving: Sales Tax,

Commission, and Discount



Most states charge a tax on certain items when purchased called a sales tax.

A 5% sales tax rate on a purchase of a $10.00 item gives a sales tax of

sales tax = 5% of $10 = 0.05 ∙ $10.00 = $0.50

Calculating Sales Tax and Total Price



The total price to the customer would be

purchase price

plus

sales tax

$10.00 + $0.50 = $10.50

Sales Tax and Total Price



sales tax = tax rate ∙ purchase price

total price = purchase price + sales tax

Sales Tax and Total Price



A wage is payment for performing work.

An employee who is paid a commission as a wage is paid a percent of his or her total sales.

commission = commission rate • sales

Calculating Commissions



amount of discount = discount rate ∙ original price

sale price = original price - amount of discount

Discount and Sale Price`


7.6

Percent and Problem Solving: Interest



Interest is money charged for using other people’s money.

Money borrowed, loaned, or invested is called the principal amount, or simply principal.

The interest rate is the percent used in computing the interest (usually per year).

Simple interest is interest computed on the original principal.

Calculating Simple Interest



simple Interest = Principal • Rate

or

I = P • R • T

where the rate is understood to be per year and time is in years.

Simple Interest



total amount (paid or received) = principal + interest

Finding the Total Amount of a Loan



Compound interest is computed on not only the principal, but also on the interest already earned in previous compounding periods.

If interest is compounded annually on an investment, this means that interest is added to the principal at the end of each year and next year’s interest is computed on this new amount.

Calculating Compound Interest



Compound Interest Formula

1n t

rA Pn

The total amount A in an account is given by

where P is the principal, r is the interest rate written as a decimal, t is the length of time in years, and n is the number of times compounded per year.



total amount = original principal • compound interest

factor

The compound interest factor comes from the compound interest table found in Appendix C of the textbook.

Finding Total Amounts with Compound Interest

© 2010 Pearson Prentice Hall. All rights reserved Chapter 1 Review of Real Numbers.

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Transcript of © 2010 Pearson Prentice Hall. All rights reserved Chapter 1 Review of Real Numbers.