Transcript Prealgebra

Chapter 1
Review of Real
Numbers
© 2010 Pearson Prentice Hall. All rights reserved
1.1
Tips for Success in
Mathematics
© 2010 Pearson Prentice Hall. All rights reserved
Getting Ready for This Course
Positive
Attitude
Believe you can succeed.
Make sure you have time
Scheduling for your classes.
Be
Prepared
Have all the materials you
need, like a lab manual,
calculator, or other
supplies.
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Martin-Gay, Prealgebra, 6ed
3
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
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General Tips for Success
Tip
Get help if you need it.
Details
Ask for help when you don’t understand
something. Know when your instructors office
hours are, and whether tutoring services are
available.
Organize your assignments, quizzes, tests, and
Organize class materials. 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.
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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.
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Continued
6
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.
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Getting Help
Tip
Details
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
Get help as good chance you won’t understand the
soon as you concepts covered in the next period.
need it.
For help try your instructor, a tutoring
center, or a math lab. A study group can
also help increase your understanding of
covered materials.
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Preparing for and Taking an Exam
Steps for Preparing for a Test
1. 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
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Preparing for and Taking an Exam
Steps for Taking Your Test
1. 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.
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Managing Your Time
Tips for Making a Schedule
1. 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.
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§ 1.2
Place Value, Names
for Numbers, and
Reading Tables
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Ten-millions
Millions
Hundred-thousands
Ten-thousands
Thousands
Hundreds
Tens
3
5
6
8
9
4
0
2
Martin-Gay, Prealgebra, 6ed
Ones
Hundred-millions
Billions
Ten-billions
Hundred-billions
Place Value
The position of each digit in a number determines its place value.
13
Writing a Number in Words
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.
Hundred-thousands
Ten-thousands
Thousands
Hundreds
Tens
3
5
6
8
9
4
0
2
Martin-Gay, Prealgebra, 6ed
Ones
Millions
Ones
Ten-millions
Thousands
Hundred-millions
Millions
Billions
Ten-billions
Hundred-billions
Billions
14
Writing a Number in Words
Ones
Tens
Hundreds
Thousands
Ten-thousands
Hundred-thousands
Millions
Ten-millions
Hundred-millions
Billions
Ten-billions
Hundred-billions
To write a whole number in words, write the number in each period
followed by the name of the period.
3 5
6
8
9
4 0
2
thirty-five million, six hundred eighty-nine thousand, four hundred two
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Helpful Hint
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.
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Expanded Form
Standard Form
4,786
Expanded Form
=
4000 + 700 + 80 + 6
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.
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Comparing Whole Numbers
We can picture whole numbers as equally spaced points
on a line called the number line.
0
1
2
3
4
5
A whole number is graphed by placing a dot on the
number line. The graph of 4 is shown.
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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.
0
1
2
3
4
5
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
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Comparing Numbers . . .
2 is less than 5
can be written in symbols as
2<5
5 is greater than 2
is written as
5>2
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Helpful Hint
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.
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Reading Tables
Most Medals Olympic Winter (1924 – 2002) Games
Gold
Silver
Bronze
Total
Germany
107
104
86
297
Russia
113
83
78
274
Norway
94
92
74
260
USA
69
71
51
191
Austria
41
57
64
162
Source: The Sydney Morning Herald, Flags courtesy of www.theodora.com/flags used with permission
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1.3
Adding and
Subtracting Whole
Numbers, and
Perimeter
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Addition Property of 0
The sum of 0 and any number is that number.
8 + 0 = 8 and 0 + 8 = 8
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Commutative Property of Addition
Changing the order of two addends does
not change their sum.
4 + 2 = 6 and 2 + 4 = 6
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Associative 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
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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
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Polygons
A polygon is a flat figure formed by line segments
connected at their ends.
Geometric figures such as triangles, squares, and
rectangles are called polygons.
triangle
square
Martin-Gay, Prealgebra, 6ed
rectangle
28
Perimeter
The perimeter of a polygon is the distance around
the polygon.
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Addition Problems
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
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Subtraction Problems
Descriptions of problems solved by subtraction may include any
of these key words or phrases:
Key Words
Examples
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
subtracted from
16 decreased by 7
5 subtracted from 9
16 – 7
9–5
Martin-Gay, Prealgebra, 6ed
Symbols
31
Helpful Hint
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.
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Helpful Hint
Since subtraction and addition are reverse
operations, don’t forget that a subtraction
problem can be checked by adding.
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Reading a Bar Graph
Number of Endangered Species
The graph shows the number of endangered species in each country.
146
89
83
73
72
64
Country
Source: The Top 10 of Everything, Russell Ash.
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1.4
Rounding and
Estimating
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Rounding
20
23
23 rounded to the nearest ten is 20.
30
40
48
48 rounded to the nearest ten is 50.
50
10
20
15
15 rounded to the nearest ten is 20.
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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.
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Estimates
Making estimates is often the quickest way
to solve real-life problems when their
solutions do not need to be exact.
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Helpful Hint
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.
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1.5
Multiplying Whole
Numbers and Area
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Multiplication
Multiplication is repeated addition with a different
notation.
4 + 4 + 4 + 4 + 4 = 5 ∙ 4 = 20
5 fours
factor product
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Multiplication Property of 0
The product of 0 and any number is 0.
90=0
06=0
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Multiplication Property of 1
The product of 1 and any number is that same number.
91=9
16=6
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Commutative Property of
Multiplication
Changing the order of two factors does not change
their product.
6  3 = 18 and 3  6 = 18
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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
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Distributive Property
Multiplication distributes over addition.
5(3 + 4) = 5  3 + 5  4
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Area
1 square inch
1
5 inches
1
3 inches
Area of a rectangle = length  width
= (5 inches)(3 inches)
= 15 square inches
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Helpful Hint
Remember that perimeter (distance around a plane figure) is
measured in units. Area (space enclosed by a plane figure) is
measured in square units.
Perimeter =
5 inches + 4 inches + 5
inches + 4 inches = 18
inches
5 inches
Rectangle
4 inches
Area = (5 inches)(4 inches) = 20 square inches
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Multiplication Words
There are several words or phrases that indicate the
operation of multiplication. Some of these are as follows:
Key Words
Examples
Symbols
multiply
multiply 4 by 3
43
product
times
product of 2 and 5
7 times 6
25
76
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1.6
Dividing Whole
Numbers
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Division
The process of separating a quantity into equal parts is
called division.
quotient
20
5
4
6
3 18
dividend
14  2  7
divisor
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Division Properties of 1
The quotient of any number, except 0, and that same
number is 1.
1
6
1 5 5
6
771
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Division Properties of 1
The quotient of any number and 1 is that same
number.
5
6
6 15
1
7 1 7
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Division Properties of 0
The quotient of 0 and any number (except 0) is 0.
0
0
0 5 0
6
070
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Division Properties of 0
The quotient of any number and 0 is not a number.
We say that
6
0
0 5
70
are undefined.
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Helpful Hint
Since division and multiplication are reverse
operations, don’t forget that a division
problem can be checked by multiplying.
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Division Words
Here are some key words and phrases that indicate the operation
of division.
Key Words
Examples
divide
divide 15 by 3
quotient
quotient of 12 and 6
divided by
8 divided by 4
divided or shared $20 divided equally
equally
among five people
Symbols
15  3
12
6
48
20  5
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Average
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.
Sum = 880
Average = 880 ÷ 10 = 88
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1.7
Exponents and
Order of Operations
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Exponents
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
base
3
5
exponent
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Exponential Notation
base
3
5
exponent
Read as “three to the fifth power” or “the fifth power of three.”
This is called exponential notation. The exponent, 5,
indicates how many times the base, 3, is a factor.
3•3•3•3•3
3 is a factor 5 times
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Reading Exponential Notation
4 = 41
is read as “four to the first power.”
4  4 = 42
is read as “four to the second power” or “four squared.”
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Reading Exponential Notation
4  4  4 = 43
is read as “four to the third power” or “four cubed.”
4  4  4  4 = 44
is read as “four to the fourth power.”
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Helpful Hint
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.
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Evaluating Exponential Expressions
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
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Helpful Hint
An exponent applies only to its base. For example,
4 • 23 means 4 • 2 • 2 • 2.
Don’t forget that 24 is not 2 • 4.
4
2 means repeated multiplication of the same factor.
24 = 2 • 2 • 2 • 2 = 16, whereas 2 • 4 = 8
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Order of Operations
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.
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1.8
Introduction to
Variables, Algebraic
Expressions, and
Equations
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Algebraic Expressions
A combination of operations on letters (variables)
and numbers is called an algebraic expression.
Algebraic Expressions
5+x
6y
3y – 4 + x
4x means 4  x
and
xy means x  y
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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.
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Evaluating Algebraic Expressions
Evaluate x + y for x = 5 and y = 2.
Replace x with 5 and y with 2 in x + y.
x+y=( 5) + ( 2)
=7
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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
x + 5 = 9
left side
right side
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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.
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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.
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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.
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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.
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Keywords and Phrases
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
increased by
subtracted
from
double
divided by amounts to
is equal to
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Translating Word Phrases
the product of 5 and a number
5x
twice a number
2x
a number decreased by 3
n–3
a number increased by 2
z+2
four times a number
4w
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Additional Word Phrases
the sum of a number and 7
x+7
three times the sum of a number and 7
3(x + 7)
the quotient of 5 and a number
5
x
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Helpful Hint
Remember that order is important when subtracting.
Study the order of numbers and variables below.
Phrase
a number
decreased by 5
a number
subtracted from 5
Translation
x–5
5–x
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Chapter 2
Integers and
Introduction to
Integers
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2.1
Introduction to
Integers
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Positive and Negative Numbers
Numbers greater than 0 are called positive numbers. Numbers
less than 0 are called negative numbers.
zero
negative numbers
positive numbers
-6 -5 -4 -3 -2 -1
0
1
2
3
4
5
6
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Integers
Some signed numbers are integers.
The integers are
{ …, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, …}
negative numbers
–6 –5 –4 –3 –2 –1
zero
0
positive numbers
1
2
3
4
5
6
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Negative and Positive Numbers
–3 indicates “negative three.”
3 and +3 both indicate “positive three.”
The number 0 is neither positive nor negative.
zero
negative numbers
positive numbers
–6 –5 –4 –3 –2 –1
0
1
2
3
4
5
6
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Comparing Integers
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”
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
86
Graphs of 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.
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
87
Absolute Value
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.
–6 –5 –4 –3 –2 –1 0 1 2 3 4
2 is 2 because –2 is 2 units from 0.
5
6
–6 –5 –4 –3 –2 –1
5
6
0
1
2
3
4
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
88
Helpful Hint
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
zero
6 =6
a positive number
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
89
Opposite Numbers
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
–6 –5 –4 –3 –2 –1
5 units
0
1
2
3
4
5
6
5 and –5 are opposites.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
90
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
91
Helpful Hint
Remember that 0 is neither positive nor
negative. Therefore, the opposite of 0 is 0.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
92
2.2
Adding Integers
© 2010 Pearson Prentice Hall. All rights reserved
Adding Two Numbers with the Same Sign
2+3=
End
Start
5
2
–6 –5 –4 –3 –2 –1 0
End
1
3
2
Start
–3
4
5
6
–2 + (–3) = –5
–2
–6 -5
–5 -4
–4 -3
–3 -2
–2 -1
–1 0
0
-6
3
1
1
2
2
Martin-Gay, Prealgebra, 6ed
3
3
4
4
5
5
6
6
94
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
Martin-Gay, Prealgebra, 6ed
95
Adding Two Numbers with Different Signs
2 + (–3) =
–1
–3
End
Start
2
–6 –5 –4 –3 –2 –1 0 1
3 End
0
3
4
5
6
5
6
–2 + 3 = 1
–2 Start
–6 –5 –4 –3 –2 –1
2
1
Martin-Gay, Prealgebra, 6ed
2
3
4
96
Adding Two Numbers with Different Signs
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
Martin-Gay, Prealgebra, 6ed
97
Helpful Hint
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.
Martin-Gay, Prealgebra, 6ed
98
Helpful Hint
Don’t forget that addition is commutative
and associative. In other words, numbers
may be added in any order.
Martin-Gay, Prealgebra, 6ed
99
Evaluating Algebraic Expressions
Evaluate x + y for x = 5 and y = –9.
Replace x with 5 and y with –9 in x + y.
x + y = ( 5 ) + (–9 )
= –4
Martin-Gay, Prealgebra, 6ed
100
2.3
Subtracting Integers
© 2010 Pearson Prentice Hall. All rights reserved
Subtracting Integers
To subtract integers, rewrite the subtraction problem as
an addition problem. Study the examples below.
9
5=4
9 + (–5) = 4
Since both expressions
equal 4, we can say
9
5 = 9 + (–5) = 4
Martin-Gay, Prealgebra, 6ed
102
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.
Martin-Gay, Prealgebra, 6ed
103
Subtracting Two Numbers
first
subtraction =
+
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
Martin-Gay, Prealgebra, 6ed
104
Adding and Subtracting Integers
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
Martin-Gay, Prealgebra, 6ed
105
Evaluating Algebraic Expressions
Evaluate x – y for x = –6 and y = 8.
Replace x with –6 and y with 8 in x – y.
x –
y
= ( –6 ) – ( 8 )
= ( –6 ) + ( –8 )
= –14
Martin-Gay, Prealgebra, 6ed
106
2.4
Multiplying and
Dividing Integers
© 2010 Pearson Prentice Hall. All rights reserved
Multiplying Integers
Consider the following pattern of products.
First factor
decreases by 1
each time.
3  5 = 15 Product
2  5 = 10 decreases by 5
each time.
15= 5
05= 0
This pattern continues as follows.
–1  5 = -5
–2  5 = -10
–3  5 = -15
This suggests that the product of a negative number and a positive
number is a negative number.
Martin-Gay, Prealgebra, 6ed
108
Multiplying Integers
Observe the following pattern.
2  (–5) = –10
1  (–5) = –5
0  (–5) = 0
Product
increases by 5
each time.
This pattern continues as follows.
–1  (–5) = 5
–2  (–5) = 10
–3  (–5) = 15
This suggests that the product of two negative numbers is a
positive number.
Martin-Gay, Prealgebra, 6ed
109
Multiplying Integers
The product of two numbers having the same
sign is a positive number.
24=8
–2  (–4) = 8
The product of two numbers having different
signs is a negative number.
2  (–4) = –8
–2  4 = –8
Martin-Gay, Prealgebra, 6ed
110
Multiplying Integers
Product of Like Signs
( + )( + ) = +
(–)(–) = +
Product of Different Signs
(–)( + ) = –
( + )(–) = –
Martin-Gay, Prealgebra, 6ed
111
Helpful Hint
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.
(–)(–)=(+)
(–)(–)(–)=(–)
(–)(–)(–)(–)=(+)
(–)(–)(–)(–)(–)=(–)
Martin-Gay, Prealgebra, 6ed
The product
of an odd
number of
negative
numbers is
a negative
result.
112
Dividing Integers
Division of integers is related to multiplication of
integers.
6
= 3 because
3· 2 =6
2
–6 =
–3 because
2
–3 · 2 = –6
6 = –3 because
–2
– 3· (–2) = 6
– 6=
3 because
–2
3 · (–2) = –6
Martin-Gay, Prealgebra, 6ed
113
Dividing Integers
The quotient of two numbers having the same
sign is a positive number.
12 ÷ 4 = 3
–12 ÷ (–4 ) = 3
The quotient of two numbers having different
signs is a negative number.
–12 ÷ 4 = –3
12 ÷ (–4) = –3
Martin-Gay, Prealgebra, 6ed
114
Dividing Numbers
Quotient of Like Signs
( )

( )
( )

( )
Quotient of Different Signs
( )

( )
( )

( )
Martin-Gay, Prealgebra, 6ed
115
2.5
Order of Operations
© 2010 Pearson Prentice Hall. All rights reserved
Order of Operations
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
117
Using the Order of Operations
Simplify 4(5 – 2) + 42.
4(5 – 2) + 42 = 4(3) + 42
Simplify inside
parentheses.
= 4(3) + 16
Write 42 as 16.
= 12 + 16
Multiply.
= 28
Add.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
118
Helpful Hint
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
119
Chapter 3
Solving Equations
and Problem Solving
© 2010 Pearson Prentice Hall. All rights reserved
3.1
Simplifying
Algebraic
Expressions
© 2010 Pearson Prentice Hall. All rights reserved
Constant and Variable Terms
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
Constant
terms
3y2 + (–4y) + 2
Variable
terms
Martin-Gay, Prealgebra, 6ed
122
Coefficients
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
Numerical
coefficient is 5.
–7y
3y2
Numerical
coefficient is –7.
Understood
numerical
coefficient is 1.
Numerical
coefficient
is 3.
Martin-Gay, Prealgebra, 6ed
123
Like Terms
Terms that are exactly the same, except that they may have
different numerical coefficients are called like terms.
Like Terms
3x, 2x
–6y, 2y, y
–3, 4
Unlike Terms
5x, x 2
7x, 7y
5y, 5
6a, ab
2ab2, –5b2a
The order of the variables
does not have to be the same.
Martin-Gay, Prealgebra, 6ed
124
Distributive Property
A sum or difference of like terms can be simplified
using the distributive property.
Distributive Property
If a, b, and c are numbers, then
ac + bc = (a + b)c
Also,
ac – bc = (a – b)c
Martin-Gay, Prealgebra, 6ed
125
Distributive Property
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.
Martin-Gay, Prealgebra, 6ed
126
Addition and Multiplication Properties
The commutative and associative properties of addition
and multiplication help simplify expressions.
Properties of Addition and Multiplication
If a, b, and c are numbers, then
Commutative Property of Addition
a+b=b+a
Commutative Property of Multiplication
a∙b=b∙a
The order of adding or multiplying two numbers can be
changed without changing their sum or product.
Martin-Gay, Prealgebra, 6ed
127
Associative Properties
The grouping of numbers in addition or multiplication
can be changed without changing their sum or product.
Associative Property of Addition
(a + b) + c = a + (b + c)
Associative Property of Multiplication
(a ∙ b) ∙ c = a ∙ (b ∙ c)
Martin-Gay, Prealgebra, 6ed
128
Helpful Hint
Examples of Commutative and Associative Properties
of Addition and Multiplication
4+3=3+4
Commutative Property of Addition
6∙9=9∙6
Commutative Property of Multiplication
(3 + 5) + 2 = 3 + (5 + 2) Associative Property of Addition
(7 ∙ 1) ∙ 8 = 7 ∙ (1 ∙ 8)
Associative Property of Multiplication
Martin-Gay, Prealgebra, 6ed
129
Multiplying Expressions
We can also use the distributive property to multiply
expressions.
The distributive property says that multiplication
distributes over addition and subtraction.
2(5 + x) = 2 ∙ 5 + 2 ∙ x = 10 + 2x
or
2(5 – x) = 2 ∙ 5 – 2 ∙ x = 10 – 2x
Martin-Gay, Prealgebra, 6ed
130
Simplifying Expressions
To simply expressions, use the distributive property
first to multiply and then combine any like terms.
Simplify: 3(5 + x) – 17
3(5 + x) – 17 = 3 ∙ 5 + 3 ∙ x + (–17)
Apply the Distributive
Property
= 15 + 3x + (–17)
Multiply
= 3x + (–2) or 3x – 2
Combine like terms
Note: 3 is not distributed to the –17 since –17
is not within the parentheses.
Martin-Gay, Prealgebra, 6ed
131
Finding Perimeter
3z feet
7z feet
9z feet
Perimeter is the distance around the figure.
Perimeter = 3z + 7z + 9z
= 19z feet
Don’t forget to insert
proper units.
Martin-Gay, Prealgebra, 6ed
132
Finding Area
(2x – 5) meters
3 meters
A = length ∙ width
= 3(2x – 5)
= 6x – 15 square meters
Don’t forget to insert
proper units.
Martin-Gay, Prealgebra, 6ed
133
Helpful Hint
Don’t forget . . .
Area:
• surface enclosed
• measured in square units
Perimeter:
• distance around
• measured in units
Martin-Gay, Prealgebra, 6ed
134
3.2
Solving Equations:
Review of the Addition
and Multiplication
Properties
© 2010 Pearson Prentice Hall. All rights reserved
Equation vs. Expression
Statements like 5 + 2 = 7 are called equations.
An equation is of the form expression = expression.
An equation can be labeled as
Equal sign
x + 5 = 9
left side
right side
Martin-Gay, Prealgebra, 6ed
136
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.
Martin-Gay, Prealgebra, 6ed
137
Multiplication Property of Equality
Let a, b, and c represent numbers and let c  0.
If a = b, then
a b
a ∙ c = b ∙ c and c = c
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.
Martin-Gay, Prealgebra, 6ed
138
Solve for x.
x4=3
To 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.
Martin-Gay, Prealgebra, 6ed
139
Check
To check, replace x with 7 in the original equation.
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.
Martin-Gay, Prealgebra, 6ed
140
Solve for x
4x = 8
To 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.
4x 8
=
4 4
1∙x = 2 or x = 2
Martin-Gay, Prealgebra, 6ed
141
Check
To check, replace x with 2 in the original
equation.
4x = 8 Original equation
4 ∙ 2 = 8 Let x = 2.
?
8 = 8 True.
The solution is 2.
Martin-Gay, Prealgebra, 6ed
142
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
4x – 6 + 6 = 10 + 6
4x = 16
Divide both sides by 4.
x=4
Martin-Gay, Prealgebra, 6ed
143
Check
To check, replace x with 4 in the original equation.
2(2x – 3) = 10
2(2 · 4 – 3) = 10
?
2(8 – 3) = 10
?
(2)5 = 10
Original equation
Let x = 4.
True.
The solution is 4.
Martin-Gay, Prealgebra, 6ed
144
Chapter 4
Fractions and Mixed
Numbers
© 2010 Pearson Prentice Hall. All rights reserved
4.1
Introduction to
Fractions and Mixed
Numbers
© 2010 Pearson Prentice Hall. All rights reserved
Parts of a Fraction
Whole numbers are used to count whole things. To refer
to a part of a whole, fractions are used.
a
A fraction is a number of the form ,
b
where a and b are integers and b is not 0.
The parts of a fraction are
numerator
denominator
a
b
Martin-Gay, Prealgebra, 6ed
fraction bar
147
Helpful Hint
4
7
Remember that the bar
in a fraction means
division. Since division by 0 is
undefined, a fraction with a
denominator of 0 is undefined.
Martin-Gay, Prealgebra, 6ed
148
Visualizing Fractions
One way to visualize fractions is to
picture them as shaded parts of a
whole figure.
Martin-Gay, Prealgebra, 6ed
149
Visualizing Fractions
Picture
Fraction
1
4
part shaded
5
6
parts shaded
7
3
parts shaded
Read as
one-fourth
equal parts
five-sixths
equal parts
seven-thirds
equal parts
Martin-Gay, Prealgebra, 6ed
150
Types of Fractions
A proper fraction is a fraction whose numerator
is less than its denominator.
1 3 2
Proper fractions have values that are less than 1. 2 , 4 , 5
An improper fraction is a fraction whose
8 5 4
numerator is greater than or equal to its
, ,
denominator.
3 5 1
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.
Martin-Gay, Prealgebra, 6ed
2 1 2
2 ,3 ,4
3 5 7
151
Fractions on Number Lines
Another way to visualize fractions is to graph them on
a number line.
3
5 equal parts
3
5
0 1
5
1
5
1
5
Martin-Gay, Prealgebra, 6ed
1
5
1
1
5
152
Fraction Properties of 1
If n is any integer other than 0, then
n
=1
n
5
=1
5
If n is any integer, then
n
=n
1
3
=3
1
Martin-Gay, Prealgebra, 6ed
153
Fraction Properties of 0
If n is any integer other than 0, then
0
0
=0
=0
n
5
If n is any integer, then
n
= undefined
0
3
= undefined
0
Martin-Gay, Prealgebra, 6ed
154
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.
2
3
4

2 ∙ 4  3
4

8
3
Martin-Gay, Prealgebra, 6ed
4

11
4
155
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.
quotient =
remainder
original denominator
Martin-Gay, Prealgebra, 6ed
156
4.2
Factors and Simplest
Form
© 2010 Pearson Prentice Hall. All rights reserved
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.
Martin-Gay, Prealgebra, 6ed
158
Helpful Hint
The natural number 1 is neither prime nor
composite.
Martin-Gay, Prealgebra, 6ed
159
Prime Factorization
A prime factorization of a number expresses
the number as a product of its factors and the
factors must be prime numbers.
Martin-Gay, Prealgebra, 6ed
160
Helpful Hints
Remember a factor is any number that
divides a number evenly (with a
remainder of 0).
Martin-Gay, Prealgebra, 6ed
161
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.
Martin-Gay, Prealgebra, 6ed
162
Divisibility Tests
A whole number is divisible by
2 if its last digit is 0, 2, 4, 6, or 8.
196 is divisible by 2
3 if the sum of its digits is divisible by 3.
117 is divisible by 3 since 1 + 1 + 7 = 9 is divisible by 3.
Martin-Gay, Prealgebra, 6ed
163
Divisibility Tests
A whole number is divisible by
5 if the ones digit is 0 or 5.
2,345 is divisible by 5.
10 if its last digit is 0.
8,470 is divisible by 10.
Martin-Gay, Prealgebra, 6ed
164
Equivalent Fractions
3
Graph
on the number line.
4
3
4
1
1 4
8
0
Graph 6 on the number line.
6
8
1
8
Martin-Gay, Prealgebra, 6ed
165
Equivalent Fractions
Fractions that represent the same portion of a
whole or the same point on the number line are
called equivalent fractions.
6 6÷2 3
=
=
8 8÷ 2 4
3 3 2 6
=
=
4 4 2 8
Martin-Gay, Prealgebra, 6ed
166
Fundamental Property of
Fractions
If a, b, and c are numbers, then
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.
Martin-Gay, Prealgebra, 6ed
167
Simplest Form
A fraction is in simplest form, or lowest terms, when
the numerator and denominator have no common
factors other than 1.
Using the fundamental principle
of fractions, divide the
numerator and denominator by
the common factor of 7.
14 14 ÷ 7 2
=
=
21 21÷ 7 3
Using the prime factorization of
the numerator and denominator,
divide out common factors.
14 7  2 7  2 2
=
=
=
21 7  3 7  3 3
Martin-Gay, Prealgebra, 6ed
168
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.
Martin-Gay, Prealgebra, 6ed
169
Helpful Hints
5
5
1
=
=
10 5 • 2 2
15 3 • 5 5
=
= =5
3
3
1
When all factors of the numerator or denominator
are divided out, don’t forget that 1 still remains in
that numerator or denominator.
Martin-Gay, Prealgebra, 6ed
170
4.3
Multiplying and
Dividing Fractions
© 2010 Pearson Prentice Hall. All rights reserved
Multiplying Fractions
1
of
2
0
3
8
3
4
3 6
4 8
is
3
8
1
The word “of” means multiplication and “is” means equal to.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
172
Multiplying Fractions
1
of
2
3
4
is
3
8
means
1 3 3
 
2 4 8
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
173
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
174
Examples
3 5
3  5 15
 

2 7
2  7 14
If the numerators have common factors with the
denominators, divide out common factors before
multiplying.
1
3 2
32
3
 

4 5 2  2  5 10
3 2
3
 
4 5 10
or
2
Martin-Gay, Prealgebra, 6ed
175
Examples
6
3x 8
3 x  42



5
4 5x
45 x
or
2
3x 8
6


5
4 5x
1
Martin-Gay, Prealgebra, 6ed
176
Helpful Hint
Recall that when the denominator of a
fraction contains a variable, such as
8
,
5x
we assume that the variable is not 0.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
177
Expressions with Fractional Bases
The base of an exponential expression can also be a
fraction.
FI
HK
2 3
2 2 2
222
8

  

3
3 3 3
333
27
Martin-Gay, Prealgebra, 6ed
178
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their
product is 1. The reciprocal of the fraction
a
b
because
is
b
a
a b a  b ab
 
 1
b a b  a ab
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Martin-Gay, Prealgebra, 6ed
179
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
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Martin-Gay, Prealgebra, 6ed
180
Helpful Hint
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
181
Helpful Hint
When dividing by a fraction, do not look for
common factors to divide out until you rewrite
the division as multiplication.
Do not try to divide out these two 2s.
1 2 1 3 3
   
2 3 2 2 4
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Martin-Gay, Prealgebra, 6ed
182
Fractional Replacement Values
If x =
5
2
and y =
, evaluate x  y .
6
5
5
2
Replace x with and y with .
6
5
5 2
5 5
25
x y     
6 5
6 2
12
Martin-Gay, Prealgebra, 6ed
183
4.4
Adding and Subtracting
Like Fractions, Least
Common Denominator,
and Equivalent Fractions
© 2010 Pearson Prentice Hall. All rights reserved
Like and Unlike 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
4
and
5
5
2
3
and
3
4
5
3
and
7
7
5
5
and
6
12
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Martin-Gay, Prealgebra, 6ed
185
Adding or Subtracting Like Fractions
If a, b, and c, are numbers and b is not 0, then
a c ac
a c a c
 
also
 
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.
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Martin-Gay, Prealgebra, 6ed
186
Adding or Subtracting Like Fractions
2 4

7 7
Start
0
6
= 7
2
4
7
7
End
6
7
1
1
7
To add like fractions, add the numerators and write the sum over
the common denominator.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
187
Helpful Hint
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
188
Equivalent Negative Fractions
2 2
2
2



3 3
3
3
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Martin-Gay, Prealgebra, 6ed
189
Least Common Denominator
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).
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Martin-Gay, Prealgebra, 6ed
190
Least Common Multiple
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.)
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
191
Least Common Denominator
5
5
and
To find the LCD of
12
18
First, write each denominator as a product of primes.
12 = 2 • 2 • 3
18 = 2 • 3 • 3
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.
LCD = 2 • 2 • 3 • 3 = 36
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
192
4.5
Adding and Subtracting
Unlike Fractions
© 2010 Pearson Prentice Hall. All rights reserved
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
194
Adding or Subtracting Unlike Fractions
1 7
Add: 9  12
Step 1: Find the LCD of 9 and 12.
9=3∙3
and
12 = 2 ∙ 2 ∙ 3
LCD = 2 ∙ 2 ∙ 3 ∙ 3 = 36
Step 2: Rewrite equivalent fractions with the LCD.
1 1 4 4


9 9  4 36
7
7  3 21


12 12  3 36
Continued.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
195
Adding or Subtracting Unlike Fractions
Continued:
Step 3: Add like fractions.
1 4 7  3
4 21 25




9  4 12  3 36 36 36
Step 4: Write the sum in simplest form.
25
36
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
196
Writing Fractions in Order
One important application of the least common
denominator is to use the LCD to help order or compare
fractions.
3 4
?
Insert < or > to form a true sentence.
5 7
The LCD for these fractions is 35.
Write each fraction as an equivalent fraction with a
denominator of 35.
3 3  7 21


5 5  7 35
4 4  5 20


7 7  5 35
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
Continued.
197
Writing Fractions in Order
Continued:
Compare the numerators of the equivalent fractions.
21 20
>
Since 21 > 20, then
35
35
3
4
>
Thus,
5
7
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
198
Evaluating Expressions
Evaluate x – y if x = 2 and y = 3 .
3
Replacing x with
2
3
4
and y with 3 ,
4
2
3
then, x – y  
3 4
2  4 33 8 9
1


  
3  4 4  3 12 12
12
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay,Martin-Gay,
Prealgebra, 5edPrealgebra, 6ed
199
199
Solving Equations Containing Fractions
1 5
Solve: x  
3 12
1
To get x by itself, add to both sides.
3
1 1 5 1
x   
3 3 12 3
5 1 4
 
12 3  4
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
Continued.
200
Solving Equations Containing Fractions
Continued:
5 4
x 
12 12
9 3
 
12 4
Write fraction in
simplest form.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
201
4.6
Complex Fractions and
Review of Order of
Operations
© 2010 Pearson Prentice Hall. All rights reserved
Complex Fraction
A fraction whose numerator or denominator or both
numerator and denominator contain fractions is
called a complex fraction.
2
3
x
4
2 3

3 5
y 1

5 7
2
4
5
7
8
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Martin-Gay, Prealgebra, 6ed
203
Method 1: Simplifying Complex Fractions
This method makes use of the fact that a fraction bar
means division.
1
3
2
3  29  3
8
3 8
4
4
1
9
When dividing fractions, multiply by the
reciprocal of the divisor.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
204
Method 1: Simplifying Complex Fractions
Recall the order of operations. Since the fraction bar is
a grouping symbol, simplify the numerator and
denominator separately. Then divide.
2
1 1
3 1
4
1 3 1



2 6  2  3 6  6 6  6  4  12  8
3 2
33 2 4
9
8
1
6 1



1
4  3 3  4 12 12 12
4 3
When dividing fractions, multiply by the
reciprocal of the divisor.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
205
Method 2: Simplifying Complex Fractions
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.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
206
Method 2: Simplifying Complex Fractions
F
H
F
H
I
K
I
K
1 1
1 1
12 

2 6
2 6 
3 2
3 2

12 
4 3
4 3
1I
1I
F
F
12
 12
H2K H6K
3I
2I
F
F
12
 12
H4K H3K
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 property
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
Continued.
207
Method 2: Simplifying Complex Fractions
Continued:
FI
HK
FI
HK
FI
HK 6  2  8  8
FI 9  8 1
HK
1
1
1 1
12
 12

2
6
2 6 
3 2
3
2

12
 12
4 3
4
3
Step 3: Multiply.
The result is the same no
matter which method is used.
Step 4: Simplify.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
208
Reviewing the Order of Operations
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
209
4.7
Operations on Mixed
Numbers
© 2010 Pearson Prentice Hall. All rights reserved
Mixed Numbers
Recall that a mixed number is the sum of a whole
number and a proper fraction.
3
0
1
4
 3 
4
5
5
2
3
19
4
5
 3
4
5
5
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Martin-Gay, Prealgebra, 6ed
211
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.
1 1
Multiply: 3  2
5 4
1 1 16 9
3 2 

5 4 5 4
Change mixed numbers
to improper fractions.
4  4 9 36
1
 
7
5 4
5
5
Write the solution
as a mixed number
if possible.
Remove common factors
and multiply.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
212
Adding or Subtracting Mixed Numbers
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
213
Adding or Subtracting Mixed Numbers
Add:
5  56
2 14
7
The LCD of 14 and 7 is 14.
5 
2 14
5
2 14
Write equivalent fractions with the LCD of 14.
12
 5 67   5 14
Add the fractions, then add the whole numbers.
17
7 14
Notice that the fractional part is improper.
Since17 is1 3 , write the sum as
14
14
17  7  1 3  8 3
7 14
14
14
Make sure the fractional
part is always proper.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
214
Adding or Subtracting Mixed Numbers
When subtracting mixed numbers, borrowing may be needed.
1
3
3
0
1
2
3
4
5
1
1
1
3 1
4
3  2 1  2 1  2    2
3
3
3
3 3
3
Borrow 1 from 3.
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Martin-Gay, Prealgebra, 6ed
215
Adding or Subtracting Mixed Numbers
3  36
Subtract: 5 14
7
3 
5 14
The LCD of 14 and 7 is 14.
3
5 14
Write equivalent fractions with the LCD of 14.
12
 3 67   3 14
To subtract the fractions, we have to borrow.
3 
5 14
3  4  1 3  4  17  4 17
5 14
14
14
14
3 
5 14
17
4 14
12   3 12
 3 67   3 14
14
5
114
Subtract the fractions, then
subtract the whole numbers.
Notice that the fractional
part is proper.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
216
4.8
Solving Equations
Containing Fractions
© 2010 Pearson Prentice Hall. All rights reserved
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
218
Multiplication Property of Equality
Let a, b, and c represent numbers and let c  0.
If a = b, then
a ∙ c = b ∙ c and
a b

c c
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.
Martin-Gay, Prealgebra, 6ed
219
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
220
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
221
Solve for x
1
5
x =
7
9
  
1
5
7 x =
7
7
9
35
x=
9
Multiply both sides by 7.
Simplify both sides.
Martin-Gay, Prealgebra, 6ed
222
Solve for x
3(y + 3)
= 2y + 6
5
3(y + 3) 

5
=  2y + 6  5
 5 
3y + 9 = 10y + 30
9 = 7y + 30
21= 7y
3 = y
Multiply both sides by 5.
Simplify both sides.
Add – 3y to both sides.
Add – 30 to both sides.
Divide both sides by 7.
Martin-Gay, Prealgebra, 6ed
223
Chapter 5
Decimals
© 2010 Pearson Prentice Hall. All rights reserved
5.1
Introduction to
Decimals
© 2010 Pearson Prentice Hall. All rights reserved
Decimal Notation
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.
16.743
Whole
number part
Decimal
part
Decimal point
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Martin-Gay, Prealgebra, 6ed
226
Place Value
hundred-thousandths
ten-thousandths
thousandths
7
hundredths
6
tenths
ones
1
tens
The position of
each digit in a
number determines
its place value.
hundreds
Place Value
3 4
1
1
1
100 10 1
10 1 1000 1 100,000
10,000
decimal point 100
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Martin-Gay, Prealgebra, 6ed
227
Place Value
1
Notice that the value of each place is
10
of the value of the place to its left.
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Martin-Gay, Prealgebra, 6ed
228
Place Value
16.734
The digit 3 is in the hundredths place, so
3
its value is 3 hundredths or
.
100
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Martin-Gay, Prealgebra, 6ed
229
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
230
Writing a Decimal in Words
Write the decimal 143.056 in words.
143.056
decimal part
whole number part
one hundred forty-three
and
fifty-six thousandths
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Martin-Gay, Prealgebra, 6ed
231
Writing Decimals in Standard Form
A decimal written in words can be written in standard
form by reversing the procedure.
Write one hundred six and five hundredths in standard form.
one hundred six and five hundredths
whole-number part
decimal
decimal part
106 . 05
5 must be in the
hundredths place
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
232
Helpful Hint
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
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
233
Writing Decimals as Fractions
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.
0.9
is read “nine tenths” and written as a fraction as
9
10
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Martin-Gay, Prealgebra, 6ed
234
Writing Decimals as Fractions
0.21 is read as
twenty-one hundredths
and written as a fraction as
0.011 is read as
eleven thousandths
and written as a fraction as
21
100
11
1000
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Martin-Gay, Prealgebra, 6ed
235
Comparing Two Positive Decimals
0 .37 
37
100
2 decimal 2 zeros
places
0 .029 
29
1000
3 decimal 3 zeros
places
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.
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Martin-Gay, Prealgebra, 6ed
236
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
0.3
0.7
3
7
10
10
1
0.3 < 0.7 or
0.7 > 0.3
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Martin-Gay, Prealgebra, 6ed
237
Comparing Two Positive Decimals
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
238
Comparing Two Positive Decimals
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.
35.638
3
35.638
35.657
<
<
5
35.657
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Martin-Gay, Prealgebra, 6ed
239
Helpful Hint
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 on
When 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
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Martin-Gay, Prealgebra, 6ed
240
Rounding Decimals
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
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Martin-Gay, Prealgebra, 6ed
241
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.
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Martin-Gay, Prealgebra, 6ed
242
Rounding Decimals to a Place Value
Round 326.4386 to the nearest tenth.
Locate the digit to the right of the tenths place.
tenths place
326.4386
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
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Martin-Gay, Prealgebra, 6ed
243
5.2
Adding and Subtracting
Decimals
© 2010 Pearson Prentice Hall. All rights reserved
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.
© 2010 Pearson Prentice Hall. All rights reserved
Martin-Gay, Prealgebra, 6ed
245
Helpful Hint
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
two 0s inserted
71.74
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Martin-Gay, Prealgebra, 6ed
246
Helpful Hint
Don’t forget that the decimal point in a
whole number is after the last digit.
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247
Estimating Operations on Decimals
Estimating sums, differences, products, and
quotients of decimal numbers is an
important skill whether you use a calculator
or perform decimal operations by hand.
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Martin-Gay, Prealgebra, 6ed
248
Estimating When Adding Decimals
Add 23.8 + 32.1.
Exact
23.8
+32.1
55.9
Estimate
rounds to
rounds to
24
32
56
This is a reasonable answer.
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249
Helpful Hint
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.
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250
Evaluating with Decimals
Evaluate x + y for x = 5.5 and y = 2.8.
Replace x with 5.5 and y with 2.8 in x + y.
x + y = ( 5.5) + (2.8)
= 8.3
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251
5.3
Multiplying Decimals
and Circumference of a
Circle
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Multiplying Decimals
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
1 decimal
place

21
3
100
2 decimal
places
=
1000
= 0.021
3 decimal places
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253
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.
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254
Estimating when Multiplying Decimals
Multiply 32.3  1.9.
Exact
32.3
1.9
290.7
323.0
61.37
Estimate
rounds to
rounds to
32
2
64
This is a reasonable answer.
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Martin-Gay, Prealgebra, 6ed
255
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.
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256
Multiplying Decimals by Powers of 10
76.543  10 = 765.43
1 zero
76.543  100 = 7654.3
2 zeros
Decimal point moved 1
place to the right.
Decimal point moved 2
places to the right.
76.543  100,000 = 7,654,300
5 zeros
Decimal point moved 5
places to the right.
The decimal point is moved the same number of places as
there are zeros in the power of 10.
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257
Multiplying Decimals by Powers 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 =
3.4305 =
343.05
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258
Multiplying Decimals by Powers of 10
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 =
008.57 =
0.0857
Notice that zeros had to be inserted.
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Martin-Gay, Prealgebra, 6ed
259
The Circumference of a Circle
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.
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260
The Circumference of a Circle
r
d
Circumference = 2·p ·radius
or
Circumference = p ·diameter
C = 2 p r or C = p d
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261
p
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 22 .
7
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Martin-Gay, Prealgebra, 6ed
262
The Circumference of a Circle
Find the circumference of a circle
whose radius is 4 inches.
4 inches
C = 2pr = 2p·4 = 8p inches
8pinches is the exact circumference of this circle.
If we replace p with the approximation 3.14, C = 8p 
8(3.14) = 25.12 inches.
25.12 inches is the approximate circumference of the
circle.
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263
5.4
Dividing Decimals
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Dividing by a Decimal
Division of decimal numbers is similar to division of
whole numbers.
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.
divisor
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0 84
63 52.92
- 504
2 52
-252
0
Martin-Gay, Prealgebra, 6ed
quotient
dividend
265
Dividing by a Decimal
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
6 . 3 52 . 92
dividend
63 . 529 . 2
84
63 52 9.2
- 504
25 2
-252
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0
Martin-Gay, Prealgebra, 6ed
266
Dividing by a Decimal
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.
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267
Estimating When Dividing Decimals
Divide 258.3 ÷ 2.8
Exact
92.25
28. 2583.
- 252
63
- 56
70
- 56
140
- 140
0
Estimate
rounds to
100
3 300
This is a reasonable answer.
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Martin-Gay, Prealgebra, 6ed
268
Dividing Decimals by Powers of 10
There are patterns that occur when dividing by powers
of 10, such as 10, 100, 1000, and so on.
456.2
 45 . 62
10
The decimal point moved 1 place
to the left.
1 zero
456.2
 0 . 4562
1 , 000
The decimal point moved 3 places
to the left.
3 zeros
The pattern suggests the following rule.
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269
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.
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
1 ,
10 such as 100 is the same as multiplying by its reciprocal 100
or 0.01.
463.7
1
 463.7 
 463.7  0.01  4.637
100
100
To divide by a number is the same as multiplying by its reciprocal.
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270
5.5
Fractions, Decimals,
and Order of
Operations
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Writing Fractions as Decimals
To write a fraction as a decimal, divide the
numerator by the denominator.
3
= 3  4 = 0.75
4
2
= 2  5 = 0.40
5
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272
Comparing Fractions and Decimals
To compare decimals and fractions, write the
fraction as an equivalent decimal.
1
.
Compare 0.125 and
4
1
= 0.25
4
Therefore, 0.125 < 0.25
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273
Order of Operations
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.
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274
Using the Order of Operations
Simplify ( –2.3)2 + 4.1(2.2 + 3.1)
( –2.3)2 + 4.1(2.2 + 3.1)
= ( –2.3)2 + 4.1(5.3)
Simplify inside
parentheses.
= 5.29 + 4.1(5.3)
Write ( –2.3)2 as 5.29.
= 5.29 + 21.73
Multiply.
= 27.02
Add.
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Martin-Gay, Prealgebra, 6ed
275
Finding the Area of a Triangle
height
1
A =
base
2
base • height
1
A =
2
bh
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276
5.6
Equations Containing
Decimals
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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.
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278
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.
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279
Solving Equations with Decimals
–0.01(5a + 4) = 0.04 – 0.01(a + 4)
–1(5a + 4) = 4 – 1(a + 4)
Multiply both sides by 100.
–5a – 4 = 4 – a – 4
Apply the distributive property.
–4a – 4 = 4 – 4
Add a to both sides.
–4a = 4
a = –1
Add 4 to both sides and simplify.
Divide both sides by 4.
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280
5.7
Decimal Applications:
Mean, Median, and
Mode
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Measures of Central Tendency
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.
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282
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.
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283
Finding the Mean
Find the mean of the following list of
numbers.
2.5
5.1
9.5
6.8
2.5
Continued.
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284
Finding the Mean
The mean is the average of the numbers:
2.5
5.1
2.5 + 5.1+ 9.5 + 6.8 + 2.5
5
9.5
6.8
= 5.28
2.5
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285
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.
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286
Finding the Median
Find the median of the following list of
numbers.
2.5
5.1
9.5
6.8
2.5
Continued.
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287
Finding the Median
List the numbers in numerical order:
2.5
2.5
Median
5.1
6.8
9.5
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288
Helpful Hint
In order to compute the median, the
numbers must first be placed in order.
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289
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.)
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290
Finding the Mode
Find the mode of the following list of
numbers.
2.5
5.1
9.5
6.8
2.5
Continued.
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291
Finding the Mode
The mode occurs the most often:
2.5
5.1
The mode is 2.5.
9.5
6.8
2.5
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292
Helpful Hint
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, 9
has no mode. There is no number or numbers
that occur more often than the others.
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293
Chapter 6
Ratio, Proportion,
and
Triangle Applications
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6.1
Ratio and Rates
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Writing Ratios as Fractions
A ratio is the quotient of two quantities.
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
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296
Ratio
The ratio of a number a to a number b is their
quotient. Ways of writing ratios are
a to b,
a : b,
a
and
b
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297
Writing Rates as Fractions
A rate is a special kind of ratio. It is used to
compare different kinds of quantities.
5 miles
1 mile

55 minutes 11 minutes
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298
Finding Unit Rates
To write a rate as a unit rate, divide the
numerator of the rate by the denominator.
314.5 miles
17 gallons
314.5 ÷ 17 = 18.5
18.5 miles
The unit rate is 1 gallon .
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Finding Unit Prices
When a unit rate is “money per item,” it is also
called a unit price.
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 ($0.23 per ounce )
unit price =

12 ounces 1 ounce
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300
6.2
Proportions
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Solving Proportions
A proportion is a statement that two ratios or
rates are equal.
c
a
If and are two ratios, then
d
b
a c
=
b d
is a proportion.
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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.
c I
F
I
F
 bd GJ
G
J
HK Hd K
Multiply both sides bd a
by the LCD, bd
b
Simplify
cross product
a
c
bc
=
b
d
ad
These are called
cross products.
ad = bc
cross product
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Determining Whether Proportions are True
3 12
Is =
a true proportion?
8 32
3 12
=
8 32
?
3  32 = 8  12
96 = 96
True proportion
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304
Finding Unknown Numbers in Proportions
26 28
Solve
= .
x 49
26  49 = 28x
1274 = 28x
45.5 = x
Cross multiply.
Simplify the left side.
Divide both sides by 28.
26
28
=
Check:
45.5 49
0.57143 = 0.57143
(Rounded)
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305
6.3
Proportions and
Problem Solving
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Solving Problems by Writing Proportions
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?
16 ounces 28 ounces

80 calories x calories
16x  80  28
16x  2240
x  140
Solve the proportion.
Cross multiply.
Simplify the right side.
Divide both side by 140.
A 28-oz Cinnamon Mocha Iced Tea has 140 calories.
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307
Helpful Hint
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 inches 7 inches

5 miles
n miles
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308
6.4
Square Roots and the
Pythagorean
Theorem
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The Square of a Number
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
(–6)2 = (–6) (–6) = 36.
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310
Square Root of a Number
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.
We use the symbol
, called a radical sign, to
indicate the positive square root.
16  4 because 42 = 16 and 4 is positive.
25  5 because 52 = 25 and 5 is positive.
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311
Square Root of a Number
The square root,
, of a positive number a is the
positive number b whose square is a. In symbols,
a  b if b  a.
2
9  3 because 3  9.
2
Also, 0  0.
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312
Helpful Hint
Remember that the radical sign
is
used to indicate the positive square root
of a nonnegative number.
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313
Perfect Squares
1
4
Numbers like , 36,
, and 1 are
4
25
called perfect squares because their square
root is a whole number or a fraction.
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314
Approximating Square Roots
A square root such as 6 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.
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Right Triangles
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.
The legs of a right triangle are the other two sides.
hypotenuse
leg
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leg
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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
c
a
a b  c
2
2
2
b
In other words,
(leg)2 + (other leg)2 = (hypotenuse)2.
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317
6.5
Congruent and
Similar Triangles
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Congruent Triangles
Two triangles are congruent when they have the same
shape and the same size. Corresponding angles are
equal, and corresponding sides are equal.
equal angles
e = 11
a=6
c = 11
d=6
b=9
f=9
equal angles
equal angles
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319
Similar Triangles
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.
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320
Similar Triangles
In similar triangles, the measures of corresponding angles
are equal and corresponding sides are in proportion.
a=3
b=5
d=6
c=8
e = 10
f = 16
Side a corresponds to side d, side b corresponds to side e, and
side c corresponds to side f.
a 3 1
 
d 6 2
b 5 1


e 10 2
c
8 1


f 16 2
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321
Chapter 7
Percents
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7.1
Percents, Decimals, and
Fractions
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Understanding Percent
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.
1
1% 
 0.01
100
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324
Writing a Decimal as a Percent
Multiply by 1 in the form of 100%.
0.65  0.65(100%)  65.% or 65%
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Writing a Percent as a Decimal
Replace the percent symbol with its decimal
equivalent, 0.01; then multiply.
43%  43(0.01)  0.43
100%  100(0.01)  1.00 or 1
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Writing a Percent as a Fraction
Replace the percent symbol with its fraction
1
equivalent,
; then multiply. Don’t forget to
100
simplify the fraction, if possible.
1
43
43%  43 

100 100
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Writing a Fraction as a Percent
Multiply by 1 in the form of 100%.
3 3
3 100
300
 • 100%  •
%
%  60%
5 5
5 1
5
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Helpful Hint
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.
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Summary
• To write a percent as a decimal, replace the %
symbol with its decimal equivalent, 0.01; then
multiply.
• To write a percent as a fraction, replace the %
1
symbol with its fraction equivalent,
; then
100
multiply.
• To write a decimal or fraction as a percent,
multiply by 100%.
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330
7.2
Solving Percent
Problems with
Equations
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Key Words
of means multiplication (∙)
is means equals ()
what (or some equivalent) means the unknown
number
Let x stand for the unknown number.
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332
Helpful Hint
Remember that an equation is simply a mathematical
statement that contains an equal sign ().
6  18x
equal sign
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333
Solving Percent Problems
20%
20%
percent
of 50

10
• 50  10
base
amount
Percent Equation
percent ∙ base  amount
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Helpful Hint
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.
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Helpful Hint
Use the following to see if your answers are reasonable.
100% of a number  the number
a percent greater than 
100%
a percent less than
100%

a number larger than the
original number
a number less than the
original number
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7.3
Solving Percent
Problems with
Proportions
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Writing Percent Problems as Proportions
To understand the proportion method, recall that
30% means the ratio of 30 to 100, or 30 .
100
30
3
30% 

100 10
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Writing Percent Problems as Proportions
30
3
Since the ratio
is equal to the ratio
, we
100
10
have the proportion
30
3 ,

100 10
called the percent proportion.
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Percent Proportion
amount percent

base
100
always 100
or
amount
base
a
p

b 100
percent
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Symbols and Key Words
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.
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Helpful Hints
Part of
Proportion
How It’s
Identified
Percent
% or percent
Base
Appears after of
Amount
Part compared to whole
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Solving Percent Proportions for the Amount
What number is 20% of 8?
amount
amount
base
percent base
a
20

8 100
percent
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Solving Percent Proportions for the Base
20 is 40% of what number?
amount
amount
base
percent
base
20 40

b 100
percent
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Solving Percent Proportions for the Percent
What percent of 40 is 8?
percent
amount
base
Helpful Hint
base amount
8
p

40 100
percent
Recall from our percent proportion that this number, p already is a
percent. Just keep the number the same and attach a % symbol.
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Helpful Hint
A ratio in a proportion may be simplified before
solving the proportion. The unknown number in
both
3 30
6 30

and 
2 b
4 b
is 20.
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7.4
Applications of Percent
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Equation Method
The freshman class of 450 students is 36% of all students
at State College. How many students go to State
College?
Equation Method
State the problem in words, then translate to an equation.
In words: 450 is
36% of what number?
Translate: 450  36%
•
x
Solve: 450  0.36x
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Proportion Method
The freshman class of 450 students is 36% of all
students at State College. How many students go to
State College?
Proportion Equation Method
State the problem in words, then translate to a proportion.
In words: 450
is 36% of what number?
amount
percent
base
450 36
=
Translate and Solve:
b
100
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Percent Increase Percent Decrease
percent increase 
amount of increase
original amount
percent decrease 
amount of decrease
original amount
In each case write the quotient as a percent.
Helpful Hint
Make sure that this number in the denominator is the original
number and not the new number.
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7.5
Percent and Problem
Solving: Sales Tax,
Commission, and Discount
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Calculating Sales Tax and Total Price
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
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Sales Tax and Total Price
The total price to the customer would be
purchase price
sales tax
plus
$10.00

$0.50

$10.50
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Sales Tax and Total Price
sales tax  tax rate ∙ purchase price
total price  purchase price  sales tax
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Calculating Commissions
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
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Discount and Sale Price`
amount of discount  discount rate ∙ original price
sale price  original price  amount of discount
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7.6
Percent and Problem
Solving: Interest
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Calculating Simple 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.
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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.
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Finding the Total Amount of a Loan
total amount (paid or received)  principal  interest
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Calculating Compound Interest
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.
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Compound Interest Formula
The total amount A in an account is given by
r

A  P 1  
 n
nt
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.
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Finding Total Amounts with Compound Interest
total amount  original principal • compound
interest factor
The compound interest factor comes from the
compound interest table found in Appendix C of
the textbook.
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