# The Language of Algebra

## Prime Factorization and the Least Common Multiple

### Find the Prime Factorization of a Composite Number

In the previous section, we found the factors of a number. Prime numbers have only two factors, the number  and the prime number itself. Composite numbers have more than two factors, and every composite number can be written as a unique product of primes. This is called the prime factorization of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes. Finding the prime factorization of a composite number will help you later in this course.
You may want to refer to the following list of prime numbers less than  as you work through this section.

### Prime Factorization Using the Factor Tree Method

One way to find the prime factorization of a number is to make a factor tree. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment—a “branch” of the factor tree.
If a factor is prime, we circle it (like a bud on a tree), and do not factor that “branch” any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree.
We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.
For example, let’s find the prime factorization of  We can start with any factor pair such as  and  We write  and  below  with branches connecting them.
The factor  is prime, so we circle it. The factor  is composite, so we need to find its factors. Let’s use  and  We write these factors on the tree under the
The factor  is prime, so we circle it. The factor  is composite, and it factors into  We write these factors under the  Since  is prime, we circle both
The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.
In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.
Note that we could have started our factor tree with any factor pair of  We chose  and  but the same result would have been the same if we had started with  and  and

### Prime Factorization Using the Ladder Method

The ladder method is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. Some people prefer the ladder method to the factor tree method, and vice versa.
To begin building the “ladder,” divide the given number by its smallest prime factor. For example, to start the ladder for  we divide  by  the smallest prime factor of
To add a “step” to the ladder, we continue dividing by the same prime until it no longer divides evenly.
Then we divide by the next prime; so we divide  by
We continue dividing up the ladder in this way until the quotient is prime. Since the quotient,  is prime, we stop here.
Do you see why the ladder method is sometimes called stacked division?
The prime factorization is the product of all the primes on the sides and top of the ladder.
Notice that the result is the same as we obtained with the factor tree method.

### Find the Least Common Multiple (LCM) of Two Numbers

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

### Listing Multiples Method

A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of  and  We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.
We see that  and  appear in both lists. They are common multiples of  and  We would find more common multiples if we continued the list of multiples for each.
The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of  and  is

### Prime Factors Method

Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of  and
We start by finding the prime factorization of each number.
Then we write each number as a product of primes, matching primes vertically when possible.
Now we bring down the primes in each column. The LCM is the product of these factors.
Notice that the prime factors of  and the prime factors of  are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that  is the least common multiple.

### Key Concepts

1. Find the prime factorization of a composite number using the tree method.

2. Find the prime factorization of a composite number using the ladder method.

3. Find the LCM by listing multiples.

4. Find the LCM using the prime factors method.

### Section Exercises

### Practice Makes Perfect

Find the Prime Factorization of a Composite Number
In the following exercises, find the prime factorization of each number using the factor tree method.
In the following exercises, find the prime factorization of each number using the ladder method.
In the following exercises, find the prime factorization of each number using any method.
Find the Least Common Multiple (LCM) of Two Numbers
In the following exercises, find the least common multiple (LCM) by listing multiples.
In the following exercises, find the least common multiple (LCM) by using the prime factors method.
In the following exercises, find the least common multiple (LCM) using any method.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

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

### Chapter Review Exercises

### Use the Language of Algebra

Use Variables and Algebraic Symbols
In the following exercises, translate from algebra to English.
Identify Expressions and Equations
In the following exercises, determine if each is an expression or equation.
Simplify Expressions with Exponents
In the following exercises, write in exponential form.
In the following exercises, write in expanded form.
In the following exercises, simplify each expression.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify.

### Evaluate, Simplify, and Translate Expressions

Evaluate an Expression
In the following exercises, evaluate the following expressions.
Identify Terms, Coefficients and Like Terms
In the following exercises, identify the terms in each expression.
In the following exercises, identify the coefficient of each term.
In the following exercises, identify the like terms.
Simplify Expressions by Combining Like Terms
In the following exercises, simplify the following expressions by combining like terms.
Translate English Phrases to Algebraic Expressions
In the following exercises, translate the following phrases into algebraic expressions.

### Solve Equations Using the Subtraction and Addition Properties of Equality

Determine Whether a Number is a Solution of an Equation
In the following exercises, determine whether each number is a solution to the equation.
Model the Subtraction Property of Equality
In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality.
Solve Equations using the Subtraction Property of Equality
In the following exercises, solve each equation using the subtraction property of equality.
Solve Equations using the Addition Property of Equality
In the following exercises, solve each equation using the addition property of equality.
Translate English Sentences to Algebraic Equations
In the following exercises, translate each English sentence into an algebraic equation.
Translate to an Equation and Solve
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
Mixed Practice
In the following exercises, solve each equation.

### Find Multiples and Factors

Identify Multiples of Numbers
In the following exercises, list all the multiples less than  for each of the following.
Use Common Divisibility Tests
In the following exercises, using the divisibility tests, determine whether each number is divisible by
Find All the Factors of a Number
In the following exercises, find all the factors of each number.
Identify Prime and Composite Numbers
In the following exercises, identify each number as prime or composite.

### Prime Factorization and the Least Common Multiple

Find the Prime Factorization of a Composite Number
In the following exercises, find the prime factorization of each number.
Find the Least Common Multiple of Two Numbers
In the following exercises, find the least common multiple of each pair of numbers.

### Everyday Math

### Chapter Practice Test

In the following exercises, translate from an algebraic equation to English phrases.
In the following exercises, identify each as an expression or equation.
In the following exercises, simplify, using the order of operations.
In the following exercises, evaluate each expression.
In the following exercises, translate each phrase into an algebraic expression.
In the following exercises, solve each equation.
In the following exercises, translate each English sentence into an algebraic equation and then solve it.


# Integers

## Introduction to Integers

At over 29,000 feet, Mount Everest stands as the tallest peak on land. Located along the border of Nepal and China, Mount Everest is also known for its extreme climate. Near the summit, temperatures never rise above freezing. Every year, climbers from around the world brave the extreme conditions in an effort to scale the tremendous height. Only some are successful. Describing the drastic change in elevation the climbers experience and the change in temperatures requires using numbers that extend both above and below zero. In this chapter, we will describe these kinds of numbers and operations using them.


# Integers

## Introduction to Integers

### Locate Positive and Negative Numbers on the Number Line

Do you live in a place that has very cold winters? Have you ever experienced a temperature below zero? If so, you are already familiar with negative numbers. A negative number is a number that is less than  Very cold temperatures are measured in degrees below zero and can be described by negative numbers. For example,  (read as “negative one degree Fahrenheit”) is  below  A minus sign is shown before a number to indicate that it is negative.  shows  which is  below
Temperatures are not the only negative numbers. A bank overdraft is another example of a negative number. If a person writes a check for more than he has in his account, his balance will be negative.
Elevations can also be represented by negative numbers. The elevation at sea level is  Elevations above sea level are positive and elevations below sea level are negative. The elevation of the Dead Sea, which borders Israel and Jordan, is about  below sea level, so the elevation of the Dead Sea can be represented as  See .
Depths below the ocean surface are also described by negative numbers. A submarine, for example, might descend to a depth of  Its position would then be  as labeled in .
Both positive and negative numbers can be represented on a number line. Recall that the number line created in Add Whole Numbers started at  and showed the counting numbers increasing to the right as shown in . The counting numbers  on the number line are all positive. We could write a plus sign,  before a positive number such as  or  but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.
Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with  at the first mark to the left of  at the next mark, and so on. See .
The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number.

### Order Positive and Negative Numbers

We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See .
Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers. Remember that we use the notation  (read  is less than ) when  is to the left of  on the number line. We write  (read  is greater than ) when  is to the right of  on the number line. This is shown for the numbers  and  in .
The numbers lines to follow show a few more examples.
ⓐ

 is to the right of  on the number line, so
 is to the left of  on the number line, so
ⓑ

 is to the left of  on the number line, so
 is to the right of  on the number line, so
ⓒ

 is to the right of  on the number line, so
 is to the left of  on the number line, so

### Find Opposites

On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers  and  are the same distance from zero, they are called opposites. The opposite of  is  and the opposite of  is  as shown in (a). Similarly,  and  are opposites as shown in (b).

### Opposite Notation

Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol  in three different ways.

### Integers

The set of counting numbers, their opposites, and  is the set of integers.
We must be very careful with the signs when evaluating the opposite of a variable.

### Simplify Expressions with Absolute Value

We saw that numbers such as  and  are opposites because they are the same distance from  on the number line. They are both five units from  The distance between  and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative.
The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of  is written as  and the absolute value of  is written as  as shown in .
We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.
Absolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.

### Translate Word Phrases into Expressions with Integers

Now we can translate word phrases into expressions with integers. Look for words that indicate a negative sign. For example, the word negative in “negative twenty” indicates  So does the word opposite in “the opposite of
As we saw at the start of this section, negative numbers are needed to describe many real-world situations. We’ll look at some more applications of negative numbers in the next example.

### Key Concepts

1. Opposite Notation

2. Absolute Value Notation

### Practice Makes Perfect

Locate Positive and Negative Numbers on the Number Line
For the following exercises, draw a number line and locate and label the given points on that number line.
Order Positive and Negative Numbers on the Number Line
In the following exercises, order each of the following pairs of numbers, using  or
Find Opposites
In the following exercises, find the opposite of each number.
In the following exercises, simplify.
In the following exercises, evaluate.
Simplify Expressions with Absolute Value
In the following exercises, simplify each absolute value expression.
In the following exercises, evaluate each absolute value expression.
In the following exercises, fill in  to compare each expression.
In the following exercises, simplify each expression.
Translate Word Phrases into Expressions with Integers
Translate each phrase into an expression with integers. Do not simplify.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Integers

## Add Integers

### Model Addition of Integers

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.
Most students are comfortable with the addition and subtraction facts for positive numbers.  But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.
The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of  Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.
Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.
We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.
If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in .
We will model four addition facts using the numbers
 and  are very similar. The first example adds  positives and  positives—both positives. The second example adds  negatives and  negatives—both negatives. In each case, we got a result of  positives or  negatives. When the signs are the same, the counters are all the same color.
Now let’s see what happens when the signs are different.

### Simplify Expressions with Integers

Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers.
For example, if you want to add  you don’t have to count out  blue counters and  red counters.
Picture  blue counters with  red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because  there are  more negative counters.
Let’s try another one. We’ll add  Imagine  red counters and  more red counters, so we have  red counters all together. This means the sum is
Look again at the results of  - .
The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.

### Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers when evaluating expressions.
Next we'll evaluate an expression with two variables.

### Translate Word Phrases to Algebraic Expressions

All our earlier work translating word phrases to algebra also applies to expressions that include both positive and negative numbers. Remember that the phrase the sum indicates addition.

### Add Integers in Applications

Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of  could be represented as  Let’s practice translating and solving a few applications.
Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.

### Key Concepts

1. Addition of Positive and Negative Integers

### Practice Makes Perfect

Model Addition of Integers
In the following exercises, model the expression to simplify.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate each phrase into an algebraic expression and then simplify.
Add Integers in Applications
In the following exercises, solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Integers

## Subtract Integers

### Model Subtraction of Integers

Remember the story in the last section about the toddler and the cookies? Children learn how to subtract numbers through their everyday experiences. Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.
We will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.
Perhaps when you were younger, you read  as five take away three. When we use counters, we can think of subtraction the same way.
We will model four subtraction facts using the numbers  and
Notice that  and  are very much alike.
1. First, we subtracted  positives from  positives to get  positives.
2. Then we subtracted  negatives from  negatives to get  negatives.
Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.
Now let’s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.

### Simplify Expressions with Integers

Do you see a pattern?  Are you ready to subtract integers without counters? Let’s do two more subtractions. We’ll think about how we would model these with counters, but we won’t actually use the counters.
1. Subtract
Think: We start with  negative counters.
We have to subtract  positives, but there are no positives to take away.
So we add  neutral pairs to get the  positives. Now we take away the  positives.
So what’s left? We have the original  negatives plus  more negatives from the neutral pair. The result is  negatives.

Notice, that to subtract  we added  negatives.
2. Subtract
Think: We start with  positives.
We have to subtract  negatives, but there are no negatives to take away.
So we add  neutral pairs to the  positives. Now we take away the  negatives.
What’s left? We have the original  positives plus  more positives from the neutral pairs. The result is  positives.

Notice that to subtract  we added
While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.
Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows:
Look at these two examples.
We see that  gives the same answer as
Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract  long ago. But knowing that  gives the same answer as  helps when we are subtracting negative numbers.
Now look what happens when we subtract a negative.
We see that  gives the same result as  Subtracting a negative number is like adding a positive.
Look again at the results of  - .

### Evaluate Variable Expressions with Integers

Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.

### Translate Word Phrases to Algebraic Expressions

When we first introduced the operation symbols, we saw that the expression  may be read in several ways as shown below.
Be careful to get  and  in the right order!

### Subtract Integers in Applications

It’s hard to find something if we don’t know what we’re looking for or what to call it. So when we solve an application problem, we first need to determine what we are asked to find. Then we can write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.
Geography provides another application of negative numbers with the elevations of places below sea level.
Managing your money can involve both positive and negative numbers. You might have overdraft protection on your checking account. This means the bank lets you write checks for more money than you have in your account (as long as they know they can get it back from you!)

### Key Concepts

1. Subtraction of Integers
2. Subtraction Property

3. Solve Application Problems

### Practice Makes Perfect

Model Subtraction of Integers
In the following exercises, model each expression and simplify.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression for the given values.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate each phrase into an algebraic expression and then simplify.
Subtract Integers in Applications
In the following exercises, solve the following applications.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Integers

## Multiply and Divide Integers

### Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.
We remember that  means add  times. Here, we are using the model shown in  just to help us discover the pattern.
Now consider what it means to multiply  by  It means subtract  times. Looking at subtraction as taking away, it means to take away  times. But there is nothing to take away, so we start by adding neutral pairs as shown in .
In both cases, we started with  neutral pairs. In the case on the left, we took away  times and the result was  To multiply  we took away  times and the result was  So we found that
Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.
When we multiply a number by  the result is the same number. What happens when we multiply a number by Let’s multiply a positive number and then a negative number by  to see what we get.
Each time we multiply a number by  we get its opposite.

### Divide Integers

Division is the inverse operation of multiplication. So,  because  In words, this expression says that  can be divided into  groups of  each because adding five three times gives  If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.
Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.
Remember, you can always check the answer to a division problem by multiplying.
Just as we saw with multiplication, when we divide a number by  the result is the same number. What happens when we divide a number by  Let’s divide a positive number and then a negative number by  to see what we get.
When we divide a number by,  we get its opposite.

### Simplify Expressions with Integers

Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.

### Evaluate Variable Expressions with Integers

Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.

### Translate Word Phrases to Algebraic Expressions

Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is product and for division is quotient.

### Key Concepts

1. Multiplication of Signed Numbers

2. Division of Signed Numbers

3. Multiplication by

4. Division by

### Practice Makes Perfect

Multiply Integers
In the following exercises, multiply each pair of integers.
Divide Integers
In the following exercises, divide.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Integers

## Solve Equations Using Integers; The Division Property of Equality

### Determine Whether a Number is a Solution of an Equation

In Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations.
The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.

### Solve Equations with Integers Using the Addition and Subtraction Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.
When you add or subtract the same quantity from both sides of an equation, you still have equality.

### Model the Division Property of Equality

All of the equations we have solved so far have been of the form  or  We were able to isolate the variable by adding or subtracting the constant term. Now we’ll see how to solve equations that involve division.
We will model an equation with envelopes and counters in .
Here, there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?
To determine the number, separate the counters on the right side into  groups of the same size. So  counters divided into  groups means there must be  counters in each group (since
What equation models the situation shown in ? There are two envelopes, and each contains  counters. Together, the two envelopes must contain a total of  counters. So the equation that models the situation is
We can divide both sides of the equation by  as we did with the envelopes and counters.
We found that each envelope contains  Does this check? We know  so it works. Three counters in each of two envelopes does equal six.
 shows another example.
Now we have  identical envelopes and  How many counters are in each envelope? We have to separate the  into  Since  there must be  in each envelope. See .
The equation that models the situation is  We can divide both sides of the equation by
Does this check? It does because

### Solve Equations Using the Division Property of Equality

The previous examples lead to the Division Property of Equality. When you divide both sides of an equation by any nonzero number, you still have equality.

### Translate to an Equation and Solve

In the past several examples, we were given an equation containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations.

### Key Concepts

1. How to determine whether a number is a solution to an equation.

2. Properties of Equalities

3. Division Property of Equality

### Section Exercises

### Practice Makes Perfect

Determine Whether a Number is a Solution of an Equation
In the following exercises, determine whether each number is a solution of the given equation.
Solve Equations Using the Addition and Subtraction Properties of Equality
In the following exercises, solve for the unknown.
Model the Division Property of Equality
In the following exercises, write the equation modeled by the envelopes and counters and then solve it.
Solve Equations Using the Division Property of Equality
In the following exercises, solve each equation using the division property of equality and check the solution.
Translate to an Equation and Solve
In the following exercises, translate and solve.
Mixed Practice
In the following exercises, solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

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

### Chapter Review Exercises

### Introduction to Integers

Locate Positive and Negative Numbers on the Number Line
In the following exercises, locate and label the integer on the number line.
Order Positive and Negative Numbers
In the following exercises, order each of the following pairs of numbers, using  or
Find Opposites
In the following exercises, find the opposite of each number.
In the following exercises, simplify.
In the following exercises, evaluate.
Simplify Absolute Values
In the following exercises, simplify.
In the following exercises, evaluate.
In the following exercises, fill in  for each of the following pairs of numbers.
In the following exercises, simplify.
Translate Phrases to Expressions with Integers
In the following exercises, translate each of the following phrases into expressions with positive or negative numbers.

### Add Integers

Model Addition of Integers
In the following exercises, model the following to find the sum.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate each phrase into an algebraic expression and then simplify.
Add Integers in Applications
In the following exercises, solve.

### Subtract Integers

Model Subtraction of Integers
In the following exercises, model the following.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Translate Phrases to Algebraic Expressions
In the following exercises, translate each phrase into an algebraic expression and then simplify.
Subtract Integers in Applications
In the following exercises, solve the given applications.

### Multiply and Divide Integers

Multiply Integers
In the following exercises, multiply.
Divide Integers
In the following exercises, divide.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.

### Solve Equations using Integers; The Division Property of Equality

Determine Whether a Number is a Solution of an Equation
In the following exercises, determine whether each number is a solution of the given equation.
Using the Addition and Subtraction Properties of Equality
In the following exercises, solve.
Model the Division Property of Equality
In the following exercises, write the equation modeled by the envelopes and counters. Then solve it.
Solve Equations Using the Division Property of Equality
In the following exercises, solve each equation using the division property of equality and check the solution.
Translate to an Equation and Solve.
In the following exercises, translate and solve.

### Everyday Math

### Chapter Practice Test

In the following exercises, compare the numbers, using
In the following exercises, find the opposite of each number.
In the following exercises, simplify.
In the following exercises, evaluate.
In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible.
In the following exercises, solve.
In the following exercises, solve.
In the following exercises, translate and solve.


# Fractions

## Introduction to Fractions

Often in life, whole amounts are not exactly what we need. A baker must use a little more than a cup of milk or part of a teaspoon of sugar. Similarly a carpenter might need less than a foot of wood and a painter might use part of a gallon of paint. In this chapter, we will learn about numbers that describe parts of a whole. These numbers, called fractions, are very useful both in algebra and in everyday life. You will discover that you are already familiar with many examples of fractions!


# Fractions

## Visualize Fractions

### Understand the Meaning of Fractions

Andy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That’s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts.
In math, we write  to mean one out of two parts.
On Tuesday, Andy and Bobby share a pizza with their parents, Fred and Christy, with each person getting an equal amount of the whole pizza. How much of the pizza does each person get? There is one whole pizza, divided evenly into four equal parts. Each person has one of the four equal parts, so each has  of the pizza.
On Wednesday, the family invites some friends over for a pizza dinner. There are a total of  people. If they share the pizza equally, each person would get  of the pizza.
A fraction is a way to represent parts of a whole. The denominator  represents the number of equal parts the whole has been divided into, and the numerator  represents how many parts are included. The denominator,  cannot equal zero because division by zero is undefined.
In , the circle has been divided into three parts of equal size. Each part represents  of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.
What does the fraction  represent? The fraction  means two of three equal parts.
In  and , we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in . Here, the whole is modeled as one long, undivided rectangular tile. Beneath it are tiles of equal length divided into different numbers of equally sized parts.
We’ll be using fraction tiles to discover some basic facts about fractions. Refer to  to answer the following questions:
It takes  twenty-fourths, so
This leads us to the Property of One.
What if we have more fraction pieces than we need for  whole? We’ll look at this in the next example.

### Model Improper Fractions and Mixed Numbers

In  (b), you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth,  so altogether you had eight fifths, which we can write as  The fraction  is one whole,  plus three fifths,  or  which is read as one and three-fifths.
The number  is called a mixed number. A mixed number consists of a whole number and a fraction.
Fractions such as  and  are called improper fractions. In an improper fraction, the numerator is greater than or equal to the denominator, so its value is greater than or equal to one. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as  and  are proper fractions.

### Convert between Improper Fractions and Mixed Numbers

In , we converted the improper fraction  to the mixed number  using fraction circles. We did this by grouping six sixths together to make a whole; then we looked to see how many of the  pieces were left. We saw that  made one whole group of six sixths plus five more sixths, showing that
The division expression  (which can also be written as ) tells us to find how many groups of  are in  To convert an improper fraction to a mixed number without fraction circles, we divide.
In , we changed  to an improper fraction by first seeing that the whole is a set of five fifths. So we had five fifths and four more fifths.
Where did the nine come from? There are nine fifths—one whole (five fifths) plus four fifths. Let us use this idea to see how to convert a mixed number to an improper fraction.

### Model Equivalent Fractions

Let’s think about Andy and Bobby and their favorite food again. If Andy eats  of a pizza and Bobby eats  of the pizza, have they eaten the same amount of pizza? In other words, does  We can use fraction tiles to find out whether Andy and Bobby have eaten equivalent parts of the pizza.
Fraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of  and extend it to include eighths, tenths, and twelfths.
Start with a  tile. How many fourths equal one-half? How many of the  tiles exactly cover the  tile?
Since two  tiles cover the  tile, we see that  is the same as  or
How many of the  tiles cover the  tile?
Since three  tiles cover the  tile, we see that  is the same as
So,  The fractions are equivalent fractions.

### Find Equivalent Fractions

We used fraction tiles to show that there are many fractions equivalent to  For example,  and  are all equivalent to  When we lined up the fraction tiles, it took four of the  tiles to make the same length as a  tile. This showed that  See .
We can show this with pizzas, too. (a) shows a single pizza, cut into two equal pieces with  shaded. (b) shows a second pizza of the same size, cut into eight pieces with  shaded.
This is another way to show that  is equivalent to
How can we use mathematics to change  into  How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we’ve described could be written as:
These models lead to the Equivalent Fractions Property, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.
When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.
So, we say that  and  are equivalent fractions.

### Locate Fractions and Mixed Numbers on the Number Line

Now we are ready to plot fractions on a number line. This will help us visualize fractions and understand their values.
Let us locate  and  on the number line.
We will start with the whole numbers  and  because they are the easiest to plot.
The proper fractions listed are  and  We know proper fractions have values less than one, so  and  are located between the whole numbers  and  The denominators are both  so we need to divide the segment of the number line between  and  into five equal parts. We can do this by drawing four equally spaced marks on the number line, which we can then label as  and
Now plot points at  and
The only mixed number to plot is  Between what two whole numbers is  Remember that a mixed number is a whole number plus a proper fraction, so  Since it is greater than  but not a whole unit greater,  is between  and  We need to divide the portion of the number line between  and  into three equal pieces (thirds) and plot  at the first mark.
Finally, look at the improper fractions  and  Locating these points will be easier if you change each of them to a mixed number.
Here is the number line with all the points plotted.
In Introduction to  Integers, we defined the opposite of a number. It is the number that is the same distance from zero on the number line but on the opposite side of zero. We saw, for example, that the opposite of  is  and the opposite of  is
Fractions have opposites, too. The opposite of  is  It is the same distance from  on the number line, but on the opposite side of
Thinking of negative fractions as the opposite of positive fractions will help us locate them on the number line. To locate  on the number line, first think of where  is located. It is an improper fraction, so we first convert it to the mixed number  and see that it will be between  and  on the number line. So its opposite,  will be between  and  on the number line.

### Order Fractions and Mixed Numbers

We can use the inequality symbols to order fractions. Remember that  means that  is to the right of  on the number line. As we move from left to right on a number line, the values increase.

### Key Concepts

1. Property of One
2. Mixed Numbers
3. Proper and Improper Fractions
4. Convert an improper fraction to a mixed number.
5. Convert a mixed number to an improper fraction.
6. Equivalent Fractions Property

### Practice Makes Perfect

In the following exercises, name the fraction of each figure that is shaded.
In the following exercises, shade parts of circles or squares to model the following fractions.
In the following exercises, use fraction circles to make wholes using the following pieces.
In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.
In the following exercises, draw fraction circles to model the given fraction.
In the following exercises, rewrite the improper fraction as a mixed number.
In the following exercises, rewrite the mixed number as an improper fraction.
In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.
In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
In the following exercises, plot the numbers on a number line.
In the following exercises, order each of the following pairs of numbers, using  or

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Fractions

## Multiply and Divide Fractions

### Simplify Fractions

In working with equivalent fractions, you saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we’ll use the fraction that is in simplified form.
A fraction is considered simplified if there are no common factors, other than  in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.
For example,
1.  is simplified because there are no common factors of  and
2.  is not simplified because  is a common factor of  and
The process of simplifying a fraction is often called reducing the fraction. In the previous section, we used the Equivalent Fractions Property to find equivalent fractions. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.
Notice that  is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.
To simplify a negative fraction, we use the same process as in . Remember to keep the negative sign.
After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: a fraction is considered simplified if there are no common factors in the numerator and denominator.
When we simplify an improper fraction, there is no need to change it to a mixed number.
Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property.
We can also simplify fractions containing variables. If a variable is a common factor in the numerator and denominator, we remove it just as we do with an integer factor.

### Multiply Fractions

A model may help you understand multiplication of fractions. We will use fraction tiles to model  To multiply  and  think  of
Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three  tiles evenly into two parts, we exchange them for smaller tiles.
We see  is equivalent to  Taking half of the six  tiles gives us three  tiles, which is
Therefore,
Look at the result we got from the model in . We found that  Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?
This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.
When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In Example 4.26 we will multiply two negatives, so the product will be positive.
When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer,  can be written as  So,  for example.

### Find Reciprocals

The fractions  and  are related to each other in a special way. So are  and  Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be
Such pairs of numbers are called reciprocals.
To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.
To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.
To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to  Is there any number  so that  No. So, the number  does not have a reciprocal.
In a previous chapter, we worked with opposites and absolute values.  compares opposites, absolute values, and reciprocals.

### Divide Fractions

Why is  We previously modeled this with counters. How many groups of  counters can be made from a group of  counters?
There are  groups of  counters. In other words, there are four s in  So,
What about dividing fractions? Suppose we want to find the quotient:  We need to figure out how many s there are in  We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in . Notice, there are three  tiles in  so
Let’s use money to model  in another way. We often read  as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in . So we can think of  as, “How many quarters are there in two dollars?” One dollar is  quarters, so  dollars would be  quarters. So again,
Using fraction tiles, we showed that  Notice that  also. How are  and  related? They are reciprocals. This leads us to the procedure for fraction division.
We need to say  and  to be sure we don’t divide by zero.

### Key Concepts

1. Equivalent Fractions Property
2. Simplify a fraction.
3. Fraction Multiplication
4. Reciprocal
5. Fraction Division

### Practice Makes Perfect

Simplify Fractions
In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.
Multiply Fractions
In the following exercises, use a diagram to model.
In the following exercises, multiply, and write the answer in simplified form.
Find Reciprocals
In the following exercises, find the reciprocal.
Divide Fractions
In the following exercises, model each fraction division.
In the following exercises, divide, and write the answer in simplified form.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Fractions

## Multiply and Divide Mixed Numbers and Complex Fractions

### Multiply and Divide Mixed Numbers

In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction. And you learned how to do that in Visualize Fractions.

### Translate Phrases to Expressions with Fractions

The words quotient and ratio are often used to describe fractions. In Subtract Whole Numbers, we defined quotient as the result of division. The quotient of  and  is the result you get from dividing  by  or  Let’s practice translating some phrases into algebraic expressions using these terms.

### Simplify Complex Fractions

Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction, which is a fraction in which the numerator or the denominator contains a fraction.
Some examples of complex fractions are:
To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction  can be written as

### Simplify Expressions with a Fraction Bar

Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction  could be the result of dividing  a negative by a positive, or of dividing  a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.
If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative.
Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example,  means  The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide.
We’ll add fraction bars to our set of grouping symbols from Use the Language of Algebra to have a more complete set here.

### Key Concepts

1. Multiply or divide mixed numbers.
2. Simplify a complex fraction.
3. Placement of negative sign in a fraction.
4. Simplify an expression with a fraction bar.

### Practice Makes Perfect

Multiply and Divide Mixed Numbers
In the following exercises, multiply and write the answer in simplified form.
In the following exercises, divide, and write your answer in simplified form.
Translate Phrases to Expressions with Fractions
In the following exercises, translate each English phrase into an algebraic expression.
Simplify Complex Fractions
In the following exercises, simplify the complex fraction.
Simplify Expressions with a Fraction Bar
In the following exercises, identify the equivalent fractions.
In the following exercises, simplify.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Fractions

## Add and Subtract Fractions with Common Denominators

### Model Fraction Addition

How many quarters are pictured? One quarter plus  quarters equals  quarters.
Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that
Let’s use fraction circles to model the same example,
So again, we see that

### Add Fractions with a Common Denominator

 shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces.

### Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into  slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or  of the pizza) left in the box. If Leonardo eats  of these remaining pieces (or  of the pizza), how much is left? There would be  pieces left (or  of the pizza).
Let’s use fraction circles to model the same example,
Start with seven  pieces. Take away two  pieces. How many twelfths are left?
Again, we have five twelfths,

### Subtract Fractions with a Common Denominator

We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.
Now lets do an example that involves both addition and subtraction.

### Key Concepts

1. Fraction Addition
2. Fraction Subtraction

### Practice Makes Perfect

Model Fraction Addition
In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.
Add Fractions with a Common Denominator
In the following exercises, find each sum.
Model Fraction Subtraction
In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.
Subtract Fractions with a Common Denominator
In the following exercises, find the difference.
Mixed Practice
In the following exercises, perform the indicated operation and write your answers in simplified form.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Fractions

## Add and Subtract Fractions with Different Denominators

### Find the Least Common Denominator

In the previous section, we explained how to add and subtract fractions with a common denominator. But how can we add and subtract fractions with unlike denominators?
Let’s think about coins again. Can you add one quarter and one dime? You could say there are two coins, but that’s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit—cents. One quarter equals  cents and one dime equals  cents, so the sum is  cents. See .
Similarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is  Since there are  cents in one dollar,  cents is  and  cents is  So we add  to get  which is  cents.
You have practiced adding and subtracting fractions with common denominators. Now let’s see what you need to do with fractions that have different denominators.
First, we will use fraction tiles to model finding the common denominator of  and
We’ll start with one  tile and  tile. We want to find a common fraction tile that we can use to match both  and  exactly.
If we try the  pieces,  of them exactly match the  piece, but they do not exactly match the  piece.
If we try the  pieces, they do not exactly cover the  piece or the  piece.
If we try the  pieces, we see that exactly  of them cover the  piece, and exactly  of them cover the  piece.
If we were to try the  pieces, they would also work.
Even smaller tiles, such as  and  would also exactly cover the  piece and the  piece.
The denominator of the largest piece that covers both fractions is the least common denominator (LCD) of the two fractions. So, the least common denominator of  and  is
Notice that all of the tiles that cover  and  have something in common: Their denominators are common multiples of  and  the denominators of  and  The least common multiple (LCM) of the denominators is  and so we say that  is the least common denominator (LCD) of the fractions  and
To find the LCD of two fractions, we will find the LCM of their denominators. We follow the procedure we used earlier to find the LCM of two numbers. We only use the denominators of the fractions, not the numerators, when finding the LCD.
To find the LCD of two fractions, find the LCM of their denominators. Notice how the steps shown below are similar to the steps we took to find the LCM.

### Convert Fractions to Equivalent Fractions with the LCD

Earlier, we used fraction tiles to see that the LCD of  when  is  We saw that three  pieces exactly covered  and two  pieces exactly covered  so
We say that  and  are equivalent fractions and also that  and  are equivalent fractions.
We can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The Equivalent Fractions Property is repeated below for reference.
To add or subtract fractions with different denominators, we will first have to convert each fraction to an equivalent fraction with the LCD. Let’s see how to change  and  to equivalent fractions with denominator  without using models.

### Add and Subtract Fractions with Different Denominators

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.
When we use the Equivalent Fractions Property, there is a quick way to find the number you need to multiply by to get the LCD. Write the factors of the denominators and the LCD just as you did to find the LCD. The “missing” factors of each denominator are the numbers you need.
The LCD,  has  factors of  and  factors of
Twelve has two factors of  but only one of —so it is ‘missing‘ one  We multiplied the numerator and denominator of  by  to get an equivalent fraction with denominator
Eighteen is missing one factor of —so you multiply the numerator and denominator  by  to get an equivalent fraction with denominator  We will apply this method as we subtract the fractions in the next example.
In the next example, one of the fractions has a variable in its numerator. We follow the same steps as when both numerators are numbers.

### Identify and Use Fraction Operations

By now in this chapter, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions

### Use the Order of Operations to Simplify Complex Fractions

In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,
Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.

### Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

### Key Concepts

1. Find the least common denominator (LCD) of two fractions.
2. Equivalent Fractions Property
3. Convert two fractions to equivalent fractions with their LCD as the common denominator.
4. Add or subtract fractions with different denominators.
5. Summary of Fraction Operations
6. Simplify complex fractions.

### Practice Makes Perfect

Find the Least Common Denominator (LCD)
In the following exercises, find the least common denominator (LCD) for each set of fractions.
Convert Fractions to Equivalent Fractions with the LCD
In the following exercises, convert to equivalent fractions using the LCD.
Add and Subtract Fractions with Different Denominators
In the following exercises, add or subtract. Write the result in simplified form.
Identify and Use Fraction Operations
In the following exercises, perform the indicated operations. Write your answers in simplified form.
Use the Order of Operations to Simplify Complex Fractions
In the following exercises, simplify.
Mixed Practice
In the following exercises, simplify.
In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?


# Fractions

## Add and Subtract Mixed Numbers

### Model Addition of Mixed Numbers with a Common Denominator

So far, we’ve added and subtracted proper and improper fractions, but not mixed numbers. Let’s begin by thinking about addition of mixed numbers using money.
If Ron has  dollar and  quarter, he has  dollars.
If Don has  dollars and  quarter, he has  dollars.
What if Ron and Don put their money together? They would have  dollars and  quarters. They add the dollars and add the quarters. This makes  dollars. Because two quarters is half a dollar, they would have  and a half dollars, or  dollars.
When you added the dollars and then added the quarters, you were adding the whole numbers and then adding the fractions.
We can use fraction circles to model this same example:

### Add Mixed Numbers

Modeling with fraction circles helps illustrate the process for adding mixed numbers: We add the whole numbers and add the fractions, and then we simplify the result, if possible.
In , the sum of the fractions was a proper fraction. Now we will work through an example where the sum is an improper fraction.
An alternate method for adding mixed numbers is to convert the mixed numbers to improper fractions and then add the improper fractions. This method is usually written horizontally.
 compares the two methods of addition, using the expression  as an example. Which way do you prefer?

### Model Subtraction of Mixed Numbers

Let’s think of pizzas again to model subtraction of mixed numbers with a common denominator. Suppose you just baked a whole pizza and want to give your brother half of the pizza. What do you have to do to the pizza to give him half? You have to cut it into at least two pieces. Then you can give him half.
We will use fraction circles (pizzas!) to help us visualize the process.
Start with one whole.
Algebraically, you would write:
What if we start with more than one whole? Let’s find out.
In the next example, we’ll subtract more than one whole.
What if you start with a mixed number and need to subtract a fraction? Think about this situation: You need to put three quarters in a parking meter, but you have only a  bill and one quarter. What could you do? You could change the dollar bill into  quarters. The value of  quarters is the same as one dollar bill, but the  quarters are more useful for the parking meter. Now, instead of having a  bill and one quarter, you have  quarters and can put  quarters in the meter.
This models what happens when we subtract a fraction from a mixed number. We subtracted three quarters from one dollar and one quarter.
We can also model this using fraction circles, much like we did for addition of mixed numbers.

### Subtract Mixed Numbers with a Common Denominator

Now we will subtract mixed numbers without using a model. But it may help to picture the model in your mind as you read the steps.
Just as we did with addition, we could subtract mixed numbers by converting them first to improper fractions. We should write the answer in the form it was given, so if we are given mixed numbers to subtract we will write the answer as a mixed number.

### Add and Subtract Mixed Numbers with Different Denominators

To add or subtract mixed numbers with different denominators, we first convert the fractions to equivalent fractions with the LCD. Then we can follow all the steps we used above for adding or subtracting fractions with like denominators.

### Key Concepts

1. Add mixed numbers with a common denominator.
2. Subtract mixed numbers with common denominators.

3. Subtract mixed numbers with common denominators as improper fractions.

### Practice Makes Perfect

Model Addition of Mixed Numbers
In the following exercises, use a model to find the sum. Draw a picture to illustrate your model.
Add Mixed Numbers with a Common Denominator
In the following exercises, add.
Model Subtraction of Mixed Numbers
In the following exercises, use a model to find the difference. Draw a picture to illustrate your model.
Subtract Mixed Numbers with a Common Denominator
In the following exercises, find the difference.
Add and Subtract Mixed Numbers with Different Denominators
In the following exercises, write the sum or difference as a mixed number in simplified form.
Mixed Practice
In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Fractions

## Solve Equations with Fractions

### Determine Whether a Fraction is a Solution of an Equation

As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations.
The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction.

### Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions.
We used the Subtraction Property of Equality in . Now we’ll use the Addition Property of Equality.
The next example may not seem to have a fraction, but let’s see what happens when we solve it.

### Solve Equations with Fractions Using the Multiplication Property of Equality

Consider the equation  We want to know what number divided by  gives  So to “undo” the division, we will need to multiply by  The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.
Let’s use the Multiplication Property of Equality to solve the equation

### Solve Equations with a Coefficient of

Look at the equation  Does it look as if  is already isolated? But there is a negative sign in front of  so it is not isolated.
There are three different ways to isolate the variable in this type of equation. We will show all three ways in .

### Solve Equations with a Fraction Coefficient

When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to
For example, in the equation:
The coefficient of  is  To solve for  we need its coefficient to be  Since the product of a number and its reciprocal is  our strategy here will be to isolate  by multiplying by the reciprocal of  We will do this in .

### Translate Sentences to Equations and Solve

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.
When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.
In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Evaluate, Simplify, and Translate Expressions.

### Key Concepts

1. Determine whether a number is a solution to an equation.
2. Addition, Subtraction, and Division Properties of Equality
3. The Multiplication Property of Equality

### Section Exercises

### Practice Makes Perfect

Determine Whether a Fraction is a Solution of an Equation
In the following exercises, determine whether each number is a solution of the given equation.
Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality
In the following exercises, solve.
Solve Equations with Fractions Using the Multiplication Property of Equality
In the following exercises, solve.
Mixed Practice
In the following exercises, solve.
Translate Sentences to Equations and Solve
In the following exercises, translate to an algebraic equation and solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

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

### Chapter Review Exercises

### Visualize Fractions

In the following exercises, name the fraction of each figure that is shaded.
In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.
In the following exercises, convert the improper fraction to a mixed number.
In the following exercises, convert the mixed number to an improper fraction.
In the following exercises, locate the numbers on a number line.
In the following exercises, order each pair of numbers, using  or

### Multiply and Divide Fractions

In the following exercises, simplify.
In the following exercises, multiply.
In the following exercises, find the reciprocal.
In the following exercises, divide.

### Multiply and Divide Mixed Numbers and Complex Fractions

In the following exercises, perform the indicated operation.
In the following exercises, translate the English phrase into an algebraic expression.
In the following exercises, simplify the complex fraction
In the following exercises, simplify.

### Add and Subtract Fractions with Common Denominators

In the following exercises, add.
In the following exercises, subtract.

### Add and Subtract Fractions with Different Denominators

In the following exercises, find the least common denominator.
In the following exercises, change to equivalent fractions using the given LCD.
In the following exercises, perform the indicated operations and simplify.
In the following exercises, evaluate.

### Add and Subtract Mixed Numbers

In the following exercises, perform the indicated operation.

### Solve Equations with Fractions

In the following exercises, determine whether the each number is a solution of the given equation.
In the following exercises, solve the equation.
In the following exercises, translate and solve.

### Chapter Practice Test

Convert the improper fraction to a mixed number.
Convert the mixed number to an improper fraction.
Locate the numbers on a number line.
In the following exercises, simplify.
Evaluate.
In the following exercises, solve the equation.


# Decimals

## Introduction to Decimals

Gasoline price changes all the time. They might go down for a period of time, but then they usually rise again. One thing that stays the same is that the price is not usually a whole number. Instead, it is shown using a decimal point to describe the cost in dollars and cents. We use decimal numbers all the time, especially when dealing with money. In this chapter, we will explore decimal numbers and how to perform operations using them.


# Decimals

## Decimals

### Name Decimals

You probably already know quite a bit about decimals based on your experience with money. Suppose you buy a sandwich and a bottle of water for lunch. If the sandwich costs , the bottle of water costs , and the total sales tax is , what is the total cost of your lunch?
The total is  Suppose you pay with a  bill and  pennies. Should you wait for change? No,  and  pennies is the same as
Because  each penny is worth  of a dollar. We write the value of one penny as  since
Writing a number with a decimal is known as decimal notation. It is a way of showing parts of a whole when the whole is a power of ten. In other words, decimals are another way of writing fractions whose denominators are powers of ten. Just as the counting numbers are based on powers of ten, decimals are based on powers of ten.  shows the counting numbers.
How are decimals related to fractions?  shows the relation.
When we name a whole number, the name corresponds to the place value based on the powers of ten. In Whole Numbers, we learned to read  as ten thousand. Likewise, the names of the decimal places correspond to their fraction values. Notice how the place value names in  relate to the names of the fractions from .
Notice two important facts shown in .
1. The “th” at the end of the name means the number is a fraction. “One thousand” is a number larger than one, but “one thousandth” is a number smaller than one.
2. The tenths place is the first place to the right of the decimal, but the tens place is two places to the left of the decimal.
Remember that  lunch? We read  as five dollars and three cents. Naming decimals (those that don’t represent money) is done in a similar way. We read the number  as five and three hundredths.
We sometimes need to translate a number written in decimal notation into words. As shown in , we write the amount on a check in both words and numbers.
The number  is read fifteen and sixty-eight hundredths.

### Write Decimals

Now we will translate the name of a decimal number into decimal notation. We will reverse the procedure we just used.
Let’s start by writing the number six and seventeen hundredths:
The second bullet in Step 2 is needed for decimals that have no whole number part, like ‘nine thousandths’. We recognize them by the words that indicate the place value after the decimal – such as ‘tenths’ or ‘hundredths.’  Since there is no whole number, there is no ‘and.’ We start by placing a zero to the left of the decimal and continue by filling in the numbers to the right, as we did above.
Before we move on to our next objective, think about money again. We know that  is the same as  The way we write  depends on the context. In the same way, integers can be written as decimals with as many zeros as needed to the right of the decimal.

### Convert Decimals to Fractions or Mixed Numbers

We often need to rewrite decimals as fractions or mixed numbers. Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. We know that  means  dollars and  cents. Since there are  cents in one dollar,  cents means  of a dollar, so
We convert decimals to fractions by identifying the place value of the farthest right digit. In the decimal  the  is in the hundredths place, so  is the denominator of the fraction equivalent to
For our  lunch, we can write the decimal  as a mixed number.
Notice that when the number to the left of the decimal is zero, we get a proper fraction. When the number to the left of the decimal is not zero, we get a mixed number.

### Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

### Order Decimals

Which is larger,  or
If you think of this as money, you know that  (forty cents) is greater than  (four cents). So,
In previous chapters, we used the number line to order numbers.
Where are  and  located on the number line?
We see that  is to the right of  So we know
How does  compare to  This doesn’t translate into money to make the comparison easy. But if we convert  and  to fractions, we can tell which is larger.
Because  we know that  Therefore,
Notice what we did in converting  to a fraction—we started with the fraction  and ended with the equivalent fraction  Converting  back to a decimal gives  So  is equivalent to  Writing zeros at the end of a decimal does not change its value.
If two decimals have the same value, they are said to be equivalent decimals.
We say  and  are equivalent decimals.
Remember, writing zeros at the end of a decimal does not change its value.

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because  lies to the right of  on the number line, we know that  Similarly, smaller numbers lie to the left on the number line. For example, because  lies to the left of  on the number line, we know that
If we zoomed in on the interval between  and  we would see in the same way that

### Round Decimals

In the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas station might post the price of unleaded gas at  per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay , because the final price would be rounded to the nearest cent. In Whole Numbers, we saw that we round numbers to get an approximate value when the exact value is not needed. Suppose we wanted to round  to the nearest dollar. Is it closer to  or to  What if we wanted to round  to the nearest ten cents; is it closer to  or to  The number lines in  can help us answer those questions.
Can we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers.

### Key Concepts

1. Name a decimal number.

2. Write a decimal number from its name.

3. Convert a decimal number to a fraction or mixed number.

4. Order decimals.

5. Round a decimal.

### Practice Makes Perfect

Name Decimals
In the following exercises, name each decimal.
Write Decimals
In the following exercises, translate the name into a decimal number.
Convert Decimals to Fractions or Mixed Numbers
In the following exercises, convert each decimal to a fraction or mixed number.
Locate Decimals on the Number Line
In the following exercises, locate each number on a number line.
Order Decimals
In the following exercises, order each of the following pairs of numbers, using
Round Decimals
In the following exercises, round each number to the nearest tenth.
In the following exercises, round each number to the nearest hundredth.
In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Decimals

## Decimal Operations

### Add and Subtract Decimals

Let’s take one more look at the lunch order from the start of Decimals, this time noticing how the numbers were added together.
All three items (sandwich, water, tax) were priced in dollars and cents, so we lined up the dollars under the dollars and the cents under the cents, with the decimal points lined up between them. Then we just added each column, as if we were adding whole numbers. By lining up decimals this way, we can add or subtract the corresponding place values just as we did with whole numbers.
How much change would you get if you handed the cashier a  bill for a  purchase? We will show the steps to calculate this in the next example.

### Multiply Decimals

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.
Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators.
So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side in . Look for a pattern.
There is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places.
How many decimal places would you expect for the product of  If you said “five”, you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors—in this case two plus three—to get the number of decimal places in the product—in this case five.
Once we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors.
The rules for multiplying positive and negative numbers apply to decimals, too, of course.
When you multiply signed decimals, first determine the sign of the product and then multiply as if the numbers were both positive. Finally, write the product with the appropriate sign.
In the next example, we’ll need to add several placeholder zeros to properly place the decimal point.

### Multiply by Powers of

In many fields, especially in the sciences, it is common to multiply decimals by powers of  Let’s see what happens when we multiply  by some powers of
Look at the results without the final zeros. Do you notice a pattern?

The number of places that the decimal point moved is the same as the number of zeros in the power of ten.  summarizes the results.
We can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of  and then move the decimal point that same of places to the right.
So, for example, to multiply  by  move the decimal point  places to the right.
Sometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let’s multiply  by  We need to move the decimal point  places to the right. Since there is only one digit to the right of the decimal point, we must write a  in the hundredths place.

### Divide Decimals

Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.
To understand decimal division, let’s consider the multiplication problem
Remember, a multiplication problem can be rephrased as a division problem. So we can write
We can think of this as “If we divide 8 tenths into four groups, how many are in each group?”  shows that there are four groups of two-tenths in eight-tenths. So
Using long division notation, we would write
Notice that the decimal point in the quotient is directly above the decimal point in the dividend.
To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder.
In everyday life, we divide whole numbers into decimals—money—to find the price of one item. For example, suppose a case of  water bottles cost  To find the price per water bottle, we would divide  by  and round the answer to the nearest cent (hundredth).

### Divide a Decimal by Another Decimal

So far, we have divided a decimal by a whole number. What happens when we divide a decimal by another decimal? Let’s look at the same multiplication problem we looked at earlier, but in a different way.
Remember, again, that a multiplication problem can be rephrased as a division problem. This time we ask, “How many times does  go into  Because  we can say that  goes into  four times. This means that  divided by  is
We would get the same answer,  if we divide  by  both whole numbers. Why is this so? Let’s think about the division problem as a fraction.
We multiplied the numerator and denominator by  and ended up just dividing  by  To divide decimals, we multiply both the numerator and denominator by the same power of  to make the denominator a whole number. Because of the Equivalent Fractions Property, we haven’t changed the value of the fraction. The effect is to move the decimal points in the numerator and denominator the same number of places to the right.
We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.
It may help to review the vocabulary for division:
Now we will divide a whole number by a decimal number.

### Use Decimals in Money Applications

We often apply decimals in real life, and most of the applications involving money. The Strategy for Applications we used in The Language of Algebra gives us a plan to follow to help find the answer. Take a moment to review that strategy now.
Be careful to follow the order of operations in the next example. Remember to multiply before you add.

### Key Concepts

1. Add or subtract decimals.

2. Multiply decimal numbers.

3. Multiply a decimal by a power of 10.

4. Divide a decimal by a whole number.

5. Divide decimal numbers.

6. Strategy for Applications

### Practice Makes Perfect

Add and Subtract Decimals
In the following exercises, add or subtract.
Multiply Decimals
In the following exercises, multiply.
Divide Decimals
In the following exercises, divide.
Mixed Practice
In the following exercises, simplify.
Use Decimals in Money Applications
In the following exercises, use the strategy for applications to solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?