# Decimals

## Decimals and Fractions

### Convert Fractions to Decimals

In Decimals, we learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar indicates division. So  can be written  or  This means that we can convert a fraction to a decimal by treating it as a division problem.

### Repeating Decimals

So far, in all the examples converting fractions to decimals the division resulted in a remainder of zero. This is not always the case. Let’s see what happens when we convert the fraction  to a decimal. First, notice that  is an improper fraction. Its value is greater than  The equivalent decimal will also be greater than
We divide  by
No matter how many more zeros we write, there will always be a remainder of  and the threes in the quotient will go on forever. The number  is called a repeating decimal. Remember that the “…” means that the pattern repeats.
How do you know how many ‘repeats’ to write? Instead of writing  we use a shorthand notation by placing a line over the digits that repeat. The repeating decimal  is written  The line above the  tells you that the  repeats endlessly. So
For other decimals, two or more digits might repeat.  shows some more examples of repeating decimals.
It is useful to convert between fractions and decimals when we need to add or subtract numbers in different forms. To add a fraction and a decimal, for example, we would need to either convert the fraction to a decimal or the decimal to a fraction.

### Order Decimals and Fractions

In Decimals, we compared two decimals and determined which was larger. To compare a decimal to a fraction, we will first convert the fraction to a decimal and then compare the decimals.
When ordering negative numbers, remember that larger numbers are to the right on the number line and any positive number is greater than any negative number.

### Simplify Expressions Using the Order of Operations

The order of operations introduced in Use the Language of Algebra also applies to decimals. Do you remember what the phrase “Please excuse my dear Aunt Sally” stands for?

### Find the Circumference and Area of Circles

The properties of circles have been studied for over  years. All circles have exactly the same shape, but their sizes are affected by the length of the radius, a line segment from the center to any point on the circle. A line segment that passes through a circle’s center connecting two points on the circle is called a diameter. The diameter is twice as long as the radius. See .
The size of a circle can be measured in two ways. The distance around a circle is called its circumference.
Archimedes discovered that for circles of all different sizes, dividing the circumference by the diameter always gives the same number. The value of this number is pi, symbolized by Greek letter  (pronounced pie). However, the exact value of  cannot be calculated since the decimal never ends or repeats (we will learn more about numbers like this in The Properties of Real Numbers.)
If we want the exact circumference or area of a circle, we leave the symbol  in the answer. We can get an approximate answer by substituting  as the value of  We use the symbol  to show that the result is approximate, not exact.
Since the diameter is twice the radius, another way to find the circumference is to use the formula
Suppose we want to find the exact area of a circle of radius  inches. To calculate the area, we would evaluate the formula for the area when  inches and leave the answer in terms of
We write  after the  So the exact value of the area is  square inches.
To approximate the area, we would substitute
Remember to use square units, such as square inches, when you calculate the area.

### Approximate  with a Fraction

Convert the fraction  to a decimal. If you use your calculator, the decimal number will fill up the display and show  But if we round that number to two decimal places, we get  the decimal approximation of  When we have a circle with radius given as a fraction, we can substitute  for  instead of  And, since  is also an approximation of  we will use the  symbol to show we have an approximate value.

### Key Concepts

1. Convert a Fraction to a Decimal To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.
2. Properties of Circles

 is the length of the radius
 is the length of the diameter
The circumference is .
The area is .

### Practice Makes Perfect

Convert Fractions to Decimals
In the following exercises, convert each fraction to a decimal.
In the following exercises, simplify the expression.
Order Decimals and Fractions
In the following exercises, order each pair of numbers, using  or
In the following exercises, write each set of numbers in order from least to greatest.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify.
Mixed Practice
In the following exercises, simplify. Give the answer as a decimal.
Find the Circumference and Area of Circles
In the following exercises, approximate the ⓐ circumference and ⓑ area of each circle. If measurements are given in fractions, leave answers in fraction form.

### 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?


# Decimals

## Solve Equations with Decimals

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

Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals.
Now that we’ve worked with decimals, we are ready to find solutions to equations involving decimals. 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, a fraction, or a decimal. We’ll list these steps here again for easy reference.

### Solve Equations with Decimals

In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.
When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

### Translate to an Equation and Solve

Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.

### Key Concepts

1. Determine whether a number is a solution to an equation.

2. Properties of Equality

### Practice Makes Perfect

Determine Whether a Decimal is a Solution of an Equation
In the following exercises, determine whether each number is a solution of the given equation.
Solve Equations with Decimals
In the following exercises, solve the equation.
Mixed Practice
In the following exercises, solve the equation. Then check your solution.
Translate to an Equation and Solve
In the following exercises, translate 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.

ⓑ 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?


# Decimals

## Averages and Probability

One application of decimals that arises often is finding the average of a set of numbers. What do you think of when you hear the word average? Is it your grade point average, the average rent for an apartment in your city, the batting average of a player on your favorite baseball team? The average is a typical value in a set of numerical data. Calculating an average sometimes involves working with decimal numbers. In this section, we will look at three different ways to calculate an average.

### Calculate the Mean of a Set of Numbers

The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. Students want to know the mean of their test scores. Climatologists report that the mean temperature has, or has not, changed. City planners are interested in the mean household size.
Suppose Ethan’s first three test scores were  To find the mean score, he would add them and divide by
His mean test score is  points.
Did you notice that in the last example, while all the numbers were whole numbers, the mean was  a number with one decimal place? It is customary to report the mean to one more decimal place than the original numbers. In the next example, all the numbers represent money, and it will make sense to report the mean in dollars and cents.

### Find the Median of a Set of Numbers

When Ann, Bianca, Dora, Eve, and Francine sing together on stage, they line up in order of their heights. Their heights, in inches, are shown in .
Dora is in the middle of the group. Her height,  is the median of the girls’ heights. Half of the heights are less than or equal to Dora’s height, and half are greater than or equal. The median is the middle value.
What if Carmen, the pianist, joins the singing group on stage? Carmen is  inches tall, so she fits in the height order between Bianca and Dora. Now the data set looks like this:
There is no single middle value. The heights of the six girls can be divided into two equal parts.
Statisticians have agreed that in cases like this the median is the mean of the two values closest to the middle. So the median is the mean of  The median height is  inches.
Notice that when the number of girls was  the median was the third height, but when the number of girls was  the median was the mean of the third and fourth heights. In general, when the number of values is odd, the median will be the one value in the middle, but when the number is even, the median is the mean of the two middle values.

### Identify the Mode of a Set of Numbers

The average is one number in a set of numbers that is somehow typical of the whole set of numbers. The mean and median are both often called the average.  Yes, it can be confusing when the word average refers to two different numbers, the mean and the median!  In fact, there is a third number that is also an average.  This average is the mode. The mode of a set of numbers is the number that occurs the most. The frequency, is the number of times a number occurs. So the mode of a set of numbers is the number with the highest frequency.
Suppose Jolene kept track of the number of miles she ran since the start of the month, as shown in .
If we list the numbers in order it is easier to identify the one with the highest frequency.
Jolene ran  miles three times, and every other distance is listed only once. So the mode of the data is  miles.
Some data sets do not have a mode because no value appears more than any other. And some data sets have more than one mode. In a given set, if two or more data values have the same highest frequency, we say they are all modes.

### Use the Basic Definition of Probability

The probability of an event tells us how likely that event is to occur. We usually write probabilities as fractions or decimals.
For example, picture a fruit bowl that contains five pieces of fruit - three bananas and two apples.
If you want to choose one piece of fruit to eat for a snack and don’t care what it is, there is a  probability you will choose a banana, because there are three bananas out of the total of five pieces of fruit. The probability of an event is the number of favorable outcomes divided by the total number of outcomes.
Converting the fraction  to a decimal, we would say there is a  probability of choosing a banana.
This basic definition of probability assumes that all the outcomes are equally likely to occur. If you study probabilities in a later math class, you’ll learn about several other ways to calculate probabilities.

### Key Concepts

1. Calculate the mean of a set of numbers.

2. Find the median of a set of numbers.

3. Identify the mode of a set of numbers.

### Practice Makes Perfect

Calculate the Mean of a Set of Numbers
In the following exercises, find the mean.
Find the Median of a Set of Numbers
In the following exercises, find the median.
Identify the Mode of a Set of Numbers
In the following exercises, identify the mode.
Use the Basic Definition of Probability
In the following exercises, express the probability as both a fraction and a decimal. (Round to three decimal places, 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?


# Decimals

## Ratios and Rate

### Write a Ratio as a Fraction

When you apply for a mortgage, the loan officer will compare your total debt to your total income to decide if you qualify for the loan. This comparison is called the debt-to-income ratio. A ratio compares two quantities that are measured with the same unit. If we compare  and , the ratio is written as
In this section, we will use the fraction notation. When a ratio is written in fraction form, the fraction should be simplified. If it is an improper fraction, we do not change it to a mixed number. Because a ratio compares two quantities, we would leave a ratio as  instead of simplifying it to  so that we can see the two parts of the ratio.

### Ratios Involving Decimals

We will often work with ratios of decimals, especially when we have ratios involving money. In these cases, we can eliminate the decimals by using the Equivalent Fractions Property to convert the ratio to a fraction with whole numbers in the numerator and denominator.
For example, consider the ratio  We can write it as a fraction with decimals and then multiply the numerator and denominator by  to eliminate the decimals.
Do you see a shortcut to find the equivalent fraction? Notice that  and  The least common denominator of  and  is  By multiplying the numerator and denominator of  by  we ‘moved’ the decimal two places to the right to get the equivalent fraction with no decimals. Now that we understand the math behind the process, we can find the fraction with no decimals like this:
You do not have to write out every step when you multiply the numerator and denominator by powers of ten. As long as you move both decimal places the same number of places, the ratio will remain the same.
Some ratios compare two mixed numbers. Remember that to divide mixed numbers, you first rewrite them as improper fractions.

### Applications of Ratios

One real-world application of ratios that affects many people involves measuring cholesterol in blood. The ratio of total cholesterol to HDL cholesterol is one way doctors assess a person's overall health. A ratio of less than  to  is considered good.

### Ratios of Two Measurements in Different Units

To find the ratio of two measurements, we must make sure the quantities have been measured with the same unit. If the measurements are not in the same units, we must first convert them to the same units.
We know that to simplify a fraction, we divide out common factors. Similarly in a ratio of measurements, we divide out the common unit.

### Write a Rate as a Fraction

Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. Examples of rates are  miles in  hours,  words in  minutes, and  dollars per  ounces.
When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator.

### Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of  This tells us that every three hours, Bob will travel  miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of  unit is referred to as a unit rate.
Unit rates are very common in our lives. For example, when we say that we are driving at a speed of  miles per hour we mean that we travel  miles in  hour. We would write this rate as  miles/hour (read  miles per hour). The common abbreviation for this is  mph. Note that when no number is written before a unit, it is assumed to be
So  miles/hour really means
Two rates we often use when driving can be written in different forms, as shown:
Another example of unit rate that you may already know about is hourly pay rate. It is usually expressed as the amount of money earned for one hour of work. For example, if you are paid  for each hour you work, you could write that your hourly (unit) pay rate is  (read  per hour.)
To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of

### Find Unit Price

Sometimes we buy common household items ‘in bulk’, where several items are packaged together and sold for one price. To compare the prices of different sized packages, we need to find the unit price. To find the unit price, divide the total price by the number of items. A unit price is a unit rate for one item.
Unit prices are very useful if you comparison shop. The better buy is the item with the lower unit price. Most grocery stores list the unit price of each item on the shelves.
Notice in  that we rounded the unit price to the nearest cent. Sometimes we may need to carry the division to one more place to see the difference between the unit prices.

### Translate Phrases to Expressions with Fractions

Have you noticed that the examples in this section used the comparison words ratio of, to, per, in, for, on, and from? When you translate phrases that include these words, you should think either ratio or rate. If the units measure the same quantity (length, time, etc.), you have a ratio. If the units are different, you have a rate. In both cases, you write a fraction.

### Practice Makes Perfect

Write a Ratio as a Fraction
In the following exercises, write each ratio as a fraction.
Write a Rate as a Fraction
In the following exercises, write each rate as a fraction.
Find Unit Rates
In the following exercises, find the unit rate. Round to two decimal places, if necessary.
Find Unit Price
In the following exercises, find the unit price. Round to the nearest cent.
In the following exercises, find each unit price and then identify the better buy. Round to three decimal places.
Translate Phrases to Expressions with Fractions
In the following exercises, translate the English phrase into an algebraic expression.

### 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?


# Decimals

## Simplify and Use Square Roots

### Simplify Expressions with Square Roots

To start this section, we need to review some important vocabulary and notation.
Remember that when a number  is multiplied by itself, we can write this as  which we read aloud as  For example,  is read as
We call  the square of  because  Similarly,  is the square of  because

### Modeling Squares

Do you know why we use the word square? If we construct a square with three tiles on each side, the total number of tiles would be nine.
This is why we say that the square of three is nine.
The number  is called a perfect square because it is the square of a whole number.
The chart shows the squares of the counting numbers  through  You can refer to it to help you identify the perfect squares.
What happens when you square a negative number?
When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.
The chart shows the squares of the negative integers from  to
Did you notice that these squares are the same as the squares of the positive numbers?

### Square Roots

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because  we say  is the square of  We can also say that  is a square root of
Notice  also, so  is also a square root of  Therefore, both  and  are square roots of
So, every positive number has two square roots: one positive and one negative.
What if we only want the positive square root of a positive number? The radical sign,  stands for the positive square root. The positive square root is also called the principal square root.
We can also use the radical sign for the square root of zero. Because  Notice that zero has only one square root.
The chart shows the square roots of the first  perfect square numbers.
Every positive number has two square roots and the radical sign indicates the positive one. We write  If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example,

### Square Root of a Negative Number

Can we simplify  Is there a number whose square is
None of the numbers that we have dealt with so far have a square that is  Why? Any positive number squared is positive, and any negative number squared is also positive. In the next chapter we will see that all the numbers we work with are called the real numbers. So we say there is no real number equal to  If we are asked to find the square root of any negative number, we say that the solution is not a real number.

### Square Roots and the Order of Operations

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.
Notice the different answers in parts ⓐ and ⓑ of . It is important to follow the order of operations correctly. In ⓐ , we took each square root first and then added them. In ⓑ , we added under the radical sign first and then found the square root.

### Estimate Square Roots

So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.
We might conclude that the square roots of numbers between  and  will be between  and  and they will not be whole numbers. Based on the pattern in the table above, we could say that  is between  and  Using inequality symbols, we write

### Approximate Square Roots with a Calculator

There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the  or  key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. The symbol for an approximation is  and it is read approximately.
Suppose your calculator has a  display. Using it to find the square root of  will give  This is the approximate square root of  When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.
You will seldom use this many digits for applications in algebra. So, if you wanted to round  to two decimal places, you would write
How do we know these values are approximations and not the exact values? Look at what happens when we square them.
The squares are close, but not exactly equal, to

### Simplify Variable Expressions with Square Roots

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?
Consider  where  Can you think of an expression whose square is
When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.

### Use Square Roots in Applications

As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!

### Square Roots and Area

We have solved applications with area before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.
If the area of the square is  square units, the length of a side is  units. See .

### Square Roots and Gravity

Another application of square roots involves gravity. On Earth, if an object is dropped from a height of  feet, the time in seconds it will take to reach the ground is found by evaluating the expression  For example, if an object is dropped from a height of  feet, we can find the time it takes to reach the ground by evaluating

It would take  seconds for an object dropped from a height of  feet to reach the ground.

### Square Roots and Accident Investigations

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is  feet, then the speed of the car can be found by evaluating

### Key Concepts

1. Square Root Notation  is read ‘the square root of ’
If , then , for .

2. Use a strategy for applications with square roots.

### Section Exercises

### Practice Makes Perfect

Simplify Expressions with Square Roots
In the following exercises, simplify.
Estimate Square Roots
In the following exercises, estimate each square root between two consecutive whole numbers.
Approximate Square Roots with a Calculator
In the following exercises, use a calculator to approximate each square root and round to two decimal places.
Simplify Variable Expressions with Square Roots
In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)
Use Square Roots in Applications
In the following exercises, solve. Round to one decimal place.

### 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

### Decimals

Name Decimals
In the following exercises, name each decimal.
Write Decimals
In the following exercises, write as a decimal.
Convert Decimals to Fractions or Mixed Numbers
In the following exercises, convert each decimal to a fraction. Simplify the answer if possible.
Locate Decimals on the Number Line
Order Decimals
In the following exercises, order each of the following pairs of numbers, using  or
Round Decimals
In the following exercises, round each number to the nearest: ⓐ hundredth ⓑ tenth ⓒ whole number.

### Decimal Operations

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.
Use Decimals in Money Applications
In the following exercises, use the strategy for applications to solve.

### Decimals and Fractions

Convert Fractions to Decimals
In the following exercises, convert each fraction to a decimal.
Order Decimals and Fractions
In the following exercises, order each pair of numbers, using  or
In the following exercises, write each set of numbers in order from least to greatest.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify
Find the Circumference and Area of Circles
In the following exercises, approximate the ⓐ circumference and ⓑ area of each circle.

### Solve Equations with Decimals

Determine Whether a Decimal is a Solution of an Equation
In the following exercises, determine whether the each number is a solution of the given equation.
Solve Equations with Decimals
In the following exercises, solve.
Translate to an Equation and Solve
In the following exercises, translate and solve.

### Averages and Probability

Find the Mean of a Set of Numbers
In the following exercises, find the mean of the numbers.
Find the Median of a Set of Numbers
In the following exercises, find the median.
Find the Mode of a Set of Numbers
In the following exercises, identify the mode of the numbers.
Use the Basic Definition of Probability
In the following exercises, solve. (Round decimals to three places.)

### Ratios and Rate

Write a Ratio as a Fraction
In the following exercises, write each ratio as a fraction. Simplify the answer if possible.
Write a Rate as a Fraction
In the following exercises, write each rate as a fraction. Simplify the answer if possible.
Find Unit Rates
In the following exercises, find the unit rate.
Find Unit Price
In the following exercises, find the unit price.
In the following exercises, find each unit price and then determine the better buy.
Translate Phrases to Expressions with Fractions
In the following exercises, translate the English phrase into an algebraic expression.

### Simplify and Use Square Roots

Simplify Expressions with Square Roots
In the following exercises, simplify.
Estimate Square Roots
In the following exercises, estimate each square root between two consecutive whole numbers.
Approximate Square Roots
In the following exercises, approximate each square root and round to two decimal places.
Simplify Variable Expressions with Square Roots
In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)
Use Square Roots in Applications
In the following exercises, solve. Round to one decimal place.

### Chapter Practice Test

In the following exercises, simplify each expression.
In the following exercises, solve.
In the following exercises, simplify.


# Percents

## Introduction to Percents

When you deposit money in a savings account at a bank, it earns additional money. Figuring out how your money will grow involves understanding and applying concepts of percents. In this chapter, we will find out what percents are and how we can use them to solve problems.


# Percents

## Understand Percent

### Use the Definition of Percent

How many cents are in one dollar? There are  cents in a dollar. How many years are in a century? There are  years in a century. Does this give you a clue about what the word “percent” means? It is really two words, “per cent,” and means per one hundred. A percent is a ratio whose denominator is  We use the percent symbol  to show percent.
According to data from the American Association of Community Colleges  about  of community college students are female. This means  out of every  community college students are female, as  shows. Out of the  squares on the grid,  are shaded, which we write as the ratio
Similarly,  means a ratio of  means a ratio of  and  means a ratio of  In words, "one hundred percent" means the total  is  and since  we see that  means  whole.

### Convert Percents to Fractions and Decimals

Since percents are ratios, they can easily be expressed as fractions. Remember that percent means per  so the denominator of the fraction is
The previous example shows that a percent can be greater than  We saw that  means  or  These are improper fractions, and their values are greater than one.
In Decimals, we learned how to convert fractions to decimals. To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal.
Let's summarize the results from the previous examples in , and look for a pattern we could use to quickly convert a percent number to a decimal number.
Do you see the pattern?
To convert a percent number to a decimal number, we move the decimal point two places to the left and remove the  sign. (Sometimes the decimal point does not appear in the percent number, but just like we can think of the integer  as  we can think of  as ) Notice that we may need to add zeros in front of the number when moving the decimal to the left.
 uses the percents in  and shows visually how to convert them to decimals by moving the decimal point two places to the left.

### Convert Decimals and Fractions to Percents

To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is  it is easy to change that fraction to a percent.
To convert a mixed number to a percent, we first write it as an improper fraction.
Let's summarize the results from the previous examples in  so we can look for a pattern.
Do you see the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.
 uses the decimal numbers in  and shows visually to convert them to percents by moving the decimal point two places to the right and then writing the  sign.
In Decimals, we learned how to convert fractions to decimals. Now we also know how to change decimals to percents. So to convert a fraction to a percent, we first change it to a decimal and then convert that decimal to a percent.
Sometimes when changing a fraction to a decimal, the division continues for many decimal places and we will round off the quotient. The number of decimal places we round to will depend on the situation. If the decimal involves money, we round to the hundredths place. For most other cases in this book we will round the number to the nearest thousandth, so the percent will be rounded to the nearest tenth.
When we first looked at fractions and decimals, we saw that some fractions converted to a repeating decimal. For example, when we converted the fraction  to a decimal, we wrote the answer as  We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.

### Key Concepts

1. Convert a percent to a fraction.

2.
Convert a percent to a decimal.

3.
Convert a decimal to a percent.

4.
Convert a fraction to a percent.

### Practice Makes Perfect

Use the Definition of Percents
In the following exercises, write each percent as a ratio.
In the following exercises, write as

Convert Percents to Fractions and Decimals
In the following exercises, convert each percent to a fraction and simplify all fractions.
In the following exercises, convert each percent to a decimal.
In the following exercises, convert each percent to

Convert Decimals and Fractions to Percents
In the following exercises, convert each decimal to a percent.
In the following exercises, convert each fraction to a percent.
In the following exercises, convert each fraction to a percent.
In the following exercises, convert each fraction to a percent.
In the following exercises, complete the table.

### Everyday Math

Amelio is shopping for textbooks online. He found three sellers that are offering a book he needs for the same price, including shipping. To decide which seller to buy from he is comparing their customer satisfaction ratings. The ratings are given in the chart.
Use the chart to answer the following questions

### 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.


# Percents

## Solve General Applications of Percent

### Translate and Solve Basic Percent Equations

We will solve percent equations by using the methods we used to solve equations with fractions or decimals. In the past, you may have solved percent problems by setting them up as proportions. That was the best method available when you did not have the tools of algebra. Now as a prealgebra student, you can translate word sentences into algebraic equations, and then solve the equations.
We'll look at a common application of percent—tips to a server at a restaurant—to see how to set up a basic percent application.
When Aolani and her friends ate dinner at a restaurant, the bill came to  They wanted to leave a  tip. What amount would the tip be?
To solve this, we want to find what amount is  of  The  is called the base. The amount of the tip would be  or  See . To find the amount of the tip, we multiplied the percent by the base.
In the next examples, we will find the amount. We must be sure to change the given percent to a decimal when we translate the words into an equation.
In the next examples, we are asked to find the base.
In the next examples, we will solve for the percent.

### Solve Applications of Percent

Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we'll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.
We will update the strategy we used in our earlier applications to include equations now. Notice that we will translate a sentence into an equation.
Now that we have the strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we'll solve involve everyday situations, you can rely on your own experience.

### Find Percent Increase and Percent Decrease

People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent.
To find the percent increase, first we find the amount of increase, which is the difference between the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.
Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference between the original amount and the final amount. Then we find what percent the amount of decrease is of the original amount.

### Key Concepts

1. Solve an application.

2. Find percent increase.

3. Find percent decrease.

### Practice Makes Perfect

Translate and Solve Basic Percent Equations
In the following exercises, translate and solve.
Solve Applications of Percents
In the following exercises, solve the applications of percents.
Find Percent Increase and Percent Decrease
In the following exercises, find the percent increase or percent decrease.

### 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?


# Percents

## Solve Sales Tax, Commission, and Discount Applications

### Solve Sales Tax Applications

Sales tax and commissions are applications of percent in our everyday lives. To solve these applications, we will follow the same strategy we used in the section on decimal operations. We show it again here for easy reference.
Remember that whatever the application, once we write the sentence with the given information (Step 2), we can translate it to a percent equation and then solve it.
Do you pay a tax when you shop in your city or state? In many parts of the United States, sales tax is added to the purchase price of an item. See . The sales tax is determined by computing a percent of the purchase price.
To find the sales tax multiply the purchase price by the sales tax rate. Remember to convert the sales tax rate from a percent to a decimal number. Once the sales tax is calculated, it is added to the purchase price. The result is the total cost—this is what the customer pays.

### Solve Commission Applications

Sales people often receive a commission, or percent of total sales, for their sales. Their income may be just the commission they earn, or it may be their commission added to their hourly wages or salary. The commission they earn is calculated as a certain percent of the price of each item they sell. That percent is called the rate of commission.
To find the commission on a sale, multiply the rate of commission by the total sales. Just as we did for computing sales tax, remember to first convert the rate of commission from a percent to a decimal.

### Solve Discount Applications

Applications of discount are very common in retail settings . When you buy an item on sale, the original price of the item has been reduced by some dollar amount. The discount rate, usually given as a percent, is used to determine the amount of the discount. To determine the amount of discount, we multiply the discount rate by the original price. We summarize the discount model in the box below.
The sale price should always be less than the original price. In some cases, the amount of discount is a fixed dollar amount. Then we just find the sale price by subtracting the amount of discount from the original price.
In , the amount of discount was a set amount,  In  the discount is given as a percent of the original price.
There may be times when you buy something on sale and want to know the discount rate. The next example will show this case.

### Solve Mark-up Applications

Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the wholesale price. The retailer then adds a mark-up to the wholesale price to get the list price, the price he sells the item for. The mark-up is usually calculated as a percent of the wholesale price. The percent is called the mark-up rate. To determine the amount of mark-up, multiply the mark-up rate by the wholesale price. We summarize the mark-up model in the box below.
The list price should always be more than the wholesale price.

### Key Concepts

1. Sales Tax The sales tax is a percent of the purchase price.

2. Commission A commission is a percentage of total sales as determined by the rate of commission.

3. Discount An amount of discount is a percent off the original price, determined by the discount rate.

4. Mark-up The mark-up is the amount added to the wholesale price, determined by the mark-up rate.

### Practice Makes Perfect

Solve Sales Tax Applications
In the following exercises, find ⓐ the sales tax and ⓑ the total cost.
In the following exercises, find the sales tax rate.
Solve Commission Applications
In the following exercises, find the commission.
In the following exercises, find the rate of commission.
Solve Discount Applications
In the following exercises, find the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent if needed.)
Solve Mark-up Applications
In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

### 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?


# Percents

## Solve Simple Interest Applications

### Use the Simple Interest Formula

Do you know that banks pay you to let them keep your money? The money you put in the bank is called the principal,  and the bank pays you interest,  The interest is computed as a certain percent of the principal; called the rate of interest,  The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable for time,  represents the number of years the money is left in the account.
The formula we use to calculate simple interest is  To use the simple interest formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information by listing all four variables and filling in the given information.
In the next example, we will use the simple interest formula to find the principal.
Now we will solve for the rate of interest.

### Solve Simple Interest Applications

Applications with simple interest usually involve either investing money or borrowing money. To solve these applications, we continue to use the same strategy for applications that we have used earlier in this chapter. The only difference is that in place of translating to get an equation, we can use the simple interest formula.
We will start by solving a simple interest application to find the interest.
There may be times when you know the amount of interest earned on a given principal over a certain length of time, but you don't know the rate. For instance, this might happen when family members lend or borrow money among themselves instead of dealing with a bank. In the next example, we'll show how to solve for the rate.
There may be times when you take a loan for a large purchase and the amount of the principal is not clear. This might happen, for instance, in making a car purchase when the dealer adds the cost of a warranty to the price of the car. In the next example, we will solve a simple interest application for the principal.
In the simple interest formula, the rate of interest is given as an annual rate, the rate for one year. So the units of time must be in years. If the time is given in months, we convert it to years.

### Key Concepts

1. Simple interest

### Practice Makes Perfect

Use the Simple Interest Formula
In the following exercises, use the simple interest formula to fill in the missing information.
In the following exercises, solve the problem using the simple interest formula.
Solve Simple Interest Applications
In the following exercises, solve the problem using the simple interest formula.

### 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?


# Percents

## Solve Proportions and their Applications

### Use the Definition of Proportion

In the section on Ratios and Rates we saw some ways they are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a proportion.
The equation  is a proportion because the two fractions are equal. The proportion  is read  is to  as  is to
If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion  we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.
Look at the proportions  and  From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?
To determine if a proportion is true, we find the cross products of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross product because of the cross formed. If, and only if, the given proportion is true, that is, the two sides are equal, then the cross products of a proportion will be equal.
Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are both equal, we have a proportion.

### Solve Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.
When the variable is in a denominator, we’ll use the fact that the cross products of a proportion are equal to solve the proportions.
We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.

### Solve Applications Using Proportions

The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the proportion, we must make sure the units are correct—the units in the numerators match and the units in the denominators match.

### Write Percent Equations As Proportions

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.
For example,  and we can simplify  Since the equation  shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:
If we restate the problem in the words of a proportion, it may be easier to set up the proportion:
We could also say:
First we will practice translating into a percent proportion. Later, we’ll solve the proportion.

### Translate and Solve Percent Proportions

Now that we have written percent equations as proportions, we are ready to solve the equations.
In the next example, the percent is more than  which is more than one whole. So the unknown number will be more than the base.
Percents with decimals and money are also used in proportions.

### Key Concepts

1. Proportion

2. Cross Products of a Proportion

3. Percent Proportion

### Section Exercises

### Practice Makes Perfect

Use the Definition of Proportion
In the following exercises, write each sentence as a proportion.
In the following exercises, determine whether each equation is a proportion.
Solve Proportions
In the following exercises, solve each proportion.
Solve Applications Using Proportions
In the following exercises, solve the proportion problem.
Write Percent Equations as Proportions
In the following exercises, translate to a proportion.
Translate and Solve Percent Proportions
In the following exercises, translate and solve using proportions.

### 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

### Understand Percent

In the following exercises, write each percent as a ratio.
In the following exercises, write as a ratio and then as a percent.
In the following exercises, convert each percent to a fraction.
In the following exercises, convert each percent to a decimal.
In the following exercises, convert each percent to ⓐ a simplified fraction and ⓑ a decimal.
In the following exercises, convert each decimal to a percent.
In the following exercises, convert each fraction to a percent.
In the following exercises, translate and solve.

### Solve General Applications of Percents

In the following exercises, solve.
In the following exercises, solve.

### Solve Sales Tax, Commission, and Discount Applications

In the following exercises, find ⓐ the sales tax ⓑ the total cost.
In the following exercises, find the sales tax rate.
In the following exercises, find the commission.
In the following exercises, find the rate of commission.
In the following exercises, find the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.
In the following exercises, find ⓐ the amount of discount ⓑ the discount rate. (Round to the nearest tenth of a percent if needed.)
In the following exercises, find

### Solve Simple Interest Applications

In the following exercises, solve the simple interest problem.

### Solve Proportions and their Applications

In the following exercises, write each sentence as a proportion.
In the following exercises, determine whether each equation is a proportion.
In the following exercises, solve each proportion.
In the following exercises, solve the proportion problem.
In the following exercises, translate to a proportion.
In the following exercises, translate and solve using proportions.
In the following exercises, convert each percent to ⓐ a decimal ⓑ a simplified fraction.
In the following exercises, convert each fraction to a percent. (Round to  decimal places if needed.)
In the following exercises, solve the percent problem.


# The Properties of Real Numbers

## Introduction to the Properties of Real Numbers

A quilt is formed by sewing many different pieces of fabric together. The pieces can vary in color, size, and shape. The combinations of different kinds of pieces provide for an endless possibility of patterns. Much like the pieces of fabric, mathematicians distinguish among different types of numbers. The kinds of numbers in an expression provide for an endless possibility of outcomes. We have already described counting numbers, whole numbers, and integers. In this chapter, we will learn about other types of numbers and their properties.


# The Properties of Real Numbers

## Rational and Irrational Numbers

### Identify Rational Numbers and Irrational Numbers

Congratulations! You have completed the first six chapters of this book! It's time to take stock of what you have done so far in this course and think about what is ahead. You have learned how to add, subtract, multiply, and divide whole numbers, fractions, integers, and decimals. You have become familiar with the language and symbols of algebra, and have simplified and evaluated algebraic expressions. You have solved many different types of applications. You have established a good solid foundation that you need so you can be successful in algebra.
In this chapter, we'll make sure your skills are firmly set. We'll take another look at the kinds of numbers we have worked with in all previous chapters. We'll work with properties of numbers that will help you improve your number sense. And we'll practice using them in ways that we'll use when we solve equations and complete other procedures in algebra.
We have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?

### Rational Numbers

What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.
All fractions, both positive and negative, are rational numbers. A few examples are
Each numerator and each denominator is an integer.
We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.
Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.
Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.
What about decimals? Are they rational? Let's look at a few to see if we can write each of them as the ratio of two integers. We've already seen that integers are rational numbers. The integer  could be written as the decimal  So, clearly, some decimals are rational.
Think about the decimal  Can we write it as a ratio of two integers? Because  means  we can write it as an improper fraction,  So  is the ratio of the integers  and  It is a rational number.
In general, any decimal that ends after a number of digits (such as  or  is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction.
Let's look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number, since  for any integer,  We can also change any integer to a decimal by adding a decimal point and a zero.

We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we just considered.

What do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.

### Irrational Numbers

Are there any decimals that do not stop or repeat? Yes. The number  (the Greek letter pi, pronounced ‘pie’), which is very important in describing circles, has a decimal form that does not stop or repeat.
Similarly, the decimal representations of square roots of whole numbers that are not perfect squares never stop and never repeat. For example,
A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number.
Let's summarize a method we can use to determine whether a number is rational or irrational.
If the decimal form of a number
1. stops or repeats, the number is rational.
2. does not stop and does not repeat, the number is irrational.
Let's think about square roots now. Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.

### Classify Real Numbers

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.
 illustrates how the number sets are related.
Does the term “real numbers” seem strange to you? Are there any numbers that are not “real”, and, if so, what could they be? For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers. You won't encounter imaginary numbers in this course, but you will later on in your studies of algebra.

### Key Concepts

1. Real numbers

### Practice Makes Perfect

Rational Numbers
In the following exercises, write as the ratio of two integers.
In the following exercises, determine which of the given numbers are rational and which are irrational.
In the following exercises, identify whether each number is rational or irrational.
Classifying Real Numbers
In the following exercises, determine whether each number is whole, integer, rational, irrational, and real.

### 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.


# The Properties of Real Numbers

## Commutative and Associative Properties

In the next few sections, we will take a look at the properties of real numbers. Many of these properties will describe things you already know, but it will help to give names to the properties and define them formally. This way we’ll be able to refer to them and use them as we solve equations in the next chapter.

### Use the Commutative and Associative Properties

Think about adding two numbers, such as  and
The results are the same.
Notice, the order in which we add does not matter. The same is true when multiplying  and
Again, the results are the same!  The order in which we multiply does not matter.
These examples illustrate the commutative properties of addition and multiplication.
The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.
What about subtraction? Does order matter when we subtract numbers? Does  give the same result as
Since changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative.
Let’s see what happens when we divide two numbers. Is division commutative?
Since changing the order of the division did not give the same result, division is not commutative.
Addition and multiplication are commutative. Subtraction and division are not commutative.
Suppose you were asked to simplify this expression.
How would you do it and what would your answer be?
Some people would think  and then  Others might start with  and then
Both ways give the same result, as shown in . (Remember that parentheses are grouping symbols that indicate which operations should be done first.)
When adding three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition.
The same principle holds true for multiplication as well. Suppose we want to find the value of the following expression:
Changing the grouping of the numbers gives the same result, as shown in .
When multiplying three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Multiplication.
If we multiply three numbers, changing the grouping does not affect the product.
You probably know this, but the terminology may be new to you. These examples illustrate the Associative Properties.
Besides using the associative properties to make calculations easier, we will often use it to simplify expressions with variables.

### Evaluate Expressions using the Commutative and Associative Properties

The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate.
Let’s do one more, this time with multiplication.

### Simplify Expressions Using the Commutative and Associative Properties

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in  part ⓑ was easier to simplify than part ⓐ because the opposites were next to each other and their sum is  Likewise, part ⓑ in  was easier, with the reciprocals grouped together, because their product is  In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.
Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is
In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.
When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.
No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be. The next example will show you how using the Associative Property of Multiplication can make your work easier if you plan ahead.
When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples.
In The Language of Algebra, we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression  by rewriting it as  and then simplified it to  We were using the Commutative Property of Addition.

### Key Concepts

1. Commutative Properties
2. Associative Properties

### Practice Makes Perfect

Use the Commutative and Associative Properties
In the following exercises, use the commutative properties to rewrite the given expression.
In the following exercises, use the associative properties to rewrite the given expression.
Evaluate Expressions using the Commutative and Associative Properties
In the following exercises, evaluate each expression for the given value.
Simplify Expressions Using the Commutative and Associative Properties
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.

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


# The Properties of Real Numbers

## Distributive Property

### Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need  that is,  dollars and  quarter. How much money do they need all together? You can think about the dollars separately from the quarters.
They need  times  so  and  times  quarter, so  cents. In total, they need
If you think about doing the math in this way, you are using the Distributive Property.
Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:
In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression  the order of operations says to work in the parentheses first. But we cannot add  and  since they are not like terms. So we use the Distributive Property, as shown in .
Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in  would look like this:
The distributive property can be used to simplify expressions that look slightly different from  Here are two other forms.
Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.
Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.
In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.
The next example will use the ‘backwards’ form of the Distributive Property,
When you distribute a negative number, you need to be extra careful to get the signs correct.
In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember,
Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

### Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works.
In the examples below, we will practice evaluating some of the expressions from previous examples; in part ⓐ , we will evaluate the form with parentheses, and in part ⓑ we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

### Key Concepts

1. Distributive Property:

### Practice Makes Perfect

Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the distributive property.
Evaluate Expressions Using the Distributive Property
In the following exercises, evaluate both expressions for the given value.

### 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?


# The Properties of Real Numbers

## Properties of Identity, Inverses, and Zero

### Recognize the Identity Properties of Addition and Multiplication

What happens when we add zero to any number? Adding zero doesn’t change the value. For this reason, we call  the additive identity.
For example,
What happens when you multiply any number by one? Multiplying by one doesn’t change the value. So we call  the multiplicative identity.
For example,

### Use the Inverse Properties of Addition and Multiplication

Notice that in each case, the missing number was the opposite of the number.
We call  the additive inverse of  The opposite of a number is its additive inverse. A number and its opposite add to  which is the additive identity.
What number multiplied by  gives the multiplicative identity,  In other words, two-thirds times what results in
What number multiplied by  gives the multiplicative identity,  In other words two times what results in
Notice that in each case, the missing number was the reciprocal of the number.
We call  the multiplicative inverse of  The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to  which is the multiplicative identity.
We’ll formally state the Inverse Properties here:

### Use the Properties of Zero

We have already learned that zero is the additive identity, since it can be added to any number without changing the number’s identity. But zero also has some special properties when it comes to multiplication and division.

### Multiplication by Zero

What happens when you multiply a number by  Multiplying by  makes the product equal zero. The product of any real number and  is

### Dividing with Zero

What about dividing with  Think about a real example: if there are no cookies in the cookie jar and three people want to share them, how many cookies would each person get? There are  cookies to share, so each person gets  cookies.
Remember that we can always check division with the related multiplication fact. So, we know that
Zero divided by any real number except zero is zero.
Now let’s think about dividing a number by zero. What is the result of dividing  by  Think about the related multiplication fact. Is there a number that multiplied by  gives
Since any real number multiplied by  equals  there is no real number that can be multiplied by  to obtain  We can conclude that there is no answer to  and so we say that division by zero is undefined.
Division by zero is undefined.
We summarize the properties of zero.

### Simplify Expressions using the Properties of Identities, Inverses, and Zero

We will now practice using the properties of identities, inverses, and zero to simplify expressions.
All the properties of real numbers we have used in this chapter are summarized in .

### Key Concepts

1. Identity Properties
2. Inverse Properties
3. Properties of Zero

### Practice Makes Perfect

Recognize the Identity Properties of Addition and Multiplication
In the following exercises, identify whether each example is using the identity property of addition or multiplication.
Use the Inverse Properties of Addition and Multiplication
In the following exercises, find the multiplicative inverse.
Use the Properties of Zero
In the following exercises, simplify using the properties of zero.
Simplify Expressions using the Properties of Identities, Inverses, and Zero
In the following exercises, simplify using the properties of identities, inverses, and zero.

### 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?


# The Properties of Real Numbers

## Systems of Measurement

In this section we will see how to convert among different types of units, such as feet to miles or kilograms to pounds. The basic idea in all of the unit conversions will be to use a form of  the multiplicative identity, to change the units but not the value of a quantity.

### Make Unit Conversions in the U.S. System

There are two systems of measurement commonly used around the world. Most countries use the metric system. The United States uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.
The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, or hours.
The equivalencies among the basic units of the U.S. system of measurement are listed in . The table also shows, in parentheses, the common abbreviations for each measurement.
In many real-life applications, we need to convert between units of measurement. We will use the identity property of multiplication to do these conversions. We’ll restate the Identity Property of Multiplication here for easy reference.
To use the identity property of multiplication, we write  in a form that will help us convert the units. For example, suppose we want to convert inches to feet. We know that  foot is equal to  inches, so we can write  as the fraction  When we multiply by this fraction, we do not change the value but just change the units.
But  also equals  How do we decide whether to multiply by  or  We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert  inches to feet. If we choose the fraction that has inches in the denominator, we can eliminate the inches.
On the other hand, if we wanted to convert  feet to inches, we would choose the fraction that has feet in the denominator.
We treat the unit words like factors and ‘divide out’ common units like we do common factors.
When we use the Identity Property of Multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.
Sometimes to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

### Use Mixed Units of Measurement in the U.S. System

Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units.

### Make Unit Conversions in the Metric System

In the metric system, units are related by powers of  The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is  meters; the prefix kilo- means thousand. One centimeter is  of a meter, because the prefix centi- means one one-hundredth (just like one cent is  of one dollar).
The equivalencies of measurements in the metric system are shown in . The common abbreviations for each measurement are given in parentheses.
To make conversions in the metric system, we will use the same technique we did in the U.S. system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.
Have you ever run a  or  race? The lengths of those races are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.
Since the metric system is based on multiples of ten, conversions involve multiplying by multiples of ten. In Decimal Operations, we learned how to simplify these calculations by just moving the decimal.
To multiply by  we move the decimal to the right  places, respectively. To multiply by  we move the decimal to the left  places respectively.
We can apply this pattern when we make measurement conversions in the metric system.
In , we changed  grams to kilograms by multiplying by  This is the same as moving the decimal  places to the left.

### Use Mixed Units of Measurement in the Metric System

Performing arithmetic operations on measurements with mixed units of measures in the metric system requires the same care we used in the U.S. system. But it may be easier because of the relation of the units to the powers of  We still must make sure to add or subtract like units.

### Convert Between U.S. and Metric Systems of Measurement

Many measurements in the United States are made in metric units. A drink may come in  bottles, calcium may come in  capsules, and we may run a  race. To work easily in both systems, we need to be able to convert between the two systems.
 shows some of the most common conversions.
We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.
The conversion factors in  are not exact, but the approximations they give are close enough for everyday purposes. In , we rounded the number of fluid ounces to the nearest tenth.

### Convert Between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for  What does that mean?
The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written  The metric system uses degrees Celsius, written   shows the relationship between the two systems.
If we know the temperature in one system, we can use a formula to convert it to the other system.

### Section Exercises

### Practice Makes Perfect

Make Unit Conversions in the U.S. System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the U.S. System
In the following exercises, solve and write your answer in mixed units.
Make Unit Conversions in the Metric System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the Metric System
In the following exercises, solve and write your answer in mixed units.
Convert Between U.S. and Metric Systems
In the following exercises, make the unit conversions. Round to the nearest tenth.
Convert between Fahrenheit and Celsius
In the following exercises, convert the Fahrenheit temperature to degrees Celsius. Round to the nearest tenth.
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

### 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

### Rational and Irrational Numbers

In the following exercises, write as the ratio of two integers.
In the following exercises, determine which of the numbers is rational.
In the following exercises, identify whether each given number is rational or irrational.
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

### Commutative and Associative Properties

In the following exercises, use the commutative property to rewrite the given expression.
In the following exercises, use the associative property to rewrite the given expression.
In the following exercises, evaluate each expression for the given value.
In the following exercises, simplify using the commutative and associative properties.

### Distributive Property

In the following exercises, simplify using the distributive property.
In the following exercises, evaluate using the distributive property.

### Properties of Identities, Inverses, and Zero

In the following exercises, identify whether each example is using the identity property of addition or multiplication.
In the following exercises, find the additive inverse.
In the following exercises, find the multiplicative inverse.
In the following exercises, simplify.

### Systems of Measurement

In the following exercises, convert between U.S. units. Round to the nearest tenth.
In the following exercises, solve, and state your answer in mixed units.
In the following exercises, convert between metric units.
In the following exercises, solve.
In the following exercises, convert between U.S. and metric units. Round to the nearest tenth.
In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

### Chapter Practice Test

In the following exercises, simplify the given expression.
In the following exercises, solve using the appropriate unit conversions.


# Solving Linear Equations

## Introduction to Solving Linear Equations

Teetering high above the floor, this amazing mobile remains aloft thanks to its carefully balanced mass. Any shift in either direction could cause the mobile to become lopsided, or even crash downward. In this chapter, we will solve equations by keeping quantities on both sides of an equal sign in perfect balance.


# Solving Linear Equations

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

We are now ready to “get to the good stuff.” You have the basics down and are ready to begin one of the most important topics in algebra: solving equations. The applications are limitless and extend to all careers and fields. Also, the skills and techniques you learn here will help improve your critical thinking and problem-solving skills. This is a great benefit of studying mathematics and will be useful in your life in ways you may not see right now.

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

We began our work solving equations in previous chapters. It has been a while since we have seen an equation, so we will review some of the key concepts before we go any further.
We said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.
In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.
We introduced the Subtraction and Addition Properties of Equality in Solving Equations Using the Subtraction and
Addition Properties of Equality. In that section, we modeled how these properties work and then applied them to solving equations with whole numbers. We used these properties again each time we introduced a new system of numbers. Let’s review those properties here.
When you add or subtract the same quantity from both sides of an equation, you still have equality.
We introduced the Subtraction Property of Equality earlier by modeling equations with envelopes and counters.  models the equation
The goal is to isolate the variable on one side of the equation. So we ‘took away’  from both sides of the equation and found the solution
Some people picture a balance scale, as in , when they solve equations.
The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.
Let’s review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.
In the original equation in the previous example,  was added to the , so we subtracted  to ‘undo’ the addition. In the next example, we will need to ‘undo’ subtraction by using the Addition Property of Equality.
Now let’s review solving equations with fractions.
In Solve Equations with Decimals, we solved equations that contained decimals. We’ll review this next.

### Solve Equations That Need to Be Simplified

In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.

### Translate an Equation and Solve

In previous chapters, we translated word sentences into equations. The first step is to look for the word (or words) that translate(s) to the equal sign.  reminds us of some of the words that translate to the equal sign.
Let’s review the steps we used to translate a sentence into an equation.
Now we are ready to try an example.

### Translate and Solve Applications

In most of the application problems we solved earlier, we were able to find the quantity we were looking for by simplifying an algebraic expression. Now we will be using equations to solve application problems. We’ll start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for.

### Key Concepts

1. Determine whether a number is a solution to an equation.
If it is true, the number is a solution.
If it is not true, the number is not a solution.

2. Subtraction and Addition Properties of Equality
3. Translate a word sentence to an algebraic equation.
4. Problem-solving strategy

### Practice Makes Perfect

Solve Equations Using the Subtraction and Addition Properties of Equality
In the following exercises, determine whether the given value is a solution to the equation.
In the following exercises, solve each equation.
Solve Equations that Need to be Simplified
In the following exercises, solve each equation.
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve.
Translate and Solve Applications
In the following exercises, translate into an 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.

ⓑ 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.


# Solving Linear Equations

## Solve Equations Using the Division and Multiplication Properties of Equality

### Solve Equations Using the Division and Multiplication Properties of Equality

We introduced the Multiplication and Division Properties of Equality in Solve Equations Using Integers; The Division Property of Equality and Solve Equations with Fractions. We modeled how these properties worked using envelopes and counters and then applied them to solving equations (See Solve Equations Using Integers; The Division Property of Equality). We restate them again here as we prepare to use these properties again.
When you divide or multiply both sides of an equation by the same quantity, you still have equality.
Let’s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to ‘undo’ the operation on the variable. In the example below the variable is multiplied by  so we will divide both sides by  to ‘undo’ the multiplication.
In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?

### Solve Equations That Need to be Simplified

Many equations start out more complicated than the ones we’ve just solved. First, we need to simplify both sides of the equation as much as possible

### Key Concepts

1. Division and Multiplication Properties of Equality

### Practice Makes Perfect

Solve Equations Using the Division and Multiplication Properties of Equality
In the following exercises, solve each equation for the variable using the Division Property of Equality and check the solution.
In the following exercises, solve each equation for the variable using the Multiplication Property of Equality and check the solution.
Solve Equations That Need to be Simplified
In the following exercises, solve the equation.

### 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?


# Solving Linear Equations

## Solve Equations with Variables and Constants on Both Sides

### Solve an Equation with Constants on Both Sides

You may have noticed that in all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we’ll see how to solve equations where the variable terms and/or constant terms are on both sides of the equation.
Our strategy will involve choosing one side of the equation to be the variable side, and the other side of the equation to be the constant side. Then, we will use the Subtraction and Addition Properties of Equality, step by step, to get all the variable terms together on one side of the equation and the constant terms together on the other side.
By doing this, we will transform the equation that started with variables and constants on both sides into the form  We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.

### Solve an Equation with Variables on Both Sides

What if there are variables on both sides of the equation? We will start like we did above—choosing a variable side and a constant side, and then use the Subtraction and Addition Properties of Equality to collect all variables on one side and all constants on the other side. Remember, what you do to the left side of the equation, you must do to the right side too.

### Solve Equations with Variables and Constants on Both Sides

The next example will be the first to have variables and constants on both sides of the equation. As we did before, we’ll collect the variable terms to one side and the constants to the other side.
We’ll summarize the steps we took so you can easily refer to them.
It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.
To solve an equation with fractions, we still follow the same steps to get the solution.
We follow the same steps when the equation has decimals, too.

### Solve Equations Using a General Strategy

Each of the first few sections of this chapter has dealt with solving one specific form of a linear equation. It’s time now to lay out an overall strategy that can be used to solve any linear equation. We call this the general strategy. Some equations won’t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.
In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.

### Key Concepts

1. Solve an equation with variables and constants on both sides
2. General strategy for solving linear equations

### Practice Makes Perfect

Solve an Equation with Constants on Both Sides
In the following exercises, solve the equation for the variable.
Solve an Equation with Variables on Both Sides
In the following exercises, solve the equation for the variable.
Solve an Equation with Variables and Constants on Both Sides
In the following exercises, solve the equations for the variable.
Solve an Equation Using the General Strategy
In the following exercises, solve the linear equation using the general strategy.

### 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?


# Solving Linear Equations

## Solve Equations with Fraction or Decimal Coefficients

### Solve Equations with Fraction Coefficients

Let’s use the General Strategy for Solving Linear Equations introduced earlier to solve the equation
This method worked fine, but many students don’t feel very confident when they see all those fractions. So we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.
We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but with no fractions. This process is called clearing the equation of fractions. Let’s solve the same equation again, but this time use the method that clears the fractions.
Notice in  that once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.
In the next example, we’ll have variables and fractions on both sides of the equation.
In , we’ll start by using the Distributive Property. This step will clear the fractions right away!
Many times, there will still be fractions, even after distributing.

### Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money and percent. But decimals are really another way to represent fractions. For example,  and  So, when we have an equation with decimals, we can use the same process we used to clear fractions—multiply both sides of the equation by the least common denominator.
The next example uses an equation that is typical of the ones we will see in the money applications in the next chapter. Notice that we will distribute the decimal first before we clear all decimals in the equation.

### Key Concepts

1. Solve equations with fraction coefficients by clearing the fractions.

### Section Exercises

### Practice Makes Perfect

Solve equations with fraction coefficients
In the following exercises, solve the equation by clearing the fractions.
Solve Equations with Decimal Coefficients
In the following exercises, solve the equation by clearing the decimals.

### 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

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

In the following exercises, determine whether the given number is a solution to the equation.
In the following exercises, solve the equation using the Subtraction Property of Equality.
In the following exercises, solve the equation using the Addition Property of Equality.
In the following exercises, solve the equation.
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
In the following exercises, translate into an algebraic equation and solve.

### Solve Equations using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the Division Property of Equality.
In the following exercises, solve each equation using the Multiplication Property of Equality.
In the following exercises, solve each equation.

### Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the equations with constants on both sides.
In the following exercises, solve the equations with variables on both sides.
In the following exercises, solve the equations with constants and variables on both sides.
In the following exercises, solve each linear equation using the general strategy.

### Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation by clearing the fractions.
In the following exercises, solve each equation by clearing the decimals.

### Chapter Practice Test

In the following exercises, solve each equation.