CHAPTER 2 Operations with Rational Numbers and Introduction to Real Numbers

2.1 Visualize Fractions

Learning Objectives

By the end of this section, you will be able to:

• Find equivalent fractions
• Simplify fractions
• Multiply fractions
• Divide fractions
• Simplify expressions written with a fraction bar
• Translate phrases to expressions with fractions

Find Equivalent Fractions

Fractions are a way to represent parts of a whole. The fraction means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See (Figure 1). The fraction represents two of three equal parts. In the fraction , the 2 is called the numerator and the 3 is called the denominator.

The circle on the left has been divided into 3 equal parts. Each part is of the 3 equal parts. In the circle on the right, of the circle is shaded (2 of the 3 equal parts).

Fraction

A fraction is written , where and

• a is the numerator and b is the denominator.

A fraction represents parts of a whole. The denominator b is the number of equal parts the whole has been divided into, and the numerator a indicates how many parts are included.

If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate pieces, or, in other words, one whole pie.

So . This leads us to the property of one that tells us that any number, except zero, divided by itself is 1

Property of One

Any number, except zero, divided by itself is one.

If a pie was cut in pieces and we ate all 6, we ate pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate pieces, or one whole pie. We ate the same amount—one whole pie.

The fractions and have the same value, 1, and so they are called equivalent fractions. Equivalent fractions are fractions that have the same value.

Let’s think of pizzas this time. (Figure 2) shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that is equivalent to . In other words, they are equivalent fractions.

Since the same amount is of each pizza is shaded, we see that is equivalent to . They are equivalent fractions.

Equivalent Fractions

Equivalent fractions are fractions that have the same value.

How can we use mathematics to change into How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into pieces instead of just 2. Mathematically, what we’ve described could be written like this as . See (Figure 3).

Cutting each half of the pizza into pieces, gives us pizza cut into 8 pieces: .

This model leads to the following property:

Equivalent Fractions Property

If are numbers where , then

If we had cut the pizza differently, we could get

So, we say are equivalent fractions.

EXAMPLE 1

Find three fractions equivalent to .

Solution

To find a fraction equivalent to , we multiply the numerator and denominator by the same number. We can choose any number, except for zero. Let’s multiply them by 2, 3, and then 5.

So, are equivalent to .

TRY IT 1.1

Find three fractions equivalent to .

TRY IT 1.2

Find three fractions equivalent to .

Simplify Fractions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.

For example,

• is simplified because there are no common factors of 2 and 3.
• is not simplified because is a common factor of 10 and 15.

Simplified Fraction

A fraction is considered simplified if there are no common factors in its numerator and denominator.

The phrase reduce a fraction means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

In Example 1, we used the equivalent fractions property to find equivalent fractions. Now we’ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.

Equivalent Fractions Property

If are numbers where ,

EXAMPLE 2

Simplify: .

Solution
 Rewrite the numerator and denominator showing the common factors. Simplify using the equivalent fractions property.

Notice that the fraction is simplified because there are no more common factors.

TRY IT 2.1

Simplify: .

TRY IT 2.2

Simplify: .

Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the equivalent fractions property.

EXAMPLE 3

How to Simplify a Fraction

Simplify: .

Solution

TRY IT 3.1

Simplify: .

TRY IT 3.2

Simplify: .

We now summarize the steps you should follow to simplify fractions.

HOW TO: Simplify a Fraction

1. Rewrite the numerator and denominator to show the common factors.
If needed, factor the numerator and denominator into prime numbers first.
2. Simplify using the equivalent fractions property by dividing out common factors.
3. Multiply any remaining factors, if needed.

EXAMPLE 4

Simplify: .

Solution
 Rewrite showing the common factors, then divide out the common factors. Simplify.

TRY IT 4.1

Simplify: .

TRY IT 4.2

Simplify: .

Multiply Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.

We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with .

Now we’ll take of .

Notice that now, the whole is divided into 8 equal parts. So .

To multiply fractions, we multiply the numerators and multiply the denominators.

Fraction Multiplication

If are numbers where , then

To multiply fractions, multiply the numerators and multiply the denominators.

When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In Example 5, we will multiply negative and a positive, so the product will be negative.

EXAMPLE 5

Multiply: .

Solution

The first step is to find the sign of the product. Since the signs are the different, the product is negative.

 Determine the sign of the product; multiply. Are there any common factors in the numerator and the demoninator? No.

TRY IT 5.1

Multiply: .

TRY IT 5.2

Multiply: .

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as . So, for example, .

EXAMPLE 6

Multiply: .

Solution

Determine the sign of the product. The signs are the same, so the product is positive.

 Write as a fraction. Multiply. Rewrite 20 to show the common factor 5 and divide it out. Simplify.

TRY IT 6.1

Multiply: .

TRY IT 6.2

Multiply: .

Divide Fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.

The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of is .

Notice that . A number and its reciprocal multiply to 1.

To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

The reciprocal of is , since .

Reciprocal

The reciprocal of is .

A number and its reciprocal multiply to one .

To divide fractions, we multiply the first fraction by the reciprocal of the second.

Fraction Division

If are numbers where , then

We need to say to be sure we don’t divide by zero!

EXAMPLE 7

Divide: .

Solution
 To divide, multiply the first fraction by the reciprocal of the second. Multiply.

TRY IT 7.1

Divide: .

TRY IT 7.2

Divide: .

EXAMPLE 8

Find the quotient: .

Solution
 To divide, multiply the first fraction by the reciprocal of the second. Determine the sign of the product, and then multiply.. Rewrite showing common factors. Remove common factors. Simplify.

TRY IT 8.1

Find the quotient: .

TRY IT 8.2

Find the quotient: .

There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.

• “To multiply fractions, multiply the numerators and multiply the denominators.”
• “To divide fractions, multiply the first fraction by the reciprocal of the second.”

Another way is to keep two examples in mind:

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

Complex Fraction

A complex fraction is a fraction in which the numerator or the denominator contains a fraction.

Some examples of complex fractions are:

To simplify a complex fraction, we remember that the fraction bar means division. For example, the complex fraction means .

EXAMPLE 9

Simplify: .

Solution
 Rewrite as division. Multiply the first fraction by the reciprocal of the second. Multiply. Look for common factors. Divide out common factors and simplify.

TRY IT 9.1

Simplify: .

TRY IT 9.2

Simplify: .

EXAMPLE 10

Simplify: .

Solution
 Rewrite as division. Multiply the first fraction by the reciprocal of the second. Multiply. Look for common factors. Divide out common factors and simplify.

TRY IT 10.1

Simplify: .

TRY IT 10.2

Simplify: .

Simplify Expressions with a Fraction Bar

The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

To simplify the expression , we first simplify the numerator and the denominator separately. Then we divide.

HOW TO: Simplify an Expression with a Fraction Bar

1. Simplify the expression in the numerator. Simplify the expression in the denominator.
2. Simplify the fraction.

EXAMPLE 11

Simplify: .

Solution
 Use the order of operations to simpliy the numerator and the denominator. Simplify the numerator and the denominator. Simplify. A negative divided by a positive is negative.

TRY IT 11.1

Simplify: .

TRY IT 11.2

Simplify: .

Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.

For any positive numbers a and b,

EXAMPLE 12

Simplify: .

Solution
 Multiply. Simplify. Divide.

TRY IT 12.1

Simplify: .

4

TRY IT 12.2

Simplify: .

2

Translate Phrases to Expressions with Fractions

Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.

The English words quotient and ratio are often used to describe fractions. Remember that “quotient” means division. The quotient of and is the result we get from dividing by , or .

EXAMPLE 13

Translate the English phrase into an algebraic expression: the quotient of the difference of m and n, and p.

Solution

We are looking for the quotient of the difference of m and n, and p. This means we want to divide the difference of .

TRY IT 13.1

Translate the English phrase into an algebraic expression: the quotient of the difference of a and b, and cd.

TRY IT 13.2

Translate the English phrase into an algebraic expression: the quotient of the sum of and , and

Key Concepts

• Equivalent Fractions Property: If are numbers where , then
and .
• Fraction Division: If are numbers where , then . To divide fractions, multiply the first fraction by the reciprocal of the second.
• Fraction Multiplication: If are numbers where , then . To multiply fractions, multiply the numerators and multiply the denominators.
• Placement of Negative Sign in a Fraction: For any positive numbers , .
• Property of One: Any number, except zero, divided by itself is one.
• Simplify a Fraction
1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.
2. Simplify using the equivalent fractions property by dividing out common factors.
3. Multiply any remaining factors.
• Simplify an Expression with a Fraction Bar
1. Simplify the expression in the numerator. Simplify the expression in the denominator.
2. Simplify the fraction.

Glossary

complex fraction
A complex fraction is a fraction in which the numerator or the denominator contains a fraction.
denominator
The denominator is the value on the bottom part of the fraction that indicates the number of equal parts into which the whole has been divided.
equivalent fractions
Equivalent fractions are fractions that have the same value.
fraction
A fraction is written , where is the numerator and is the denominator. A fraction represents parts of a whole. The denominator is the number of equal parts the whole has been divided into, and the numerator indicates how many parts are included.
numerator
The numerator is the value on the top part of the fraction that indicates how many parts of the whole are included.
reciprocal
The reciprocal of is . A number and its reciprocal multiply to one: .
simplified fraction
A fraction is considered simplified if there are no common factors in its numerator and denominator.

Practice Makes Perfect

Find Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

 1 2 3 4

Simplify Fractions

In the following exercises, simplify.

 5 6 7 8 9 10 11 12 13 14

Multiply Fractions

In the following exercises, multiply.

 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Divide Fractions

In the following exercises, divide.

 31 32 33 34 35 36 37 38 39 40 41 42 43 44

In the following exercises, simplify.

 45 46 47 48 49 50

Simplify Expressions Written with a Fraction Bar

In the following exercises, simplify.

 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

 71. the quotient of r and the sum of s and 10 72. the quotient of A and the difference of 3 and B 73. the quotient of the difference of 74. the quotient of the sum of

Everyday Math

 75. Baking. A recipe for chocolate chip cookies calls for cup brown sugar. Imelda wants to double the recipe. a) How much brown sugar will Imelda need? Show your calculation. b) Measuring cups usually come in sets of cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe. 76. Baking. Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs cup of condensed milk. a) How much condensed milk will Nina need? Show your calculation. b) Measuring cups usually come in sets of cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for pans of fudge. 77. Portions Don purchased a bulk package of candy that weighs pounds. He wants to sell the candy in little bags that hold pound. How many little bags of candy can he fill from the bulk package? 78. Portions Kristen has yards of ribbon that she wants to cut into equal parts to make hair ribbons for her daughter’s 6 dolls. How long will each doll’s hair ribbon be?

Writing Exercises

 79. Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn’t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael’s reasoning. 80. Give an example from everyday life that demonstrates how . 81. Explain how you find the reciprocal of a fraction. 82. Explain how you find the reciprocal of a negative number.