Fractions and Rational Expressions

Objectives

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Fractions and Rational Expressions

1. Name the fraction represented by a shaded region.2. Graph fractions on a number line.3. Simplify fractions.4. Write equivalent fractions.5. Use, <, >, or = to write a true statement.6. Write improper fractions as mixed numbers.7. Write mixed numbers as improper fractions.

5.1

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.1 - 2

Objective 1 Name the fraction represented by a shaded region.


Definition Fraction: A number that describes a part of a whole.

We can describe the three lots that have been sold out of the five total lots using the fraction , which is read “three fifths.”Fractions have numerators (the number 3) and denominators (the number 5 in this example).

35


Definitions Numerator: The number written in the top position of the fraction.Denominator: The number written in the bottom position of a fraction.

The denominator, 5, is the number of equal-sized divisions.The numerator, 3, is the number of those division we are interested in working with.

35

Numerator

Denominator


Definition Rational number: A number that can be expressed in the form , where a and b are integers and b ≠ 0.

…is a rational number because 3 and 5 are integers and the denominator, 5, is not 0.

35

ab


Example 1 Name the fraction represented by the shaded region.

a.

b.


Example 2 In a group of 35 people at a conference, 17 are wearing glasses. What fraction of the people at the conference are wearing glasses? What fraction are not wearing glasses?


Objective 2 Graph fractions on a number line.


Example 3 Graph the fraction on a number line.

a. 34

0 1


Objective 3 Simplify fractions.


Definitions Simplify: The process of writing an equivalent expression with fewer symbols or smaller numbers.Simplest form: An equivalent expression written with the fewest symbols and the smallest numbers possible.


Example 4 Simplify. 51


If the denominator of a fraction is 1, the fraction can be simplified to the numerator.

Rule

In math language:

, when is any number.1n

n n




If the numerator of a fraction is 0, and the denominator is any number other than 0, the fraction can be simplified to 0.

Rule

In math language:

00 0, when 0.n n

n


Example 6 Simplify.50


If the denominator of a fraction is 0, and the numerator is any number other than 0, we say the fraction is undefined.

Rule

In math language:

, is undefined, when 0.0n

n




A fraction with the same nonzero numerator and nonzero denominator can be simplified to 1.

Rules

In math language: 1, when 0.n

nn

In math language:

If the numerator and denominator of a fraction are both 0, the fraction is indeterminate.

0is indeterminate.

0


Objective 4 Write equivalent fractions.


Definition Equivalent fractions: Fractions that name the same number.

1

22

12

14

24

34

44

28

48

68

88

18

38

58

78


To write an equivalent fraction, multiply or divide both the numerator and denominator by the same nonzero number.

Procedure


Example 8 Fill in the blank so that the fractions are equivalent.

3 ?8 16

20 524 ?


Objective 5 Use <, >, or = to write a true statement.


We can easily compare fractions with the same denominator.

2 35 5

If two fractions don’t have the same denominator, we can draw a picture and compare them.

12

13

1 12 3

We could also write fractions so that they have a common denominator by using multiples.


Definition Multiple: A number that is evenly divisible by a given number.

Multiples of 2 are 2, 4, 6, 8, 10,…Multiples of 3 are 3, 6, 9, 12, 15,…

Notice the common multiple for 2 and 3 is 6…it appears in both lists.

To upscale 1/2, we multiply numerator and denominator by 3. To upscale 1/3, we, multiply numerator and denominator by 2.


To compare two fractions:Procedure

1. Write equivalent fractions that have a common denominator.

2. Compare the numerators of the equivalent fractions.


Example 10 Use <, >, or = to write a true statement.

5 4?

8 7a.

7 9?

8 11

b.

7 9?

8 11


Objective 6 Write improper fractions as mixed numbers.


Definition Improper fraction: A fraction in which the absolute value of the numerator is greater than or equal to the absolute value of the denominator.

94

72 8

8


Definition Mixed number: An integer combined with a fraction

When we say combined, we literally mean added.

1 12 2

4 4 Note: 2 ¼ is read “two and one-

forth” and means two wholes plus ¼ of another whole.

How does this apply to negative mixed numbers?

3

58

358

3 58

358

Note: The negative sign applies to both the integer and the fraction.


To write an improper fraction as a mixed number:

Procedure

1. Divide the denominator into the numerator.2. Write the result in the following form.

minremainder

quotientoriginal deno ator


Example 11 Write the improper fraction as a mixed number.

203

a. 417

b.


Objective 7 Write mixed numbers as improper fractions.


To write a mixed number as an improper fraction:

Procedure

1. Multiply the integer by the denominator.2. Add the resulting product to the numerator

to find the numerator of the improper fraction.

3. Keep the same denominator.


Example 13 Write the mixed number as an improper fraction.

75

8a.

39

7b.

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Fractions and Rational Expressions