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Objectives
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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
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Objective 1 Name the fraction represented by a shaded region.
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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).
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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.
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Numerator
Denominator
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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.
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ab
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Example 1 Name the fraction represented by the shaded region.
a.
b.
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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?
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Objective 2 Graph fractions on a number line.
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Example 3 Graph the fraction on a number line.
a. 34
0 1
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Objective 3 Simplify fractions.
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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.
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Example 4 Simplify. 51
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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
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Example 5 Simplify. 09
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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
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Example 6 Simplify.50
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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
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Example 7 Simplify. 44
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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
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Objective 4 Write equivalent fractions.
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Definition Equivalent fractions: Fractions that name the same number.
1
22
12
14
24
34
44
28
48
68
88
18
38
58
78
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To write an equivalent fraction, multiply or divide both the numerator and denominator by the same nonzero number.
Procedure
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Example 8 Fill in the blank so that the fractions are equivalent.
3 ?8 16
20 524 ?
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Objective 5 Use <, >, or = to write a true statement.
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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.
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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.
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To compare two fractions:Procedure
1. Write equivalent fractions that have a common denominator.
2. Compare the numerators of the equivalent fractions.
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Example 10 Use <, >, or = to write a true statement.
5 4?
8 7a.
7 9?
8 11
b.
7 9?
8 11
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Objective 6 Write improper fractions as mixed numbers.
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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
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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.
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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
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Example 11 Write the improper fraction as a mixed number.
203
a. 417
b.
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Objective 7 Write mixed numbers as improper fractions.
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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.
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Example 13 Write the mixed number as an improper fraction.
75
8a.
39
7b.