Fractions and Rational Expressions

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Fractions and Rational Expressions. Name the fraction represented by a shaded region. Graph fractions on a number line. Simplify fractions. Write equivalent fractions. Use, <, >, or = to write a true statement. Write improper fractions as mixed numbers.

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### Fractions and Rational Expressions

Name the fraction represented by a shaded region.

Graph fractions on a number line.

Simplify fractions.

Write equivalent fractions.

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

Write improper fractions as mixed numbers.

Write mixed numbers as improper fractions.

5.1

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).

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.

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.

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.

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.

Rule

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

In math language:

Example 5

Simplify.

Rule

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

In math language:

Example 6

Simplify.

Rule

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

In math language:

Example 7

Simplify.

Rules

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

In math language:

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

In math language:

Objective 4

Write equivalent fractions.

Definition

Equivalent fractions: Fractions that name the same number.

1

Procedure

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

Example 8

Fill in the blank so that the fractions are equivalent.

Objective 5

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

We can easily compare fractions with the same denominator.

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

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.

Procedure

To compare two fractions:

• Write equivalent fractions that have a common denominator.
• Compare the numerators of the equivalent fractions.

Example 10

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

b.

a.

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.

Definition

Mixed number: An integer combined with a fraction

When we say combined, we literally mean added.

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?

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

Procedure

To write an improper fraction as a mixed number:

• Divide the denominator into the numerator.
• Write the result in the following form.

Example 11

Write the improper fraction as a mixed number.

b.

a.

Objective 7

Write mixed numbers as improper fractions.

Procedure

To write a mixed number as an improper fraction:

• Multiply the integer by the denominator.
• Add the resulting product to the numerator to find the numerator of the improper fraction.
• Keep the same denominator.

Example 13

Write the mixed number as an improper fraction.

b.

a.