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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

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Fractions .

Find equivalent fractions

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

Two circles are shown, each divided into three equal pieces by lines. The left hand circle is labeled “one third” in each section. Each section is shaded. The circle on the right is shaded in two of its three sections.
The circle on the left has been divided into 3 equal parts. Each part is 1 3 of the 3 equal parts. In the circle on the right, 2 3 of the circle is shaded (2 of the 3 equal parts).
Doing the Manipulative Mathematics activity “Model Fractions” will help you develop a better understanding of fractions, their numerators and denominators.

Fraction

A fraction is written a b , where b 0 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 6 6 pieces, or, in other words, one whole pie.

A circle is shown and is divided into six section. All sections are shaded.

So 6 6 = 1 . This leads us to the property of one that tells us that any number, except zero, divided by itself is 1.

Property of one

a a = 1 ( a 0 )

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

Doing the Manipulative Mathematics activity “Fractions Equivalent to One” will help you develop a better understanding of fractions that are equivalent to one.

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

The fractions 6 6 and 8 8 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. [link] 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 1 2 is equivalent to 4 8 . In other words, they are equivalent fractions    .

A circle is shown that is divided into eight equal wedges by lines. The left side of the circle is a pizza with four sections making up the pizza slices. The right side has four shaded sections. Below the diagram is the fraction four eighths.
Since the same amount is of each pizza is shaded, we see that 1 2 is equivalent to 4 8 . They are equivalent fractions.

Equivalent fractions

Equivalent fractions are fractions that have the same value.

How can we use mathematics to change 1 2 into 4 8 ? 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 8 pieces instead of just 2. Mathematically, what we’ve described could be written like this as 1 · 4 2 · 4 = 4 8 . See [link] .

A circle is shown and is divided in half by a vertical black line. It is further divided into eighths by the addition of dotted red lines.
Cutting each half of the pizza into 4 pieces, gives us pizza cut into 8 pieces: 1 · 4 2 · 4 = 4 8 .

This model leads to the following property:

Practice Key Terms 7

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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