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By the end of this section, you will be able to:
  • Understand the meaning of fractions
  • Model improper fractions and mixed numbers
  • Convert between improper fractions and mixed numbers
  • Model equivalent fractions
  • Find equivalent fractions
  • Locate fractions and mixed numbers on the number line
  • Order fractions and mixed numbers

Before you get started, take this readiness quiz.

  1. Simplify: 5 · 2 + 1 .
    If you missed this problem, review Use the Language of Algebra .
  2. Fill in the blank with < or >: −2 __ −5
    If you missed this problem, review Introduction to Integers .

Understand the meaning of fractions

Andy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That’s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts.

In math, we write 1 2 to mean one out of two parts.

An image of a round pizza sliced vertically down the center, creating two equal pieces. Each piece is labeled as one half.

On Tuesday, Andy and Bobby share a pizza with their parents, Fred and Christy, with each person getting an equal amount of the whole pizza. How much of the pizza does each person get? There is one whole pizza, divided evenly into four equal parts. Each person has one of the four equal parts, so each has 1 4 of the pizza.

An image of a round pizza sliced vertically and horizontally, creating four equal pieces. Each piece is labeled as one fourth.

On Wednesday, the family invites some friends over for a pizza dinner. There are a total of 12 people. If they share the pizza equally, each person would get 1 12 of the pizza.

An image of a round pizza sliced into twelve equal wedges. Each piece is labeled as one twelfth.

Fractions

A fraction is written a b , where a and b are integers and b 0 . In a fraction, a is called the numerator and b is called the denominator.

A fraction is a way to represent parts of a whole. The denominator b represents the number of equal parts the whole has been divided into, and the numerator a represents how many parts are included. The denominator, b , cannot equal zero because division by zero is undefined.

In [link] , the circle has been divided into three parts of equal size. Each part represents 1 3 of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.

A circle is divided into three equal wedges. Each piece is labeled as one third.
Doing the Manipulative Mathematics activity Model Fractions will help you develop a better understanding of fractions, their numerators and denominators.

What does the fraction 2 3 represent? The fraction 2 3 means two of three equal parts.

A circle is divided into three equal wedges. Two of the wedges are shaded.

Name the fraction    of the shape that is shaded in each of the figures.

In part “a”, a circle is divided into eight equal wedges. Five of the wedges are shaded. In part “b”, a square is divided into nine equal pieces. Two of the pieces are shaded.

Solution

We need to ask two questions. First, how many equal parts are there? This will be the denominator . Second, of these equal parts, how many are shaded? This will be the numerator .


How many equal parts are there? There are eight equal parts . How many are shaded? Five parts are shaded .

Five out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is 5 8 .


How many equal parts are there? There are nine equal parts . How many are shaded? Two parts are shaded .

Two out of nine parts are shaded. Therefore, the fraction of the square that is shaded is 2 9 .

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Name the fraction of the shape that is shaded in each figure:

In part “a”, a circle is divided into eight equal wedges. Three of the wedges are shaded. In part “b”, a square is divided into nine equal pieces. Four of the pieces are shaded.

  1. 3 8
  2. 4 9

Got questions? Get instant answers now!

Name the fraction of the shape that is shaded in each figure:

In part “a”, a circle is divided into five equal wedges. Three of the wedges are shaded. In part “b”, a square is divided into four equal pieces. Three of the pieces are shaded.

  1. 3 5
  2. 3 4

Got questions? Get instant answers now!

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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