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Mathematics LibreTexts

4.1: Visualize Fractions (Part 1)

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Learning Objectives
  • 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
be prepared!

Before you get started, take this readiness quiz.

  1. Simplify: 52+1. If you missed this problem, review Example 2.1.8.
  2. Fill in the blank with < or >: 2__5. If you missed this problem, review Example 3.1.2.

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 12 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.

Figure 4.1.1

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 14 of the pizza.

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

Figure 4.1.2

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 112 of the pizza.

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

Figure 4.1.3

Definition: Fractions

A fraction is written ab, where a and b are integers and b0. 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 Figure 4.1.4, the circle has been divided into three parts of equal size. Each part represents 13 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.

Figure 4.1.4

What does the fraction 23 represent? The fraction 23 means two of three equal parts.

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

Figure 4.1.5

Example 4.1.1: name the fraction

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

  1. 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.
  2. 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 58.

How many equal parts are there? There are nineequal 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 29.

Exercise 4.1.1

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.

Answer a

38

Answer b

49

Exercise 4.1.2

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.

Answer a

35

Answer b

34

Example 4.1.2:

Shade 34 of the circle.

An image of a circle.

Solution

The denominator is 4, so we divide the circle into four equal parts (a). The numerator is 3, so we shade three of the four parts (b).

In “a”, a circle is shown divided into four equal pieces. An arrow points from “a” to “b”. In “b”, the same image is shown with three of the pieces shaded.

34 of the circle is shaded.

Exercise 4.1.3

Shade 68 of the circle.

A circle is divided into eight equal pieces.

Answer

Exercise 4.1.3.png

Exercise 4.1.4

Shade 25 of the rectangle.

A rectangle is divided vertically into five equal pieces.

Answer

Exercise 4.1.4.png

In Example 4.1.1 and Example 4.1.2, we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in Figure 4.1.6. Here, the whole is modeled as one long, undivided rectangular tile. Beneath it are tiles of equal length divided into different numbers of equally sized parts.

One long, undivided rectangular tile is shown, labeled “1”. Below it is a rectangular tile of the same size and shape that has been divided vertically into two equal pieces, each labeled as one half. Below that is another rectangular tile that has been divided into three equal pieces, each labeled as one third. Below that is another rectangular tile that has been divided into four equal pieces, each labeled as one fourth. Below that is another rectangular tile that has been divided into six pieces, each labeled as one sixth.

Figure 4.1.6

We’ll be using fraction tiles to discover some basic facts about fractions. Refer to Figure 4.1.6 to answer the following questions:

How many 12 tiles does it take to make one whole tile? It takes two halves to make a whole, so two out of two is 22 = 1.
How many 13 tiles does it take to make one whole tile? It takes three thirds, so three out of three is 33 = 1.
How many 14 tiles does it take to make one whole tile? It takes four fourths, so four out of four is 44 = 1.
How many 15 tiles does it take to make one whole tile? It takes six sixths, so six out of six is 66 = 1.
What if the whole were divided into 24 equal parts? (We have not shown fraction tiles to represent this, but try to visualize it in your mind.) How many 124 tiles does it take to make one whole tile? It takes 24 twenty-fourths, so 2424 = 1.

It takes 24 twenty-fourths, so 2424=1. This leads us to the Property of One.

Definition: Property of One

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

aa=1(a0)

Example 4.1.3: fraction circles to form wholes

Use fraction circles to make wholes using the following pieces:

  1. 4 fourths
  2. 5 fifths
  3. 6 sixths

Solution

Three circles are shown. The circle on the left is divided into four equal pieces. The circle in the middle is divided into five equal pieces. The circle on the right is divided into six equal pieces. Each circle says “Form 1 whole” beneath it.

Exercise 4.1.5

Use fraction circles to make wholes with the following pieces: 3 thirds.

Answer

Exercise 4.1.5.png

Exercise 4.1.6

Use fraction circles to make wholes with the following pieces: 8 eighths.

Answer

Exercise 4.1.6.png

What if we have more fraction pieces than we need for 1 whole? We’ll look at this in the next example.

Example 4.1.4: fraction circles to form whole

Use fraction circles to make wholes using the following pieces:

  1. 3 halves
  2. 8 fifths
  3. 7 thirds

Solution

  1. 3 halves make 1 whole with 1 half left over.

Two circles are shown, both divided into two equal pieces. The circle on the left has both pieces shaded and is labeled as “1”. The circle on the right has one piece shaded and is labeled as one half.

  1. 8 fifths make 1 whole with 3 fifths left over.

Two circles are shown, both divided into five equal pieces. The circle on the left has all five pieces shaded and is labeled as “1”. The circle on the right has three pieces shaded and is labeled as three fifths.

  1. 7 thirds make 2 wholes with 1 third left over.

Three circles are shown, all divided into three equal pieces. The two circles on the left have all three pieces shaded and are labeled with ones. The circle on the right has one piece shaded and is labeled as one third.

Exercise 4.1.7

Use fraction circles to make wholes with the following pieces: 5 thirds.

Answer

Exercise 4.1.7.png

Exercise 4.1.8

Use fraction circles to make wholes with the following pieces: 5 halves.

Answer

Exercise 4.1.8.png

Model Improper Fractions and Mixed Numbers

In Example 4.1.4b, you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, 15, so altogether you had eight fifths, which we can write as 85. The fraction 85 is one whole, 1, plus three fifths, 35, or 135, which is read as one and three-fifths.

The number 135 is called a mixed number. A mixed number consists of a whole number and a fraction.

Definition: Mixed Numbers

A mixed number consists of a whole number a and a fraction bc where c0. It is written as follows.

abcc0

Fractions such as 54, 32, 55, and 73 are called improper fractions. In an improper fraction, the numerator is greater than or equal to the denominator, so its value is greater than or equal to one. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as 12, 37, and 1118 are proper fractions.

Definition: Proper and Improper Fractions

The fraction ab is a proper fraction if a<b and an improper fraction if ab.

Example 4.1.5: improper fraction

Name the improper fraction modeled. Then write the improper fraction as a mixed number.

Two circles are shown, both divided into three equal pieces. The circle on the left has all three pieces shaded. The circle on the right has one piece shaded.

Solution

Each circle is divided into three pieces, so each piece is 13 of the circle. There are four pieces shaded, so there are four thirds or 43. The figure shows that we also have one whole circle and one third, which is 113. So, 43=113.

Exercise 4.1.9

Name the improper fraction. Then write it as a mixed number.

Two circles are shown, both divided into three equal pieces. The circle on the left has all three pieces shaded. The circle on the right has two pieces shaded.

Answer

53=123

Exercise 4.1.10

Name the improper fraction. Then write it as a mixed number.

Two circles are shown, both divided into eight equal pieces. The circle on the left has all eight pieces shaded. The circle on the right has five pieces shaded.

Answer

138=158

Example 4.1.6: model a fraction

Draw a figure to model 118.

Solution

The denominator of the improper fraction is 8. Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have 11 eighths. We must shade three of the eight parts of another circle.

Two circles are shown, both divided into eight equal pieces. The circle on the left has all eight pieces shaded and is labeled as eight eighths. The circle on the right has three pieces shaded and is labeled as three eighths. The diagram indicates that eight eighths plus three eighths is one plus three eighths.

So, 118=138.

Exercise4.1.11

Draw a figure to model 76.

Answer

Exercise 4.1.11.png

Exercise 4.1.12

Draw a figure to model 65.

Answer

Exercise 4.1.12.png

Example 4.1.7: model a fraction

Use a model to rewrite the improper fraction 116 as a mixed number.

Solution

We start with 11 sixths (116). We know that six sixths makes one whole.

66=1

That leaves us with five more sixths, which is 56 (11 sixths minus 6 sixths is 5 sixths). So, 116=156.

Two circles are shown, both divided into six equal pieces. The circle on the left has all six pieces shaded and is labeled as six sixths. The circle on the right has five pieces shaded and is labeled as five sixths. Below the circles, it says one plus five sixths, then six sixths plus five sixths equals eleven sixths, and one plus five sixths equals one and five sixths. It then says that eleven sixths equals one and five sixths.

Exercise 4.1.13

Use a model to rewrite the improper fraction as a mixed number: 97.

Answer

127

Exercise 4.1.14

Use a model to rewrite the improper fraction as a mixed number: 74.

Answer

134

Example 4.1.8: model a fraction

Use a model to rewrite the mixed number 145 as an improper fraction.

Solution

The mixed number 145 means one whole plus four fifths. The denominator is 5, so the whole is 55. Together five fifths and four fifths equals nine fifths. So, 145=95.

Two circles are shown, both divided into five equal pieces. The circle on the left has all five pieces shaded and is labeled as 5 fifths. The circle on the right has four pieces shaded and is labeled as 4 fifths. It then says that 5 fifths plus 4 fifths equals 9 fifths and that 9 fifths is equal to one plus 4 fifths.

Exercise 4.1.15

Use a model to rewrite the mixed number as an improper fraction: 138.

Answer

118

Exercise 4.1.16

Use a model to rewrite the mixed number as an improper fraction: 156.

Answer

116

Convert between Improper Fractions and Mixed Numbers

In Example 4.1.7, we converted the improper fraction 116 to the mixed number 156 using fraction circles. We did this by grouping six sixths together to make a whole; then we looked to see how many of the 11 pieces were left. We saw that 116 made one whole group of six sixths plus five more sixths, showing that 116=156.

The division expression 116 (which can also be written as 6¯)11) tells us to find how many groups of 6 are in 11. To convert an improper fraction to a mixed number without fraction circles, we divide.

Example 4.1.9:

Convert 116 to a mixed number.

Solution

Divide the denominator into the numerator. Remember 116 means 11 ÷ 6.
Identify the quotient, remainder and divisor. CNX_BMath_Figure_04_01_031_img-01.png
Write the mixed number as quotientremainderdivisor. 156

So, 116=156.

Exercise 4.1.17

Convert the improper fraction to a mixed number: 137.

Answer

167

Exercise 4.1.18

Convert the improper fraction to a mixed number: 149.

Answer

159

HOW TO: CONVERT AN IMPROPER FRACTION TO A MIXED NUMBER

Step 1. Divide the denominator into the numerator.

Step 2. Identify the quotient, remainder, and divisor.

Step 3. Write the mixed number as quotientremainderdivisor.

Example 4.1.10:

Convert the improper fraction 338 to a mixed number.

Solution

Divide the denominator into the numerator. Remember, 338 means 8¯)33.
Identify the quotient, remainder, and divisor. CNX_BMath_Figure_04_01_032_img-01.png
Write the mixed number as quotientremainderdivisor. 418

So, 338=418.

Exercise 4.1.19

Convert the improper fraction to a mixed number: 237.

Answer

327

Exercise 4.1.20

Convert the improper fraction to a mixed number: 4811.

Answer

4411

In Example 4.1.8, we changed 145 to an improper fraction by first seeing that the whole is a set of five fifths. So we had five fifths and four more fifths.

55+45=95

Where did the nine come from? There are nine fifths—one whole (five fifths) plus four fifths. Let us use this idea to see how to convert a mixed number to an improper fraction.

Example 4.1.11: convert

Convert the mixed number 423 to an improper fraction.

Multiply the whole number by the denominator. 423
The whole number is 4 and the denominator is 3. CNX_BMath_Figure_04_01_068_img-01.png
Simplify. CNX_BMath_Figure_04_01_068_img-02.png
Add the numerator to the product.  
The numerator of the mixed number is 2. CNX_BMath_Figure_04_01_068_img-03.png
Simplify. CNX_BMath_Figure_04_01_068_img-04.png
Write the final sum over the original denominator.  
The denominator is 3. 143
Exercise 4.1.21

Convert the mixed number to an improper fraction: 357.

Answer

267

Exercise 4.1.22

Convert the mixed number to an improper fraction: 278.

Answer

238

HOW TO: CONVERT A MIXED NUMBER TO AN IMPROPER FRACTION

Step 1. Multiply the whole number by the denominator.

Step 2. Add the numerator to the product found in Step 1.

Step 3. Write the final sum over the original denominator.

Example 4.1.12:

Convert the mixed number 1027 to an improper fraction.

Multiply the whole number by the denominator. 1027
The whole number is 10 and the denominator is 7. CNX_BMath_Figure_04_01_069_img-01.png
Simplify. CNX_BMath_Figure_04_01_069_img-02.png
Add the numerator to the product.  
The numerator of the mixed number is 2. CNX_BMath_Figure_04_01_069_img-03.png
Simplify. CNX_BMath_Figure_04_01_069_img-04.png
Write the final sum over the original denominator.  
The denominator is 7. 727
Exercise 4.1.23

Convert the mixed number to an improper fraction: 4611.

Answer

5011

Exercise 4.1.24

Convert the mixed number to an improper fraction: 1113.

Answer

343

Contributors and Attributions


This page titled 4.1: Visualize Fractions (Part 1) is shared under a not declared license and was authored, remixed, and/or curated by OpenStax.

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