1.2: Visualize Fractions

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Learning Objectives

After completing this section, you should 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

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 $\frac{1}{2}$ 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 $\frac{1}{4}$ 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 $\frac{1}{12}$ 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 $\frac{a}{b},$ where $a$ and $b$ are integers and $b \neq 0 .$ 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 Figure 1.2.1, the circle has been divided into three parts of equal size. Each part represents $\frac{1}{3}$ 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. — Figure 1.2.1

What does the fraction $\frac{2}{3}$ represent? The fraction $\frac{2}{3}$ means two of three equal parts.

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

Example 1.2.1

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

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

Answer

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.

ⓐ
$\begin{matrix} How many equal parts are there? & There are eight equal parts . \\ How many are shaded? & Five parts are shaded . \end{matrix}$

Five out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is $\frac{5}{8} .$

ⓑ
$\begin{matrix} How many equal parts are there? & There are nine equal parts . \\ How many are shaded? & Two parts are shaded . \end{matrix}$

Two out of nine parts are shaded. Therefore, the fraction of the square that is shaded is $\frac{2}{9} .$

Your Turn 1.2.1

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

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

Example 1.2.2

Shade $\frac{3}{4}$ of the circle.

$An image of a circle.$

Answer

The denominator is $4,$ so we divide the circle into four equal parts ⓐ.

The numerator is $3,$ so we shade three of the four parts ⓑ.

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

$\frac{3}{4}$ of the circle is shaded.

Your Turn 1.2.2

Shade $\frac{6}{8}$ of the circle.

$A circle is divided into eight equal pieces.$

In Example 1.2.1 and Example 1.2.2, we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in Figure 1.2.2. 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 1.2.2

We’ll be using fraction tiles to discover some basic facts about fractions. Refer to Figure 1.2.2 to answer the following questions:

How many $\frac{1}{2}$ tiles does it take to make one whole tile?	It takes two halves to make a whole, so two out of two is $\frac{2}{2} = 1 .$
How many $\frac{1}{3}$ tiles does it take to make one whole tile?	It takes three thirds, so three out of three is $\frac{3}{3} = 1 .$
How many $\frac{1}{4}$ tiles does it take to make one whole tile?	It takes four fourths, so four out of four is $\frac{4}{4} = 1 .$
How many $\frac{1}{6}$ tiles does it take to make one whole tile?	It takes six sixths, so six out of six is $\frac{6}{6} = 1 .$
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 $\frac{1}{24}$ tiles does it take to make one whole tile?	It takes $24$ twenty-fourths, so $\frac{24}{24} = 1 .$

This leads us to the Property of One.

Property of One

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

$\frac{a}{a} = 1 (a \neq 0)$

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

Example 1.2.3

Use fraction circles to make wholes using the following pieces:

ⓐ $3$ halves
ⓑ $8$ fifths
ⓒ $7$ thirds

Answer

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

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

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

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

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

Your Turn 1.2.3

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

Model Improper Fractions and Mixed Numbers

In Example 1.2.3 (b), 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, $\frac{1}{5},$ so altogether you had eight fifths, which we can write as $\frac{8}{5} .$ The fraction $\frac{8}{5}$ is one whole, $1,$ plus three fifths, $\frac{3}{5},$ or $1 \frac{3}{5},$ which is read as one and three-fifths.

The number $1 \frac{3}{5}$ is called a mixed number. A mixed number consists of a whole number and a fraction.

Mixed Numbers

A mixed number consists of a whole number $a$ and a fraction bcbc where $c \neq 0 .$ It is written as follows.

abcabc

Fractions such as $\frac{5}{4}, \frac{3}{2}, \frac{5}{5},$ and $\frac{7}{3}$ 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 $\frac{1}{2}, \frac{3}{7},$ and $\frac{11}{18}$ are proper fractions.

Proper and Improper Fractions

The fraction abab is a proper fraction if $a < b$ and an improper fraction if $a \geq b .$

Example 1.2.4

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

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

Answer: Each circle is divided into three pieces, so each piece is $\frac{1}{3}$ of the circle. There are four pieces shaded, so there are four thirds or $\frac{4}{3} .$ The figure shows that we also have one whole circle and one third, which is $1 \frac{1}{3} .$ So, $\frac{4}{3} = 1 \frac{1}{3} .$

Your Turn 1.2.4

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

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

Example 1.2.5

Draw a figure to model $\frac{11}{8} .$

Answer

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, $\frac{11}{8} = 1 \frac{3}{8} .$

Your Turn 1.2.5

Draw a figure to model $\frac{7}{6} .$

Example 1.2.6

Use a model to rewrite the improper fraction $\frac{11}{6}$ as a mixed number.

Answer

We start with $11$ sixths $(\frac{11}{6}) .$ We know that six sixths makes one whole.

$\frac{6}{6} = 1$

That leaves us with five more sixths, which is $\frac{5}{6} (11 sixths minus 6 sixths is 5 sixths.)$

So, $\frac{11}{6} = 1 \frac{5}{6} .$

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

Your Turn 1.2.6

Use a model to rewrite the improper fraction as a mixed number: $\frac{9}{7} .$

Example 1.2.7

Use a model to rewrite the mixed number $1 \frac{4}{5}$ as an improper fraction.

Answer

The mixed number $1 \frac{4}{5}$ means one whole plus four fifths. The denominator is $5,$ so the whole is $\frac{5}{5} .$ Together five fifths and four fifths equals nine fifths.

So, $1 \frac{4}{5} = \frac{9}{5} .$

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

Your Turn 1.2.7

Use a model to rewrite the mixed number as an improper fraction: $1 \frac{3}{8} .$

Convert between Improper Fractions and Mixed Numbers

In Example 1.2.6, we converted the improper fraction $\frac{11}{6}$ to the mixed number $1 \frac{5}{6}$ 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 $\frac{11}{6}$ made one whole group of six sixths plus five more sixths, showing that $\frac{11}{6} = 1 \frac{5}{6} .$

The division expression $\frac{11}{6}$ (which can also be written as $611$ ) 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 1.2.8

Convert $\frac{11}{6}$ to a mixed number.

Answer

	$\frac{11}{6}$
Divide the denominator into the numerator.	Remember $\frac{11}{6}$ means $11 \div 6$ .
	$.$
Identify the quotient, remainder and divisor.
Write the mixed number as quotient remainderdivisorremainderdivisor.	$1 \frac{5}{6}$
So, $\frac{11}{6} = 1 \frac{5}{6}$

Your Turn 1.2.8

Convert the improper fraction to a mixed number: $\frac{13}{7} .$

How To

Convert an improper fraction to a mixed number.

Divide the denominator into the numerator.
Identify the quotient, remainder, and divisor.
Write the mixed number as quotient remainderdivisorremainderdivisor.

Example 1.2.9

Convert the improper fraction $\frac{33}{8}$ to a mixed number.

Answer

	$\frac{33}{8}$
Divide the denominator into the numerator.	Remember, $\frac{33}{8}$ means $833$ .
Identify the quotient, remainder, and divisor.	$.$
Write the mixed number as quotient remainderdivisorremainderdivisor.	$4 \frac{1}{8}$
	So, $\frac{33}{8} = 4 \frac{1}{8}$

Your Turn 1.2.9

Convert the improper fraction to a mixed number: $\frac{23}{7} .$

In Example 1.2.7, we changed $1 \frac{4}{5}$ 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.

$\frac{5}{5} + \frac{4}{5} = \frac{9}{5}$

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 1.2.10

Convert the mixed number $4 \frac{2}{3}$ to an improper fraction.

Answer

	$4 \frac{2}{3}$
Multiply the whole number by the denominator.
The whole number is 4 and the denominator is 3.	$.$
Simplify.	$.$
Add the numerator to the product.
The numerator of the mixed number is 2.	$.$
Simplify.	$.$
Write the final sum over the original denominator.
The denominator is 3.	$\frac{14}{3}$

Your Turn 1.2.10

Convert the mixed number to an improper fraction: $3 \frac{5}{7} .$

How To

Convert a mixed number to an improper fraction.

Multiply the whole number by the denominator.
Add the numerator to the product found in Step 1.
Write the final sum over the original denominator.

Example 1.2.11

Convert the mixed number $10 \frac{2}{7}$ to an improper fraction.

Answer

	$10 \frac{2}{7}$
Multiply the whole number by the denominator.
The whole number is 10 and the denominator is 7.	$.$
Simplify.	$.$
Add the numerator to the product.
The numerator of the mixed number is 2.	$.$
Simplify.	$.$
Write the final sum over the original denominator.
The denominator is 7.	$\frac{72}{7}$

Your Turn 1.2.11

Convert the mixed number to an improper fraction: $4 \frac{6}{11} .$

Model Equivalent Fractions

Let’s think about Andy and Bobby and their favorite food again. If Andy eats $\frac{1}{2}$ of a pizza and Bobby eats $\frac{2}{4}$ of the pizza, have they eaten the same amount of pizza? In other words, does $\frac{1}{2} = \frac{2}{4} ?$ We can use fraction tiles to find out whether Andy and Bobby have eaten equivalent parts of the pizza.

Equivalent Fractions

Equivalent fractions are fractions that have the same value.

Fraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of Figure 1.2.2 and extend it to include eighths, tenths, and twelfths.

Start with a $\frac{1}{2}$ tile. How many fourths equal one-half? How many of the $\frac{1}{4}$ tiles exactly cover the $\frac{1}{2}$ tile?

$One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into four pieces, each labeled as one fourth.$

Since two $\frac{1}{4}$ tiles cover the $\frac{1}{2}$ tile, we see that $\frac{2}{4}$ is the same as $\frac{1}{2},$ or $\frac{2}{4} = \frac{1}{2} .$

How many of the $\frac{1}{6}$ tiles cover the $\frac{1}{2}$ tile?

$One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into six pieces, each labeled as one sixth.$

Since three $\frac{1}{6}$ tiles cover the $\frac{1}{2}$ tile, we see that $\frac{3}{6}$ is the same as $\frac{1}{2} .$

So, $\frac{3}{6} = \frac{1}{2} .$ The fractions are equivalent fractions.

Example 1.2.12

Use fraction tiles to find equivalent fractions. Show your result with a figure.

ⓐ How many eighths equal one-half?
ⓑ How many tenths equal one-half?

Answer

ⓐ It takes four $\frac{1}{8}$ tiles to exactly cover the $\frac{1}{2}$ tile, so $\frac{4}{8} = \frac{1}{2} .$

$One long, undivided rectangle is shown, labeled 1. Below it is an identical rectangle divided vertically into two pieces, each labeled 1 half. Below that is an identical rectangle divided vertically into eight pieces, each labeled 1 eighth.$

ⓑ It takes five $\frac{1}{10}$ tiles to exactly cover the $\frac{1}{2}$ tile, so $\frac{5}{10} = \frac{1}{2} .$

$One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into ten pieces, each labeled as one tenth.$

$One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into twelve pieces, each labeled as one twelfth.$

Suppose you had tiles marked $\frac{1}{20} .$ How many of them would it take to equal $\frac{1}{2} ?$ Are you thinking ten tiles? If you are, you’re right, because $\frac{10}{20} = \frac{1}{2} .$

We have shown that $\frac{1}{2}, \frac{2}{4}, \frac{3}{6}, \frac{4}{8}, \frac{5}{10}, \frac{6}{12},$ and $\frac{10}{20}$ are all equivalent fractions.

Your Turn 1.2.12

Use fraction tiles to find equivalent fractions: How many eighths equal one-fourth?

Find Equivalent Fractions

We used fraction tiles to show that there are many fractions equivalent to $\frac{1}{2} .$

We can show this with pizzas, too. Figure 1.2.3(a) shows a single pizza, cut into two equal pieces with $\frac{1}{2}$ shaded. Figure 1.2.3(b) shows a second pizza of the same size, cut into eight pieces with $\frac{4}{8}$ shaded.

Two pizzas are shown. The pizza on the left is divided into 2 equal pieces. 1 piece is shaded. The pizza on the right is divided into 8 equal pieces. 4 pieces are shaded. — Figure 1.2.3

This is another way to show that $\frac{1}{2}$ is equivalent to $\frac{4}{8} .$

How can we use mathematics to change $\frac{1}{2}$ into $\frac{4}{8} ?$ How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we’ve described could be written as:

$1 times 4 over 2 times 4 is written with the 4s in red. This is set equal to 4 over 8.$

These models lead to the Equivalent Fractions Property, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.

Equivalent Fractions Property

If $a, b,$ and $c$ are numbers where $b \neq 0$ and $c \neq 0,$ then

$\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$

When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.

$The top line says that 1 times 3 over 2 times 3 equals 3 over 6, so one half equals 3 sixths. The next line says that 1 times 2 over 2 times 2 equals 2 over 4, so one half equals 2 fourths. The last line says that 1 times 10 over 2 times 10 equals 10 over 20, so one half equals 10 twentieths.$

So, we say that $\frac{1}{2}, \frac{2}{4}, \frac{3}{6},$ and $\frac{10}{20}$ are equivalent fractions.

Example 1.2.13

Find three fractions equivalent to $\frac{2}{5} .$

Answer

To find a fraction equivalent to $\frac{2}{5},$ we multiply the numerator and denominator by the same number (but not zero). Let us multiply them by $2, 3,$ and $5 .$

$On the left, we see that 2 times 2 over 5 times 2 equals 4 over 10. In the middle, we see that 2 times 3 over 5 times 3 equals 6 over 15. On the right, we see that 2 times 5 over 5 times 5 equals 10 over 25.$

So, $\frac{4}{10}, \frac{6}{15},$ and $\frac{10}{25}$ are equivalent to $\frac{2}{5} .$

Your Turn 1.2.13

Find three fractions equivalent to $\frac{3}{5} .$

Example 1.2.14

Find a fraction with a denominator of $21$ that is equivalent to $\frac{2}{7} .$

Answer

To find equivalent fractions, we multiply the numerator and denominator by the same number. In this case, we need to multiply the denominator by a number that will result in $21 .$

Since we can multiply $7$ by $3$ to get $21,$ we can find the equivalent fraction by multiplying both the numerator and denominator by $3 .$

$2 over 7 equals 2 time 3 over 7 times 3. The 3s are shown in red. This is set equal to 6 over 21.$

Your Turn 1.2.14

Find a fraction with a denominator of $21$ that is equivalent to $\frac{6}{7} .$