4.1 Visualize 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.

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.

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.

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 4.2, 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.

Figure 4.2

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

Example 4.1

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

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.

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

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

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

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

Try It 4.1

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

Try It 4.2

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

Example 4.2

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

Solution

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 ⓑ.

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

Try It 4.3

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

Try It 4.4

Shade $\frac{2}{5}$ of the rectangle.

In Example 4.1 and Example 4.2, we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in Figure 4.3. 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.

Figure 4.3

We’ll be using fraction tiles to discover some basic facts about fractions. Refer to Figure 4.3 to answer the following questions:

It takes $24$ twenty-fourths, so $\frac{24}{24}=1.$

This leads us to the Property of One.

Example 4.3

Use fraction circles to make wholes using the following pieces:

1. ⓐ $4$ fourths
2. ⓑ $5$ fifths
3. ⓒ $6$ sixths

Solution

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

Example 4.4

Use fraction circles to make wholes using the following pieces:

1. ⓐ $3$ halves
2. ⓑ $8$ fifths
3. ⓒ $7$ thirds

Solution

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

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

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

Model Improper Fractions and Mixed Numbers

In Example 4.4 (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.

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.

Example 4.5

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

Solution

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, $\frac{4}{3}=1\frac{1}{3}.$

Try It 4.9

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

Try It 4.10

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

Example 4.6

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

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.

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

Example 4.7

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

Solution

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

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

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

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

Example 4.8

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

Solution

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

Convert between Improper Fractions and Mixed Numbers

In Example 4.7, 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 $6\overline{)11}$) 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 4.9

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

Solution

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

Example 4.10

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

Solution

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

In Example 4.8, 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.

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 4.11

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

Solution

	$4\frac{2}{3}$
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.	$\frac{14}{3}$

Example 4.12

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

Solution

	$10\frac{2}{7}$
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.	$\frac{72}{7}$

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.

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 4.3 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?

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?

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 4.13

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

1. ⓐ How many eighths equal one-half?
2. ⓑ How many tenths equal one-half?
3. ⓒ How many twelfths equal one-half?

Solution

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

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

ⓒ It takes six $\frac{1}{12}$ tiles to exactly cover the $\frac{1}{2}$ tile, so $\frac{6}{12}=\frac{1}{2}.$

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.

Find Equivalent Fractions

We used fraction tiles to show that there are many fractions equivalent to $\frac{1}{2}.$ For example, $\frac{2}{4},\frac{3}{6},$ and $\frac{4}{8}$ are all equivalent to $\frac{1}{2}.$ When we lined up the fraction tiles, it took four of the $\frac{1}{8}$ tiles to make the same length as a $\frac{1}{2}$ tile. This showed that $\frac{4}{8}=\frac{1}{2}.$ See Example 4.13.

We can show this with pizzas, too. Figure 4.4(a) shows a single pizza, cut into two equal pieces with $\frac{1}{2}$ shaded. Figure 4.4(b) shows a second pizza of the same size, cut into eight pieces with $\frac{4}{8}$ shaded.

Figure 4.4

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:

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.

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.

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

Example 4.14

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

Solution

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 $5.$

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

Example 4.15

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

Solution

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 $3.$

Locate Fractions and Mixed Numbers on the Number Line

Now we are ready to plot fractions on a number line. This will help us visualize fractions and understand their values.

Let us locate $\frac{1}{5},\frac{4}{5},3,3\frac{1}{3},\frac{7}{4},\frac{9}{2},5,$ and $\frac{8}{3}$ on the number line.

We will start with the whole numbers $3$ and $5$ because they are the easiest to plot.

The proper fractions listed are $\frac{1}{5}$ and $\frac{4}{5}.$ We know proper fractions have values less than one, so $\frac{1}{5}$ and $\frac{4}{5}$ are located between the whole numbers $0$ and $1.$ The denominators are both $5,$ so we need to divide the segment of the number line between $0$ and $1$ into five equal parts. We can do this by drawing four equally spaced marks on the number line, which we can then label as $\frac{1}{5},\frac{2}{5},\frac{3}{5},$ and $\frac{4}{5}.$

Now plot points at $\frac{1}{5}$ and $\frac{4}{5}.$

The only mixed number to plot is $3\frac{1}{3}.$ Between what two whole numbers is $3\frac{1}{3}?$ Remember that a mixed number is a whole number plus a proper fraction, so $3\frac{1}{3}>3.$ Since it is greater than $3,$ but not a whole unit greater, $3\frac{1}{3}$ is between $3$ and $4.$ We need to divide the portion of the number line between $3$ and $4$ into three equal pieces (thirds) and plot $3\frac{1}{3}$ at the first mark.

Finally, look at the improper fractions $\frac{7}{4},\frac{9}{2},$ and $\frac{8}{3}.$ Locating these points will be easier if you change each of them to a mixed number.

Here is the number line with all the points plotted.

Example 4.16

Locate and label the following on a number line: $\frac{3}{4},\frac{4}{3},\frac{5}{3},4\frac{1}{5},$ and $\frac{7}{2}.$

Solution

Start by locating the proper fraction $\frac{3}{4}.$ It is between $0$ and $1.$ To do this, divide the distance between $0$ and $1$ into four equal parts. Then plot $\frac{3}{4}.$

Next, locate the mixed number $4\frac{1}{5}.$ It is between $4$ and $5$ on the number line. Divide the number line between $4$ and $5$ into five equal parts, and then plot $4\frac{1}{5}$ one-fifth of the way between $4$ and $5$.

Now locate the improper fractions $\frac{4}{3}$ and $\frac{5}{3}$.

It is easier to plot them if we convert them to mixed numbers first.

$$\frac{4}{3}=1\frac{1}{3},\frac{5}{3}=1\frac{2}{3}$$

Divide the distance between $1$ and $2$ into thirds.

Next let us plot $\frac{7}{2}.$ We write it as a mixed number, $\frac{7}{2}=3\frac{1}{2}$. Plot it between $3$ and $4.$

The number line shows all the numbers located on the number line.

In Introduction to Integers, we defined the opposite of a number. It is the number that is the same distance from zero on the number line but on the opposite side of zero. We saw, for example, that the opposite of $7$ is $\mathrm{-7}$ and the opposite of $\mathrm{-7}$ is $7.$

Fractions have opposites, too. The opposite of $\frac{3}{4}$ is $-\frac{3}{4}.$ It is the same distance from $0$ on the number line, but on the opposite side of $0.$

Thinking of negative fractions as the opposite of positive fractions will help us locate them on the number line. To locate $-\frac{15}{8}$ on the number line, first think of where $\frac{15}{8}$ is located. It is an improper fraction, so we first convert it to the mixed number $1\frac{7}{8}$ and see that it will be between $1$ and $2$ on the number line. So its opposite, $-\frac{15}{8},$ will be between $\mathrm{-1}$ and $\mathrm{-2}$ on the number line.

Example 4.17

Locate and label the following on the number line: $\frac{1}{4},-\frac{1}{4},1\frac{1}{3},\mathrm{-1}\frac{1}{3},\frac{5}{2},$ and $-\frac{5}{2}.$

Solution

Draw a number line. Mark $0$ in the middle and then mark several units to the left and right.

To locate $\frac{1}{4},$ divide the interval between $0$ and $1$ into four equal parts. Each part represents one-quarter of the distance. So plot $\frac{1}{4}$ at the first mark.

To locate $-\frac{1}{4},$ divide the interval between $0$ and $\mathrm{-1}$ into four equal parts. Plot $-\frac{1}{4}$ at the first mark to the left of $0.$

Since $1\frac{1}{3}$ is between $1$ and $2,$ divide the interval between $1$ and $2$ into three equal parts. Plot $1\frac{1}{3}$ at the first mark to the right of $1.$ Then since $\mathrm{-1}\frac{1}{3}$ is the opposite of $1\frac{1}{3}$ it is between $\mathrm{-1}$ and $\mathrm{-2}.$ Divide the interval between $\mathrm{-1}$ and $\mathrm{-2}$ into three equal parts. Plot $\mathrm{-1}\frac{1}{3}$ at the first mark to the left of $\mathrm{-1}.$

To locate $\frac{5}{2}$ and $-\frac{5}{2},$ it may be helpful to rewrite them as the mixed numbers $2\frac{1}{2}$ and $\mathrm{-2}\frac{1}{2}.$

Since $2\frac{1}{2}$ is between $2$ and $3,$ divide the interval between $2$ and $3$ into two equal parts. Plot $\frac{5}{2}$ at the mark. Then since $\mathrm{-2}\frac{1}{2}$ is between $\mathrm{-2}$ and $\mathrm{-3},$ divide the interval between $\mathrm{-2}$ and $\mathrm{-3}$ into two equal parts. Plot $-\frac{5}{2}$ at the mark.

Order Fractions and Mixed Numbers

We can use the inequality symbols to order fractions. Remember that $a>b$ means that $a$ is to the right of $b$ on the number line. As we move from left to right on a number line, the values increase.

Example 4.18

Order each of the following pairs of numbers, using $<$ or $>:$

1. ⓐ$-\frac{2}{3}\_\_\_\_\mathrm{-1}$
2. ⓑ$\mathrm{-3}\frac{1}{2}\_\_\_\_\mathrm{-3}$
3. ⓒ$-\frac{3}{7}\_\_\_\_-\frac{3}{8}$
4. ⓓ$\mathrm{-2}\_\_\_\_\frac{\mathrm{-16}}{9}$

Solution

ⓐ $-\frac{2}{3}>\mathrm{-1}$

ⓑ $\mathrm{-3}\frac{1}{2}<\mathrm{-3}$

ⓒ $-\frac{3}{7}\text{<}-\frac{3}{8}$

ⓓ $\mathrm{-2}<\frac{\mathrm{-16}}{9}$