Locating and Ordering Fractions and Mixed Numbers on the Number Line

Learning Outcomes

Locate fractions and mixed numbers on the number line
Order fractions and mixed numbers using lists and inequalities

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

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

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

A number line is shown with the numbers 3, 4, and 5. There are red dots at 3 and at 5.
The proper fractions listed are ${\dfrac{1}{5}}$ and ${\dfrac{4}{5}}$ . We know proper fractions have values less than one, so ${\dfrac{1}{5}}$ and ${\dfrac{4}{5}}$ 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 ${\dfrac{1}{5},\dfrac{2}{5},\dfrac{3}{5}}$ , and ${\dfrac{4}{5}}$ .

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

A number line is shown. It shows 0, 1 fifth, 2 fifths, 3 fifths, 4 fifths, and 1. There are red dots at 1 fifth and at 4 fifths.
The only mixed number to plot is $3{\dfrac{1}{3}}$ . Between what two whole numbers is $3{\dfrac{1}{3}}$ ? Remember that a mixed number is a whole number plus a proper fraction, so $3{\dfrac{1}{3}}>3$ . Since it is greater than $3$ , but not a whole unit greater, $3{\dfrac{1}{3}}$ 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 $3{\dfrac{1}{3}}$ at the first mark.

A number line is shown with whole number 0 through 5. Between 3 and 4, 3 and 1 third and 3 and 2 thirds are labeled. There is a red dot at 3 and 1 third.
Finally, look at the improper fractions ${\dfrac{7}{4},\dfrac{9}{2}}$ , and ${\dfrac{8}{3}}$ . Locating these points will be easier if you change each of them to a mixed number.

${\dfrac{7}{4}}=1{\dfrac{3}{4}},{\dfrac{9}{2}}=4{\dfrac{1}{2}},{\dfrac{8}{3}}=2{\dfrac{2}{3}}$

Here is the number line with all the points plotted.

A number line is shown with whole numbers 0 through 6. Between 0 and 1, 1 fifth and 4 fifths are labeled and shown with red dots. Between 1 and 2, 7 fourths is labeled and shown with a red dot. Between 2 and 3, 8 thirds is labeled and shown with a red dot. Between 3 and 4, 3 and 1 third is labeled and shown with a red dot. Between 4 and 5, 9 halves is labeled and shown with a red dot.

Example

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

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

A number line is shown. It shows 0, 1 fourth, 2 fourths, 3 fourths, and 1. There is a red dot at 3 fourths.
Next, locate the mixed number $4{\dfrac{1}{5}}$ . 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 $4{\dfrac{1}{5}}$ one-fifth of the way between $4$ and $5$ .

A number line is shown. It shows 4, 4 and 1 fifth, 4 and 2 fifths, 4 and 3 fifths, 4 and 4 fifths, and 5. There is a red dot at 4 and 1 fifth.
Now locate the improper fractions ${\dfrac{4}{3}}$ and ${\dfrac{5}{3}}$ . It is easier to plot them if we convert them to mixed numbers first.

${\dfrac{4}{3}}=1{\dfrac{1}{3}},{\dfrac{5}{3}}=1{\dfrac{2}{3}}$
Divide the distance between $1$ and $2$ into thirds.

A number line is shown. It shows 1, 1 and 1 third, 1 and 2 thirds, and 2. Below 1 it says 3 thirds. Below 1 and 1 third it says 4 thirds. Below 1 and 2 thirds it says 5 thirds. Below 2 it says 6 thirds. There are red dots at 1 and 1 third and 1 and 2 thirds.
Next let us plot ${\dfrac{7}{2}}$ . We write it as a mixed number, ${\dfrac{7}{2}}=3{\dfrac{1}{2}}$ . Plot it between $3$ and $4$ .

A number line is shown. It shows 3, 3 and 1 half, and 4. Below 3 it says 6 halves. Below 3 and 1 half it says 7 halves. Below 4 it says 8 halves. There is a red dot at 3 and 1 half.
The number line shows all the numbers located on the number line.

try it

Watch the following video to see more examples of how to locate fractions on a number line.

We previously 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 $-7$ and the opposite of $-7$ is $7$ .

A number line is shown. It shows the numbers negative 7, 0 and 7. There are red dots at negative 7 and 7. The space between negative 7 and 0 is labeled as 7 units. The space between 0 and 7 is labeled as 7 units.
Fractions have opposites, too. The opposite of ${\dfrac{3}{4}}$ is $-{\dfrac{3}{4}}$ . It is the same distance from $0$ on the number line, but on the opposite side of $0$ .

A number line is shown. It shows the numbers negative 1, negative 3 fourths, 0, 3 fourths, and 1. There are red dots at negative 3 fourths and 3 fourths. The space between negative 3 fourths and 0 is labeled as 3 fourths of a unit. The space between 0 and 3 fourths is labeled as 3 fourths of a unit.
Thinking of negative fractions as the opposite of positive fractions will help us locate them on the number line. To locate $-{\dfrac{15}{8}}$ on the number line, first think of where ${\dfrac{15}{8}}$ is located. It is an improper fraction, so we first convert it to the mixed number $1{\dfrac{7}{8}}$ and see that it will be between $1$ and $2$ on the number line. So its opposite, $-{\dfrac{15}{8}}$ , will be between $-1$ and $-2$ on the number line.

Example

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

Show Solution

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

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

A number line is shown. It shows the numbers negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, and 4. There are 4 tick marks between negative 1 and 0. There are 4 tick marks between 0 and 1. The first tick mark between 0 and 1 is labeled as 1 fourth and marked with a red dot. The first tick mark between 0 and negative 1 is labeled as negative 1 fourth and marked with a red dot.
Since $1{\dfrac{1}{3}}$ is between $1$ and $2$ , divide the interval between $1$ and $2$ into three equal parts. Plot $1{\dfrac{1}{3}}$ at the first mark to the right of $1$ . Then since $-1{\dfrac{1}{3}}$ is the opposite of $1{\dfrac{1}{3}}$ it is between $-1$ and $-2$ . Divide the interval between $-1$ and $-2$ into three equal parts. Plot $-1{\dfrac{1}{3}}$ at the first mark to the left of $-1$ .

A number line is shown. The integers from negative 2 to 2 are labeled. Between negative 2 and negative 1, negative 1 and 1 third is labeled and marked with a red dot. Between 1 and 2, 1 and 1 third is labeled and marked with a red dot.
To locate ${\dfrac{5}{2}}$ and $-{\dfrac{5}{2}}$ , it may be helpful to rewrite them as the mixed numbers $2{\dfrac{1}{2}}$ and $-2{\dfrac{1}{2}}$ .

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

A number line is shown. The integers from negative 4 to 4 are labeled. Between negative 3 and negative 2, negative 5 halves is labeled and marked with a red dot. Between 2 and 3, 5 halves is labeled and marked with a red dot.

Try it

In the next video we give more examples of how to locate negative and positive fractions on a number line.

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

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

$-{\dfrac{2}{3}}$ ____ $- 1$
$-3{\dfrac{1}{2}}$ ____ $- 3$
$-{\dfrac{3}{7}}$ ____ ${\dfrac{3}{8}}$
$-2$ ____ ${\dfrac{-16}{9}}$

Show Solution

Solution:

1. $-{\dfrac{2}{3}}>-1$

A number line is shown. The integers from negative 3 to 3 are labeled. Negative 1 is marked with a red dot. Between negative 1 and 0, negative 2 thirds is labeled and marked with a red dot.

2. $-3{\dfrac{1}{2}}<-3$ A number line is shown. The integers from negative 4 to 4 are labeled. There is a red dot at negative 3. Between negative 4 and negative 3, negative 3 and one half is labeled and marked with a red dot.

3. $-{\dfrac{3}{7}}<-{\dfrac{3}{8}}$ A number line is shown. The numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3 are labeled. Between negative 1 and 0, negative 3 sevenths and negative 3 eighths are labeled and marked with red dots.

4. $-2<{\dfrac{-16}{9}}$ A number line is shown. The numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3 are labeled. There is a red dot at negative 2. Between negative 2 and negative 1, negative 16 over 9 is labeled and marked with a red dot.

Try it

In the following video we show another example of how to order integers, fractions and mixed numbers using inequality symbols.

Module 4: Fractions