Locating and Ordering Fractions and Mixed Numbers on the Number Line

Learning Outcomes

Locate and label improper and proper fractions on a number line
Order fractions and mixed numbers on a number line
Use inequality symbols to compare fractions and mixed numbers

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

Let us locate ${\Large\frac{1}{5},\frac{4}{5}},3,3{\Large\frac{1}{3},\frac{7}{4},\frac{9}{2}},5$ , and ${\Large\frac{8}{3}}$ 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 ${\Large\frac{1}{5}}$ and ${\Large\frac{4}{5}}$ . We know proper fractions have values less than one, so ${\Large\frac{1}{5}}$ and ${\Large\frac{4}{5}}$ 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 ${\Large\frac{1}{5},\frac{2}{5},\frac{3}{5}}$ , and ${\Large\frac{4}{5}}$ .

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

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 $3{\Large\frac{1}{3}}$ . Between what two whole numbers is $3{\Large\frac{1}{3}}$ ? Remember that a mixed number is a whole number plus a proper fraction, so $3{\Large\frac{1}{3}}>3$ . Since it is greater than $3$ , but not a whole unit greater, $3{\Large\frac{1}{3}}$ 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 $3{\Large\frac{1}{3}}$ 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 ${\Large\frac{7}{4},\frac{9}{2}}$ , and ${\Large\frac{8}{3}}$ . Locating these points will be easier if you change each of them to a mixed number.

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

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: ${\Large\frac{3}{4},\frac{4}{3},\frac{5}{3}},4{\Large\frac{1}{5}}$ , and ${\Large\frac{7}{2}}$ .

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

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 $4{\Large\frac{1}{5}}$ . 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 $4{\Large\frac{1}{5}}$ 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 ${\Large\frac{4}{3}}$ and ${\Large\frac{5}{3}}$ . It is easier to plot them if we convert them to mixed numbers first.

${\Large\frac{4}{3}}=1{\Large\frac{1}{3}},{\Large\frac{5}{3}}=1{\Large\frac{2}{3}}$
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 ${\Large\frac{7}{2}}$ . We write it as a mixed number, ${\Large\frac{7}{2}}=3{\Large\frac{1}{2}}$ . 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.

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 $-7$ 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 ${\Large\frac{3}{4}}$ is $-{\Large\frac{3}{4}}$ . 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 $-{\Large\frac{15}{8}}$ on the number line, first think of where ${\Large\frac{15}{8}}$ is located. It is an improper fraction, so we first convert it to the mixed number $1{\Large\frac{7}{8}}$ and see that it will be between $1$ and $2$ on the number line. So its opposite, $-{\Large\frac{15}{8}}$ , will be between $-1$ and $-2$ on the number line.

Example

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

Show Solution

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

To locate ${\Large\frac{1}{4}}$ , divide the interval between $0$ and $1$ into four equal parts. Each part represents one-quarter of the distance. So plot ${\Large\frac{1}{4}}$ 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 $1{\Large\frac{1}{3}}$ is between $1$ and $2$ , divide the interval between $1$ and $2$ into three equal parts. Plot $1{\Large\frac{1}{3}}$ at the first mark to the right of $1$ . Then since $-1{\Large\frac{1}{3}}$ is the opposite of $1{\Large\frac{1}{3}}$ it is between $-1$ and $-2$ . Divide the interval between $-1$ and $-2$ into three equal parts. Plot $-1{\Large\frac{1}{3}}$ 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 ${\Large\frac{5}{2}}$ and $-{\Large\frac{5}{2}}$ , it may be helpful to rewrite them as the mixed numbers $2{\Large\frac{1}{2}}$ and $-2{\Large\frac{1}{2}}$ .

Since [latex]2{\Large\frac{1}{2}[/latex] is between $2$ and $3$ , divide the interval between $2$ and $3$ into two equal parts. Plot ${\Large\frac{5}{2}}$ at the mark. Then since $-2{\Large\frac{1}{2}}$ is between $-2$ and $-3$ , divide the interval between $-2$ and $-3$ into two equal parts. Plot $-{\Large\frac{5}{2}}$ 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 $>:$

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

Show Solution

Solution:

1. $-{\Large\frac{2}{3}}>-1$

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. $-3{\Large\frac{1}{2}}<-3$ 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. $-{\Large\frac{3}{7}}<-{\Large\frac{3}{8}}$ 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. $-2<{\Large\frac{-16}{9}}$ 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 3: Fractions