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. Doing the Manipulative Mathematics activity "Number Line Part

3

" will help you develop a better understanding of the location of fractions on the number line. 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. 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

\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}

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\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. 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.

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

\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.

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

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:

\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}

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\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

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

\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. 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.

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

\frac{7}{2}

. We write it as a mixed number,

\frac{7}{2}=3\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.

A number line is shown. It shows the whole numbers 0 through 5. Between any 2 numbers are 10 tick marks. Between 0 and 1, between the 7th and 8th tick mark, 3 fourths is labeled and shown with a red dot. Between 1 and 2, 4 thirds and 5 thirds are labeled and shown with red dots. Between 3 and 4, 7 halves is labeled and shown with a red dot. Between 4 and 5, 4 and 1 fifth is labeled and shown with a red dot.

try it

#146007 [ohm_question height="270"]146007[/ohm_question] #146008 [ohm_question height="270"]146008[/ohm_question]

Watch the following video to see more examples of how to locate fractions on a number line. https://youtu.be/EIdmdTRQWTE 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

-7

and the opposite of

-7

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

\frac{3}{4}

-\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

-\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

-1

and

-2

on the number line.

A number line is shown. It shows the numbers negative 2, negative 1, 0, 1, and 2. Between negative 2 and negative 1, negative 1 and 7 eighths is labeled and marked with a red dot. The distance between negative 1 and 7 eighths and 0 is marked as 15 eighths units. Between 1 and 2, 1 and 7 eighths is labeled and marked with a red dot. The distance between 0 and 1 and 7 eighths is marked as 15 eighths units.

Example

Locate and label the following on the number line:

\frac{1}{4},-\frac{1}{4},1\frac{1}{3},-1\frac{1}{3},\frac{5}{2}

, and

-\frac{5}{2}

Answer: 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. 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. To locate $-\frac{1}{4}$ , divide the interval between $0$ and $-1$ into four equal parts. Plot $-\frac{1}{4}$ at the first mark to the left of $0$ . 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 $1\frac{1}{3}$ 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 $-1\frac{1}{3}$ is the opposite of $1\frac{1}{3}$ it 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 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 $\frac{5}{2}$ and $-\frac{5}{2}$ , it may be helpful to rewrite them as the mixed numbers $2\frac{1}{2}$ and $-2\frac{1}{2}$ . 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 $-2\frac{1}{2}$ is between $-2$ and $-3$ , divide the interval between $-2$ and $-3$ into two equal parts. Plot $-\frac{5}{2}$ at the mark. 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

#146009 [ohm_question height="270"]146009[/ohm_question] #146011 [ohm_question height="270"]146011[/ohm_question]

In teh next video we give more examples of how to locate negative and positive fractions on a number line. https://youtu.be/nbSlAAZVQV4

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

>:

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

Answer: Solution: 1. $-\frac{2}{3}>-1$ 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. $-3\frac{1}{2}<-3$ 3. $-\frac{3}{7}\text{<}-\frac{3}{8}$ 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. $-2<\frac{-16}{9}$

Try it

#146013 [ohm_question height="270"]146013[/ohm_question] #146012 [ohm_question height="270"]146012[/ohm_question]

In the following video we show another example of how to order integers, fractions and mixed numbers using inequality symbols. https://youtu.be/Phsf_fJgerc

Locating and Ordering Fractions and Mixed Numbers on the Number Line

Learning Outcomes

Example

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Example

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Order Fractions and Mixed Numbers

Example

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Study Guides > Prealgebra

Locating and Ordering Fractions and Mixed Numbers on the Number Line

Learning Outcomes

Example

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Example

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Order Fractions and Mixed Numbers

Example

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Licenses & Attributions

CC licensed content, Original

CC licensed content, Shared previously

CC licensed content, Specific attribution