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 15,45,3,313,74,92,5, and 83 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 15 and 45. We know proper fractions have values less than one, so 15 and 45 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 15,25,35, and 45.

Now plot points at 15 and 45.

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 313. Between what two whole numbers is 313? Remember that a mixed number is a whole number plus a proper fraction, so 313>3. Since it is greater than 3, but not a whole unit greater, 313 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 313 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 74,92, and 83. Locating these points will be easier if you change each of them to a mixed number.

74=134,92=412,83=223

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: 34,43,53,415, and 72.

Solution:
Start by locating the proper fraction 34. It is between 0 and 1. To do this, divide the distance between 0 and 1 into four equal parts. Then plot 34.

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 415. 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 415 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 43 and 53. It is easier to plot them if we convert them to mixed numbers first.

43=113,53=123
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 72. We write it as a mixed number, 72=312 . 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

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 34 is 34. 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 158 on the number line, first think of where 158 is located. It is an improper fraction, so we first convert it to the mixed number 178 and see that it will be between 1 and 2 on the number line. So its opposite, 158, 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: 14,14,113,113,52, and 52.

 

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 >:

  1. 23 ____ 1
  2. 312 ____ 3
  3. 37 ____ 38
  4. 2 ____ 169

 

Try it

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