### 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 [latex]{\Large\frac{1}{5},\frac{4}{5}},3,3{\Large\frac{1}{3},\frac{7}{4},\frac{9}{2}},5[/latex], and [latex]{\Large\frac{8}{3}}[/latex] on the number line.

We will start with the whole numbers [latex]3[/latex] and [latex]5[/latex] because they are the easiest to plot.

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

Now plot points at [latex]{\Large\frac{1}{5}}[/latex] and [latex]{\Large\frac{4}{5}}[/latex].

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

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

[latex]{\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}}[/latex]

Here is the number line with all the points plotted.

### Example

Locate and label the following on a number line: [latex]{\Large\frac{3}{4},\frac{4}{3},\frac{5}{3}},4{\Large\frac{1}{5}}[/latex], and [latex]{\Large\frac{7}{2}}[/latex].

Solution:

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

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

Now locate the improper fractions [latex]{\Large\frac{4}{3}}[/latex] and [latex]{\Large\frac{5}{3}}[/latex]. It is easier to plot them if we convert them to mixed numbers first.

[latex]{\Large\frac{4}{3}}=1{\Large\frac{1}{3}},{\Large\frac{5}{3}}=1{\Large\frac{2}{3}}[/latex]

Divide the distance between [latex]1[/latex] and [latex]2[/latex] into thirds.

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

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 [latex]7[/latex] is [latex]-7[/latex] and the opposite of [latex]-7[/latex] is [latex]7[/latex].

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

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

### Example

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

### 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 [latex]a>b[/latex] means that [latex]a[/latex] is to the right of [latex]b[/latex] 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 [latex]<[/latex]; or [latex]>:[/latex]

- [latex]-{\Large\frac{2}{3}}[/latex] ____ [latex]- 1[/latex]
- [latex]-3{\Large\frac{1}{2}}[/latex] ____ [latex]- 3[/latex]
- [latex]-{\Large\frac{3}{7}}[/latex] ____ [latex]{\Large\frac{3}{8}}[/latex]
- [latex]-2[/latex] ____ [latex]{\Large\frac{-16}{9}}[/latex]

### Try it

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