We are ready now to combine what we have learned about directed numbers with our knowledge of fractions. Just like we saw with integers, we can order and compare fractions and perform addition and subtraction of fractions using the number line.

On the number line below, each tick is labelled with a multiple of the fraction \dfrac15. We can see that the point furthest to the left is plotted at the fraction -\dfrac35, and the point furthest to the right is plotted at the fraction \dfrac65.

Number line from negative seven fifths to seven fifths. Points are plotted at negative three fifths and six fifths.

It is common to see number lines where only the integers are labelled, with ticks between each integer that represent a fraction of one whole.

A point has been plotted on this number line. Can you see which fraction the point lies on?

We can find the denominator of the fraction using the number of ticks between each integer. To go from 0 to 1 we need to move up 3 ticks, so each tick represents \dfrac13. To find the numerator of the fraction we can count the ticks from 0 to the point, which gives 8. This means that the point is plotted at the fraction \dfrac83.

Another way to identify the fraction is to see that the point is two thirds to the right of the integer 2. So it lies on the number 2+\dfrac23, which we can write as the mixed number 2\dfrac23.

Exploration

Use the following applet to plot fractions on a number line.

Follow the directions in the applet and click New Numbers to explore more fractions on the number line.

Loading interactive...

The number of spaces between the numbers is the denominator and the spaces between the point and 0 is the numerator.

Fractions and mixed numerals can be plotted on the number line.

Examples

Example 1

Where is the point plotted on the number line?

Worked Solution

Create a strategy

The number of spaces between the numbers is the denominator and the spaces between the point and 0 is the numerator.

Apply the idea

There are 5 spaces between the numbers which means the denominator is 5.

For the numerator, add the spaces from 0 to 1 and the spaces between -1 and the point:5+3=8

But the point is to the left of 0 which means it is negative.

The plotted point is -\dfrac85.

Idea summary

The number of spaces between the numbers is the denominator and the spaces between the point and 0 is the numerator of the plotted fraction on a number line.

Fractions with different denominators

If we look at any two integers, it is simple to see which is greater. But if we are given two fractions, it can be less obvious to see which is greater if the fractions do not have the same denominator.

The point on this number line is -\dfrac56.

If we wanted to plot another point at the fraction -\dfrac23, we need to first write \dfrac23 as an equivalent fraction with a denominator of 6.

This image shows a number line from negative 2 to 1 with plotted fractions. Ask your teacher for more information.

Since \dfrac13 is the same as \dfrac{1\times2}{3\times2}=\dfrac26, then -\dfrac23 is the same as -\dfrac{2\times 2}{3\times 2}=-\dfrac{4}{6}.

Here are both points plotted on the same number line.

Now let's compare -\dfrac13 and -\dfrac25. In this case we need to rewrite both fractions so that they have a common denominator. Let's choose a denominator of 15, which is the lowest common denominator of the two fractions. We rewrite -\dfrac13 as -\dfrac{1\times 5}{3\times 5}=-\dfrac{5}{15} and we rewrite -\dfrac25 as -\dfrac{2\times 3}{5\times 3}=-\dfrac{6}{15}.

This image shows a number line from negative 1 to 0 with plotted fractions. Ask your teacher for more information.

Here are both fractions plotted on the same number line.

Examples

Example 2

Which number is greatest?

-\dfrac53

-\dfrac52

-\dfrac73

Worked Solution

Create a strategy

Rewite the fractions with their common denominator and plot them on a number line.

Apply the idea

The lowest common denominator of the fractions is 3\times2=6.

Option A

\displaystyle -\dfrac53	\displaystyle =	\displaystyle -\dfrac{5\times 2}{3\times 2}	Multiply both parts by 2
	\displaystyle =	\displaystyle -\dfrac{10}{6}	Evaluate

Option B

\displaystyle -\dfrac52	\displaystyle =	\displaystyle -\dfrac{5\times3}{2\times3}	Multiply both parts by 3
	\displaystyle =	\displaystyle -\dfrac{15}{6}	Evaluate

Option C

\displaystyle -\dfrac73	\displaystyle =	\displaystyle -\dfrac{7\times2}{3\times2}	Multiply both parts by 2
	\displaystyle =	\displaystyle -\dfrac{14}{6}	Evaluate

Plotting the fractions, we have:

Since -\dfrac{10}{6}=-\dfrac53 is the furthest right, we can say that -\dfrac53 is the greatest number.

So the answer is option A.

Idea summary

We need to find the lowest common denominator of the fractions to compare them.

Arithmetic with directed fractions

Adding and subtracting positive and negative fractions works in the same way as adding and subtracting positive and negative integers.

Examples

Example 3

Find the value of \,2\dfrac{2}{9}-\left(-\dfrac{5}{9}\right).

Worked Solution

Create a strategy

Recall that adjacent same signs results to a positive sign when combined.

Use a number line using the denominator as the number of spaces between two whole numbers.

Apply the idea

\displaystyle 2\dfrac29-\left(-\dfrac59\right)

\displaystyle =

\displaystyle 2\dfrac29+\dfrac59

Combine the adjacent signs

The right-hand side tells us that we need to add 5 units to 2\dfrac29 on the number line.

We landed on 2\dfrac79.

2\dfrac{2}{9}-\left(-\dfrac{5}{9}\right)=2\dfrac79

Reflect and check

We could also find this answer using arithmetic:

\displaystyle 2\dfrac29-\left(-\dfrac59\right)	\displaystyle =	\displaystyle 2\dfrac29+\dfrac59	Combine the adjacent signs
	\displaystyle =	\displaystyle 2+\dfrac29+\dfrac59	Group the fraction parts
	\displaystyle =	\displaystyle 2+\dfrac{7}{9}	Add the numerators
	\displaystyle =	\displaystyle 2\dfrac{7}{9}	Write as a mixed numeral

Idea summary

Addition and subtraction of directed fractions can be done the same way as the addition and subtraction with positive and negative integers.

Adjacent opposite signs results in a negative sign when combined.
Adjacent same signs results in a positive sign when combined.

Outcomes

MA4-4NA

compares, orders and calculates with integers, applying a range of strategies to aid computation

7.04 Fractions and the number line

Ideas

Fractions on the number line

Exploration

Examples

Example 1

Fractions with different denominators

Examples

Example 2

Arithmetic with directed fractions

Examples

Example 3

Outcomes

MA4-4NA

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