5.05 Compare fractions

Lesson

Let's practice finding  relatable denominators  to help us in this lesson.

Examples

Example 1

Which fraction has a denominator that is a multiple of the denominator in \dfrac{2}{4}.

A
\dfrac{6}{7}
B
\dfrac{8}{9}
C
\dfrac{7}{8}
Worked Solution
Create a strategy

List the multiples of 4 to see if the other denominators are multiples of 4.

Apply the idea

The first 3 multiples of 4 are 4,\,8,\,12.

Among the choices, the fraction \dfrac{7}{8} has a denominator 8 that is in the list of multiples. So, the correct answer is Option C.

Idea summary

Look for common multiples between the denominators of two different fractions as this will help us to find common denominators.

Compare fractions using area models

We can compare fractions using area models.

Examples

Example 2

Which fraction is smaller?

A
B
Worked Solution
Create a strategy

Compare the number of parts shaded.

Apply the idea

Both options have the same total number of parts. Option A has 2 shaded parts and option B has 3 shaded parts.

Since 2 \lt 3, \, \,\dfrac{2}{6} is smaller than \dfrac{3}{6}.

The correct option is A.

Idea summary
• When comparing fractions, if the denominators are the same, then we can compare the numerators.

• The denominator also tells us how many parts make up one whole.

Compare fractions greater than one

We can compare improper fractions and mixed numbers.

Examples

Example 3

Think about the fractions \dfrac{8}{6} and \dfrac{8}{3}.

a

Plot the number \dfrac{8}{6} on the number line.

Worked Solution
Create a strategy

The numerator tells us how many spaces to jump to the right from 0.

Apply the idea

Each of the 6 spaces represents \dfrac{1}{6}.

Count 8 spaces to the right from 0 and plot \dfrac{8}{6}.

b

Plot the number \dfrac{8}{3} on the number line.

Worked Solution
Apply the idea

Each of the 3 spaces represents \dfrac{1}{3}.

Count 8 spaces to the right from 0 and plot the point \dfrac{8}{3}.

c

Which fraction is bigger?

A
\dfrac{8}{3}
B
\dfrac{8}{6}
Worked Solution
Create a strategy

Compare the plotted points from parts (a) and (b). The larger number is further to the right on the number line.

Apply the idea

Here is \dfrac{8}{6} on the number line:

Here is \dfrac{8}{3} on the number line:

We can see that \dfrac{8}{3} is further to the right, so it is the bigger number.

Idea summary

To plot a proper fraction on a number line:

• Start the number line at 0 and end it at 1.

• Divide the number line into the number of parts equal to the denominator.

• From 0, count to the right the number of parts equal to the numerator.

• Plot the point.

To compare fractions on a number line, the fraction furthest to the right is the largest.

Compare fractions with different denominators

Let's see now how to compare fractions that have different denominators.

Examples

Example 4

Let's compare the fractions \dfrac{1}{2} and \dfrac{7}{8}.

a

Change \dfrac{1}{2} into eighths.

Worked Solution
Create a strategy

Count how many \dfrac{1}{8} bricks have the same length as one \dfrac{1}{2} brick.

Apply the idea

Based on the diagram, one \dfrac{1}{2} brick has the same length as four \dfrac{1}{8} bricks. So:\dfrac{1}{2}=\dfrac{4}{8}

b

Which of the following is the correct number sentence?

A
\dfrac{1}{2}<\dfrac{7}{8}
B
\dfrac{1}{2}=\dfrac{7}{8}
C
\dfrac{1}{2}>\dfrac{7}{8}
Worked Solution
Create a strategy

Compare the result from part (a) to \dfrac{7}{8}.

Apply the idea

From part (a) \dfrac{1}{2}=\dfrac{4}{8}. Since \dfrac{4}{8} and \dfrac{7}{8} have the same denominator, we can compare their numerators.

Since 4\lt 7:\dfrac{4}{8} < \dfrac{7}{8}So, the correct answer is Option A.

Idea summary

When comparing fractions that are close in value, it helps for them to have a common denominator. Drawing a fraction rectangle shows the value of the fraction.

Outcomes

VCMNA211

Compare fractions with related denominators and locate and represent them on a number line