Topics
4. Fractions
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AustraliaNSW
Stage 3

4.05 Compare fractions

Lesson
Practice
Lesson

Are you ready?

Do you remember how to  name a fraction  using a model?

Examples

Example 1

Which of the following shows \dfrac{1}{10} of the area of the shape shaded?

A
A rectangle divided into 10 equal parts. 2 parts are shaded.
B
A rectangle divided into 11 equal parts. 1 part is shaded.
C
A rectangle divided into 10 equal parts. 1 parts are shaded.
D
A rectangle divided into 9 equal parts. 1 part is shaded.
Worked Solution
Create a strategy

The numerator tells us how many parts should be shaded. The denominator tells us how many parts to divide the shape into.

Apply the idea

The fraction \dfrac{1}{10} is asking for one part of the shape to be shaded out of 10 total parts. The shape in option C has 10 total parts with 1 shaded part.

The answer is option C.

Idea summary

  • The numerator (top number) is the number of parts shaded to represent the fraction.

  • The denominator (bottom number) is the number of equal parts the shape is divided into.

Fraction benchmarks

Benchmarks are numbers that we use to make things easier, with fractions we often benchmark with the numbers 0, \, \dfrac{1}{2} or 1. This video shows how to see if fractions are closer to 0, \, \dfrac{1}{2} or 1.

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Examples

Example 2

Is the fraction \dfrac{1}{4} closer to 0 or 1?

Worked Solution
Create a strategy

Plot the fraction on a number line.

Apply the idea

To plot \dfrac{1}{4} on a number line, divide the line between 0 and 1 into 4 spaces and jump right 1 space from 0.

0\frac{1}{4}\frac{1}{2}\frac{3}{4}1

We can see that \dfrac{1}{4} is less than \dfrac{1}{2}, so it is closer 0.

Idea summary

Fractions are often benchmarked with the numbers 0, \, \dfrac{1}{2} or 1.

The halfway point between 0 and 1 is \dfrac{1}{2}.

If the fraction is less than halfway, then it is closer 0.

If the fraction is more than halfway, then it is closer to 1.

Compare fractions

Now that you have seen how to benchmark fractions, let's see how to use benchmarks to help us compare fraction sizes.

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Examples

Example 3

Let's compare the fractions \dfrac{2}{5} and \dfrac{3}{4}.

a

Is the fraction \dfrac{2}{5} closer to 0 or 1?

Worked Solution
Create a strategy

Plot the fraction on a number line.

Apply the idea

To plot \dfrac{2}{5} on the number line, we divide the line into 5 spaces and jump 2 spaces to the right from 0.

0\frac{1}{5}\frac{2}{5}\frac{3}{5}\frac{4}{5}1

We can see that \dfrac{2}{5} is 2 spaces away from 0 and 3 spaces away from 1. So \dfrac{2}{5} is closer 0.

b

Is the fraction \dfrac{3}{4} closer to 0 or 1?

Worked Solution
Create a strategy

Plot the fraction on a number line.

Apply the idea

To plot \dfrac{3}{4} on the number line, we divide the line into 4 spaces and jump 3 spaces to the right from 0.

0\frac{1}{4}\frac{1}{2}\frac{3}{4}1

We can see that \dfrac{3}{4} is 3 spaces away from 0 and 1 space away from 1. So \dfrac{3}{4} is closer 1.

c

Use greater than (\gt) or less than (\lt) symbol in the box to make this number sentence true:\dfrac{2}{5} \, ⬚ \, \dfrac{3}{4}

Worked Solution
Create a strategy

Compare the plotted points. The larger number is closer to 1.

Apply the idea

In parts (a) and (b), we found that \dfrac{2}{5} is closer 0 and \dfrac{3}{4} is closer 1.

So \dfrac{2}{5} is smaller than \dfrac{3}{4}.\dfrac{2}{5} \, \lt \, \dfrac{3}{4}

Idea summary

When comparing fractions, if the denominator is the same, then we compare the numerator.

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

We can use the benchmarks of 0 and 1 to compare fractions.

Fractions with large denominators

What if the denominator of the fraction makes it hard to know how big it is? Benchmarking can help us with that as well.

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Examples

Example 4

Is the fraction \dfrac{9}{32} closer to 0 or \dfrac{1}{2}?

Worked Solution
Create a strategy

Plot the fraction on a number line.

Apply the idea

Here is the number line from 0 to 1. Each space represents \dfrac{1}{32}.

0\frac{1}{2}1

Count 9 spaces to the right from 0 to plot the number \dfrac{9}{32}.

0\frac{1}{2}1

We can see that \dfrac{9}{32} is 9 spaces away from 0 and 7 spaces away from \dfrac{1}{2}. So \dfrac{9}{32} is closer \dfrac{1}{2}.

Idea summary

We can plot a given fraction and the benchmarks 0, \, \dfrac{1}{2} and 1 on a number line to see which benchmark the fraction is closest to.

Outcomes

MA3-7NA

compares, orders and calculates with fractions, decimals and percentages

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