Topics
8. Fractions
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AustraliaNSW
Stage 2

8.16 Equivalent fractions

Lesson
Practice
Lesson

Are you ready?

Can you identify the number and size of equal parts?

Examples

Example 1

Here is a shape divided into parts, use it to answer the following questions.

A square divided into 4 equal parts.
a

This shape has ⬚ equal parts.

Worked Solution
Create a strategy

Count the number of smaller squares that make up the big square.

Apply the idea

This shape has 4 equal parts.

b

Each part is \dfrac{⬚}{⬚} of the whole.

Worked Solution
Create a strategy

We can write this fraction as:

A fraction with parts explained. Ask your teacher for more information.
Apply the idea

There is 1 shaded part out of 4. Each part is \dfrac{1}{4} of the whole.

Idea summary

For area models:

  • 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.

Equivalent fractions

How can we identify equivalent fractions? This video will show us.

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Examples

Example 2

Fill in the blank to find an equivalent fraction to \dfrac{1}{3}:

\dfrac{1}{3}= \dfrac{⬚}{6}

Worked Solution
Create a strategy

Use fraction area models.

Apply the idea

On the left of the equals sign we have \dfrac{1}{3} which looks like this:

A rectangle divided into 3 equal parts with 1 shaded part.

1 out of the 3 squares are shaded. We want to write this as a fraction of 6. Dividing the model into 6 parts would look like this:

A rectangle divided into 6 equal parts with 2 parts shaded.

We can see that 2 out of 6 parts are shaded to get the same area. So:

\displaystyle \frac{1}{3}\displaystyle =\displaystyle \frac{2}{6}
Reflect and check

We also could have multiplied the numerator and denominator by 2 since 3\times 2=6.

\displaystyle \dfrac{1}{3}\displaystyle =\displaystyle \dfrac{1 \times 2}{3 \times 2}Multiply the numerator and denominator by 2
\displaystyle =\displaystyle \dfrac{2}{6}

Compare fractions with different denominators

Here we look at comparing using equivalent fractions.

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Exploration

Using the geogebra applet, slide the slider to see what some fractions out of 8 look like.

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The number of shaded parts equals the top number (numerator).

Examples

Example 3

We are going to compare the two fractions \dfrac{1}{8} and \dfrac{1}{4}.

a

Turn \dfrac{1}{4} into a fraction in eighths.

Worked Solution
Create a strategy

Multiply the numerator and denominator by 2.

Apply the idea

Since 4 \times 2 =8, we should multiply the numerator and denominator by 2 to change the fraction into eighths.

\displaystyle \dfrac{1}{4}\displaystyle =\displaystyle \dfrac{1}{4} \times \dfrac{2}{2}Multiply the top and bottom by 2
\displaystyle =\displaystyle \dfrac{2}{8}Perform the multiplications
b

Which fraction is larger?

A
The fractions are the same size.
B
\dfrac{1}{8}
C
\dfrac{1}{4}
Worked Solution
Create a strategy

Remember that \dfrac{1}{4} = \dfrac{2}{8} as found in part (a).

Apply the idea

\dfrac{1}{4}=\dfrac{2}{8} which is shown in the model below:

\dfrac{1}{8} is shown in the model below:

The model for \dfrac{1}{4} has more shaded parts than the model for \dfrac{1}{8}.

So \dfrac{1}{4} is the larger fraction, the answer is option C.

Idea summary

To compare fractions, model both fractions using area models with the same number of parts. Then count the number of shaded parts in each model.

Whole numbers as fractions

This next video shows us that whole numbers can also be written as fractions.

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Examples

Example 4

Fill in the blank to find an equivalent fraction to 5:

5= \dfrac{⬚}{2}

Worked Solution
Create a strategy

Write 5 as a fraction and then multiply the numerator and denominator to find the equivalent fraction.

Apply the idea

5 as a fraction is \dfrac{5}{1}.

But we want a denominator of 2. So we need to multiply the denominator and the numerator by 2.

\displaystyle \dfrac{5}{1}\displaystyle =\displaystyle \dfrac{5 \times 2}{1 \times 2}Multiply by 2
\displaystyle =\displaystyle \dfrac{10}{2}

5=\dfrac{10}{2}

Idea summary

Equivalent fractions represent the same size, but have different numerators and denominators.

To find an equivalent fraction we can multiply the numerator and denominator by the same number.

Outcomes

MA2-7NA

represents, models and compares commonly used fractions and decimals

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