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2.02 Equivalent and simplified fractions

Lesson

Introduction

Sometimes multiple fractions can represent the same amount.

Consider the shaded area of the hexagon below.

A hexagon divided into 6 equal parts with 3  shaded.

We can see that 3 of the 6 parts have been shaded in, so the area represents \dfrac{3}{6} of the whole shape. But we can also see that half the shape has been shaded in, which we can write this as \dfrac{1}{2} of the whole shape.

Since the same area is shaded for both \dfrac{3}{6} and \dfrac{1}{2}, these two fractions must be equal.

These are called equivalent fractions, since they have different numerators and denominators but are still equal.

Equivalent fractions

We can see equivalent fractions in action by representing them visually and seeing how we can convert from one to the other.

Consider the grid below.

A rectangle with 3 out of 18 grid squares  shaded.

Of the 18 grid squares, 3 of them are shaded. This means that we can represent the shaded area with the fraction \dfrac{3}{18}.

But what about equivalent fractions?

Notice that the placement of shaded squares is the same on each row. So, by removing some of the grid lines, we get:

A rectangle with 1 out of 6 parts shaded. Ask your teacher for more information.

As we can see, the same fraction of the grid is shaded but now it is 1 out of 6 parts, so we can also represent this area as \dfrac{1}{6} of the grid.

This shows us that \dfrac{1}{16} and \dfrac{3}{18} are equivalent fractions, which we can express using the equivalence:\dfrac{1}{6} = \dfrac{3}{18}

Exploration

Explore the following applet to investigate equivalent fractions further.

Loading interactive...

For each fraction, we can find an equivalent fraction by multiplying both the numerator and denominator by the same number.

We can increase the denominator by multiplying it by any whole number. However, if we do this we also have to multiply the numerator by the same number.

Looking at \dfrac{1}{6} we can change 6 to 18 by multiplying it by 3. So we get:

\displaystyle \dfrac{1}{6}\displaystyle =\displaystyle \dfrac{1\times3}{6\times3}Multiply the numerator and denominator by 3
\displaystyle =\displaystyle \dfrac{3}{18}Evaluate the multiplications

Which gives us the same equivalent fractions as before.

We can also decrease the denominator by cancelling common factors in both the numerator and denominator. Since we know that\dfrac{3}{18} = \dfrac{1\times3}{6\times3}

And from cancelling out the 3s we get:\dfrac{3}{18} = \dfrac{1}{6}

We can write each step of the process like this. Notice that we could also follow the process in reverse to increase the denominator.

An image on how to simplify the fraction 3 over 18. Ask your teacher for more information.

Examples

Example 1

Select the two fractions which are equivalent to \dfrac{8}{12}.

A
\dfrac{20}{30}
B
\dfrac{7}{3}
C
\dfrac{2}{6}
D
\dfrac{2}{3}
Worked Solution
Create a strategy

Decrease the denominator by cancelling common factors in both the numerator and denominator. Increase the denominator by multiplying a number in both the numerator and denominator.

Apply the idea

The numerator, 8, and the denominator, 12 has a common factor of 4. So we can decrease the denominator by cancelling 4 in both the numerator and denominator.

\displaystyle \dfrac{8}{12}\displaystyle =\displaystyle \dfrac{2\times4}{3\times4}Take out the factor of 4
\displaystyle =\displaystyle \dfrac{2}{3}Cancel out 4

So \dfrac{2}{3} is equivalent to \dfrac{8}{12}.

Now, we can also increase \dfrac{2}{3} by multiplying the numerator and denominator by 10.

\displaystyle \dfrac{2}{3}\displaystyle =\displaystyle \dfrac{2\times10}{3\times10}Multiply the numerator and denominator by 10
\displaystyle =\displaystyle \dfrac{20}{30}Evaluate

So \dfrac{20}{30} is also equivalent to \dfrac{8}{12} since it has the same simplified form of \dfrac{2}{3}.

So options A and D are the correct answers.

Idea summary

The same amount of a whole can be written as many different fractions.

When two fractions represent the same amount of a whole they are equivalent fractions.

We can convert a fraction into an equivalent fraction by multiplying the numerator and denominator by the same number, or cancelling a common factor from the numerator and denominator.

Simplify fractions

When we cancel common factors from the numerator and denominator of a fraction, the numbers become smaller. We call this simplifying the fraction.

Examples

Example 2

In the grid below, \dfrac{20}{32} parts have been shaded. Use the grid to simplify \dfrac{20}{32}.

A grid 4 rows of 8 squares. The first 5 squares in each row are shaded. A total of 20 out of 32 squares are shaded.
Worked Solution
Create a strategy

Find the fraction of each row that is shaded.

Apply the idea

The grid can be divided into 4 identical rows. One row is pictured below.

A row of 8 squares with 5 shaded.

In one row, 5 out of 8 parts are shaded. So \dfrac{5}{8} are shaded.\dfrac{20}{32} = \dfrac{5}{8}

Reflect and check

We also could have simplified the fraction by writing the numerator and denominator as a multiple of a common factor, and then cancelling out the common factor.

The common factor of 20 and 32 is 4 so we can write both as multiples of 4:

\displaystyle \dfrac{20}{32}\displaystyle =\displaystyle \dfrac{5\times 4}{8 \times 4}Write both numbers as multiples of 4
\displaystyle =\displaystyle \dfrac{5}{8}Cancel out the 4
Idea summary

When we find an equivalent fraction by cancelling common factors from the numerator and denominator we are simplifying the fraction.

When the fraction has no common denominators between the numerator and denominator (other than 1) it is a fully simplified fraction.

Consequently, two fractions are equivalent when they can be simplified to the same fraction.

Compare fractions

In order to compare fractions with different denominators we can find equivalent fractions with the same denominator and compare the equivalent fractions.

For example, which is larger, \dfrac{5}{12} or \dfrac{3}{8}?

We can try comparing these fractions visually:

A rectangle divided into 12 equal parts with 5 shaded.

5 parts out of 12

A rectangle divided into 8 equal parts with 3 shaded.

3 parts out of 8

Because the size of a twelfth is different from the size of an eighth, we have to find equivalent fractions with the same denominator in order to compare these two fractions.

But what denominator should we try to get?

When comparing fractions we want to make their new denominators equal to the lowest common multiple of their current denominators.

Since 24 is the lowest common multiple of 12 and 8, we want to find the equivalent fractions of \dfrac{5}{12} and \dfrac{3}{8} that have a denominator of 24.

We can do this by multiplying the numerator and denominator of \dfrac{5}{12} by a factor of 2, and the numerator and denominator of \dfrac{3}{8} by a factor of 3,

\text{Multiply the numerator and denominator by }2\text{Multiply the numerator and denominator by }3
\dfrac{5}{12}\to\dfrac{5\times2}{12\times2}\to\dfrac{10}{24}\dfrac{3}{8}\to\dfrac{3\times3}{8\times3}\to\dfrac{9}{24}

Since \dfrac{10}{24} is larger than \dfrac{9}{24} we can see that \dfrac{5}{12} is larger than \dfrac{3}{8}.

We could also add some gridlines to the fraction grids so that they both have 24 equal parts in order to compare them.

A rectangle divided into 24 equal parts with 10 parts shaded.

10 parts out of 24

A rectangle divided into 24 equal parts with 9 parts shaded.

9 parts out of 24

We can see that after adding the gridlines, the grid for \dfrac{5}{12} now has \dfrac{10}{24} shaded parts and the grid for \dfrac{3}{8} now has \dfrac{9}{24} shaded parts. This tells us that \dfrac{5}{12} is greater than \dfrac{3}{8}.

Examples

Example 3

Arrange the following fractions from biggest to smallest. \dfrac{2}{3},\, \dfrac{3}{7},\, \dfrac{4}{5}

Worked Solution
Create a strategy

Find the lowest common denominator of the fractions.

Apply the idea

The lowest common multiple of 3,\,7, and 5 is 3\times 5 \times 7=105.

\displaystyle \dfrac{2}{3}\displaystyle =\displaystyle \dfrac{2\times35}{3\times35}Multiply the numerator and denominator by 35
\displaystyle =\displaystyle \dfrac{70}{105}Evaluate
\displaystyle \dfrac{3}{7}\displaystyle =\displaystyle \dfrac{3\times15}{7\times15}Multiply the numerator and denominator by 15
\displaystyle =\displaystyle \dfrac{45}{105}Evaluate
\displaystyle \dfrac{4}{5}\displaystyle =\displaystyle \dfrac{4\times21}{5\times21}Multiply the numerator and denominator by 21
\displaystyle =\displaystyle \dfrac{84}{105}Evaluate

Arranging these from biggest to smallest, we have: \dfrac{84}{105},\,\dfrac{70}{105},\,\dfrac{45}{105} So we have the following order for the original fractions: \dfrac{4}{5},\,\dfrac{2}{3},\,\dfrac{3}{7}

Idea summary

To compare fractions with different denominators:

  • Find the lowest common denominator

  • Find the equivalent fraction with this denominator

  • Compare the numerators of the equivalent fractions

Outcomes

MA4-5NA

operates with fractions, decimals and percentages

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