2.02 Equivalent and simplified 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.

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:

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}

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:

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.

Examples

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}.

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.

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

Watch question walkthrough

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:

5 parts out of 12

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,

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.

10 parts out of 24

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