2.02 Equivalent and simplified fractions

Sometimes multiple fractions can represent the same amount.

Consider the shaded area of the hexagon below.

We can see that $3$3 of the $6$6 parts have been shaded in, so the area represents $\frac{3}{6}$36​ of the whole shape.

But we can also see that half the shape has been shaded in, which we can write as $\frac{1}{2}$12​ of the whole shape.

Since the same area is shaded for both $\frac{3}{6}$36​ and $\frac{1}{2}$12​, 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.

Of the $18$18 grid squares, $3$3 of them are coloured. This means that we can represent the shaded area with the fraction $\frac{3}{18}$318​.

But what are equivalent fractions?

Notice that the placement of coloured 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$1 out of $6$6 parts, so we can also represent this area as $\frac{1}{6}$16​ of the grid.

This shows us that $\frac{1}{6}$16​ and $\frac{3}{18}$318​ are equivalent fractions, which we can express using the equivalence:

$\frac{1}{6}=\frac{3}{18}$16​=318​

Changing the denominator

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 $\frac{1}{6}$16​ we can change $6$6 to $18$18 by multiplying it by $3$3. So we get:

$\frac{1}{6}=\frac{1\times3}{6\times3}$16​=1×36×3​

Then we can evaluate the two multiplications:

$\frac{1\times3}{6\times3}=\frac{3}{18}$1×36×3​=318​

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

$\frac{3}{18}=\frac{1\times3}{6\times3}$318​=1×36×3​

And from cancelling out the $3$3s we get:

$\frac{1\times3}{6\times3}$1×36×3​

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

Simplifying fractions

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

Worked examples

Example 1

Simplify $\frac{8}{24}$824​.

Think: First we want to find any common factors between $8$8 and $24$24. $4$4 is a common factor between both these numbers so we can cancel this first.

Do: Cancel $4$4 from both the numerator and denominator.

Cancelling a factor of $4$4 from $\frac{8}{24}$824​

So $\frac{2}{6}$26​ is a simplified equivalent fraction of $\frac{8}{24}$824​.

Since there is still a common factor of $2$2 between the numerator and denominator of $\frac{2}{6}$26​ we can cancel one more time to get:

Cancelling a factor of $2$2 from $\frac{2}{6}$26​

So $\frac{1}{3}$13​ is an even simpler equivalent fraction of $\frac{8}{24}$824​.

Reflect: Since $1$1 and $3$3 have no common factors, we cannot cancel any further. This means that $\frac{1}{3}$13​ is a fully simplified fraction.

Example 2

Fully simplify $\frac{42}{63}$4263​.

Think: In order to fully simplify the fraction we cancel common factors from the top and bottom of the fraction until there are no common factors left.

Do: Notice that $42$42 and $63$63 both belong to the $7$7 times tables. We can write $42$42 as $6\times7$6×7, and we can write $63$63 as $9\times7$9×7.

Cancelling the common factor of $7$7 gives us:

Cancelling a factor of $7$7 from $\frac{42}{63}$4263​

We have simplified the fraction, but it is not fully simplified since $6$6 and $9$9 still have a common factor of $3$3.

Cancelling the common factor of $3$3 gives us:

Cancelling a factor of $3$3 from $\frac{6}{9}$69​

Since $2$2 and $3$3 have no common factors (except for $1$1), this is the fully simplified fraction we are looking for.

Reflect: In order to simplify the fraction we looked for common factors between the numerator and denominator and then cancelled by that number. We reached the fully simplified fraction when there were no more common factors to cancel.

Notice here that, by simplifying, we have found that $\frac{42}{63}$4263​, $\frac{6}{9}$69​ and $\frac{2}{3}$23​ are all equivalent fractions. Since fractions like $\frac{4}{6}$46​ and $\frac{20}{30}$2030​ also simplify to give $\frac{2}{3}$23​, these fractions are also equivalent to the three we have just found.

Comparing 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, $\frac{5}{12}$512​ or $\frac{3}{8}$38​?

We can try comparing these fractions visually:

$5$5 parts out of $12$12

$3$3 parts out of $8$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$24 is the lowest common multiple of $12$12 and $8$8, we want to find the equivalent fractions of $\frac{5}{12}$512​ and $\frac{3}{8}$38​ that have a denominator of $24$24.

We can do this by multiplying the numerator and denominator of $\frac{5}{12}$512​ by a factor of $2$2, and the numerator and denominator of $\frac{3}{8}$38​ by a factor of $3$3,

Multiplying the numerator and denominator by $2$2

Multiplying the numerator and denominator by $3$3

Since $\frac{10}{24}$1024​ is larger than $\frac{9}{24}$924​ we can see that $\frac{5}{12}$512​ is larger than $\frac{3}{8}$38​.

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

$10$10 parts out of $24$24

$9$9 parts out of $24$24

We can see that after adding the gridlines, the grid for $\frac{5}{12}$512​ now has $\frac{10}{24}$1024​ shaded parts and the grid for $\frac{3}{8}$38​ now has $\frac{9}{24}$924​ shaded parts. This tells us that $\frac{5}{12}$512​ is greater than $\frac{3}{8}$38​.

Practice questions

Question 1

Question 2

Question 3