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.

 

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:

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,

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.

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