Comparing fractions can sometimes be really easy. Take for example these two fractions – which one is larger?
Example 1
It’s obvious that the fraction on the left is larger than the fraction on the right, isn’t it? Because 4 is bigger than 3.
Take a look at the diagrams below. To the left is the first fraction, to the right the second. When splitting a circle into 5 equal parts, clearly 4 parts are bigger than 3 parts.
However, sometimes it’s not as easy as that… take a look at the next example.
Example 2
Take these two fractions – which one is larger?
This one is far from obvious.
Why is the example 2 so much harder than example 1?
Well, it’s because in example 1 both fractions have the same denominator. In example 2, the fractions have different denominators.
So, the way to compare fractions is to change them so they have the same denominator (I will refer to this as the common denominator). Then it will be really easy.
The easiest way to find the common denominator of two fractions is to multiply the two denominators together.
7 × 11 = 77
We now need to change both fractions so that they have a denominator of 77.
Changing the first fraction, we are changing the denominator from 11 to 77, so we are multiplying by 7. If we multiply the denominator by 7, we must also multiply the numerator by 7. Whatever we do to the denominator we must do the same to the numerator, and vice versa.
Doing so, the fraction will change:
Now let’s do the same to the second fraction. This time, to get the denominator from 7 to 77, we multiplied by 11. So we must multiply the numerator by 11 also.
Now the two fractions have the same denominator, so it’s super easy to say which one is bigger. Clearly 56 > 55 so again, like in example 1, the fraction on the left is larger.
So, the important thing to do when comparing fractions is to make the denominators the same.
Example 3 – comparing more than 2 fractions
It becomes slightly more difficult when there are more than 2 fractions. For example, take these three fractions.
We don’t want to multiply the three denominators together to get the common denominator this time (we could, but it’s not the most efficient way). Instead, we want to find the lowest common multiple of the denominators (12, 4 and 3). This is the smallest number that are in each of the numbers’ times tables. You may be able to spot straight away that 12 is the lowest common multiple. If you can’t, don’t worry, we’ll go into this in more detail in example 4.
So let’s change all three fractions so they have a denominator of 12 (the first one doesn’t need to change as it is already 12).
So now the three fractions have the same denominator:
So clearly the second fraction is largest, followed by the third and then first.
There is a little trick for getting the lowest common multiple of multiple numbers. Let’s look at example 4.
Example 4
Put these fractions into order, smallest to largest:
This time there are four fractions, and it’s not as clear-cut as Example 3 what the common denominator should be.
The trick I use for finding this number is as follows:
- Try the largest denominator first and see if it is a multiple of each of the other denominators
- If that number doesn’t work, keep trying the multiples of that number until one of them works
So, the largest denominator is 20. But 20 does not work as it is not a multiple of 6. So we move on to the next multiple of 20, which is 40.
40 does not work either as, like 20, it is not a multiple of 6. So we move on to the next multiple of 20, which is 60.
60 works! Because 60 ÷ 12 = 5, 60 ÷ 10 = 6, 60 ÷ 6 = 10 and 60 ÷ 3 = 20. So 60 will be our common denominator.
Let’s change all the fractions.
So here are the fractions now:
So here are the original fractions in order from smallest to largest:
I hope you have found that useful. Comparing fractions comes up a lot in secondary school maths, not just in questions such as example 4, but the concept also appears in many other types of questions.
You can also find some more questions like this in our fractions-decimals-percentages worksheet. For more targeted practice, CorbettMaths have an ordering fractions worksheet.
If your child needs help with this topic or any aspect of mathematics, book in a free taster session.
