What you need to know

Fractions are very useful little things – they appear everywhere. When you cut a pizza (or a cake, if you’re more of a dessert person) into slices, you cut it up into fractions. If you cut it up into 8 slices, then each slice is \dfrac{1}{8} of the whole pizza (or cake), and so on.

Because they appear so much, we must be very comfortable with them, and know how to add, subtract, multiply, divide, and simplify them. If you have \frac{1}{8} of one pizza, and \text{1}{6} of another, how much pizza do you have? The answer is important, of course, but pizza is just the beginning – fractions really are everywhere.

Simplifying Fractions

When simplifying fractions, the aim is to make the numbers on the numerator (the top of the fraction) and the denominator (the bottom of the fraction) smaller, without actually changing the value of the fraction. To do this, we have to be aware of the all-important rule: if you multiply/divide the top and bottom of a fraction by the same number, the value of the fraction is unchanged.

So, when simplifying fractions, we are going to be looking for common factors in the top and bottom to give us some number we can divide them both by. In other words, we will cancel said factor.

Example: Write \dfrac{12}{30} in its simplest form.

Okay, so immediately we can see that both 12 and 30 are even numbers, meaning they both have a factor of 2. Therefore, we get

\dfrac{12}{30} = \dfrac{6 \times 2}{15 \times 2} = \dfrac{6}{15}

Are there more common factors? Yes, both 6 and 15 are multiplies of 3. Therefore, we get

\dfrac{6}{15} = \dfrac{2 \times 3}{5 \times 3} = \dfrac{2}{5}

This time, there are no more common factors so we have fully simplified the fraction.

Adding & Subtracting Fractions

If you’re given two fractions, it isn’t always clear how to add them together. If I have half of something and then a third of the same thing, how much do I have in total? It’s not obvious.

So, to do this, we are going to use the rule mentioned above – we’re going to be multiplying the top and bottom of a fraction (or rather, multiple fractions) by the same thing in order to make it so the fractions that we are adding/subtracting have the same denominator – a common denominator.

Example: Work out \dfrac{1}{6} + \dfrac{3}{4}. Write your answer in its simplest form.

So, we want both denominators to be the same. The easiest way to choose what this denominator will be is to use the product of both denominators, here: 6 \times 4 = 24. Alternatively, if you can spot it, you can make your calculation (slightly) simpler by using the lowest common multiple of both the denominators, but if you aren’t sure what this is then don’t worry – just choose the product. Now, in order to make the bottom of the first fraction 24 it needs to be multiplied by 4, so we must also multiply the top by 4.

\dfrac{1}{6} = \dfrac{1 \times 4}{6 \times 4} = \dfrac{4}{24}

To make the bottom of the second fraction 24 it must be multiplied by 6, so we must also multiply the top by 6.

\dfrac{3}{4} = \dfrac{3 \times 6}{4 \times 6} = \dfrac{18}{24}

Now, adding them together is made easy – because the denominators are the same, all we do is add the numerators. Think about it, if you have one quarter of a pizza and you take 2 more quarters, how many do you have? 3 quarters – you’re just adding the numerators. Let’s do it.

\dfrac{1}{6} + \dfrac{3}{4} = \dfrac{4}{24} + \dfrac{18}{24} = \dfrac{22}{24}

Now all that remains is to simplify it. Cancelling out a factor of 2, it becomes \frac{11}{12}, which cannot be simplified further.

Multiplying Fractions

Multiplying fractions is just about the easiest thing you can do to them – simply multiply the numerators together and multiply the denominators. Piece of cake (or pizza, if you aren’t the dessert type).

Example: Work out \dfrac{4}{5} \times \dfrac{7}{12}. Write your answer in its simplest form.

As mentioned, all we’re going to do is multiply across the top and across the bottom. Thus, our calculation looks like

\dfrac{4}{5} \times \dfrac{7}{12} = \dfrac{4 \times 7}{5 \times 12} = \dfrac{28}{60}

See, not so bad at all. It just comes down to multiplication in the end. Now, we must simplify our answer. Both top and bottom have a factor of 4, so cancelling the 4 we get: \dfrac{7}{15}. There are no more common factors, so this is our final answer.

Dividing Fractions

Lucky for us, dividing fractions isn’t too bad either! You just have to remember the rule: Keep, Change, Flip. What this means, is that to do a division, you must keep the first fraction as it is, change the division sign into a multiplication, and flip the second fraction. At that point, you just work out the multiplication as you now know how. Just make sure you remember all 3 steps.

Example: Work out \dfrac{1}{2} \div \dfrac{5}{9}. Write your answer in its simplest form.

We’re going to keep the first fraction the same, change the symbol to a multiplication, and flip the second fraction. This leaves us with.

\dfrac{1}{2} \div \dfrac{5}{9} = \dfrac{1}{2} \times \dfrac{9}{5}

Now, doing the multiplication we get

\dfrac{1}{2} \times \dfrac{9}{5} = \dfrac{1 \times 9}{2 \times 5} = \dfrac{9}{10}

This cannot be made any simpler, so we are done.

Example Questions

It’s a multiplication, so we multiply across the top and multiply across the bottom.

 

\dfrac{6}{13} \times \dfrac{4}{3} = \dfrac{6 \times 4}{3 \times 13} = \dfrac{24}{39}

 

Now, simplify – both top and bottom have a factor of 3, which can cancel. So, we get

 

\dfrac{24}{39} = \dfrac{8}{13}

 

This cannot be simplified any further.

We require a common denominator. We will use 3 \times 10 = 30. So, to give both fractions this denominator, we need to 1) multiply the top and bottom of the first fraction by 3, and 2) multiply the top and bottom of the second fraction by 10. Then we are able to subtract the numerator. So, we get

 

\dfrac{7}{10} - \dfrac{8}{3} = \dfrac{21}{30} - \dfrac{80}{30} = -\dfrac{59}{30}

 

Now, simplify – both top and bottom have a factor of 3 that can cancel, leaving us with

 

-\dfrac{59}{30}

 

This cannot be simplified further, so we are done.

To divide fractions, we need to Keep, Change, and Flip. Changing the division sign to a multiplication, and flipping the second fraction we get

 

\dfrac{9}{11} \div \dfrac{6}{7} = \dfrac{9}{11} \times \dfrac{7}{6}

 

Doing the multiplication, we get

 

\dfrac{9}{11} \times \dfrac{7}{6} = \dfrac{9 \times 7}{11 \times 6} = \dfrac{63}{66}

 

Now, simplify – both top and bottom have a factor of 3 which we can cancel, leaving us with

 

\dfrac{63}{66} = \dfrac{21}{22}

 

This cannot be simplified further, so we are done.

For GCSE Maths Tutors and teachers looking for KS3 and GCSE Maths fraction resources then you are welcome to use the selection of online tests and fraction worksheets above. At Maths Made easy we are in the process of building a super resource where you can get all Maths resources in one place. For more GCSE Maths revision resources visit our main page.

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