This page contains the GCSE AQA Mathematics Fractions Questions and their answers for revision and understanding Fractions.
A recurring decimal is a decimal number which has a pattern than repeats over and over after the decimal place. Every recurring decimal can also be written as a fraction. In this topic, we’ll look at how to go from a recurring decimal to and fraction and vice versa.
Two examples of recurring decimals, one quite common and one less common, are
1/3 = 0.3 = 0.33333…31=0.3˙=0.33333…
6/11 = 0.54 = 0.54545454…
Notice the dot on top of some of the digits; this tells us what is repeated. The first dot denotes the start of the repeated section, and the second dot denotes the end of it.
Convert Fractions to Recurring DecimalsTo convert a fraction to a recurring decimal we must treat the fraction like it is a division and use some method of division to divide the numerator by the denominator. Here we will use short division, also known as the “bus stop method” (I highly recommend this method of division for all purposes). In this example, we’ll see how it soon becomes obvious that the result of your division is a recurring decimal.
Example: Write 7/33 as a decimal.
So, we’re going to try dividing 7 by 33. When setting up the bus stop method, you should put in a whole lot of zeros after the decimal place – chances are you’ll only need a few, but it’s better to put more than you need. So, the set up short division should look like
Doing the short division (which you can brush up
Since the last remainder was 7, we looked for how many times 33 goes into 70, but we already did that once. The answer is twice, and then the remainder is 4, which means we’re now going to be asking the question of how many times 33 goes into 40, but we’ve already done that, too. Carrying this on, the pattern becomes obvious.
Therefore, we must have that 7/33 = 0.21. Practice a few of these and you’ll quickly get used to spotting when they start repeating.
Convert Recurring Decimals to Fractions
This is really what this topic is about. Let’s dive into an example to see how it goes.
Example: Write 0.14 as a fraction.
Firstly, set x = 0.14=0.1˙4˙, the thing we want to convert to a fraction. Then,
100x=14.1˙4˙.
Now that we have two numbers, x and 100x with the same digits after the decimal point, if we subtract one from another, the numbers after the decimal point will cancel.
100x – x = 99x = 14.4 – 0.4 = 14
Removing the working out steps from this line, we have
99x=14
Then, if we divide both sides of this by 99, we get
x = 14/99.
Okay, so how does this method work? Firstly, always assign the thing your converting to be x. Once you’ve done this, the aim is to end up with two numbers (both will be some multiple of ten times by x) that have exactly the same recurring digits after the decimal point. This really is key, because then when you subtract one from the other (in the main step of the process), the digits after the decimal point will cancel and you’ll be left with a nice whole number.
At that point, it’s just a case of solving a straightforward linear equation by doing one division, and you’ve got your answer.
Let’s see another example – remember, the aim is to get two multiples of x both with the same thing after the decimal place.
Example: Write 0.83˙ as a fraction.
Firstly, notice how the 8 has no dot above it so isn’t repeating. This will make a difference.
So, let x = 0.83˙. This time, if we multiply this by 10, 100, 1,000 etc, we’re not going to end up with something that has the same digits after the decimal point as xx.
However, if instead we take 10x = 8.3˙ and 100x = 83.3˙, then we have two multiplies of xx that do have the same digits after the decimal point.
So, subtracting one from the other, we get
100x−10x=90x=83.3˙−8.3˙=83−8=75
Removing the working out steps from this line, we have
90x=75
Dividing both sides by 90, we get that
x=75/90 =5/6.
This time we were unable to subtract x from anything to get the desired outcome, so we had to be clever and multiply x by 10 and 100 before doing the subtraction. You will need to do something like this anytime there’s a decimal digit in your number that isn’t involved in the recurring part.
Here we will be going through how to convert between fractions, decimals, and percentages (in all directions).
Decimals and Percentages
Converting between decimals and percentages is nice and straightforward.
• Convert decimal to percentage – multiply by 100 (shift decimal point right two places).
• Convert percentage to decimal – divide by 100 (shift decimal point left two places).
Example: a) Write 37% as a decimal.
b) Write 0.548 as a percentage.
a) To convert this to a decimal we will divide by 100, so we get
37 div 100 = 0.3737÷100=0.37
b) To convert this to a percentage we will times by 100, so we get
0.548×100=54.8%
Decimals and Fractions
Converting between decimals and fractions is a little more work.
• Convert fraction to decimal – treat the fraction like a division and divide the number on the top by the number on the bottom. There are ways we can make this process easier, which we’ll see in the examples below.
• Convert decimal to fraction – write the decimal as a fraction with 1 on the bottom. Then, keep multiplying top and bottom by 10 until the decimal becomes a whole number.
Example: a) Write 12/25 as a decimal.
b) Write 11/8 as a decimal.
a) Dividing 12 by 25 doesn’t sound too pleasant, but there is a way we can change the fraction (before dividing top by bottom) to make life easier. Notice that 25×4=100 and dividing by 100 is straight forward. So, if we times top and bottom by 4 we get
12/25=48/100=0.48
So, we have successfully converted this fraction to a decimal.
b) In this case, there is no nice shortcut. We’ll just have to divide 11 by 8 in whichever way feels most comfortable – here, we’ll go for the bus stop method (make sure to put lots of zeroes after your decimal point!) Doing this, we get the picture shown on the right, and we see that the result is
11/8=1.375
Example: Write 4.56 as a fraction in its simplest form.
Any number divided by 1 is equal to itself, so we can write 4.56 /1 Now, if we multiply top and bottom by 100 (or multiply by 10 twice, if you’re not sure why we choose 100), and we’ll see that we get
14.56/1=4.56×100/100=456/100
Success! We’ve written the decimal as a fraction. All that remains now is to simplify it. Cancelling out factors (until there are no common factors left), we get
456/100=228/50=114/25
Fractions and Percentages
We’ll use the tools we’ve already learned to help us convert between fractions and percentages.
• Convert percentages to fractions – a percentage is already out of 100, so we must put the value in a fraction over 100. Then, if necessary, multiply top and bottom by powers of 10 to make the values into whole numbers.
• Convert fractions to percentages – firstly convert the fraction to a decimal (using the method we’ve seen), then convert that decimal to a percentage (also using the method we’ve seen).
Example: Write 48.1% as a fraction.
As mentioned, percentages are already out of 100 so
48.1%=48.1/100
Then, multiplying top and bottom by 10 (to make the numbers whole) we get
48%=481/1000
Example: Write 4/5 as a percentage.
Firstly, let’s convert it to a decimal. Notice that if we times top and bottom by 2, the fraction becomes
4/5=4*2/5*2=8/10
Now, dividing by 10 isn’t too tricky: 0.88÷10=0.8. Then, to convert this decimal to a percentage, we times by 100:
0.8×100=80%
NOTE: Alternatively, if you can write a fraction with 100 on the denominator, then the value on the top immediately gives you what the fraction would be as a percentage. For example,
34/50 =68/100=68%
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 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 1/8 of one pizza, and 16 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 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
12/30=6×2 /15×2=6/15
Are there more common factors? Yes, both 6 and 15 are multiplies of 3. Therefore, we get
6/15=2×3/5×3=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 1/6+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: 6×4=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.
=1/6=1×4/6×4=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.
3/4=4×6/3×6=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.
1/6+3/4=4/24+18/24=22/24
Now all that remains is to simplify it. Cancelling out a factor of 2, it becomes 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 4/5×7/12 . 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
4/5×7/12=4×7/5×12=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: . 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 1/2÷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.
1/2÷5/9=1/2×9/5
Now, doing the multiplication we get
1/2×9/5=1×9/2×5=9/10
This cannot be made any simpler, so we are done.