This page contains the GCSE AQA Mathematics Ratios Questions and their answers for revision and understanding Ratios.
Ratios are something we use to compare amounts of things. For example, if you have a recipe that requires twice as much sugar as it does butter, then we can say that the ratio of sugar to butter is 2:1, which we say like “2 to 1”. Ratios can be scaled up/cancelled down like fractions, i.e. as long as you multiply/divide both parts of a ratio by the same number (sometimes referred to as a scale factor) then the ratio stays the same. For example, if your recipe contains 400g of sugar and 200g of butter, then this is still the ratio 2:1, because if we divide both the values by 200, then we get
400:200=(400÷200):(200÷200)=2:1
In this case, we say that the ratio of 2:1 was scaled up by a scale factor of 200. Often, we break a ratio up into parts. In this case we might say the ratio is “2 parts sugar to 1 part butter”, so, in total, this ratio is made up of 2+1=3 parts. As a result, the recipe is
2/3 sugar, and 1/3butter.
As you can see, ratios have a lot in common with fractions. This is a good thing! It means that if you feel comfortable with one of those things, then you’ll probably be in a good spot with the other, too. Almost all of the ratio questions you’ll see with involve either scaling up/down a given ratio or breaking up the ratio into “parts of a whole” as we just saw, so those ideas are very important to be familiar with.
You will also see ratios with more than 2 parts. For example, if you want to make something more exciting than buttercream, you might need 3 ingredients split according to the ratio 2:3:5. All the same rules still apply, you just have to multiply/divide all 3 numbers in the ratio by the same value in order to ensure the ratio stays the same.
Example: Liv and Laura win a lottery of £350,000 and decide to split their winnings according to the ratio 3:4. Work out how much each person receives.
We’ll go through two (admittedly quite similar) methods for answering this question.
Method 1: In total, this ratio is made up of 3+4=7 parts. This means that Liv receives 3/7 of the total winnings, and Laura receives 4/7 of the total. So, we get
Liv’s winnings =3/7×350,000=£150,000
Laura’s winnings =4/7×350,000=£200,000
Method 2: In total, this ratio is made up of 3+4=7 parts. The scale factor required to scale a total of 7 up a total of 350,000 is
350,000÷7=50,000.
Since all numbers in the ratio must be multiplied by the same scale factor, the ratio becomes
3:4=(3×50,000):(4×50,000)=150,000:200,000
Therefore, Liv receives £150,000 and Laura receives £200,000. Another way to think of this is that 1 part in the ratio is worth £50,000, so since Liv and Laura have 3 and 4 parts in the ratio respectively, we need to multiply £50,000 by 3 and then 4 to get their respective winnings.
Example: Sharwend, Barney, and Ronnie share sweets according to the ratio 5:3:7. Ronnie gets 20 more sweets than Barney. Work out how many sweets Barney received.
So, because we aren’t satisfied by a ratio with 3 values, we’re going to add another! The 4th value is going to be the difference between Barney and Ronnie. In the ratio, Ronnie has 7 parts and Barney has 3, so the difference between them is 4. Thus, the new ratio is
Barney : Ronnie : difference =5:3:7:4.
Now, the actual difference between the number of sweets Barney and Ronnie receive (as we’re told in the question) is 20. Therefore, the scale factor that we need can be calculated by dividing the actual difference by the difference that we added on to the ratio. So, we get
scale factor =20÷4=5.
Then, since we must multiply each bit of a ratio by the same number (scale factor), to find the number of sweets Barney received, we multiply the number of parts he has in the ratio by 5:
3×5=15 sweets.
Adding an extra value to a ratio to account for a difference is very useful technique, partly because these types of questions are quite common. Additionally, we could’ve applied this method in the previous example by adding another value to the ratio for the total. So. Many. Methods.
If we have two ratios that are equal, say 15:1015:10 and 3:23:2, then because they only differ by a scale factor, we get that
15/10=3/2, and that 15/3=10/2.
If you cancel down these fractions, you can see that these things are indeed equal, like I say. In general, if a:b=c:d, then ba=dc andca=db. Let’s see an example of how this might be useful.
Example: The ratio of Eddy’s salary to Stu’s salary is 11:9. If E is Eddy’s salary, and SS is Stu’s salary, find an expression for E in terms of S.
So, we have that E:S=11:9. We now know that this can be expressed like
e/s=11/9
Then, if we multiply both sides of this equation by S, we get
E=11/9,
which is an expression for E in terms of S. This is useful because it acts like a formula for figuring out what Eddy’s salary is once you know Stu’s. If Stu earns £25,650, then
Eddy’s salary =E=911×25,650=£31,350.
As you can see, ratios can be useful in a lot of different ways and can appear in a huge variety of questions, so it’s important to make sure you’re really comfortable with the core ideas behind ratios as well as practising to get an idea of the common (and less common) types of questions asked.