## What you need to know

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\div200):(200\div200)=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

\dfrac{2}{3}\text{ sugar, and }\dfrac{1}{3}\text{ butter.}

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 \dfrac{3}{7} of the total winnings, and Laura receives \dfrac{4}{7} of the total. So, we get

\text{Liv’s winnings }=\dfrac{3}{7}\times350,000=\pounds150,000

\text{Laura’s winnings }=\dfrac{4}{7}\times350,000=\pounds200,000

**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\div7=50,000.

Since all numbers in the ratio must be multiplied by the same scale factor, the ratio becomes

3:4=(3\times50,000):(4\times50,000)=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

\text{Sharwend : 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

\text{scale factor }=20\div4=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\times5=15\text{ 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 15:10 and 3:2, then because they only differ by a scale factor, we get that

\dfrac{15}{10}=\dfrac{3}{2}\text{, and that }\dfrac{15}{3}=\dfrac{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 \frac{a}{b}=\frac{c}{d} and \frac{a}{c}=\frac{b}{d}. 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 S 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

\dfrac{E}{S}=\dfrac{11}{9}

Then, if we multiply both sides of this equation by S, we get

E=\dfrac{11}{9}S,

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

\text{Eddy’s salary }=E=\dfrac{11}{9}\times25,650=\pounds31,350.

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.

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### Example Questions

1) Divide 35 into the ratio 2:5.

There are 2+5=7 parts in total in this ratio, and to divide 35 into this ratio we need the total to be 35. Therefore, we need to scale everything up a factor of 35\div7=5. Scaling up the ratio by multiplying each value by 5, we get

2:5=(2\times5):(5\times5)=10:25

So, to divide 35 in the ratio 2:5 we must split it into 10 and 25.

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2) Lucy is tiling her bathroom. She buys white and blue tiles according to the ratio 13:2. Blue tiles cost £2.80 whilst white tiles cost £2.35. She buys 16 blue tiles in total. Work out how much spends on tiles.

Firstly, we need to work out how many tiles she buys, and then we can worry about how much Lucy is spending.

The ratio is 2 parts blue, and she buys 16 tiles – therefore, we need to scale up the ratio by a factor of 16\div2=8. So, the number of white tiles she buys is

13\times8=104

Then, the cost of 16 blue tiles is

16\times2.80=\pounds44.80

The cost of 104 white tiles is

104\times2.35=\pounds244.40

Therefore, Lucy’s total spend on tiles is

44.80+244.40=\pounds289.20

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3) Alieke, Jon, and Kate all tracked the number of books they read last year. Jon read twice as many books as Kate, but Alieke read 4 times as many books as Jon.

a) Write down a ratio of the number of books read by Alieke to the number of books read by Jon to the number of books read by Kate. (HINT: if you’re stuck, try letting x be the number of books read by Kate in the last year)

b) Alieke read 63 more books than Kate last year. Work out how many books Alieke, Jon, and Kate read in total last year.

a) We’re going to go with the hint and say that x is the number of books read by Kate. Then, we know that Jon read twice as many, so the number of books Jon read is 2x. Furthermore, Alieke read four times as many books as Jon, so she read 4\times2x=8x books in the last year. Therefore, the ratio books read is

\text{Alieke : Jon : Kate }=8x:2x:x

All these values in the ratio have a factor of x, so we can simplify by dividing through by x. Doing so, we get the ratio

8:2:1

b) This question gives us information on the difference between two values, so we’re going to add a 4th value to our ratio: the **difference** between Alieke and Kate. In the ratio, Alieke has 8 parts and Kate has 1, so the difference is 8-1=7, and thus our new ratio is

\text{Alieke : Jon : Kate : difference}=8:2:1:7

The question tells us that the difference between Kate’s number and Alieke’s is 63, whilst the value of the difference in the ratio is 7. Therefore, the ratio must be scaled up by a factor of 63\div7=9.

So, scaling up the ratio by a factor of 9 (ignoring the 7 since that was only there to aid the process), we get

8:2:1=(8\times9):(2\times9):(1\times9)=72:18:9

Therefore, the total number of books read by the 3 people is

72+18+9=99\text{ books}.

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