## What you need to know

### Ratio and Proportion

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.

Having a good knowledge of the following topics will help with ratio:

- Fractions
- Factors and multiples, LCM and HCF

### Example 1: Ratio and Proportion

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.

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

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 2: Ratio and Proportion

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}.

### Example 3: Ratio and Proportion

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.

### Example Questions

**Question 1:** In a school, the ratio of the number of students with blond hair to the number of students with brown hair is 5 : 4.

**a)** What fraction of students have blond hair?

**b)** If there are 450 students in the school, how many of them have brown hair?

**a)** In order to work out the fraction of students that have blond hair, we need to add up the ratio parts. The sum of the ratio is 5 + 4 = 9. This means that we are dealing with 9ths. Since the ratio share for blond students is 4, this means that the fraction of blond students is \dfrac{4}{9}

**b)** We know from the previous question that \dfrac{4}{9} of the students have blond hair. Therefore, the fraction of students with brown hair is \dfrac{5}{9}. If there is a total of 450 students in the school, we need to work out what \dfrac{5}{9} of 450 is:

\dfrac{5}{9} \times 450 = 250 \text{ students}

**Question 2:** Divide 35 in the ratio 2 : 5.

In total, there are 7 parts to this ratio (2 + 5 = 7). If 7 shares have a value of 35, then 1 share has a value of 5 (35 \div7 = 5).

Since 1 share has a value of 5, then 2 shares will have a value of 10 (2 \times 5 = 10).

Since 1 share has a value of 5, then 5 shares will have a value of 25 (5 \times 5 = 25).

**Question 3:** Lucy is tiling her bathroom. She buys white and blue tiles in the ratio of 13 : 2. Blue tiles cost £2.80 whilst white tiles cost £2.35. If she buys 16 blue tiles, how much does Lucy spend on tiles in total?

To work out the total cost of the tiles Lucy buys, we need to work out how many white tiles she buys. In order to work out the number of white tiles, we need to work out the total number of tiles she buys.

The ratio is 2 parts blue to 13 parts white. If Lucy buys 16 blue tiles, this is 8 times more than the figure for blue tiles in the ratio (16\div2=8). If the number of blue tiles she buys is 8 times more than the blue tiles figure given in the ratio, then the number of white tiles she buys must also be 8 times more than the white tiles figure in the ratio. Therefore, the number of white tiles she buys is:

13\times8 = 104 white tiles

Now that we know how many tiles of each colour she has bought, we can calculate the total cost of the tiles. 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

**Question 4: **Steve receives £200 each month from his parents as an allowance. 20% of this amount is spent on his football magazine subscription, and the rest he spends on football stickers, sweets and fizzy drinks in the ratio of 5 : 2 : 1. How much does Steve spend on football stickers?

The first thing we need to do is to deduct the 20% spent on the magazine subscription so that we can work out how much of his allowance Steve has left over. 20% of £200 can be calculated as follows:

0.2 \times \pounds200 = \pounds 40

You may prefer to calculate the 20% in your head. 10% of £200 is £20, so 20% is 2 \times \pounds20 = \pounds40

Then deduct this from £200:

\pounds200 - \pounds40 = \pounds160

Steve therefore has £160 pounds remaining which he spends on sweets, football stickers and fizzy drinks in the ratio of 5 : 2 : 1.

By adding up the ratio, we know that we are dealing with eighths. (We know we are dealing in eighths because 5 + 2 + 1 = 8). We know from the ratio that the share he spends on football stickers is 5, meaning that Steve spends \frac{5}{8} of the remaining allowance on football stickers.

The actual amount that Steve spends on football stickers can be calculated as follows:

\dfrac{5}{8} \times \pounds160 = \pounds100

**Question 5:** Alieke, Jon, and Kate tracked the number of books they read last year. Jon read twice as many books as Kate, but Alieke reads 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.

**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 are told that Jon reads twice as many books as Kate. As a ratio, this can be written as 2 : 1.

We are also told that Alieke reads 4 times as many books as Jon. As a ratio, this can be written as 4 : 1.

The issue we have now is that in the Jon : Kate ratio, Jon’s share is 2, while in the Alieke : Jon ratio, Jon’s share is 1. B y doubling the 4 : 1 ratio for Aleike : Jon to 8 : 2, Jon’s share is now the same in both ratios.

This means that we can express this is a 3-way ratio as follows:

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

**b)** We are told that Alieke read 63 more books than Kate last year, and we know from the previous question that the Alieke : Kate reading ratio is 8 : 1. In this ratio, the share is 8 parts to 1 part, so we can conclude that the difference between the ratio share is 7 parts (8 - 1 = 7).

If the difference in the ratio share is 7 parts, and the difference in the number of books read is 63, then we can work out the number of books read that 1 share of the ratio represents:

63 \div 7 = 9\text{ books}

If one share of the ratio represents 7 books read, we can now work out how many books were read in total by the three people. By adding up the ratio, we know that the total number of shares is 11 (8 + 2 + 1 = 11), so the total number of books read can be calculated as follows:

11 \times 9\text{ books} = 99\text{ books}

### Worksheets and Exam Questions

#### (NEW) Ratios (Foundation) Exam Style Questions - MME

Level 4-5#### (NEW) Ratios (Higher) Exam Style Questions - MME

Level 6-9#### Ratio Foundation

Level 4-5#### Ratio and Percentages

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