R4, R5, R6, R8Ratio, Proportion & Rates of ChangeFoundation & HigherGrades 2–6

Ratio

Q: Does the order of a ratio matter?

Yes. 2: 3 and 3: 2 describe different situations — always follow the order of the words in the question.

Q: What is a unit ratio?

A ratio in the form 1: n or n: 1. Map scales are usually unit ratios, such as 1: 25,000.

A ratio compares quantities: a recipe with 2 parts flour to 3 parts sugar has flour and sugar in the ratio $2 : 3$ . Ratio questions are among the most common on AQA papers, and almost all of them yield to one idea — work out the value of one part.

40 min Video by Maths Genie AQA GCSE Maths

The big picture

A ratio describes shares, not amounts. $2 : 3$ could mean 2 kg and 3 kg, or 20 g and 30 g. What stays fixed is that there are 5 equal parts, and flour is $\frac{2}{5}$ of the total. Converting between ratios and fractions of the whole is the key to the harder problems.

Bar models make every ratio question visual. Draw one box per part, label what you know, and the value of one box follows — whether you are given the total, one share, or the difference between shares.

Sharing in a RatioCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Simplify ratios, including with different units, and write them in the form $1 : n$
Share an amount in a given ratio
Solve ratio problems given one share or the difference between shares
Convert between ratios and fractions, and combine two ratios

Simplifying ratios

Divide every part by the highest common factor. If the quantities have units, convert to the same unit first, then drop the units.

The form $1 : n$ divides both parts by the first number, so $n$ may be a decimal: $4 : 10 = 1 : 2.5$ . It is useful for comparing ratios and for scales on maps.

Example 1Simplify

45

: 2

1Same units:

45 : 200

2HCF is 5:

9 : 40

Answer

9 : 40

Your turnSame method, new numbers — try it before moving on.

Write $750$ g $: 2$ kg in its simplest form.

Your answer

Type it like 3:5.

Show worked solution

Solution

1Same units: $750 : 2000$ .
2Divide by 250: $3 : 8$ .

Answer: $3 : 8$

Sharing in a ratio

Add the parts to find the total number of parts. Divide the amount by the total parts to find one part, then multiply for each share.

Check that the shares add up to the original amount.

Sharing £120 in the ratio

3 : 5

: Amy gets £45 and Ben gets £75.

Example 1Share £120 between Amy and Ben in the ratio

3 : 5

1Total parts:

3 + 5 = 8

2One part:

120 \div 8 = 15

3Amy:

3 \times 15 = 45

; Ben:

5 \times 15 = 75

Answer

Amy £45, Ben £75

Your turnSame method, new numbers — try it before moving on.

£270 is shared in the ratio $2 : 3 : 4$ . How much is the largest share, in pounds?

Your answer

Show worked solution

Solution

1Total parts: $9$ . One part: $270 \div 9 = 30$ .
2Largest share: $4 \times 30 = £120$ .

Answer: £120

Given one share or a difference

If you are told one share, divide it by its number of parts to find one part. If you are told the difference between two shares, divide it by the difference in parts.

Example 1Cats and dogs at a rescue centre are in the ratio

4 : 7

. There are 21 more dogs than cats. How many animals are there?

1Difference in parts:

7 - 4 = 3

3

parts

= 21

, so one part

= 7

3Total:

11

parts

= 77

Answer

77

animals

Your turnSame method, new numbers — try it before moving on.

Red and blue beads are in the ratio $5 : 2$ . There are 35 red beads. How many beads are there altogether?

Your answer

Show hint

5 parts are 35 beads.

Show worked solution

Solution

1One part: $35 \div 5 = 7$ .
2Total: $7$ parts $= 49$ .

Answer: $49$

Tip — Label the bar model with the given information before calculating — it shows whether you have been given a total, a share or a difference.

Ratios, fractions and combining ratios

In a ratio $a : b$ , the first quantity is $\frac{a}{a + b}$ of the total. The first is also $\frac{a}{b}$ times the second.

To combine $a : b$ and $b : c$ , make the $b$ parts the same using their LCM, then read off $a : b : c$ .

Example 1In a class, boys : girls

= 4 : 5

. What fraction of the class are girls?

1Total parts:

9

2Girls: 5 of those 9 parts.

Answer

\frac{5}{9}

Example 2

A : B = 2 : 3

and

B : C = 4 : 5

. Find

A : B : C

1The

B

parts are 3 and 4; their LCM is 12.

A : B = 8 : 12

and

B : C = 12 : 15

Answer

8 : 12 : 15

Your turnSame method, new numbers — try it before moving on.

Tea : coffee $= 3 : 2$ and coffee : juice $= 5 : 4$ . Find tea : juice in its simplest form.

Your answer

Type it like 3:5.

Show hint

Make the coffee parts equal: the LCM of 2 and 5 is 10.

Show worked solution

Solution

1Tea : coffee $= 15 : 10$ ; coffee : juice $= 10 : 8$ .
2So tea : coffee : juice $= 15 : 10 : 8$ , and tea : juice $= 15 : 8$ .

Answer: $15 : 8$

Part-to-part versus part-to-whole

The fraction $\frac{4}{9}$ and the ratio $4 : 5$ describe the same situation but compare different things: the fraction compares a part with the $\textbf{whole}$ , the ratio compares a part with the $\textbf{other part}$ . Mixing them up is the most common error on multi-step ratio questions.

Think like an examiner

Write the value of one part clearly — it usually earns the first method mark.
Answer the question asked: if it wants the total, a single share is not enough.
Keep ratios in the order given: "boys to girls" must be written boys first.

Ratio methods

\text{one part} = \dfrac{\text{total}}{\text{total parts}}

Sharing an amount.

a : b \Rightarrow \text{first} = \tfrac{a}{a+b} \text{ of total}

Ratio to fraction.

\text{one part} = \dfrac{\text{difference}}{\text{difference in parts}}

Given a difference.

Watch out for these

Sharing £120 in

3 : 5

120 \div 3

and

120 \div 5

Divide by the total parts, 8, first.

Simplifying

45

: 2

m to

45 : 2

Convert to the same units first:

45 : 200 = 9 : 40

Writing boys : girls

= 4 : 5

as "

\frac{4}{5}

are boys".

Boys are

\frac{4}{9}

of the class.

Combining

2 : 3

and

4 : 5

2 : 3 : 5

Match the shared term first:

8 : 12 : 15

Stretch yourself

A bag contains red and green counters in the ratio $3 : 5$ . Six more red counters are added, and the ratio becomes $3 : 4$ . How many counters were in the bag originally?

Hint — Let the original numbers be $3k$ and $5k$ . Form an equation using the new ratio.

Reveal solution

SolutionRed

= 3k

, green

= 5k

. After adding:

\frac{3k + 6}{5k} = \frac{3}{4}

, so

4(3k + 6) = 15k

, giving

12k + 24 = 15k

and

k = 8

. Originally there were

24

red and

40

green:

64

counters. Check:

30 : 40 = 3 : 4

✓.

\blacksquare

Questions students ask

Can ratios contain decimals or fractions?

Yes, but simplest form uses whole numbers. Multiply to clear them:

1.5 : 2 = 3 : 4

and

\frac{1}{2} : \frac{1}{3} = 3 : 2

Does the order of a ratio matter?

Yes.

2 : 3

and

3 : 2

describe different situations — always follow the order of the words in the question.

What is a unit ratio?

A ratio in the form

1 : n

n : 1

. Map scales are usually unit ratios, such as

1 : 25{,}000

Key takeaways

Simplify by dividing by the HCF, after matching units.
Share: total parts, then one part, then each share.
Given a share or difference, divide by the matching number of parts.
Part of the whole: $\frac{a}{a + b}$ .
Combine ratios by matching the shared term with the LCM.

Exam-style questions

Practise it the way it is examined

3 original questions · 9 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 23 marks

(a)Simplify $18 : 30$ .[1]
Show mark scheme (a)
- B1 $3 : 5$
Answer: $3 : 5$
(b)Share 91 sweets between Ali and Beth in the ratio $4 : 3$ .[2]
Show mark scheme (b)
- M1 $91 \div 7 = 13$
- A1Ali 52, Beth 39
Answer: Ali 52, Beth 39

Question 2Grade 43 marks

Concrete is made from cement, sand and gravel in the ratio $1 : 2 : 4$ by mass. A builder has 30 kg of cement, 50 kg of sand and 110 kg of gravel.

What is the greatest mass of concrete the builder can make?[3]
Show mark scheme
- M1Considers how many batches each material allows, e.g. sand $50 \div 2 = 25$ parts
- M1Identifies sand as the limiting material: one part $= 25$ kg (cement 30, gravel $110 \div 4 = 27.5$ )
- A1 $7 \times 25 = 175$ kg
Answer: $175$ kg

Examiner note

Using all 30 kg of cement would need 60 kg of sand — more than the builder has. Always check each ingredient.

Question 3Grade 63 marks

The ratio of Kim's age to Leo's age is $2 : 5$ . In 8 years' time the ratio will be $2 : 3$ . How old is Leo now?[3]
Show mark scheme
- M1Forms an equation, e.g. $\frac{2k + 8}{5k + 8} = \frac{2}{3}$
- M1 $6k + 24 = 10k + 16$
- A1Leo is 10 ( $k = 2$ )
Answer: $10$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Simplify $12 : 20$ .
- $3 : 5$
- $6 : 10$
- $2 : 3$
- $4 : 5$
Show answer and explanation
Answer: $3 : 5$
Divide by 4.
Share £40 in the ratio $1 : 3$ . What is the smaller share?
- £10
- £13.33
- £30
- £4
Show answer and explanation
Answer: £10
4 parts: one part is £10.
Write $5 : 8$ in the form $1 : n$ .
- $1 : 1.6$
- $1 : 3$
- $1 : 0.625$
- $1 : 40$
Show answer and explanation
Answer: $1 : 1.6$
$8 \div 5 = 1.6$ .
Flour : sugar $= 5 : 2$ . There is 350 g of flour. How much sugar?
- 140 g
- 875 g
- 70 g
- 100 g
Show answer and explanation
Answer: 140 g
One part is 70 g.
Boys : girls $= 3 : 7$ . What fraction are boys?
- $\frac{3}{10}$
- $\frac{3}{7}$
- $\frac{7}{10}$
- $\frac{1}{3}$
Show answer and explanation
Answer: $\frac{3}{10}$
3 parts out of 10.
Simplify $1.2$ m $: 80$ cm.
- $3 : 2$
- $12 : 8$
- $1.2 : 80$
- $2 : 3$
Show answer and explanation
Answer: $3 : 2$
$120 : 80 = 3 : 2$ .
Ratio $2 : 7$ . The difference between the shares is 45. What is the total?
- $81$
- $63$
- $45$
- $90$
Show answer and explanation
Answer: $81$
5 parts = 45, so 9 parts = 81.
$x : y = 3 : 4$ and $y : z = 6 : 5$ . Find $x : z$ .
- $9 : 10$
- $3 : 5$
- $18 : 20$
- $1 : 1$
Show answer and explanation
Answer: $9 : 10$
$9 : 12 : 10$ .

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