1MA1 · R4–R8Ratio, Proportion & Rates of ChangeFoundation & HigherGrades 2–6

Ratio

Q: Does the order of a ratio matter?

Yes. "Boys to girls is 2: 3" is different from "girls to boys is 2: 3".

Q: Can ratios have decimals?

Yes, especially in the form 1: n, but otherwise simplify to whole numbers.

A recipe needs flour and sugar in the ratio $3 : 1$ . A map uses a scale of $1 : 25\,000$ . Two friends split a prize $3 : 5$ . A ratio compares quantities by their relative size, and nearly every ratio problem comes down to one step: find the value of $\textbf{one part}$ .

45 min Ratio & Proportion

The big picture

A ratio $a : b$ says that for every $a$ of one thing there are $b$ of the other, so the whole is made of $a + b$ equal parts. Once you know what one part is worth — from the total, from one share, or from the difference between shares — every other quantity follows.

Edexcel likes ratio problems that give information indirectly: "Sam gets £36 more than Tia", or a ratio that changes after something is added. Drawing the parts as bars keeps these clear.

What you'll learn

Write and simplify ratios, including with different units
Share a quantity in a given ratio
Solve problems using the difference between parts or a single known share
Combine ratios and connect ratios to fractions

Simplifying ratios

Divide all parts by their highest common factor: $18 : 24 = 3 : 4$ .

Convert to the same units first: 45 cm to 2 m is $45 : 200 = 9 : 40$ .

To write a ratio in the form $1 : n$ , divide both parts by the first: $4 : 10 = 1 : 2.5$ .

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

Write $2.5$ kg $: 750$ g in its simplest form.

Your answer

Type it like 3:5.

Show worked solution

Solution

1Same units: $2500 : 750$ .
2Divide by 250: $10 : 3$ .

Answer: $10 : 3$

Tip — A ratio has no units once simplified, but the parts must be measured in the same units before you simplify.

Sharing in a ratio

Add the parts to find the total number of shares, divide the amount by that total to find one part, then multiply.

Check that the shares add up to the original amount.

Example 1Share £240 in the ratio

3 : 5

1Total parts:

3 + 5 = 8

2One part:

240 \div 8 = 30

3Shares:

3 \times 30

and

5 \times 30

Answer£90 and £150

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

Share £360 in the ratio $4 : 5$ . How much is the larger share, in pounds?

Your answer

Show worked solution

Solution

19 parts: one part $= £40$ .
2Larger share $= 5 \times 40 = £200$ .

Answer: £200

Using a difference or one share

If you know the difference between two shares, that difference corresponds to the difference in parts.

If you know one share, divide it by its number of parts to find one part.

Example 1Aled and Bea share money in the ratio

4 : 7

. Bea receives £36 more than Aled. How much do they share altogether?

1Difference in parts:

7 - 4 = 3

parts

= 36

, so one part is 12.

2Total parts: 11, so total

= 11 \times 12

Answer£132 (Aled £48, Bea £84)

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

Two amounts are in the ratio $3 : 8$ . The larger is 45 more than the smaller. What is the total?

Your answer

Show worked solution

Solution

1Difference: 5 parts $= 45$ , so one part $= 9$ .
2Total: 11 parts $= 99$ .

Answer: $99$

Why this works

The difference of £36 is not a share of anything on its own — it is the value of the three extra parts Bea has. Matching money to parts, not to people, is the key move.

Combining ratios and fractions

To combine $a : b = 2 : 3$ and $b : c = 4 : 5$ , make the shared quantity match. $b$ is 3 in one and 4 in the other; the LCM is 12, so scale to $8 : 12$ and $12 : 15$ , giving $a : b : c = 8 : 12 : 15$ .

A ratio converts to fractions of the whole: in $2 : 3$ , the first quantity is $\tfrac25$ of the total.

Example 1The ratio of red to blue counters in a bag is

3 : 5

. After 12 red counters are added, the ratio is

1 : 1

. How many counters were there originally?

1Originally

3k

red and

5k

blue.

2After adding:

3k + 12 = 5k

, so

k = 6

3Original counters:

8k

Answer48 counters (18 red and 30 blue)

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

In a club, boys : girls $= 5 : 3$ . When 12 more girls join, the ratio becomes $5 : 4$ . How many members were there originally?

Your answer

Show hint

Let boys $= 5k$ and girls $= 3k$ .

Show worked solution

Solution

1 $\frac{3k + 12}{5k} = \frac{4}{5}$ , so $15k + 60 = 20k$ and $k = 12$ .
2Originally $8k = 96$ members.

Answer: $96$

Tip — When a ratio changes, use a letter for one part and write both quantities in terms of it. The number of blue counters did not change, which gives the equation.

Think like an examiner

Show the value of one part explicitly.
Check that shares add to the total.
Give answers in context with units.

Ratio

a : b \Rightarrow \text{one part} = \tfrac{\text{total}}{a + b}

Sharing.

a : b \Rightarrow \tfrac{a}{a + b} \text{ of the whole}

Ratio to fraction.

Watch out for these

Dividing £240 by 3 and by 5 for the ratio

3 : 5

Divide by the total number of parts, 8.

Simplifying 45 cm : 2 m as

45 : 2

Convert units first:

45 : 200

Saying that in

2 : 3

the first part is

\tfrac23

of the whole.

It is

\tfrac25

;

\tfrac23

compares the parts to each other.

Combining ratios without matching the shared term.

Scale so the common quantity has the same number.

Stretch yourself

Three sisters share an inheritance in the ratio of their ages, $5 : 7 : 8$ . The eldest receives £4200 more than the youngest. How much is the inheritance, and what does the middle sister receive?

Hint — Find the difference in parts between the eldest and the youngest.

Reveal solution

SolutionThe eldest has 8 parts and the youngest 5, a difference of 3 parts

=

£4200, so one part is £1400. The total is

20

parts

=

£28 000, and the middle sister receives

7 \times 1400 =

£9800. Check:

7000 + 9800 + 11\,200 = 28\,000

✓.

\blacksquare

Questions students ask

Does the order of a ratio matter?

Yes. "Boys to girls is

2 : 3

" is different from "girls to boys is

2 : 3

Can ratios have decimals?

Yes, especially in the form

1 : n

, but otherwise simplify to whole numbers.

How are ratios and fractions related?

A ratio compares parts to parts; a fraction compares a part to the whole. In

a : b

, the first part is

\tfrac{a}{a + b}

of the whole.

Key takeaways

Simplify by dividing by the HCF, after matching units.
Share by finding one part: total ÷ sum of parts.
Use a difference between shares by matching it to the difference in parts.
Combine ratios by matching the common term.
In $a : b$ , the first quantity is $\tfrac{a}{a + b}$ of the whole.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 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 21 mark

Write $45 : 60$ in its simplest form.[1]
Show mark scheme
- B1 $3 : 4$
Answer: $3 : 4$

Question 2Grade 43 marks

Orange paint is made by mixing red and yellow paint in the ratio $2 : 3$ . Ali has 12 litres of red and 15 litres of yellow.

What is the greatest amount of orange paint Ali can make?[3]
Show mark scheme
- P1Checks both colours: red allows one part = 6 litres, yellow allows one part = 5 litres
- P1Yellow limits: one part = 5 litres
- A1 $25$ litres
Answer: $25$ litres

Question 3Grade 63 marks

$a : b = 2 : 3$ and $b : c = 4 : 5$ . $a + b + c = 105$ . Work out $c$ .[3]
Show mark scheme
- M1 $a : b : c = 8 : 12 : 15$
- M1One part $= 105 \div 35 = 3$
- A1 $c = 45$
Answer: $45$

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.

Next step

Keep practising

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