GCSE Maths · Ratio, proportion and rates of change

Ratio GCSE Questions, Worked Examples and Answers

A ratio compares quantities multiplicatively. Put the quantities in the stated order and the same units, decide what one ratio part is worth, then scale every part by the same factor.

Edexcel · AQA · OCRFoundation & Higher15 original questions

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Order, units, parts

What you need to know about GCSE ratio

The ratio 3 : 5 means that for every three parts of the first quantity there are five parts of the second. The numbers describe relative size, not the quantities themselves.

Keep the scale factor consistent

Sharing starts with the total number of parts

To split £84 in the ratio 3 : 4, count seven parts, find one part, then build both shares.

Count3 + 4 = 7seven equal ratio parts altogether

Value one part£84 ÷ 7 = £12each part has the same value

Build shares£36 and £48multiply £12 by 3 and by 4

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Simplify a ratio

What the problem asks: A ratio is given and simplest whole-number form is required, possibly with decimals or fractions.

How to solve it: Put quantities in the same units. Clear decimals or fractions if needed, then divide every part by the highest common factor.

$2.5:1.5=25:15=5:3$

Write a ratio in the form 1 : n

What the problem asks: The question specifies 1 : n or n : 1 rather than simplest whole-number form.

How to solve it: Divide every part by the number that must become 1. The other part may be a decimal.

$5:8=1:1.6$

Share a total

What the problem asks: A total amount and a part : part ratio are both given.

How to solve it: Add the ratio numbers to count all parts, divide the total by that sum, then multiply for each share.

$\text{one part}=\frac{\text{total}}{a+b}$

Find an amount from one known share

What the problem asks: The ratio is known and one corresponding quantity is given.

How to solve it: Divide the known quantity by its matching ratio number to find one part, then multiply by the ratio number you need.

$\text{one part}=\frac{\text{known quantity}}{\text{its ratio number}}$

Use a recipe or mixture

What the problem asks: All ingredients must grow or shrink together.

How to solve it: Find one scale factor from the old number of servings to the new number, then multiply every ingredient by it.

Connect ratio and fractions

What the problem asks: A part of a combined whole is requested.

How to solve it: For ratio a : b, the whole has a + b parts, so the first fraction is a/(a+b).

$a:b\quad\Rightarrow\quad\frac{a}{a+b}\text{ and }\frac{b}{a+b}$

Combine two ratios

What the problem asks: Two ratios share a middle quantity, such as A : B and B : C.

How to solve it: Scale both ratios until the shared quantity has the same number of parts, then join them.

Solve an algebraic ratio

What the problem asks: Expressions or variables appear as ratio parts, or one variable must be written in terms of another.

How to solve it: Represent each part with one scale factor, or use equivalent fractions and cross-multiply. Then solve the resulting equation.

A reliable routine

A dependable ratio routine

Write labels above or beside the ratio so the order cannot silently swap.

Copy the ratio in the same order as the named quantities.
Convert quantities to the same units before comparing or simplifying.
Decide whether you need the total parts, one part, or a scale factor.
Apply the same multiplier or divisor to every ratio part and check the result against the context.

Check: For a sharing question, the shares must add to the original total. For an equivalent ratio, every part must have been scaled by the same factor.

Fully worked

Ratio GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Simplify a ratio

2 marks

Question

Simplify $42:56$ .

Recognise it: both parts are whole numbers and the ratio must be reduced.

Why this method: divide both parts by their highest common factor, $14$ .

$42:56=(42\div14):(56\div14)=\boxed{3:4}$

Example 2

Convert units before simplifying

3 marks

Question

Simplify $1.5\text{ m}:60\text{ cm}$ .

Recognise it: the quantities use different units, so the raw numbers cannot yet be compared.

Why this method: convert metres to centimetres first.

$1.5\text{ m}=150\text{ cm}$

$150:60=15:6=\boxed{5:2}$

Example 3

Share an amount

3 marks

Question

Share £420 in the ratio $3:4$ .

Recognise it: one total must be split into two shares.

Why this method: the ratio contains $3+4=7$ equal parts.

$\text{one part}=420\div7=60$

$3\times60=180,\qquad4\times60=240$

$\boxed{\text{£}180\text{ and £}240}$

Check: $180+240=420$ .

Example 4

Use one known quantity

3 marks

Question

Red and blue beads are in the ratio $5:7$ . There are $63$ blue beads. Find the number of red beads and the total number of beads.

Recognise it: seven ratio parts correspond to $63$ blue beads.

Why this method: find one part, then build the five red parts.

$\text{one part}=63\div7=9$

$\text{red}=5\times9=45$

$\text{total}=45+63=\boxed{108}$

Example 5

Interpret a part-to-whole ratio

3 marks

Question

In a box, the ratio of orange counters to all counters is $3:8$ . There are $144$ counters altogether. How many are not orange?

Recognise it: $8$ is the whole number of parts, not a second colour.

Why this method: three eighths are orange, so five eighths are not orange.

$\text{one part}=144\div8=18$

$\text{not orange}=5\times18=\boxed{90}$

Example 6

Scale a recipe

3 marks

Question

A recipe for $4$ people uses $300$ g of flour and $120$ g of sugar. Find the amounts needed for $10$ people.

Recognise it: the number of servings changes and every ingredient must keep the same ratio.

Why this method: use the scale factor $10\div4=2.5$ .

$300\times2.5=750\text{ g}$

$120\times2.5=300\text{ g}$

$\boxed{750\text{ g flour and }300\text{ g sugar}}$

Example 7

Combine linked ratios

Harder4 marks

Question

$A:B=2:3$ and $B:C=4:5$ . Find $A:B:C$ .

Recognise it: the two ratios share $B$ , but it has three parts in one and four in the other.

Why this method: make both versions of $B$ equal. The lowest common multiple of $3$ and $4$ is $12$ .

$A:B=2:3=8:12$

$B:C=4:5=12:15$

$\boxed{A:B:C=8:12:15}$

Example 8

Solve an algebraic ratio

Higher only4 marks

Question

The ratio $(x+5):(x-1)$ is equal to $3:2$ . Find $x$ .

Recognise it: two algebraic expressions form a ratio equal to $3:2$ .

Why this method: equivalent ratios form equal fractions, so cross-multiply and solve the linear equation.

$\frac{x+5}{x-1}=\frac32$

$2(x+5)=3(x-1)$

$2x+10=3x-3$

$\boxed{x=13}$

Check: $(13+5):(13-1)=18:12=3:2$ .

15 original questions · total 43 marks

Ratio GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 50 minutes · show the value of one part and include units · answers start collapsed

Simplify a ratio

1 mark

Simplify $36:48$ .

Show worked answer

Recognise it: divide both parts by their highest common factor, $12$ .

$36:48=\boxed{3:4}$

Make the units match

2 marks

Simplify $2.4\text{ kg}:600\text{ g}$ .

Show worked answer

Recognise it: convert to one unit first.

$2.4\text{ kg}=2400\text{ g}$

$2400:600=\boxed{4:1}$

Write a ratio in the form 1 : n

2 marks

Write $5:8$ in the form $1:n$ .

Show worked answer

Recognise it: the first part must become $1$ , so divide both parts by $5$ .

$5:8=(5\div5):(8\div5)$

$\boxed{1:1.6}$

Share money

3 marks

Share £198 in the ratio $4:5$ .

Show worked answer

Recognise it: split one total into $4+5=9$ parts.

$198\div9=22$

$4\times22=88,\qquad5\times22=110$

$\boxed{\text{£}88\text{ and £}110}$

Find a group and the total

3 marks

The ratio of boys to girls in a club is $7:9$ . There are $72$ girls. Find the number of boys and the total membership.

Show worked answer

Recognise it: nine parts equal $72$ , so one part is $8$ .

$\text{boys}=7\times8=56$

$\text{total}=56+72=\boxed{128}$

Write a ratio as a fraction

2 marks

Green and yellow tiles are in the ratio $5:3$ . What fraction of the tiles are green?

Show worked answer

Recognise it: the whole has $5+3=8$ parts and green uses $5$ of them.

$\boxed{\frac58}$

Scale a recipe

3 marks

A recipe uses $450$ g of rice for $6$ people. How much rice is needed for $14$ people?

Show worked answer

Recognise it: the amount per person stays constant.

$450\div6=75\text{ g per person}$

$75\times14=\boxed{1050\text{ g}}$

Use a map scale

3 marks

A map has scale $1:25{,}000$ . Two points are $7.2$ cm apart on the map. Find the real distance in kilometres.

Show worked answer

Recognise it: one map centimetre represents $25{,}000$ real centimetres.

$7.2\times25{,}000=180{,}000\text{ cm}$

Since $100{,}000$ cm is $1$ km,

$\boxed{1.8\text{ km}}$

Simplify ratios with decimals and fractions

4 marks

Simplify each ratio.

(a) $2.5:1.5$

(b) $\frac13:\frac14$

Show worked answer

Recognise it: clear the decimals or fractions first so both parts become whole numbers.

(a) Multiply both parts by $10$ , then simplify by $5$ .

$2.5:1.5=25:15=\boxed{5:3}$

(b) Multiply both parts by the lowest common multiple of the denominators, $12$ .

$\frac13:\frac14=4:3$

$\boxed{4:3}$

Compare two ratios

2 marks

Class A has $6$ laptops for $9$ pupils. Class B has $14$ laptops for $21$ pupils. Which class has the greater ratio of laptops to pupils, or are the ratios equal?

Show worked answer

Recognise it: simplify both ratios to compare the same relationship.

$6:9=2:3$

$14:21=2:3$

$\boxed{\text{Neither; the ratios are equal.}}$

Combine two ratios

Harder3 marks

$a:b=3:5$ and $b:c=10:7$ . Find $a:b:c$ .

Show worked answer

Recognise it: make the shared $b$ equal. Double the first ratio so $b=10$ .

$a:b=6:10$

$b:c=10:7$

$\boxed{a:b:c=6:10:7}$

Change one group

Harder4 marks

Cats and dogs at a shelter are in the ratio $5:3$ . There are $64$ animals. Ten dogs are adopted. Find the new ratio of cats to dogs.

Show worked answer

Recognise it: turn the original ratio into actual counts before changing dogs.

$64\div8=8$

$\text{cats}=40,\qquad\text{dogs}=24$

New dog count: $24-10=14$ .

$40:14=\boxed{20:7}$

Write one ratio quantity in terms of another

Higher only3 marks

Given $a:b=3:7$ , write $b$ in terms of $a$ .

Show worked answer

Recognise it: both quantities are multiples of the same scale factor.

Let $a=3k$ and $b=7k$ .

$k=\frac a3$

Substitute this into $b=7k$ .

$b=7\left(\frac a3\right)=\boxed{\frac{7a}{3}}$

Use the difference between ratio parts

Harder3 marks

Two lengths are in the ratio $7:4$ . Their difference is $27$ cm. Find their total length.

Show worked answer

Recognise it: the difference $27$ cm corresponds to $7-4=3$ parts.

$\text{one part}=27\div3=9\text{ cm}$

The total is $7+4=11$ parts.

$11\times9=\boxed{99\text{ cm}}$

Update a drink ratio

Harder5 marks

A drink contains squash and water in the ratio $2:7$ . There are $1.8$ litres of drink. Then $300$ ml of water is added. Find the new ratio of squash to water in its simplest form.

Show worked answer

Recognise it: first find the original quantities, then change only the water. Use millilitres throughout.

$1.8\text{ litres}=1800\text{ ml}$

There are $9$ parts, so one part is $1800\div9=200$ ml.

$\text{squash}=400\text{ ml},\qquad\text{water}=1400\text{ ml}$

New water amount: $1400+300=1700$ ml.

$400:1700=\boxed{4:17}$

Examiner-style feedback

Common ratio mistakes

Comparing different units

Convert first. The ratio 1 m : 50 cm is 100 : 50 = 2 : 1, not 1 : 50.

Changing only one side

Equivalent ratios multiply or divide every part by the same scale factor.

Dividing by one ratio number

When sharing in a : b, divide the total by a + b, because that counts all parts.

Swapping the order

If the question says red : blue, your first number must describe red. Add labels while working.

Treating part : whole as part : part

In orange : all = 3 : 8, the non-orange part is 5, because the eight parts already include orange.

Updating the ratio numbers directly

When people or objects are added, find the actual starting quantities first; the ratio numbers are not counts.

Forcing every answer into whole numbers

Simplest ratio and 1 : n are different instructions. In 1 : n form, the second number may be a decimal, such as 1 : 1.6.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Keep quantities in the named order and use matching units.
For sharing, add the parts and find the value of one part.
For equivalent ratios, use the same scale factor everywhere.
For 1 : n form, divide every part by the part that must become 1.
Check that shares total correctly and that the simplified ratio has no common factor.

Quick answers

Ratio FAQ

How do I simplify a ratio?

Convert all quantities to the same units, then divide every part by their highest common factor.

How do I share an amount in a ratio?

Add the ratio parts, divide the total by that sum to find one part, then multiply by each ratio number.

What is the difference between a ratio and a fraction?

A ratio can compare part with part or part with whole. A fraction of a whole uses the total number of ratio parts as its denominator.

How do I combine two ratios?

Scale both ratios until the shared middle quantity has the same value, then join the three quantities in order.

Do ratio quantities need the same units?

Yes, when the ratio compares measurements. Convert to one common unit before simplifying.

How do I simplify a ratio containing fractions or decimals?

Multiply every part by a common power of 10 or common denominator to create whole numbers, then divide by any common factor.

How do I write a ratio in the form 1 : n?

Divide every part by the first ratio number. Unlike simplest whole-number form, n may be a decimal.

Build connected skills

What to revise next

Connect to a whole

Fractions

Interpret ratio parts as exact fractions of a combined total.

Revise fractions

Compare per hundred

Percentages

Express one quantity as a percentage of another and use multipliers.

Revise percentages

Repeat a multiplier

Compound interest

Apply proportional growth to a changing balance over time.

Revise compound interest

Content standards

Curriculum and rights review

Curriculum references checked 4 September 2026. Ratio notation, 1 : n form, non-integer ratios, sharing, scaling, mixtures, fractions and linked ratios are included across GCSE Mathematics specifications. Algebraic ratio questions are labelled Higher only. All questions, values, contexts and solution wording are original Pass an Exam content.

Official specification references