Write a ratio from a context, simplify it fully, and rewrite it in the form 1 : n or n : 1. It is on Foundation and Higher, aimed at grade 3. Free questions with answers are on this page.
A ratio compares amounts of the same type, in order. 12:18 simplifies by dividing both sides by 6 to 2:3. The order matters: 2:3 is not 3:2. If the units differ, convert first (cm and m, minutes and hours) so you are not simplifying 2:80 when you meant 200:80.
A ratio 2:3 means 2 parts to 3 parts, five parts altogether. Writing a ratio as a fraction is 32 (first to second) or 52 (first to total) — those are different, and the question will say which it wants. Equivalent ratios are the same as equivalent fractions: multiply or divide all parts by the same number.
Same units first, then divide both parts by the HCF. 1:n means divide both parts by the left number. A ratio is not a fraction of the whole until you add the parts.
Objectives
2:3 is not the same as 3:2.
Divide both parts by the same number.
40 cm to 1 m is 40 cm to 100 cm, which is 2:5.
Key points
Write a ratio in the order the question gives the quantities.
Convert to the same units before simplifying.
Simplify by dividing every part by the highest common factor.
Write a ratio in the form 1:n (n may be a decimal).
Worked examples
Example 1
Write the ratio 4:6 in its simplest form.
2 marks
The highest common factor of 4 and 6 is 2.
4 ÷ 2 = 2 and 6 ÷ 2 = 3.
The simplest form is 2:3.
Answer: 2:3
Common slips
Slip: The highest common factor of 4 and 6 is 1. The number 1 is a common factor, but it is not the highest one.
Slip: 4 ÷ 2 = 2 and 6 stays 6. Both parts must be divided by the same number.
Slip: The simplest form is 2:6. The second part was not divided, so the ratio has changed.
Example 2
Write the ratio 40 cm to 1 m in its simplest form.
3 marks
Change 1 m into centimetres: 1 × 100 = 100 cm.
The ratio is 40:100, and the highest common factor is 20.
Dividing both parts by 20 gives 2:5.
Answer: 2:5
Common slips
Slip: Change 1 m into centimetres: 1 × 10 = 10 cm. A metre is 100 centimetres, not 10.
Slip: The ratio is 40:1, with the units left as they are. Both lengths must be in the same unit before you write the ratio.
Slip: Dividing only the first part by 20 gives 2:100. Both parts must be divided by the same number.
Example 3
Write the ratio 8:20 in its simplest form.
2 marks
The highest common factor of 8 and 20 is 4.
8 ÷ 4 = 2 and 20 ÷ 4 = 5.
The simplest form is 2:5.
Answer: 2:5
Common slips
Slip: The highest common factor of 8 and 20 is 1. The number 1 is a common factor, but it is not the highest one.
Slip: 8 ÷ 4 = 2 and 20 stays 20. Both parts must be divided by the same number.
Slip: The simplest form is 2:20. The second part was not divided, so the ratio has changed.
Questions and answers
GCSE Maths writing & simplifying ratio questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Write the ratio 15:25 in its simplest form.
Show answer
Answer: 3:5
The highest common factor of 15 and 25 is 5.
15 ÷ 5 = 3 and 25 ÷ 5 = 5.
M1 — The highest common factor of 15 and 25 is 5.
A1 — Answer 3:5
Question 2 Easy2 marks
Write the ratio 30 cm to 2 m in its simplest form.
Show answer
Answer: 3:20
Change 2 m into centimetres: 2 × 100 = 200 cm.
The ratio is 30:200, and the highest common factor is 10.
M1 — Change 2 m into centimetres: 2 × 100 = 200 cm.
A1 — Answer 3:20
Question 3 Medium2 marks
Write the ratio 60 cm to 1 m in its simplest form.
Show answer
Answer: 3:5
Change 1 m into centimetres: 1 × 100 = 100 cm.
The ratio is 60:100, and the highest common factor is 20.
M1 — Change 1 m into centimetres: 1 × 100 = 100 cm.
A1 — Answer 3:5
Question 4 Medium2 marks
Write the ratio 14:18 in its simplest form.
Show answer
Answer: 7:9
The highest common factor of 14 and 18 is 2.
14 ÷ 2 = 7 and 18 ÷ 2 = 9.
M1 — The highest common factor of 14 and 18 is 2.
A1 — Answer 7:9
Question 5 Hard2 marks
Write the ratio 35:40 in its simplest form.
Show answer
Answer: 7:8
The highest common factor of 35 and 40 is 5.
35 ÷ 5 = 7 and 40 ÷ 5 = 8.
M1 — The highest common factor of 35 and 40 is 5.
A1 — Answer 7:8
Question 6 Hard2 marks
Write the ratio 8:12 in its simplest form.
Show answer
Answer: 2:3
The highest common factor of 8 and 12 is 4.
8 ÷ 4 = 2 and 12 ÷ 4 = 3.
M1 — The highest common factor of 8 and 12 is 4.
A1 — Answer 2:3
Question 7 Exam-style3 marks
Write the ratio 50 cm to 2 m in its simplest form.
Show answer
Answer: 1:4
Change 2 m into centimetres: 2 × 100 = 200 cm.
The ratio is 50:200, and the highest common factor is 50.
Dividing both parts by 50 gives 1:4.
M1 — Change 2 m into centimetres: 2 × 100 = 200 cm.
M1 — The ratio is 50:200, and the highest common factor is 50.
A1 — Answer 1:4
Common mistakes
Most lost marks on this topic are one slip.
Dividing only one part.
Every part is divided by the same number.
Mixing centimetres and metres in one ratio.
Convert to the same unit first.
Swapping the order.
Keep the order of the words in the question.
Spot the slip
The highest common factor of 6 and 9 is
3
6
18
Show the slip
3 3 divides both, and no larger number does.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Show the factor you divided by — a single line '÷6' next to the ratio is enough.
In 1:n questions the answer is allowed to have a decimal; do not round unless told to.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.