What is a ratio?
A ratio compares two or more quantities, showing how much of one thing there is compared to another. Ratios are written using a colon, such as 3:2, and are read as "3 to 2".
Two things matter with every ratio:
- Order matters — the ratio 3:2 is not the same as 2:3.
- Units must match — before comparing two quantities in a ratio, they need to be in the same units (e.g. both in cm, or both in minutes).
Writing a ratio
Worked Example 1
A fruit bowl contains 8 apples and 12 oranges. Write the ratio of apples to oranges in its simplest form.
1
Write the ratio in the order asked for: apples : oranges = 8 : 12
2
Find the highest common factor of 8 and 12, which is 4
3
Divide both sides by 4: 8 ÷ 4 = 2, and 12 ÷ 4 = 3
Answer2 : 3
Thinking of a ratio as equal-sized "parts" makes both simplifying and sharing problems much easier.
Equivalent ratios
Just like fractions, a ratio has infinitely many equivalent forms. Multiplying or dividing both sides of a ratio by the same number gives an equivalent ratio.
Worked Example 2
A recipe uses flour and sugar in the ratio 3:2. If you use 300g of flour, how much sugar is needed?
1
Find the scale factor from the flour amounts: 300 ÷ 3 = 100
2
Multiply the sugar side of the ratio by the same scale factor: 2 × 100 = 200
Answer200g of sugar
Ratio as a fraction
A ratio can be converted into fractions by treating the total number of parts as the denominator.
Worked Example 3
A class has boys and girls in the ratio 3:5. What fraction of the class are girls?
1
Find the total number of parts: 3 + 5 = 8
2
Girls are 5 parts out of the 8 total parts
Answer5⁄8
Ratios with three or more parts
Ratios aren't limited to two quantities. The same rules for equivalent ratios apply — whatever you do to one part, you must do to every part.
Worked Example 4
Three siblings share sweets in the ratio 2:3:5. If the smallest share is 6 sweets, how many sweets does the largest share get?
1
The smallest ratio part is 2, and this corresponds to 6 sweets, so find the scale factor: 6 ÷ 2 = 3
2
Multiply the largest part (5) by the scale factor: 5 × 3 = 15
Answer15 sweets
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Write the ratio 6:9 in its simplest form.Foundation
Q2A bag has 5 red counters and 15 blue counters. Write the ratio of red to blue in simplest form.Foundation
Q3The ratio of cats to dogs at a shelter is 2:7. If there are 4 cats, how many dogs are there?Foundation
Q4Write the ratio 40p : £1.20 in its simplest form. (Hint: convert to the same units first.)Foundation
Q5A drink is made from squash and water in the ratio 1:4. What fraction of the drink is squash?Foundation
Higher (Grade 5–7)
Q6A model car is built to a scale of 1:20. If the model is 9cm long, how long is the real car, in metres?Higher
Q7Paint is mixed from blue and white in the ratio 5:3. How much white paint is needed for 600ml of blue paint?Higher
Q8Three friends share a prize in the ratio 2:3:4. What fraction of the total does the friend with the largest share receive?Higher
Q9The ratio of adults to children on a coach is 3:8. There are 33 people on the coach. How many are children?Higher
Q10Write 2.5 : 4 as a ratio of whole numbers in its simplest form.Higher
Higher — Hard (Grade 8–9)
Q11a : b = 2 : 5 and b : c = 3 : 4. Find a : b : c.Grade 8–9
Q12A alloy is made from copper, zinc and tin in the ratio 5:3:2. If the alloy contains 45kg of copper, find the total mass of the alloy.Grade 8–9
Q13The ratio of x to y is 3:7. Find the ratio of (x + y) to y.Grade 8–9
Q14A map has a scale of 1:25,000. Two towns are 8.4km apart in real life. How far apart are they on the map, in centimetres?Grade 8–9
Q15In a school, the ratio of Year 10 to Year 11 students is 6:5. If Year 11 has 20 fewer students than Year 10, how many students are in each year group?Grade 8–9
Answers
Foundation (Q1–Q5)
Q12 : 3
Q21 : 3
Q314 dogs(scale factor 2, 7 × 2)
Q41 : 3(40p : 120p)
Q51⁄5
Higher (Q6–Q10)
Q61.8m(9 × 20 = 180cm)
Q7360ml(scale factor 120, 3 × 120)
Q84⁄9
Q924 children(33 ÷ 11 = 3, 8 × 3)
Q105 : 8(2.5:4 → 5:8 after doubling)
Higher — Hard (Q11–Q15)
Q116 : 15 : 20(scale a:b to 6:15, then b:c to 15:20)
Q1290kg(scale factor 9, total parts 10, 9 × 10)
Q1310 : 7
Q1433.6cm(840,000cm ÷ 25,000)
Q15Year 10: 120, Year 11: 100(1 part = 20, from the difference of 1 part; 6×20, 5×20)
Common mistakes
Common Mistake 1
Mixing up the order of a ratio
"Boys to girls" means boys first — writing girls : boys instead flips the ratio and gives a completely different answer. Always check the order stated in the question.
Common Mistake 2
Comparing quantities in different units
40p : £1.20 cannot be simplified directly — convert both amounts to the same unit (e.g. pence) before simplifying, otherwise the ratio is meaningless.
Common Mistake 3
Using the wrong denominator when converting to a fraction
For a ratio of 3:5, girls are not 5⁄3 of the class — the denominator must be the total number of parts (3 + 5 = 8), not the other part of the ratio.
Common Mistake 4
Only scaling one side of the ratio
When finding an equivalent ratio, the same scale factor must be applied to every part — scaling just one side breaks the ratio and gives an incorrect answer.
Exam tips
💡 Exam Tip 1
Always check the units first
Before writing or simplifying a ratio, check whether the quantities are already in the same units — if not, convert one of them first.
💡 Exam Tip 2
Find the scale factor before scaling
Work out what one part of the ratio has been multiplied by, then apply that same scale factor to the other part(s) — this avoids arithmetic slips.
💡 Exam Tip 3
Add the parts for "fraction of total" questions
Whenever a question asks for a fraction or percentage of the whole, start by adding all parts of the ratio together to find the total number of parts.
💡 Exam Tip 4
Combine ratios by matching the shared quantity
When combining two ratios like a:b and b:c, scale both so the shared letter (b) has the same value in each — then merge them into a single a:b:c ratio.
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