Ratio and proportion are fundamental concepts in GCSE Maths, appearing consistently across all major exam boards like AQA, Edexcel, and OCR. They form the backbone of many real-world problems, from scaling recipes to calculating best buys. This guide breaks down the key techniques you need to know, complete with worked examples and diagrams.
What are Ratio and Proportion?
A ratio shows the relative sizes of two or more values. It tells us how much of one thing there is compared to another. We write ratios using a colon (e.g., 2:5) or as a fraction.
Proportion describes the relationship between two ratios. If two ratios are equivalent, they are in proportion.
Core Skill 1: Simplifying Ratios
The golden rule: treat a ratio like a fraction. To simplify, ÷ all parts by the highest common factor (HCF).
Example: Simplify the ratio 18:24.
Find the HCF of 18 and 24. This is 6.
Divide both parts by 6: 18 ÷ 6 = 3, 24 ÷ 6 = 4.
- The simplified ratio is 3:4.
For ratios involving decimals or fractions, multiply all parts by a number to eliminate them first.
Core Skill 2: Sharing in a Ratio
This is a classic GCSE question: "Share £120 in the ratio 3:2".
The Unitary Method (Step-by-Step):
Add the parts: 3 + 2 = 5 parts in total.
Find the value of one part: Divide the total amount by the total number of parts. £120 ÷ 5 = £24 per part.
Multiply:
3 parts: 3 × £24 = £72
- 2 parts: 2 × £24 = £48
You can visualise this process with a simple flowchart.
flowchart TD
A["Start: Total Amount and Ratio"] --> B["Add ratio parts
to find total parts"]
B --> C["Divide total amount
by total parts to find
value of 1 part"]
C --> D["Multiply value of 1 part
by each part of the ratio"]
D --> E["Finish: Shares found"]
Core Skill 3: Combining Ratios
Sometimes you need to combine two ratios that have a common element. For example, the ratio of apples to oranges is 3:2, and the ratio of oranges to bananas is 1:4. Find the ratio of apples:oranges:bananas.
Method: Make the 'linking' value (oranges) the same in both ratios.
Apples:Oranges = 3:2
Oranges:Bananas = 1:4 is equivalent to 2:8 (we multiplied by 2 to make the oranges '2').
- Now we can combine: Apples:Oranges:Bananas = 3:2:8.
Understanding Direct and Inverse Proportion
This is where proportion comes into its own. Recognising the type is half the battle.
Direct Proportion
As one value increases, the other increases at the same rate. The relationship can be written as ( y \propto x ), which becomes ( y = kx ), where ( k ) is the 'constant of proportionality'.
Example: If 5 pencils cost £1.20, how much do 8 cost?
Find ( k ) (cost per pencil): £1.20 ÷ 5 = £0.24.
- Multiply by the new number: 8 × £0.24 = £1.92.
Inverse Proportion
As one value increases, the other decreases. The relationship is ( y \propto 1x ), which becomes ( y = kx ).
Example: It takes 6 builders 10 days to build a wall. How long would it take 4 builders? (Assume inverse proportion).
Find ( k ) (the total "builder-days"): ( k = y × x = 6 × 10 = 60 ).
- Apply to new number: ( y = kx = 604 = 15 ) days.
Use this decision chart to identify the problem type.
flowchart TD
A["Proportion Problem"] --> B["Does one quantity
increase as the other increases?"]
B -- Yes --> C["Direct Proportion
Use y = kx"]
B -- No --> D["Does one quantity
increase as the other decreases?"]
D -- Yes --> E["Inverse Proportion
Use y = k/x"]
D -- No --> F["Not a simple proportion
Check the question!"]
Tackling Scale Diagrams and Maps
A scale is a ratio of length on the drawing : real length. A common map scale is 1:25,000, meaning 1 cm on the map represents 25,000 cm (or 250 m) in real life.
To find a real distance: Map Distance × Scale Factor.
To find a map distance: Real Distance ÷ Scale Factor.
Remember: Always convert measurements to the same units before calculating!
Exam Tips and Common Pitfalls
Units are Crucial: Ensure all parts of a ratio are in the same unit before you start. Convert 50p to £0.50, or 1.2m to 120cm.
Order Matters: The ratio 2:3 is not the same as 3:2. Pay attention to what each part represents.
Check Your Answer: After sharing in a ratio, add your shares together. They should equal the original total.
Spot the Proportion Type: Keywords help. Direct: "per", "for every". Inverse: "takes longer with fewer people", "speed and time".
- Show Your Working: Even for simple ratio division, writing "120 ÷ 5 = 24" secures a method mark.
Worked GCSE-Style Question
Question: Paint is mixed from red and yellow in the ratio 3:7 to make orange.
I have 4.5 litres of red paint. How much yellow paint do I need? How much orange paint will I make?
Step-by-Step Solution:
The ratio Red:Yellow is 3:7.
We know 3 parts of red equals 4.5 litres.
Find one part: 4.5 litres ÷ 3 = 1.5 litres per part.
Find yellow paint (7 parts): 7 × 1.5 litres = 10.5 litres.
- Find total orange: Total parts = 3 + 7 = 10. 10 × 1.5 litres = 15 litres.
Answer: I need 10.5 litres of yellow paint and will make 15 litres of orange paint.
Final Thoughts
Mastering ratio and proportion is about practising the step-by-step methods until they become second nature. Start by identifying what the question is asking for—simplifying, sharing, combining, or a proportion relationship—then apply the correct technique. Use the flowcharts in this guide as a mental checklist. With consistent practice, you'll find these problems are a reliable source of marks in your GCSE Maths exam.
- Good luck with your revision!