Proportion (AQA GCSE Maths): Revision Notes
Proportion
What is proportion?
Proportion describes the relationship between two quantities and how they change in relation to each other. Understanding proportion is essential for solving problems involving costs, rates, time, and many real-world situations you'll encounter in your GCSE exam.
Proportion problems appear frequently in GCSE Mathematics exams because they connect mathematical concepts to real-world applications that students encounter daily.
There are two main types of proportion you need to understand clearly.
Direct proportion
Direct proportion occurs when two quantities change in the same direction at a consistent rate. When one quantity increases, the other increases by the same factor. When one decreases, the other decreases proportionally.
You'll recognise direct proportion in situations involving:
- Number of items purchased and total cost
- Time worked and wages earned
- Distance travelled at constant speed and time taken
Understanding direct proportion with examples
Consider buying cinema tickets: if 3 tickets cost £135, then 9 tickets will cost £405. Both quantities (number of tickets and total cost) have been multiplied by 3, showing they're directly proportional.
Solving direct proportion problems
The key method involves two clear steps:
- Find the unit rate first - work out the cost or value for one item
- Multiply by the required amount - scale up to find your answer
Worked Example: Picture Frame Costs
If 4 picture frames cost £11.40, find the cost of 7 frames.
Step 1: Find the unit rate Cost of 1 frame = £11.40 ÷ 4 = £2.85
Step 2: Multiply by required amount
Cost of 7 frames = £2.85 × 7 = £19.95
Inverse proportion
Inverse proportion occurs when two quantities change in opposite directions. As one quantity increases, the other decreases at a predictable rate. The key characteristic is that when you multiply the two quantities together, you always get the same result.
Common inverse proportion situations include:
- Speed and journey time (for a fixed distance)
- Number of people working and time to complete a task
- Sharing items among different numbers of people
Understanding inverse proportion with examples
If you travel at 40 km/h, a journey takes 2 hours. If you double your speed to 80 km/h, the journey time halves to 1 hour. As speed increases, time decreases proportionally.
Solving inverse proportion problems
The method requires you to think about the total amount of work needed:
- Calculate the total work - multiply the original quantities together
- Divide by the new quantity - this gives you the answer
Worked Example: Wall Building
If 6 people can build a wall in 4 days, how long will it take 8 people?
Step 1: Calculate total work needed Total work = 6 × 4 = 24 person-days
Step 2: Divide by new number of people Time for 8 people = 24 ÷ 8 = 3 days
Deciding when to multiply or divide
Use logical reasoning to determine your approach:
Multiply when the relationship is direct:
- More items bought = higher total cost
- More hours worked = more wages earned
- Longer journey = more fuel needed
Divide when the relationship is inverse:
- More people working = less time needed
- Faster speed = less journey time
- More people sharing = smaller individual portions
Key Decision Rule:
Inverse proportion problems often involve time, where having more resources (people, speed, power) reduces the time required. When you see time as one of your quantities, consider whether inverse proportion applies.
Exam guidance
Essential Exam Tips:
When tackling proportion problems in your exam:
- Calculate step by step - always find the unit rate first for direct proportion
- Show your working clearly - examiners award marks for method even if the final answer is wrong
- Use appropriate units - include £, hours, km/h etc. in your answers
- Round money carefully - give monetary answers to 2 decimal places
- Check your answer makes sense - does having more input give you a logical output?
- Choose your calculation method wisely - work in either pounds or pence, not both
Key Points to Remember:
- Direct proportion: quantities change in the same direction - both increase or decrease together
- Inverse proportion: quantities change in opposite directions - one goes up as the other goes down
- Direct proportion method: find the unit rate, then multiply by the required amount
- Inverse proportion method: calculate total work needed, then divide by the new quantity
- Use common sense to check whether your answer is reasonable before writing your final response