Proportion
Proportion helps us understand how quantities relate to each other. It’s a key skill for scaling recipes, comparing prices, and solving a huge range of real-world problems where things change in a fair, predictable way.
The Unitary Method
The most reliable way to solve proportion problems is the unitary method. It’s a simple, three-step process to find any unknown quantity.
1. Start with what you know
Write down the known relationship between the two quantities. E.g., “4 cookies need 100g of flour”.
2. Find the value of ‘one unit’
Divide to find the value for a single unit. E.g., “1 cookie needs $100\text{g} \div 4 = 25\text{g}$ of flour”.
3. Scale up to the required amount
Multiply the value of one unit by the number you need. E.g., “12 cookies need $25\text{g} \times 12 = 300\text{g}$ of flour”.
Worked Examples
Example 1: Scaling a Recipe
A recipe for 4 cookies needs 100g of flour. How much flour is needed for 12 cookies?
- Value of one unit: Flour for 1 cookie = $100\text{g} \div 4 = 25\text{g}$.
- Scale up: Flour for 12 cookies = $25\text{g} \times 12 = 300\text{g}$.
Answer: 300g of flour
Example 2: Best Buy Problem
Which is better value: a 500ml bottle for £1.20, or a 750ml bottle for £1.70?
We need to find a common unit to compare, like the cost per 100ml.
- Offer A (500ml):
Cost per 1ml = £1.20 ÷ 500 = £0.0024.
Cost per 100ml = £0.24 or 24p. - Offer B (750ml):
Cost per 1ml = £1.70 ÷ 750 ≈ £0.00227.
Cost per 100ml ≈ £0.227 or 22.7p.
Answer: Offer B is better value as it costs less per 100ml.
Tutor Insights
🤔 Common Misunderstandings
- Not finding the value of one unit. Students often get mixed up about what to multiply or divide by. The unitary method avoids this by always bringing it back to ‘one’.
- Forgetting to convert units, for example, when comparing prices for items measured in grams and kilograms.
📝 Common Exam Mistakes
- Not showing working. You must show your steps, like how you found the unit value, to get full marks.
- Calculation errors, especially with decimals.
- Not fully answering the question. For best-buy problems, you must state which offer is better value at the end.
Practice Questions
- If 3 cans of fizzy drink cost £1.80, how much would 5 cans cost?
- A car travels 120 miles on 10 litres of petrol. How far can it travel on 25 litres of petrol?
- A painter charges £60 for 3 hours of work. How much would they charge for 7 hours?
- Which pack of sweets is better value? Pack A: 150g for £1.20 or Pack B: 220g for £1.65.
Show Answers
- Working: Cost of 1 can = £1.80 ÷ 3 = £0.60. Cost of 5 cans = £0.60 × 5 = £3.00.
Answer: £3.00. - Working: Miles per litre = 120 ÷ 10 = 12 miles. Distance on 25 litres = $12 \times 25 = 300$ miles.
Answer: 300 miles. - Working: Charge per hour = £60 ÷ 3 = £20. Charge for 7 hours = £20 × 7 = £140.
Answer: £140. - Working: Pack A cost per 100g = (£1.20 / 150) * 100 = 80p. Pack B cost per 100g = (£1.65 / 220) * 100 = 75p.
Answer: Pack B is better value.
FAQs
What’s the difference between ratio and proportion?
A ratio compares quantities (e.g., $1:2$). Proportion is a statement that two ratios are equal (e.g., $1:2 = 2:4$). Most problems you’ll solve use the concept of direct proportion.
Do I always have to find the value of one unit?
For most GCSE questions, the unitary method (finding the value of ‘one’) is the simplest and most reliable way to solve proportion problems. Sometimes you can spot a direct multiplication factor, but the unitary method always works.
