Ratio and Proportion
O Level and IGCSE Mathematics Cheat Sheet – 1.11 Ratio and Proportion
Ratios – Key Points
- Compares quantities in a fixed relationship.
- Can be written as a : b or a to b.
- Simplify by dividing all terms by their highest common factor (HCF).
Example: 20 : 30 : 40
HCF = 10 → Simplest form = 2 : 3 : 4
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Dividing a Quantity in a Given Ratio
- Add the parts in the ratio.
- Find value of 1 part = Total ÷ Total parts.
- Multiply to get each share.
Example: Divide $60 in the ratio 2 : 3
Parts = 2 + 3 = 5
1 part = 60 ÷ 5 = 12
Shares: 2 × 12 = $24, 3 × 12 = $36
Proportion – Key Idea
- Two quantities are in proportion if they increase or decrease at the same rate.
- Direct proportion: as one increases, the other increases at constant ratio.
Formula: y = kx - Inverse proportion: as one increases, the other decreases at constant product.
Formula: xy = k
Proportional Reasoning Examples
- Recipe Scaling: If a recipe for 4 people needs 300g flour, for 10 people:
Scale factor = 10 ÷ 4 = 2.5 → 300 × 2.5 = 750g - Map Scales: If scale is 1 : 50,000, then 1 cm on map = 50,000 cm in reality = 500 m.
- Best Value: Compare cost per unit (price ÷ quantity) — lower = better value.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Exam Special Tips
- Always simplify ratios before comparing.
- Convert mixed units before writing a ratio (e.g., 50 cm : 2 m → 50 cm : 200 cm → 1 : 4).
- For inverse proportion, remember product is constant.
- When dividing money or items, parts must be whole numbers in the ratio.
Common Mistakes & Confusions
- Forgetting to simplify ratios.
- Mixing direct and inverse proportion formulas.
- Not converting to the same units before forming ratio.
- Assuming ratio parts are amounts instead of proportions.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course