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This lesson covers DfE content statements L2.3 and L2.11 — using ratio and proportion to solve problems, and substituting values into formulae expressed in words and symbols.
A ratio compares two or more quantities. It shows the relative size of each part.
The ratio 3 : 2 means "for every 3 of the first, there are 2 of the second."
Divide all parts by their Highest Common Factor (HCF).
| Original Ratio | HCF | Simplified |
|---|---|---|
| 15 : 25 | 5 | 3 : 5 |
| 24 : 36 : 12 | 12 | 2 : 3 : 1 |
| 400 : 600 | 200 | 2 : 3 |
| 0.5 : 1.5 | (× 2 first) | 1 : 3 |
| 1/4 : 3/4 | (× 4 first) | 1 : 3 |
Exam Tip: Ratios must always be written using whole numbers. If you have decimals or fractions, multiply through to clear them before simplifying.
Method: Add the ratio parts to find the total number of shares. Divide the quantity by the total shares to find one share. Then multiply each ratio part by the value of one share.
Scenario: Two business partners share profits in the ratio 3 : 5. The total profit is £4,800. How much does each receive?
Total shares = 3 + 5 = 8 One share = £4,800 ÷ 8 = £600 Partner A = 3 × £600 = £1,800 Partner B = 5 × £600 = £3,000
Check: £1,800 + £3,000 = £4,800 ✓
Scenario: Concrete is mixed in the ratio 1 : 2 : 3 (cement : sand : gravel). You need 1,200 kg of concrete. How much of each material is needed?
Total shares = 1 + 2 + 3 = 6 One share = 1,200 ÷ 6 = 200 kg Cement = 1 × 200 = 200 kg Sand = 2 × 200 = 400 kg Gravel = 3 × 200 = 600 kg
Scenario: A recipe uses flour and sugar in the ratio 5 : 2. If you have 400g of sugar, how much flour do you need?
2 shares = 400g → 1 share = 200g → Flour = 5 × 200 = 1,000g (1 kg)
Two quantities are in direct proportion if they increase or decrease at the same rate — when one doubles, the other doubles.
Find the value of one item first, then scale to the amount you need.
Scenario: A recipe for 6 flapjacks uses 240g of oats, 120g of butter, and 90g of golden syrup. How much of each ingredient is needed for 15 flapjacks?
Step 1: Find the amount for 1 flapjack.
Step 2: Multiply by 15.
| Ingredient | For 1 | For 15 |
|---|---|---|
| Oats | 40g | 600g |
| Butter | 20g | 300g |
| Syrup | 15g | 225g |
Scenario: Washing-up liquid: 500 ml for £1.20 or 750 ml for £1.65. Which is better value?
Method — find the price per 100 ml:
The 750 ml bottle is better value.
Two quantities are in inverse proportion if, when one increases, the other decreases — and their product stays constant.
Scenario: It takes 6 workers 10 days to paint a building. How long would it take 4 workers (assuming they all work at the same rate)?
Workers × Days = constant → 6 × 10 = 60 4 workers: 60 ÷ 4 = 15 days
Scenario: A journey takes 3 hours at 60 mph. How long at 45 mph?
Speed × Time = Distance (constant) → 60 × 3 = 180 miles At 45 mph: 180 ÷ 45 = 4 hours
Exam Tip: Look for the clue: if increasing one quantity means the other must decrease (e.g. more workers → less time; faster speed → less time), it is inverse proportion. Multiply the two given values to find the constant.
At Level 2, you must be able to substitute numbers into formulae given in words or symbols.
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