EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Ratio and proportion
Fractions, ratio and percentages · Lesson 3 of 5
Last Lesson and Before
Answer each one, then check.
1. Last lesson: share 20 in the ratio 1 : 4.
4 and 16
2. Work out 12 × 1.5.
18
3. Work out 450 ÷ 6.
75
4. Last lesson: write 4 : 10 in the form 1 : n.
1 : 2.5
Learning Objectives
1. Solve direct proportion problems with the unitary method.
2. Compare best buys.
3. Convert between currencies.
4. Solve inverse proportion problems.
5. Recognise direct and inverse proportion from a graph.
Direct Proportion
Two quantities are in direct proportion when they increase at the same rate: double one and the other doubles.
▸ The unitary method. Find the value of ONE, then multiply: if 6 pens cost £4.20, one costs 70p, so 10 cost £7.
▸ Scaling. Multiply both by the same number: 6 people to 9 people is × 1.5.
▸ The graph. A straight line through the origin.
▸ Examples. Cost and number of items, recipe amounts and servings, pounds and euros.
A Recipe
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A recipe for 6 people uses 450 g of flour. How much flour is needed for 10 people? |
1. Find the amount for 1 person
450 ÷ 6 = 75 g
2. Multiply by 10
75 × 10 = 750 g
Answer: 750 g of flour
Best Buy
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A 500 g bag of rice costs £1.85. A 750 g bag costs £2.70. Which is better value? |
1. Find the cost of the same amount of each, say 100 g
185 ÷ 5 = 37p and 270 ÷ 7.5 = 36p
2. Compare
36p is less than 37p
Answer: The 750 g bag is better value (36p per 100 g).
Currency Conversion
Exchange rates are direct proportion.
▸ Multiply to convert from pounds. If £1 = €1.15, then £240 is 240 × 1.15 = 276 euros, €276.
▸ Divide to convert back. €92 is 92 ÷ 1.15 = 80 pounds, £80.
▸ Check the size. Each pound buys more than one euro, so the euro amount should be bigger.
Inverse Proportion
Two quantities are in inverse proportion when one goes up as the other goes down, and their product stays the same.
▸ The idea. Double the number of workers and the job takes half the time.
▸ The method. Find the total "work": 6 painters × 8 days = 48 painter-days. Then divide.
▸ The graph. A curve that gets closer and closer to both axes.
▸ Examples. Workers and time, speed and journey time, people sharing a prize.
An Inverse Proportion Problem
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6 painters take 8 days to paint a building. How long would 4 painters take, working at the same rate? |
1. Find the total work
6 × 8 = 48 painter-days
2. Divide by the new number of painters
48 ÷ 4 = 12
Answer: 12 days
Direct and Inverse Proportion Graphs
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The shape of a graph tells you which kind of proportion you have. A straight line through the origin means direct proportion. A curve falling towards both axes means inverse proportion. |
Direct: a straight line through the origin. Inverse: a curve towards both axes. |
Key Terms
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Direct proportion Two quantities that increase at the same rate; their ratio stays the same. |
Inverse proportion As one quantity increases, the other decreases; their product stays the same. |
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Unitary method Finding the value of one item first, then scaling up. |
Exchange rate How much of one currency you get for one unit of another. |
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Best buy The option that gives the most for your money. |
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Your Task: Shopping Detective
12 minutes
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Which is the best buy? (a) Cereal: 500 g for £2.40 or 750 g for £3.45. (b) Juice: 1 litre for £1.10 or 1.5 litres for £1.62. (c) Eggs: 6 for £1.35 or 15 for £3.30. Then: 3 friends share a prize and get £40 each. How much would each get if 5 friends shared it? 1. Compare the same amount of each. 2. Show the unit price. 3. Decide, and say why. |
A good answer shows: (a) 48p vs 46p per 100 g: 750 g box. (b) 11p vs 10.8p per 100 ml: 1.5 litres. (c) 22.5p vs 22p per egg: 15 eggs. Prize: 3 × 40 = 120; 120 ÷ 5 = £24 each.
Can I...?
☐ Use the unitary method.
☐ Scale a recipe.
☐ Compare best buys.
☐ Convert between currencies.
☐ Solve inverse proportion problems.
☐ Recognise direct and inverse proportion graphs.
Summary
✓ Direct proportion: find one, then multiply.
✓ Best buy: compare the price of the same amount.
✓ Currency: multiply one way, divide the other.
✓ Inverse proportion: the product stays the same.
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EXAM FOCUS It takes 5 machines 12 hours to make an order. How long would 3 machines take? (2 marks) Ask first: if one quantity doubles, does the other double (direct) or halve (inverse)? Getting that right is most of the marks. |