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Ratio and proportion describe relationships between quantities. They appear in map scales, recipes, currency exchange, speed, and many other real-life contexts.
A ratio compares two or more quantities of the same type. The ratio 3 : 5 means "for every 3 parts of A, there are 5 parts of B."
Divide all parts by their HCF. Example: 18 : 24 → HCF = 6 → 3 : 4
Add the parts, find the value of one part, then multiply.
Example: Share £72 in the ratio 3 : 5. Total parts = 8; one part = £72 ÷ 8 = £9 → 3 × £9 = £27 and 5 × £9 = £45
The ratio 3 : 5 means the first quantity is 3/8 of the total; the second is 5/8.
If the ratio of red to blue counters is 2 : 7, and there are 56 blue counters, how many red? 2/7 = r/56 → r = 16
A scale of 1 : 50,000 means 1 cm on the map represents 50,000 cm = 500 m in real life.
Example: Two cities are 4.5 cm apart on a 1 : 200,000 map. Real distance = 4.5 × 200,000 cm = 900,000 cm = 9 km
Scale factors in geometry: if a shape is enlarged by scale factor k, all lengths multiply by k and the area multiplies by k².
Two quantities are in direct proportion if, when one doubles, the other doubles. y is directly proportional to x means y = kx for some constant k.
Example: 5 pens cost £3.75. How much do 8 pens cost? k = 3.75 ÷ 5 = 0.75; cost of 8 = 8 × 0.75 = £6.00
Direct proportion gives a straight-line graph through the origin.
Two quantities are in inverse proportion if, when one doubles, the other halves. y is inversely proportional to x means y = k/x.
Example: 6 workers take 10 days to complete a job. How long do 4 workers take? k = 6 × 10 = 60; days = 60 ÷ 4 = 15 days
Inverse proportion gives a curved (reciprocal) graph.
% change = (change ÷ original) × 100
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