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Welcome to GCSE Edexcel Maths revision.

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Topic M 24: Direct and inverse proportion.

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This video covers Higher tier.

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It includes the shared content and the labelled Higher extensions.

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Direct proportion means y changes by the same factor as x to y equals kx for constant k.

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The graph is a straight line through the origin.

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A straight line with a non-zero intercept is not direct proportion.

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Direct and inverse proportion

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Worked example: Five notebooks cost 12 pounds and 50 pence at a constant unit price.

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One costs 2 pounds and 50 pence, so eight cost 20 pounds.

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Here cost equals 2.5 multiplied by number of notebooks.

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For direct proportion, the fraction with numerator open bracket y close bracket and denominator open bracket x close bracket , end fraction is constant.

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Doubling x doubles y; tripling x triples y.

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Test ratios rather than differences.

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Inverse proportion means y equals the fraction with numerator open bracket k close bracket and denominator open bracket x close bracket , end fraction , so x y is constant.

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For positive quantities, doubling x halves y.

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Its graph is a reciprocal curve, not a straight line.

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Worked example: Four equally productive workers take 9 hours for a fixed job.

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Six workers take open bracket 4 multiplied by 9 close bracket divided by 6 equals 6 hours, assuming work is shared perfectly and each worker's rate is unchanged.

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For a fixed distance, time is inversely proportional to speed.

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This needs a constant journey length; a general time-versus-speed situation may not be inverse proportion.

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To use a proportional equation, find the constant from known values, then substitute the new input.

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Keep the complete relation, not just the value of k.

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Worked example: y equals the fraction with numerator open bracket 24 close bracket and denominator open bracket x close bracket , end fraction .

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When x equals 3, y equals 8; when x equals 8, y equals 3. x equals 0 is not allowed.

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Read the context carefully: a fixed starting fee, changing productivity or a changing total can invalidate a proportional model.

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If y is directly proportional to x squared , write y equals kx squared .

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If y is inversely proportional to x squared ,

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write y equals the fraction with numerator open bracket k close bracket and denominator open bracket x squared close bracket ,

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end fraction .

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State which power the question specifies.

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Worked example: y is proportional to x squared , and y equals 18 when x equals 3.

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Then 18 equals 9 k, so k equals 2 and y equals 2 x squared .

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At x equals 5, y equals 50.

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Worked example: y is proportional to the fraction with numerator open bracket 1 close bracket and denominator open bracket x squared close bracket ,

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end fraction ,

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and y equals 12 when x equals 2.

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Then k equals 48, so y equals the fraction with numerator open bracket 48 close bracket and denominator open bracket x squared close bracket , end fraction .

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At x equals 4, y equals 3.

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If y is proportional to square root of x and y equals 15 at x equals 9, then k equals 5 and y equals 5 multiplied by square root of x.

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To find x when y equals 20, square root of x equals 4, so x equals 16.

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Plotting y against x squared gives a straight line through the origin for y equals kx squared .

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Its gradient is k.

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Plotting against the fraction with numerator open bracket 1 close bracket and denominator open bracket x close bracket , end fraction does the same for inverse proportion.

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That completes Direct and inverse proportion.

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Revisit the notes and test yourself on the revision website.