IGCSE Maths / Direct and inverse proportion
Inverse proportion
Edexcel International GCSE Maths (4MA1), chapter 21: Direct and inverse proportion, specification point 2.5a. Free lesson.
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Inverse proportion means the product of two quantities stays constant, so as one increases, the other decreases in a fixed way. You model it with an equation and use that equation to find unknown values.
What inverse proportion means
Two quantities are inversely proportional when multiplying them always gives the same number, called k. This is why doubling one quantity halves the other. A statement that one goes up while the other goes down is not enough on its own to show inverse proportion. For y inversely proportional to x, write y = k/x. The x is underneath k. This is what makes the relationship inverse rather than direct.
How to find k and use the equation
Once you have written the equation, use a given pair of values to find k. In y = k/x, multiplying both sides by x gives k = xy, so multiply the given values. Then substitute k into the equation and use it to find the unknown value.
How inverse proportion appears on a graph
The graph of y = k/x is a curve that slopes down as x increases. It gets close to both axes but never touches either one. Each pair of values on the curve has the same product, k. If x increases, y decreases so that xy stays constant.
When x is squared
Sometimes y is inversely proportional to x², so the equation is y = k/x². The square and the division both matter. Find k by multiplying y by x², then square the new x value before dividing k by it. For y = k/x², k = yx². The graph drops more steeply than the graph of y = k/x.
A quick check and the common mistake
For y inversely proportional to x, use y = k/x and multiply the given pair to get k. Do not divide to find k: division is the direct proportion method. Always check whether the equation uses x or x². For x = 3 and y = 8, k = 3 × 8 = 24. When x = 12, y = 24/12 = 2.