IGCSE Maths / Direct and inverse proportion
Direct proportion revision notes
Edexcel International GCSE Maths (4MA1), chapter 21: Direct and inverse proportion, specification point 2.5a. Free lesson.
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In direct proportion, one quantity is always a fixed number times another, so doubling one doubles the other when the proportion is to x. Write the relationship using k, find k from the values you know, then use the equation to find a new value.
What direct proportion means
Two quantities are directly proportional when one is always the same fixed number times the other. That fixed number is the constant of proportionality, written as k. If y is directly proportional to x, doubling x doubles y and trebling x trebles y. Writing y = kx, where k is a constant, states the relationship precisely. Saying only that the quantities increase together is too vague for the mark.
How to find a new value when y ∝ x
The proportion sign, ∝, records the relationship in the question. Replace it with an equals sign and k to make an equation. Find k using the pair of values you are given, then substitute the new x value. For y ∝ x, the equation is y = kx. The graph is a straight line through the origin.
How a power changes the proportion
When y is proportional to x², the equation is y = kx². The square applies to x only, not to k. The graph is a curve that gets steeper as x grows. If x doubles, y becomes four times as large because 2² = 4. With y ∝ x³, doubling x makes y eight times as large because 2³ = 8. Check the power before deciding how the quantities scale.
Use the equation instead of guessing
Scaling by eye can give the wrong result when there is a power on x. For y ∝ x², doubling x does not just double y. Find k and substitute into the equation so the power is used correctly. For the values x = 3 and y = 45, k = 5, so the equation is y = 5x². At x = 6, y = 5 × 6² = 180, which is four times 45.
The method to remember
The same steps work for direct proportion questions. Find k before using a new value, because without k you do not yet have an equation to calculate with. Write the relationship with the proportion sign. Replace the sign with an equation containing k. Find k from the given pair, then check the power before substituting.