Proportion
Direct and inverse proportion, including finding and using the constant in y = kx and y = k/x.
Watch the lesson
Every point below, worked through in one video, and each point links straight to its own part.
Learn
Proportion, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Recognise direct and inverse proportion from tables, graphs and descriptions
Direct proportion: double one, the other doubles - the graph is a straight line through the origin. Inverse proportion: double one, the other HALVES, like speed and journey time - the graph is a curve that never touches the axes. Check a table by testing whether stays constant (direct) or x × y does (inverse).
Solve proportion problems using the unitary method - recipes, prices, scaling
Go down to one unit, then up to what you need. If 6 cakes need 270 g of flour, one cake needs 45 g, so 10 cakes need 450 g. For inverse problems flip the logic: 4 builders take 6 days, so 1 builder takes 24 days, and 8 builders take 3.
Interpret graphs of quantities in direct or inverse proportion
A straight line through the origin says "directly proportional", and its gradient is the constant - the unit price, the exchange rate. A reciprocal-shaped curve says "inversely proportional". A straight line that misses the origin is NOT direct proportion, however straight it looks.
Set up and use proportion equations like y = kx and y = , including squares and cubesHigher only
Translate the sentence into an equation with a constant k: "y is proportional to x²" means y = kx². Use the given pair of values to find k, then answer the question. If y = 48 when x = 4, then k = 3, so y = 3x² and x = 5 gives y = 75.
Common mistakes in Proportion
The slips that cost marks year after year, and what to do instead.
Treating inverse proportion as direct, so if 4 builders take 6 days, 8 builders take 12.
More builders means less time. 4 builders take 6 days, so 1 builder takes 24 days and 8 builders take 24 ÷ 8 = 3 days.
Calling any straight-line graph direct proportion, even one that misses the origin.
Direct proportion is a straight line through (0, 0). The line y = 2x + 3 is straight, but doubling x does not double y, so it is not proportional.
Finding k with x instead of x² when y is proportional to x², so y = 48 at x = 4 gives k = 12.Higher only
Write the equation first, y = kx², then substitute: 48 = k × 16, so k = 3 and y = 3x².
Writing y = kx when the question says y is inversely proportional to x.Higher only
Inverse proportion puts x on the bottom: y = . If y = 6 when x = 4, then k = 6 × 4 = 24, so y = .
Practice
Try a Proportion question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
Answer: 45
Find k = y ÷ x, then use y = kx for the new value of x.
- Direct proportion means y = kx.
- Find k: k = 18 ÷ 4 = 4.5.
- Use x = 10: y = 4.5 × 10 = 45.
Ready to practise Proportion?
Sign up free to practise this topic - questions that adapt to how you're doing, with instant encouraging feedback on every answer. No card needed.
Already have an account? Log in