Direct and Inverse Proportion
Two quantities are in direct proportion when doubling one doubles the other: more pens, more cost. They're in inverse proportion when doubling one halves the other: more workers, less time. Each has a signature graph and a one-line formula.
National curriculum KS3: solve problems involving direct and inverse proportion, including graphical and algebraic representations; relate the algebraic solution to graphical representation1 · Direct proportion: a constant ratio
Pens cost 40p each. 1 pen is 40p, 2 pens 80p, 5 pens £2. Cost ÷ number is always 40: that constant is the whole story. Change the price and the number of pens and watch two things stay true: the ratio cost/pens doesn't move, and the graph is a straight line through the origin (zero pens cost nothing).
Not every straight line is direct proportion. A taxi with a £3 flag fare plus £2 per mile is linear but not proportional: 0 miles costs £3, not £0, and doubling the miles doesn't double the fare. Direct proportion is the special case y = kx with no "+ c". The graph must pass through (0, 0).
2 · The unitary method
Most proportion questions give you one pair of values and ask for another. The reliable route is through one: find what one unit is worth, then multiply up. "Find for 1, then find for many."
There's a second route that's sometimes quicker: find the scale factor between the two quantities you know (10 ÷ 6 = 1.67) and apply it to the other. Both give the same answer because both rely on the constant ratio.
3 · Inverse proportion: a constant product
A wall takes 12 hours for 1 painter. With 2 painters, 6 hours; with 3, 4 hours; with 4, 3 hours. More painters, less time, and painters × hours is always 12. That constant product is the signature of inverse proportion. The graph is a curve that falls steeply and then flattens, never touching either axis.
Why the product is constant: rectangles
Think of x × y as the area of a rectangle. Inverse proportion says the area is fixed. Make the rectangle twice as wide and it must be half as tall. Every rectangle below has the same area.
4 · Telling them apart
Given a table of values, which is it? Test both signatures: is y ÷ x constant (direct), or is x × y constant (inverse), or neither?
| Direct | Inverse | |
|---|---|---|
| Rule | y = kx (y ÷ x = k) | y = k/x (x × y = k) |
| Double x | y doubles | y halves |
| Graph | Straight line through the origin | Curve, never reaches the axes |
| Words | "per", "each", "at a rate of" | "shared between", "the more … the less" |
| Examples | cost vs quantity; distance vs time at fixed speed; exchange rates | speed vs time for a fixed distance; workers vs time; pressure vs volume |
5 · Recipes: proportion in the kitchen
A recipe for 4 people lists ingredients. Scale to any number of people: every ingredient is multiplied by the same factor, people ÷ 4. This is direct proportion applied to a whole list at once.
6 · Where proportion shows up
Currency
£1 = €1.17: pounds and euros are directly proportional, k = 1.17. The exchange graph is a line through the origin; a "commission" would add a +c and break it.
Physics
Ohm's law V = IR: voltage directly proportional to current. Boyle's law PV = constant: pressure inversely proportional to volume. Both are the two formulas on this page with different letters.
Travel
Fixed speed: distance ∝ time (direct). Fixed distance: speed and time are inverse: double your speed, halve your journey.
Computing
More servers, less time per request (inverse, until coordination costs bite). Download time is inversely proportional to bandwidth; file size scales directly with resolution.
Best buys
Unit pricing is the unitary method in the supermarket: price ÷ quantity gives the constant k for each offer, then compare.
Sharing out
A fixed prize shared between n winners: each gets prize ÷ n. More winners, smaller share. Inverse.
Worked examples
8 pencils cost £3.20. How much do 15 cost?
6 workers take 10 days to build a wall. How long for 4 workers?
y is directly proportional to x, and y = 18 when x = 4. Find y when x = 10.
Exam-style: a journey takes 3 hours at 60 km/h. How long at 45 km/h?
Quick check
What you've learned
Direct proportion: y ÷ x is constant, y = kx, a straight line through the origin, and doubling one doubles the other. Inverse proportion: x × y is constant, y = k/x, a falling curve, and doubling one halves the other. The unitary method (find one, then find many) handles direct problems; the fixed-product idea (total work, total distance) handles inverse ones. A table tells you which by testing the ratio and the product. Before calculating, ask whether the answer should go up or down.