GCSE · Ratio, proportion and rates of change

Proportion Algebraically

At KS3 you solved proportion problems with tables and the unitary method. At GCSE you write the relationship as a formula: y = kx for direct proportion, y = k/x for inverse proportion, and (Higher) relationships like y = kx² or y = k/x². One number, k, holds the whole relationship, and a four-step method finds it every time.

GCSE subject content: solve problems involving direct and inverse proportion, including graphical and algebraic representations; recognise and interpret graphs that illustrate direct and inverse proportion; understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; (Higher) construct and interpret equations that describe direct and inverse proportion

Before you start: a quick recap

Direct proportion: two quantities go up in the same ratio. Double one, the other doubles. 3 pens cost £6, so 6 pens cost £12.
Inverse proportion: one goes up as the other goes down in the same ratio. Double the number of workers and the job takes half the time.
Revise both, with the unitary method and recipes, on the KS3 Direct and Inverse Proportion page. For the Higher sections you'll also need squares, cubes and roots (Powers and Roots) and solving simple equations (Linear Equations).

1 · Direct proportion: y = kx

Petrol costs £1.50 a litre. 2 litres cost £3, 4 litres cost £6, 10 litres cost £15. Divide any cost by its number of litres and you always get the same answer, 1.5. That fixed number is called the constant of proportionality, written k. So

y ÷ x = k so y = kx

We write "y is directly proportional to x" as y ∝ x. The symbol ∝ means "is proportional to". Drag the plum point along the line and watch y ÷ x stay the same. Then tick the box to add a fixed charge and see what breaks.

k = add a fixed charge of 3

How to recognise direct proportion

In a table, y ÷ x is the same for every pair.
The graph is a straight line through the origin (0, 0). A straight line that doesn't go through (0, 0), like a taxi fare with a fixed charge, is not proportion: doubling the distance doesn't double the fare.
k is the gradient of the line (see Linear Graphs).

2 · The four-step method

Exam questions describe the relationship in words and give you one pair of values. These four steps work for every proportion question, including the Higher ones later on:

Write it with ∝. "y is directly proportional to x" → y ∝ x.
Replace ∝ with "= k". y ∝ x becomes y = kx.
Find k. Substitute the pair of values you're given and solve for k.
Use the formula. Write it out with the number for k, then substitute to answer the question.

Pick a problem and step through it.

problem

Try it: direct proportion

k = answer

Common mistake: k upside down. From y = kx, k = y ÷ x, not x ÷ y. If y = 12 when x = 4, then 12 = k × 4, so k = 3. Always write the substitution as an equation and solve it; don't guess which way to divide.

3 · Inverse proportion: y = k/x

A field needs 12 hours of digging. 1 person takes 12 hours; 2 people take 6 hours; 3 people take 4 hours; 4 people take 3 hours. Multiply the two numbers in each pair and you always get 12. For inverse proportion, the product is constant:

x × y = k so y = k/x

We write "y is inversely proportional to x" as y ∝ 1/x: y is directly proportional to one over x. Picture it as a rectangle with width x and height y. Its area x × y is always k. Drag the corner: as the rectangle gets wider it must get shorter, and the corner traces out the curve.

k =

Inverse proportion facts and traps

If x is multiplied by a number, y is divided by the same number. Double x → halve y.
The graph is a curve that gets close to both axes but never touches them: x can't be 0 (you can't divide by 0), and y never reaches 0.
"Inverse" does not mean y = k − x. With y = 12 − x, doubling x from 2 to 4 changes y from 10 to 8, not to 5. Check: is x × y constant? If not, it isn't inverse proportion.

4 · Proportion with powers and roots Higher

y can be proportional to something built from x, not just x itself. "The distance fallen is proportional to the square of the time" means d ∝ t², so d = kt². The same four steps work; you just square (or cube, or square-root) x before you multiply by k. Pick a relationship and see its graph and table.

y ∝ k =

In words	With ∝	Formula
y is proportional to the square of x	y ∝ x²	y = kx²
y is proportional to the cube of x	y ∝ x³	y = kx³
y is proportional to the square root of x	y ∝ √x	y = k√x
y is inversely proportional to x	y ∝ 1/x	y = k/x
y is inversely proportional to the square of x	y ∝ 1/x²	y = k/x²

Exam trap: working backwards. If y = 3x² and you're told y = 75, then x² = 25, so x = 5. Don't stop at 25: you found x², not x. For y = k√x you square at the end instead.

5 · What happens when x is scaled? Higher

Because of the power, scaling x scales y by a different amount. If y = kx² and x doubles, then the new y is k(2x)² = k × 4x² = 4 × kx². So y is multiplied by 2² = 4. In general, if x is multiplied by s, then y is multiplied by s to the same power as x appears in the formula. k doesn't matter: it cancels.

y ∝ x is multiplied by

Try it: what happens to y?

y is multiplied by

6 · Recognising the graphs Higher

Each relationship has its own shape. Direct proportion (y ∝ x) is a straight line through the origin; inverse proportion (y ∝ 1/x) is a curve that never touches the axes. (Every GCSE student needs those two.) At Higher, y ∝ x² and y ∝ x³ curve upwards from the origin more and more steeply (x³ more steeply still); y ∝ √x rises from the origin but flattens; y ∝ 1/x² falls even faster than 1/x. Match each graph to its relationship.

Shape clues

Through the origin and straight → y ∝ x. Straight but not through the origin → not proportion.
Through the origin, bending upwards → y ∝ x² (or x³: steeper still after x = 1, flatter before it).
Through the origin, bending over (rising more slowly) → y ∝ √x.
Falling, never touching the axes → y ∝ 1/x or y ∝ 1/x². Test a point: if doubling x quarters y, it's 1/x².

7 · Mixed practice: the four steps with any relationship Higher

Each question is new. Write down the four steps on paper. Give exact answers, or round to 3 s.f. if they don't come out exactly. The feedback tries to spot which step went wrong.

relationship

answer

Worked examples

y is directly proportional to x. When x = 4, y = 12. Find y when x = 10.

1. With ∝: y ∝ x.

2. With k: y = kx.

3. Find k: 12 = k × 4, so k = 12 ÷ 4 = 3.

4. Use it: y = 3x. When x = 10, y = 3 × 10 = 30.

Check: x went from 4 to 10 (× 2.5), and y from 12 to 30 (× 2.5) ✓

Is this table direct proportion? x: 2, 5, 8 and y: 7, 17.5, 28.

Test y ÷ x for each pair: 7 ÷ 2 = 3.5; 17.5 ÷ 5 = 3.5; 28 ÷ 8 = 3.5.

All the same, so yes: y is directly proportional to x, with y = 3.5x.

Compare: x: 2, 5 and y: 7, 13 gives 3.5 and 2.6. Not equal, so that one is not proportion (it's y = 2x + 3).

The time T days to build a wall is inversely proportional to the number of builders n. 6 builders take 10 days. How long do 4 builders take?

1. With ∝: T ∝ 1/n.

2. With k: T = k/n.

3. Find k: 10 = k/6, so k = 10 × 6 = 60. (k is the total work: 60 builder-days.)

4. Use it: T = 60/n. When n = 4, T = 60 ÷ 4 = 15 days.

Sense check: fewer builders, so it should take longer ✓

Higher d is proportional to the square of t. d = 20 when t = 2. Find d when t = 6, and t when d = 125.

1–2. d ∝ t², so d = kt².

3. Find k: 20 = k × 2² = 4k, so k = 5.

4a. d = 5t². When t = 6, d = 5 × 36 = 180.

4b. When d = 125: 125 = 5t², so t² = 25 and t = 5 (time is positive, so ignore −5).

Higher y is inversely proportional to the square of x. y = 8 when x = 3. Find y when x = 6.

1–2. y ∝ 1/x², so y = k/x².

3. Find k: 8 = k/3² = k/9, so k = 72.

4. y = 72/x². When x = 6, y = 72 ÷ 36 = 2.

Check with scaling: x doubled (3 → 6), so y is divided by 2² = 4: 8 ÷ 4 = 2 ✓

Higher y is proportional to the square root of x. y = 12 when x = 9. Find y when x = 25, and x when y = 10.

1–2. y ∝ √x, so y = k√x.

3. Find k: 12 = k × √9 = 3k, so k = 4.

4a. y = 4√x. When x = 25, y = 4 × 5 = 20.

4b. When y = 10: 10 = 4√x, so √x = 2.5 and x = 2.5² = 6.25. (Undo the square root by squaring.)

Higher The power P of a wind turbine is proportional to the cube of the wind speed v. By what percentage does P increase if v increases by 20%?

Scaling: v is multiplied by 1.2, and P ∝ v³, so P is multiplied by 1.2³.

Calculate: 1.2³ = 1.728.

Interpret: × 1.728 is an increase of 0.728 = 72.8%. (Not 60%: you multiply, you don't add 3 × 20%.)

Quick check

What you've learned

Direct proportion y ∝ x means y = kx: y ÷ x is constant, and the graph is a straight line through the origin with gradient k.
Inverse proportion y ∝ 1/x means y = k/x: x × y is constant, and the graph is a curve that never touches the axes. Multiply x by a number and y is divided by it.
The four steps: write with ∝; replace ∝ with = k; substitute the given pair to find k; write the formula with k and use it.
Higher: y ∝ x², y ∝ x³, y ∝ √x and y ∝ 1/x² become y = kx², y = kx³, y = k√x and y = k/x². When working backwards, undo the power (square-root, cube-root or square).
Higher: if x is multiplied by s, y is multiplied by s to the same power: s² for x², s³ for x³, √s for √x, 1/s for 1/x and 1/s² for 1/x².
Higher: recognise each graph by its shape: straight through the origin; curving up; bending over; falling towards the axes.