Key Stage 4 maths · Ratio, proportion and rates of change

GCSE ratio and proportion: compound units, similarity and growth

In Years 10 and 11, ratio and proportion becomes one of the most applied parts of GCSE maths. Pupils compare lengths, areas and volumes of similar shapes, convert compound units such as speed and density, understand direct and inverse proportion and their graphs, read the gradient of a line as a rate of change, and set up growth and decay problems including compound interest.

Compound units, proportion and growth, explained

Similar shapes have the same shape but different sizes, and every length on one is multiplied by the same number, the scale factor k, to get the other. Areas are multiplied by k² and volumes by k³. A model made at a scale of 1 : 50 has areas in the ratio 1 : 2,500 and volumes in the ratio 1 : 125,000. This also underlies trigonometry: right-angled triangles with the same angles are similar, so the ratio of two sides depends only on the angle, and sine, cosine and tangent record exactly that.

Compound units combine two measures: speed in metres per second, density in grams per cubic centimetre, pressure in newtons per square metre, pay in pounds per hour, prices in pounds per kilogram. Pupils convert between them in numbers and in algebra. A speed of 72 km/h is 72,000 m in 3,600 s, which is 20 m/s, and in algebra s km/h equals 5s ÷ 18 m/s. A density of 8 g/cm³ is 8,000 kg/m³. Care with area and volume units matters, because 1 m² is 10,000 cm², not 100.

Two quantities are in direct proportion when one is a fixed multiple of the other: y = kx, whose graph is a straight line through the origin. They are in inverse proportion when their product is fixed, so X is inversely proportional to Y means X is proportional to 1/Y, with an equation like y = 12 ÷ x and a curved graph that never touches the axes. Pupils interpret equations such as y = 5x or y = 12 ÷ x, and recognise and explain the graphs.

The gradient of a straight-line graph is a rate of change, and its units come from the axes. On a distance-time graph, the gradient in metres per second is the speed. On a graph of cost against weight, it is the price per kilogram. For direct proportion, y = kx, the gradient is k itself. Writing the units in the working often shows whether a calculation makes sense before any numbers are checked.

Growth and decay are repeated percentage changes. An increase of 3% is the same as multiplying by 1.03, and a decrease of 15% is multiplying by 0.85. Doing this for n periods means multiplying by the multiplier n times, so it becomes (multiplier)ⁿ. This is the idea behind compound interest, population growth and the loss in value of a car. Pupils set up the calculation, solve it and interpret what the answer means in the context.

Words your child will hear

Scale factor: The number each length is multiplied by in an enlargement or between similar shapes.
Compound unit: A unit made from two others, such as km/h, g/cm³ or £ per hour.
Direct proportion: Two quantities where one is a fixed multiple of the other, so doubling one doubles the other.
Inverse proportion: Two quantities whose product is fixed, so doubling one halves the other.
Rate of change: How quickly one quantity changes compared with another. On a graph it is the gradient.
Multiplier: The number you multiply by for a percentage change: 1.03 for +3%, 0.85 for −15%.
Compound interest: Interest worked out on the original amount plus interest already added.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios)
Convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts
Understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; interpret equations that describe direct and inverse proportion
Interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion
Set up, solve and interpret the answers in growth and decay problems, including compound interest

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

£2,000 is invested at 3% compound interest per year. How much is it worth after 4 years, and how much interest has been earned?

A 3% increase means the new amount is 100% + 3% = 103% of the old one. The multiplier is 1.03.
Compound interest repeats this each year, so after 4 years multiply by 1.03 four times: 2,000 × 1.03⁴.
On a calculator, 1.03⁴ = 1.12550881, so 2,000 × 1.12550881 = 2,251.0176.
Money is rounded to the nearest penny, so the value is £2,251.02. The interest earned is 2,251.02 − 2,000 = £251.02.
Sense check: simple interest would be 3% of 2,000 = £60 a year, so £240 in 4 years. Compound interest gives a little more, £11.02, because interest earns interest.

After 4 years the investment is worth £2,251.02, which is £251.02 of interest.

Common mistakes and how to help

Doubling the lengths of a shape and thinking the area doubles

Pupils use the scale factor 2 for everything because it is the only number in the question.

What helps: Draw a 1 cm square and a 2 cm square: the second holds four of the first. A 2 cm cube holds eight 1 cm cubes. Areas use k² and volumes use k³.

Converting area units with the length factor, or speeds with only the time factor

Pupils know 1 m = 100 cm and write 1 m² = 100 cm². Others divide by 60 to convert km/h to m/s.

What helps: Write it out: 1 m² is 100 cm × 100 cm = 10,000 cm². For speeds, convert distance and time separately: 72 km/h is 72,000 m per 3,600 s.

Calling anything that goes down “inverse proportion”

The word inverse sounds like “opposite”, so any relationship where one value falls as the other rises is taken to qualify.

What helps: Check that the product stays constant. If 6 people paint a fence in 4 hours, that is 24 person-hours, so 8 people take 3 hours. But y = 10 − x also falls and is not inverse proportion.

Treating compound growth or decay as one single change

For a 15% fall each year for 3 years, many take off 45% in one go, or use 0.15 as the multiplier.

What helps: A car worth £12,000 is worth 12,000 × 0.85³ = £7,369.50 after three years. Taking off 45% gives £6,600, which is too low because each year’s loss is 15% of a smaller amount.

How to help at home

Scale a recipe on paper. Take a recipe for four and rewrite it for six, then for ten. Then ask what happens to the base area if the tin’s sides are twice as long: it is four times as big, so the same mixture would spread much thinner.
Compare unit prices. In a shop, work out the price per 100 g or per litre of two sizes of the same item. Writing “£ per 100 g” builds the habit of thinking in compound units.
Convert a walking speed. A brisk walk is about 5 km/h. Ask your teenager to convert that to metres per second (about 1.4), then work out how far you would walk in a minute.
Watch money grow. Ask what £100 becomes in 10 years at 4% compound interest. On a calculator, 100 × 1.04¹⁰ is about £148. Compare it with simple interest, which would give £140.
Find a rate of change you use. Phone data used per day, savings per month or a part-time wage per hour are all rates. Sketch the line and show that the gradient is the rate.

Builds on

Key Stage 3: Units, scale and fractions of quantities
Key Stage 3: Ratio and sharing in a ratio
Key Stage 3: Percentage change, proportion, compound units