Direct and inverse proportion: what they mean and how to use them

What is a proportion in maths?

Two amounts are in proportion when they're linked by multiplying: change one, and the other changes by the same scale. The GCSE content describes proportion as "equality of ratios". If 2 tickets cost £10 and 6 tickets cost £30, the ratios 2:10 and 6:30 are equal, so the cost is in proportion to the number of tickets.

Children first meet the idea in Year 6, where the national curriculum notes mention recipes and similar shapes: a recipe for 8 people needs twice as much of everything as a recipe for 4. At key stage 3 they solve problems involving direct and inverse proportion, using tables, graphs and simple formulas. It builds on ratio, so it's worth being confident with that first.

What does directly proportional mean?

Directly proportional means that when one amount is multiplied by a number, the other is multiplied by the same number. Double one and the other doubles; divide one by 3 and the other is divided by 3. If one is zero, so is the other.

Pencils at 30p each are a simple example:

Pencils	Cost
1	30p
2	60p
5	£1.50
10	£3.00

Divide the cost by the number of pencils and you always get the same answer: 150 ÷ 5 = 30, and 300 ÷ 10 = 30. That fixed number is what makes it direct proportion.

Worked example: a recipe

A recipe for 4 people uses 300 g of pasta. How much for 10 people?

1. Find the amount for 1 person: 300 ÷ 4 = 75, so 75 g each.
2. Multiply by the number of people: 75 × 10 = 750.

So 10 people need 750 g of pasta. Finding the amount for one first works for any number of people.

Worked example: buying fabric

8 metres of fabric cost £36. What do 5 metres cost?

1. Find the cost of 1 metre: 36 ÷ 8 = 4.5, so £4.50 a metre.
2. Multiply: 4.5 × 5 = 22.5.

So 5 metres cost £22.50. A quick check: 5 metres is a bit more than half of 8 metres, and £22.50 is a bit more than half of £36. Multiplying decimals covers the second step.

What does inversely proportional mean?

Inversely proportional means that when one amount is multiplied by a number, the other is divided by the same number. Double one and the other halves. Multiplying the two amounts together always gives the same answer.

Worked example: painting a fence

3 people take 8 hours to paint a fence. How long would 6 people take, and 4 people, working at the same speed?

1. Find the total work: 3 × 8 = 24, so the job is 24 hours of work for one person.
2. Share it between 6 people: 24 ÷ 6 = 4, so 4 hours.
3. Share it between 4 people: 24 ÷ 4 = 6, so 6 hours.

More people means less time, and the number of people times the hours is always 24.

Worked example: speed and time

A journey is 120 miles. At 60 miles an hour it takes 2 hours, because 60 × 2 = 120. How long does it take at 40 miles an hour? Speed times time is always 120, so 120 ÷ 40 = 3, and it takes 3 hours. The slower the speed, the longer the time.

Direct and inverse proportion formulas

Both kinds of proportion can be written as a formula using a fixed number, usually called k.

• Direct proportion: y = kx. To find k, divide y by x.
• Inverse proportion: y = k/x, which means k divided by x. To find k, multiply x by y.

For direct proportion, say y = 12 when x = 3. Then k is 12 ÷ 3 = 4, so the formula is y = 4x. When x = 7, y is 4 × 7 = 28.

For inverse proportion, say y = 6 when x = 4. Then k is 6 × 4 = 24, so the formula is y = 24/x. When x = 8, y is 24 ÷ 8 = 3.

The key stage 4 programme of study puts it this way: saying y is inversely proportional to x is the same as saying y is directly proportional to 1/x. It lists interpreting these equations for everyone, and setting them up as extra content for more highly attaining pupils. Algebra basics and solving equations cover the algebra this uses.

Direct and inverse proportion graphs

A direct proportion graph is a straight line through the origin, the point where both amounts are zero. The steeper the line, the bigger k is. For the pencils, the line goes up 30p for every pencil.

An inverse proportion graph is a curve. It starts high on the left, drops quickly, then levels out, getting closer and closer to both axes without ever touching them. That's because one amount can get very small, but never reaches zero, however big the other gets.

Recognising and reading these two shapes of graph is part of the GCSE content.

How to tell direct from inverse proportion

Ask what happens when one amount doubles:

• If the other doubles too, it's direct proportion.
• If the other halves, it's inverse proportion.
• If neither happens, it isn't in proportion at all.

Plenty of things aren't in proportion. A child's age and height both go up, but not at the same rate. A taxi with a fixed starting charge plus a cost per mile isn't in direct proportion either, because a journey of zero miles still costs something.

Common mistakes with proportion

• Assuming everything is direct. If more workers take less time, it's inverse: multiply, then divide, rather than scaling up.
• Adding instead of multiplying. If 4 people need 300 g of pasta, 10 people don't need 306 g. The amounts grow by multiplying, not adding.
• Getting the formula upside down. For direct proportion divide y by x to find k; for inverse proportion multiply them.
• Mixing units. Hours with minutes, or grams with kilograms, give wrong answers. Change them first.
• Not checking whether the answer makes sense. More pencils should cost more; more painters should take less time.

How to help at home

• Use shopping. If 3 apples cost 90p, ask what 5 would cost. Then ask what you'd do first.
• Scale recipes. Halving or tripling a recipe is direct proportion.
• Talk about journeys. Ask how long a trip would take if you drove faster or slower, which is inverse proportion.
• Ask "direct, inverse or neither?" Everyday pairs, like hours of TV and minutes left for homework, make good discussion. The percentage change page is a good next step, since percentages are proportions too.

Proportion questions to try

1. 6 pens cost £4.20. What do 10 pens cost?
2. y is directly proportional to x, and y = 15 when x = 5. Find y when x = 8.
3. 4 people take 6 hours to paint a fence. How long would 3 people take?
4. y is inversely proportional to x, and y = 10 when x = 2. Find y when x = 5.
5. Direct, inverse or neither: the number of people sharing a pizza and the number of slices each person gets.
6. Direct, inverse or neither: the number of £5 tickets bought and the total cost.

Answers

1. £7 (one pen costs 4.2 ÷ 6 = 0.7, so 70p, and 0.7 × 10 = 7)
2. 24 (k is 15 ÷ 5 = 3, and 3 × 8 = 24)
3. 8 hours (4 × 6 = 24 hours of work, and 24 ÷ 3 = 8)
4. 4 (k is 10 × 2 = 20, and 20 ÷ 5 = 4)
5. Inverse: twice as many people means half as many slices each
6. Direct: twice as many tickets cost twice as much

For the rest of what children learn at this stage, see the KS3 maths guide.