Inverse Proportion

What you will learn

• How to recognise inverse proportion from words, tables and graphs.
• How to use proportional notation such as y ∝ 1 ÷ x.
• How to find the constant of proportionality, k.
• How to use equations such as y = k ÷ x to calculate missing values.
• How to check inverse proportion using the constant product x × y.
• How to decide whether inverse proportion is a sensible model for a context.

Why this matters

Inverse proportion appears when a fixed amount is shared between changing quantities: more workers can finish a fixed job in less time, higher speed means less time for a fixed journey, and more identical taps can fill a tank faster.

It also helps you spot when direct proportion would give the wrong answer. If one value goes up while the other goes down, you should pause and test the relationship.

Prior knowledge

You should already be comfortable with:

• multiplying and dividing with whole numbers, decimals and fractions,
• using direct proportion and constants of proportionality,
• substituting values into simple formulae,
• rearranging equations involving multiplication and division,
• reading coordinates and simple curved graphs.

Clear explanation

What inverse proportion means

Two quantities are in inverse proportion when multiplying one quantity by a scale factor divides the other by the same scale factor. If one doubles, the other halves. If one is multiplied by 3, the other is divided by 3.

The job is fixed. More workers means less time, and the product stays constant: workers × hours = 24.

Proportional notation

The notation y ∝ 1 ÷ x means y is inversely proportional to x. This can be written as an equation:

The letter k is the constant of proportionality. For inverse proportion, it is the constant product of the two quantities.

Higher-tier questions can change the denominator. If y ∝ 1 ÷ x², use y = k ÷ x², so x² × y is the constant. Copy this part before substituting.

Finding the constant of proportionality

To find k, multiply a matching pair of values.

Tables and graphs

In an inverse proportion table, the value of x × y stays constant. On a graph, inverse proportion is shown by a decreasing curve, not a straight line through the origin.

A simple visual check

This graph represents y = 12 ÷ x. The labelled points all have product 12, so they fit the same inverse proportion relationship.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For inverse proportion, marks usually come from spotting that the product stays constant, finding k using multiplication, writing the relationship in the correct form such as y = k ÷ x, and checking that answers move in the opposite direction when x changes. For y ∝ 1 ÷ x², the constant is x²y, not xy. In work-rate or journey questions, explain the fixed job or fixed distance so the inverse relationship is justified.

Common mistakes

• Treating it like direct proportion: inverse proportion uses y = k ÷ x, not y = kx.
• Finding k the wrong way: for y = k ÷ x, use k = x × y.
• Checking y ÷ x instead of x × y: inverse proportion needs the product to stay constant.
• Expecting a straight-line graph: an inverse proportion graph is a decreasing curve.
• Ignoring powers: if y ∝ 1 ÷ x², the equation is y = k ÷ x², not y = k ÷ x.

Extension challenge

r is inversely proportional to the square root of s. When s = 16, r = 9. Find r when s = 81.

Exam-board guidance

Inverse proportion is common across GCSE Maths. Expect questions involving proportional formulae, tables, reciprocal-style curves, rates, measures and real-life contexts where one quantity increases as another decreases for a fixed total or fixed job. Higher-tier wording may use x² in the denominator, so the constant check changes.

Next lesson

Next, use proportion in diagrams and maps with Scale Drawings and Map Scales.