Direct and inverse proportion

Mathematics (1MA1) · Pearson Edexcel GCSE (9-1) · 4 min read · Updated: 7 Sept 2026

Introduction to Direct and Inverse Proportion

In mathematics, proportion describes how two quantities relate to each other as they change. Have you ever noticed that the more hours you work, the more you get paid? Or that the faster you cycle, the less time it takes to reach your friend's house? These are real-life examples of proportion.

In this chapter, we will look at how to describe these relationships using words, graphs, and algebra. This topic is a key part of the Ratio, proportion and rates of change section of your Edexcel GCSE.

1. Direct Proportion

When two quantities are in direct proportion, they increase or decrease at the same rate. If you double one, the other doubles too. If you halve one, the other halves.

How it looks in words

We say "\(y\) is directly proportional to \(x\)" or "\(y\) varies directly as \(x\)".

The Mathematical Rule

For direct proportion, the relationship is written as:
\(y = kx\)

The letter \(k\) is called the constant of proportionality. It is just a fixed number that connects the two quantities. For example, if apples cost \(\text{£}0.50\) each, the cost (\(C\)) is directly proportional to the number of apples (\(n\)), so \(C = 0.5n\). Here, \(k = 0.5\).

Direct Proportion Graphs

A graph showing direct proportion is always a straight line that passes through the origin \((0,0)\). This makes sense: if you buy zero apples, it costs zero pounds!

Quick Tip: If a straight-line graph does not pass through \((0,0)\), the variables are not in direct proportion.

Key Takeaway: In direct proportion, as one variable goes up, the other goes up at a constant rate. The formula is always \(y = kx\).

2. Inverse Proportion

Inverse proportion is the opposite. When one quantity goes up, the other goes down at the same rate.

Real-World Example

Think about painters painting a fence. If you have more painters, it takes less time. If you double the number of painters, the time taken is halved.

The Mathematical Rule

We say "\(y\) is inversely proportional to \(x\)" or "\(y\) is proportional to \(1/x\)".
The relationship is written as:
\(y = \frac{k}{x}\) or \(xy = k\)

Inverse Proportion Graphs

The graph of an inverse proportion relationship is a reciprocal curve. It never touches the \(x\) or \(y\) axes.

Key Takeaway: In inverse proportion, as one variable goes up, the other goes down. The formula is always \(y = \frac{k}{x}\).

3. Higher Tier: Constructing Equations (H)

Note: This section is for Higher Tier students who need to build the formulas themselves.

At the Higher Tier, you may see the symbol \(\propto\), which means "is proportional to". You will also need to deal with squares, cubes, and roots.

Step-by-Step: Finding the Equation

Step 1: Write the relationship using the \(\propto\) symbol.
Step 2: Replace \(\propto\) with "\( = k\)".
Step 3: Use the numbers given in the question to solve for \(k\).
Step 4: Rewrite the equation with your value for \(k\).

Example: Proportion with Squares

Question: \(y\) is directly proportional to the square of \(x\). When \(x = 3, y = 18\). Find the equation for \(y\) in terms of \(x\).

1. Write the relationship: \(y \propto x^2\)
2. Change to an equation: \(y = kx^2\)
3. Substitute the values: \(18 = k \times (3)^2\) \(\implies 18 = 9k\) \(\implies k = 2\)
4. Final equation: \(y = 2x^2\)

Summary of Higher Relationships

\(y\) is proportional to \(x^2\): \(y = kx^2\)
\(y\) is proportional to \(\sqrt{x}\): \(y = k\sqrt{x}\)
\(y\) is inversely proportional to \(x^2\): \(y = \frac{k}{x^2}\)

4. Common Mistakes to Avoid

Forgetting to find \(k\): Many students just multiply the numbers they see. Always find the value of \(k\) first!
Mixing up Direct and Inverse: Read the question carefully. "Direct" means multiply by \(k\); "Inverse" means divide \(k\) by the variable.
Squares and Roots: In Higher Tier questions, don't forget to square or square root the number before calculating \(k\).
The Origin: Remember, direct proportion must be a straight line through \((0,0)\). If it hits the \(y\)-axis anywhere else, it's just a linear relationship, not direct proportion.

5. Chapter Summary Checklist

Foundation and Higher students should be able to:

Identify direct proportion from a table or a graph (straight line through the origin).
Identify inverse proportion (as one goes up, the other goes down).
Use a given formula like \(y = kx\) or \(y = \frac{k}{x}\) to find missing values.
Understand that inverse proportion to \(x\) is the same as direct proportion to \(1/x\).

Higher Tier students should also be able to:

Use the \(\propto\) symbol to set up equations.
Find the value of the constant \(k\) using given coordinates or values.
Solve problems where proportion involves squares, cubes, or square roots.

For more information on related topics, see the chapters on "Units and compound measures" and "Growth, decay and iterative processes".

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the definition of direct proportion between two quantities?

Direct proportion means that as one quantity increases or decreases, the other changes at the same rate, such that their ratio remains constant.

What is the general algebraic formula for direct proportion?

The formula for direct proportion is \(y = kx\), where \(k\) is the constant of proportionality.

What are the key characteristics of a direct proportion graph?

A direct proportion graph is always a straight line that passes through the origin \((0,0)\).

What is the definition of inverse proportion?

Inverse proportion is a relationship where as one quantity increases, the other decreases at the same rate.

What is the general algebraic formula for inverse proportion?

The formula is \(y = \frac{k}{x}\) or \(xy = k\), where \(k\) is the constant of proportionality.

What does the graph of an inverse proportion relationship look like?

The graph of an inverse proportion relationship is a reciprocal curve that never touches the axes.

What does the symbol \(\propto\) represent in mathematics?

The symbol \(\propto\) means "is proportional to".

How do you find the constant of proportionality \(k\) in an exam question?

Substitute the given pair of \(x\) and \(y\) values into your proportion equation and solve for \(k\).

How do you set up an equation where \(y\) is proportional to the square of \(x\)?

The equation is written as \(y = kx^2\).

What is a common mistake when solving inverse proportion problems with squares?

A common mistake is forgetting to square the variable before calculating or using the constant \(k\).

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