Direct and Inverse Proportion Revision Notes for Edexcel GCSE Maths

Direct and inverse proportion

Understanding proportional relationships is a fundamental skill in algebra. When we talk about proportion, we're examining how two variables are connected and how they change in relation to each other.

What is proportion?

Proportion describes a special relationship between two variables where one variable changes in a predictable way when the other changes. The symbol $\propto$ means "is proportional to" and helps us express these relationships mathematically.

In proportion problems, you'll typically work with two variables (often called $x$ and $y$ ) that are linked together. Your job is to figure out their relationship and use it to find missing values when given information about one variable.

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The proportionality symbol $\propto$ is a powerful tool that allows us to express relationships before we know the exact mathematical equation. Think of it as a way to say "these variables are connected" before we figure out exactly how.

Simple proportions

The two most common types of proportional relationships you'll encounter are direct proportion and inverse proportion. Understanding these forms the foundation for tackling more complex problems.

Direct proportion

Direct proportion occurs when two variables increase or decrease together at the same rate. This means that as one variable gets larger, the other also gets larger in a predictable way.

The key characteristics of direct proportion are:

Both variables move in the same direction
When one doubles, the other doubles too
The relationship can be written as $y \propto x$
This converts to the equation $y = kx$ (where $k$ is a constant)
The graph is always a straight line passing through the origin

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Remember that for a relationship to be directly proportional, the line must pass through the origin (0,0). If it doesn't go through the origin, it's not a direct proportion.

Inverse proportion

Inverse proportion describes a relationship where one variable increases as the other decreases. This creates a balancing effect between the two variables.

The key features of inverse proportion are:

Variables move in opposite directions
When one variable doubles, the other halves
The relationship is written as $y \propto \frac{1}{x}$
This becomes the equation $y = \frac{k}{x}$ (where $k$ is a constant)
The graph forms a curved shape called a hyperbola

More complex proportional relationships

Beyond simple direct and inverse relationships, you'll encounter trickier proportions involving squares, cubes, square roots, and other mathematical functions.

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These more complex relationships follow the same basic principles. The key insight is that you can always convert a proportionality statement into an equation by replacing the $\propto$ symbol with $= k$ , where $k$ represents an unknown constant.

These more complex relationships follow the same basic principles. For example:

If $y$ is proportional to $x^2$ , then $y \propto x^2$ and $y = kx^2$
If $t$ is proportional to the square root of $h$ , then $t \propto \sqrt{h}$ and $t = k\sqrt{h}$
If $D$ varies with the cube of $t$ , then $D \propto t^3$ and $D = kt^3$
If $V$ is inversely proportional to $r$ cubed, then $V \propto \frac{1}{r^3}$ and $V = \frac{k}{r^3}$

Solving proportion problems step by step

When tackling proportion questions, follow this systematic five-step approach:

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The 5-Step Method for Proportion Problems:

Convert the sentence into a proportionality statement using the $\propto$ symbol
Replace the $\propto$ with $= k$ to create an equation
Find the value of the constant $k$ using the given information about both variables
Substitute the value of $k$ back into the equation
Use the complete equation to find the unknown value

Let's work through this method with a practical example:

Worked example

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Worked Example: Inverse Proportion Problem

Problem: $G$ is inversely proportional to the square root of $H$ . When $G = 2$ , $H = 16$ . Find an equation for $G$ in terms of $H$ , and use it to work out the value of $G$ when $H = 36$ .

Solution:

Step 1: Convert to proportionality $G \propto \frac{1}{\sqrt{H}}$

Step 2: Replace $\propto$ with $= k$ $G = \frac{k}{\sqrt{H}}$

Step 3: Find $k$ using the given values ( $G = 2$ , $H = 16$ ) $2 = \frac{k}{\sqrt{16}}$ $2 = \frac{k}{4}$ $k = 8$

Step 4: Put $k$ back into the equation $G = \frac{8}{\sqrt{H}}$

Step 5: Use the equation to find $G$ when $H = 36$ $G = \frac{8}{\sqrt{36}} = \frac{8}{6} = \frac{4}{3}$

Therefore, when $H = 36$ , $G = \frac{4}{3}$ .

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Key Points to Remember:

Direct proportion means both variables increase together, forming a straight line through the origin with equation $y = kx$
Inverse proportion means one variable increases while the other decreases, creating a curved hyperbola with equation $y = \frac{k}{x}$
Always convert proportionality statements ( $\propto$ ) into equations by replacing the symbol with $= k$
Use the five-step method: convert, replace, find $k$ , substitute, and solve
More complex proportions can involve squares, cubes, roots, but follow the same basic principles

Direct and Inverse Proportion (Edexcel GCSE Maths): Revision Notes