Proportion and graphs Revision Notes for AQA GCSE Maths

Proportion and graphs

Understanding proportion

When two quantities are related in a specific mathematical way, we say they are proportional. You can show this relationship using the special symbol $\propto$ and represent it clearly on graphs.

There are two main types of proportional relationships you need to understand:

Direct proportion

Direct proportion occurs when two quantities increase or decrease together at the same rate.

Key characteristics of direct proportion

When $y$ is directly proportional to $x$ :

You write this relationship as $y \propto x$
You can express this as the equation $y = kx$ , where $k$ represents a constant number
The graph showing this relationship creates a straight line that passes through the origin (0,0)
As one quantity increases, the other increases proportionally
As one quantity decreases, the other decreases proportionally

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The straight line graph makes direct proportion easy to spot - it always goes through the origin and maintains a constant gradient.

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Key Formula for Direct Proportion: $y = kx$ where $k$ is the constant of proportionality.

Inverse proportion

Inverse proportion occurs when one quantity increases while the other decreases, and their product remains constant.

Key characteristics of inverse proportion

When $y$ is inversely proportional to $x$ :

You write this relationship as $y \propto \frac{1}{x}$ (y is directly proportional to the reciprocal of x)
You can express this as the equation $y = \frac{k}{x}$ , where $k$ represents a constant number
The graph showing this relationship creates a curved line called a reciprocal graph
As one quantity increases, the other decreases
As one quantity decreases, the other increases
The curve never touches either axis

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The reciprocal curve has a distinctive shape that gets closer to both axes but never actually reaches them.

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Key Formula for Inverse Proportion: $y = \frac{k}{x}$ where $k$ is the constant of proportionality.

Solving proportion problems

Working with inverse proportion

When solving problems involving inverse proportion, remember that you can rearrange the equation $y = \frac{k}{x}$ to find the constant $k$ .

For example, if $p$ and $q$ are inversely proportional, you need to identify which equation correctly describes their relationship. Look for equations in the form $y = \frac{k}{x}$ rather than linear equations.

Working with direct proportion

Direct proportion appears frequently in real-world situations, such as electrical calculations.

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Worked Example: Electrical Power and Current

When electrical appliances follow the rule "Power (watts) $\propto$ Current (amps)", you can:

Step 1: Set up the proportion using known values

Electric drill: 2.2A current, 528W power

Step 2: Calculate the constant of proportionality

$k = \frac{528}{2.2} = 240$

Step 3: Use this to find unknown values

Television with 132W power
Current = $\frac{132}{240} = 0.55A$

Reading proportion graphs

Conversion graphs show direct proportional relationships between different units of measurement.

When using a conversion graph:

Identify the scales on both axes
Use the straight line to convert between units
Read values carefully by drawing lines from one axis to the graph, then to the other axis
Remember that the relationship works in both directions

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For example, a graph converting inches to centimetres allows you to convert 12.5 inches to centimetres by finding 12.5 on the inches axis, drawing a line up to the graph line, then across to read the centimetre value.

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Key Points to Remember:

Direct proportion creates straight line graphs through the origin and uses the equation $y = kx$
Inverse proportion creates curved reciprocal graphs and uses the equation $y = \frac{k}{x}$
The symbol $\propto$ means "is proportional to"
In direct proportion, both quantities change in the same direction
In inverse proportion, quantities change in opposite directions
Conversion graphs are practical examples of direct proportion relationships

Proportion and graphs (AQA GCSE Maths): Revision Notes