Rates of ChangeAQA GCSE Maths: Revision notes

Section 1

What Is Direct and Inverse Proportion?

Two quantities $X$ and $Y$ can vary together in different ways:

Direct proportion: $X$ increases as $Y$ increases at a constant rate ( $X = kY$ )
Inverse proportion: $X$ is inversely proportional to $Y$ means $X$ is proportional to $\frac{1}{Y}$ ( $X = \frac{k}{Y}$ ) — as one increases, the other decreases

Equations describing direct and inverse proportion can be interpreted (Additional Foundation) and, at Higher tier, constructed from given information.

Key termsdirect proportioninverse proportion

Example

If $X$ is inversely proportional to $Y$ and $X=10$ when $Y=2$ , then $k = XY = 20$ , so $X = \frac{20}{Y}$ .

Section 2

How Do Graphs Represent Rates of Change?

The gradient of a straight-line graph represents a rate of change — for example, a conversion graph's gradient gives the exchange rate. At Higher tier, the gradient at a point on a curve gives the instantaneous rate of change at that exact point (found using a tangent).

Key termsgradientinstantaneous rate of change

Section 3

How Do We Convert Between Standard and Compound Units?

You must be able to change freely between related standard units (time, length, area, volume/capacity, mass) and compound units (units made from two or more measures, such as speed, rates of pay, and prices), in both numerical and algebraic contexts.

Compound units you need to know and use include:

Speed (distance ÷ time)
Density (mass ÷ volume)
Pressure (force ÷ area)
Rates of pay and unit pricing

Metric conversion factors (for length, area, volume, capacity) must be known; any imperial/metric conversions will be given in the question.

Key termscompound unitdensitypressure

Common mistake

When converting compound units (e.g. km/h to m/s), remember to convert both the top and bottom parts of the unit, not just one.

Section 4

How Do We Solve Real-World Rate Problems (Exchange Rates and Best Buys)?

Ratio and proportion techniques are applied to real-world rate problems:

Currency conversion, using a given exchange rate as a multiplier
Best-buy comparisons, by finding a rate (price per unit) for each option and comparing

These often combine with percentage skills: expressing one quantity as a percentage of another, and solving percentage change or original value problems involving rates such as interest.

Key termsexchange rate

Example

If £1 = $1.25, then £80 converts to $80 \times 1.25 = \text{\textdollar}100$ .

Must Know

$X$ inversely proportional to $Y$ means $X$ is proportional to $\frac{1}{Y}$ , i.e. $X = \frac{k}{Y}$
The gradient of a straight-line graph is a constant rate of change; the gradient of a tangent gives the instantaneous rate of change on a curve
Convert confidently between standard units and compound units such as speed, density, pressure and rates of pay
Compound units combine two measures — convert both parts when changing units (e.g. km/h to m/s)
Use exchange rates as multipliers to convert between currencies
Solve best-buy problems by comparing a rate (e.g. price per unit) across options

That's the notes covered.

Carry on to the next subtopic.