Ratio, proportion and rates of change

Overview

Ratio, Proportion and Rates of Change is a fundamental pillar of GCSE Mathematics. This topic bridges the gap between abstract algebra and real-world applications. Whether you're calculating the speed of a moving vehicle, determining the density of a new material, or figuring out how much a bank investment will grow over five years, you are using rates of change.

Examiners love this topic because it tests multiple Assessment Objectives simultaneously: your ability to recall formulas (AO1), apply them to contexts (AO2), and solve complex, multi-step problems (AO3). It connects deeply with graphing skills, percentages, and algebraic rearrangement.

In your exam, expect to see straightforward calculation questions alongside challenging 'show that' proofs and multi-step contextual problems where you must interpret what a calculated value actually means in the real world.

Listen to the companion podcast for an audio walkthrough of these concepts:

Key Concepts

Concept 1: Gradient as a Rate of Change

The gradient of a straight line is a measure of its steepness, but more importantly, it represents a rate of change. It tells you how much the y-variable changes for every 1-unit increase in the x-variable.

To calculate the gradient (m) from two points (x_1, y_1) and (x_2, y_2), we use the formula:
m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}

Why this works: Think of it as finding the 'unit rate'. If you travel 100 miles in 2 hours, dividing 100 by 2 gives you 50 miles per 1 hour. The gradient formula does exactly this graphically.

Example: Find the gradient of the line passing through (2, 5) and (6, 17).
m = \frac{17 - 5}{6 - 2} = \frac{12}{4} = 3

Concept 2: Compound Interest and Repeated Percentage Change

Unlike simple interest, which adds a fixed amount each year, compound interest calculates interest on the new total each year. This leads to exponential growth.

We use a multiplier method. For a 5% increase, the multiplier is 1.05. For a 20% decrease (depreciation), the multiplier is 0.80.

Why this works: Instead of calculating 5% and adding it on (two steps), multiplying by 1.05 calculates 105% of the value directly. Raising it to a power simply repeats this multiplication for the number of years.

Example: £3000 is invested at 4% compound interest for 3 years.
Calculation: 3000 \times 1.04^3 = £3374.59

Concept 3: Compound Measures

Compound measures are units made by combining two or more other units. The three most common are Speed, Density, and Pressure.

Speed: How fast distance is covered over time.
Density: How much mass is packed into a specific volume.
Pressure: How much force is spread over a specific area.

Why this works: These formulas are all ratios. Density is the ratio of mass to volume. By dividing mass by volume, we find the mass of exactly one unit of volume.

Example: A gold bar has a mass of 386g and a volume of 20cm³. Density = 386 \div 20 = 19.3 g/cm³.

Mathematical Relationships

• Gradient: m = \frac{y_2 - y_1}{x_2 - x_1} (Must memorise)
• Compound Interest: A = P(1 \pm \frac{r}{100})^n (Must memorise)
• Speed: S = \frac{D}{T} (Must memorise)
• Density: D = \frac{M}{V} (Must memorise)
• Pressure: P = \frac{F}{A} (Must memorise)

Practical Applications

• Finance: Mortgages, savings accounts, and car depreciation all rely on compound percentage formulas.
• Engineering: Designing structural supports requires precise pressure calculations (P=F/A).
• Materials Science: Identifying unknown substances by calculating their density and comparing it to known values.