Ratio, proportion and rates of change — Edexcel GCSE Study Guide

## Overview  Ratio, Proportion, and Rates of Change form a cornerstone of the GCSE Mathematics specification. This topic is about understanding how quantities relate to one another and how they change over time or space. Whether you are scaling up a recipe, calculating the best value for money in a supermarket, or determining the speed of a vehicle, you are applying the principles of ratio and proportion. Examiners frequently use this topic to test your problem-solving skills (AO3), often embedding these concepts within real-world contexts. It connects deeply with fractions, percentages, and algebraic graphing. Typical exam questions might ask you to divide an amount into a given ratio, calculate compound interest over several years, or interpret the gradient of a velocity-time graph. Mastering these concepts will not only secure you substantial marks but also build a solid foundation for further mathematical study. Listen to the companion podcast for a detailed walkthrough of these concepts:  ## Key Concepts ### Concept 1: Simplifying and Sharing in a Ratio A ratio compares the sizes of different parts of a whole. To simplify a ratio, you must find the highest common factor (HCF) of all parts and divide through, just as you would with a fraction. When sharing an amount in a given ratio, the most reliable approach is the **unitary method**. This involves finding the value of a single 'part' before calculating the required amounts. **Example**: Share £350 in the ratio 2:5. 1. Add the parts to find the total number of parts: $2 + 5 = 7$ parts. 2. Find the value of one part by dividing the total amount by the total parts: $£350 \div 7 = £50$. 3. Multiply each part of the ratio by the value of one part: $2 \times £50 = £100$ and $5 \times £50 = £250$. 4. Check your answer: $£100 + £250 = £350$. ### Concept 2: Percentage Change and Multipliers  Calculating percentage increases and decreases using multipliers is far more efficient than finding the percentage and adding or subtracting it. A multiplier is the decimal equivalent of the new percentage. For an increase, add the percentage to 100% and convert to a decimal. For a 15% increase, the multiplier is $1.15$ ($100\% + 15\% = 115\%$). For a decrease, subtract the percentage from 100% and convert to a decimal. For a 20% decrease, the multiplier is $0.80$ ($100\% - 20\% = 80\%$). This method is crucial for **reverse percentage** problems. If you know the final value and the percentage change, you must *divide* by the multiplier to find the original value. Never calculate the percentage of the new value and add/subtract it—this is a fundamental error that examiners penalise heavily. ### Concept 3: Direct and Inverse Proportion  Proportion describes the mathematical relationship between two variables. In **direct proportion**, as one variable increases, the other increases at the same rate. The ratio between them remains constant. The equation is $y = kx$, where $k$ is the constant of proportionality. Graphically, this is a straight line passing through the origin $(0,0)$. In **inverse proportion**, as one variable increases, the other decreases. The *product* of the two variables remains constant. The equation is $y = \frac{k}{x}$. Graphically, this produces a reciprocal curve (hyperbola) that approaches but never touches the axes. ### Concept 4: Compound Measures  Compound measures involve two or more different units combined, such as speed (distance/time), density (mass/volume), and pressure (force/area). The most critical step in any compound measure calculation is ensuring **unit consistency** before you calculate. If a question gives distance in kilometres and time in minutes, but asks for speed in km/h, you must convert the time into hours first. ## Mathematical/Scientific Relationships * **Direct Proportion**: $y \propto x \Rightarrow y = kx$ * $y$ and $x$ are variables; $k$ is the constant of proportionality. * **Inverse Proportion**: $y \propto \frac{1}{x} \Rightarrow y = \frac{k}{x}$ * Used when one quantity decreases as the other increases. * **Compound Interest**: $A = P(1 + r)^n$ * $A$ = Final Amount, $P$ = Principal (original amount), $r$ = interest rate as a decimal, $n$ = number of time periods. (Must memorise) * **Speed, Distance, Time**: $S = \frac{D}{T}$ * (Must memorise) * **Density, Mass, Volume**: $D = \frac{M}{V}$ * (Must memorise) * **Pressure, Force, Area**: $P = \frac{F}{A}$ * (Must memorise) ## Practical Applications * **Currency Conversion**: Using exchange rates to convert between currencies is an application of direct proportion. * **Best Buy Problems**: Comparing the cost per unit (e.g., cost per 100g) to determine the best value for money in supermarkets. * **Scaling Recipes**: Adjusting the quantities of ingredients based on the number of people you are cooking for uses ratio. * **Population Growth/Decay**: Using compound multipliers to model bacterial growth or the depreciation of a car's value.