Ratio, proportion and rates of change — AQA GCSE Mathematics

Ratio, proportion and rates of change covers comparing quantities using multiplicative relationships, scaling recipes and maps, sharing amounts in given ratios, and solving problems involving speed, density, pressure and unit conversion. It is a high-weight AQA GCSE topic tested across all tiers, often through multi-step worded problems and ratio-to-fraction or ratio-to-equation methods.

Ratio, proportion and rates of change is a fundamental AQA GCSE Mathematics topic that explores multiplicative relationships between quantities. You will learn to simplify and compare ratios, share amounts in given ratios, solve direct and inverse proportion problems, and work with rates such as speed, density and pressure. This topic appears in both Foundation and Higher tiers and carries significant weight in exams, often combining with algebra, geometry and statistics.

Mastering this topic is essential because it underpins many real-world applications, from scaling recipes and interpreting maps to calculating fuel efficiency and currency exchange. It also develops proportional reasoning, a key skill for solving multi-step problems and algebraic manipulation. A strong grasp of ratio and proportion will boost your confidence across the entire GCSE Mathematics curriculum.

• →A ratio compares quantities of the same kind; it can be simplified by dividing all parts by their highest common factor.
• →To share an amount in a given ratio, add the parts to find the total number of parts, divide the amount by this total to find one part, then multiply by each ratio part.
• →Direct proportion means as one quantity increases, the other increases at the same rate (e.g. cost per item); inverse proportion means as one increases, the other decreases (e.g. more workers take less time).
• →Rates of change compare two different units, such as speed (distance/time), density (mass/volume) and pressure (force/area).
• →Unit conversion is crucial: convert all quantities to consistent units before calculating, and always include units in your final answer.

• Add the parts of a part : part ratio to find the total number of shares.
• Divide the quantity by the total number of shares to find the value of one share.
• Multiply the value of one share by each part to find the two required amounts.
• For a part : whole ratio, subtract the given part from the whole to find the other part before sharing.
• Express a division of a quantity into two parts as a ratio by comparing the two amounts and simplifying.
• Apply ratio to real contexts such as unit conversion, comparison, scaling, mixing and concentrations.

• 💡Write down the total number of shares before dividing the quantity.
• 💡Check that the two amounts add back to the original quantity.
• 💡In real contexts, identify the units and convert them before setting up the ratio.
• 💡Always show your working, especially when sharing in a ratio: write the total number of parts and the value of one part. This secures method marks even if you make an arithmetic slip.
• 💡In worded problems, underline the key information and units. Convert all units to a consistent system before calculating to avoid losing accuracy marks.
• 💡For 'show that' or 'prove' questions, set up an equation using the given ratio and solve algebraically. Do not just substitute numbers; show each step clearly.

• Dividing the quantity by the number of parts instead of the total number of shares: for example, sharing £60 in the ratio 2 : 3 by dividing by 2 and by 3. Correction: add 2 + 3 = 5 shares, so one share is £12, giving £24 and £36.
• Treating a part : whole ratio as a part : part ratio: for example, reading 2 : 5 as two parts to five parts. Correction: the whole is 5 shares, so the parts are 2 shares and 3 shares.
• Forgetting to convert units before applying a ratio in a real context: for example, mixing 200 ml with 1 litre as 200 : 1. Correction: convert to the same unit, giving 200 : 1000, then simplify to 1 : 5.
• Students often treat a ratio as a fraction, e.g. writing 2:3 as 2/3 instead of recognising it as 2 parts to 3 parts. Correction: a ratio compares parts, while a fraction compares a part to the whole.
• When solving inverse proportion, students may use direct proportion methods, leading to incorrect scaling. Correction: in inverse proportion, the product of the two quantities is constant, so use y = k/x.
• Students forget to convert units before calculating rates, e.g. mixing minutes and hours. Correction: always convert to the same unit (e.g. hours) before dividing.

1. 1Day 1-2: Revise the basics of ratio: simplifying, equivalent ratios, and sharing in a given ratio. Complete practice questions from your textbook or online resources.
2. 2Day 3-4: Focus on direct and inverse proportion. Learn the formulas y = kx and y = k/x, and practise identifying which type of proportion a problem involves.
3. 3Day 5-6: Study rates of change: speed, density, pressure, and unit conversion. Solve problems that require converting between units (e.g. km/h to m/s).
4. 4Day 7-8: Attempt mixed exam-style questions, including multi-step problems that combine ratio with algebra or geometry. Time yourself to build exam stamina.
5. 5Day 9-10: Review your mistakes using a mark scheme. Create a flashcard of key formulas and common pitfalls, then do a final past paper under timed conditions.

• 📋Sharing an amount in a given ratio: often a 3-4 mark question. Advice: always show the total number of parts and the value of one part.
• 📋Direct and inverse proportion word problems: may involve scaling recipes, currency conversion, or work rates. Advice: identify the type of proportion first and use the unitary method or formula.
• 📋Rates of change calculations: speed, density, pressure, or unit conversion. Advice: convert units before calculating and state the correct unit in your answer.
• 📋Ratio to equation problems: e.g. 'The ratio of boys to girls is 3:4. There are 12 more girls than boys. How many students are there?' Advice: let the parts be 3x and 4x, form an equation, and solve.