AQA GCSE Mathematics: Ratio, Proportion and Rates of Change (8300)

This guide covers Topic 3: Ratio, proportion and rates of change in AQA GCSE Mathematics (8300), which follows the site’s existing guides to Topic 1 (Number) and Topic 2 (Algebra). Ratio and proportion questions appear across both Foundation and Higher tier papers and are commonly combined with other strands, particularly algebra and geometry, in longer multi-step problems.

Where this fits in 8300

Mathematics 8300 is assessed by three equally weighted papers (Paper 1 non-calculator, Papers 2 and 3 calculator), each 1 hour 30 minutes, at either Foundation or Higher tier. The specification states that “all content can be assessed on any of the three question papers”, so ratio, proportion and rates of change questions – like every other topic – can appear on any paper, calculator or non-calculator.

Syllabus coverage

AQA GCSE MATHEMATICS (8300) – TOPIC 3: RATIO, PROPORTION AND RATES OF CHANGE

• Percentages, including expressing one quantity as a percentage of another, percentage change, and reverse percentage problems
• Simple interest and compound interest, and growth/decay problems modelled with repeated percentage change
• Direct proportion, expressed algebraically and represented graphically as a straight line through the origin
• Inverse proportion, expressed algebraically and represented graphically as a reciprocal curve
• Compound units, including speed, density, pressure, rates of pay, unit pricing, and converting between related compound units
• Ratio expressed in its simplest form, dividing a quantity into a given ratio, and using ratio to compare quantities of different units after conversion

How to approach it

Ratio and proportion questions reward careful reading of what quantity is being asked for and in what form (a ratio, a fraction, a percentage, or an absolute amount), since a numerically correct answer given in the wrong form still loses marks. Building fluency in converting between ratios, fractions and percentages – rather than treating them as separate skills – pays off across this entire topic.

Official syllabus

AQA GCSE Mathematics (8300) specification, first teaching September 2015 – aqa.org.uk.

Direct vs inverse proportion, side by side

Direct proportion means two quantities increase or decrease together at a constant rate – doubling one doubles the other – and is written algebraically as y = kx, producing a straight-line graph through the origin. Inverse proportion means one quantity increases as the other decreases – doubling one halves the other – written as y = k/x, producing a curve (a reciprocal graph) rather than a straight line. Mixing up these two relationships is one of the most common errors on this topic, so always check: does increasing one quantity make the other one bigger (direct) or smaller (inverse)?

Reverse percentage problems

Reverse percentage questions give a value after a percentage change and ask for the original value – these are consistently harder than forward percentage questions because the temptation is to simply apply the percentage to the given (already-changed) figure rather than working backwards from it. If a price after a 20% increase is £60, the original price is not 80% of £60; it is found by recognising that £60 represents 120% of the original, so the original is £60 ÷ 1.2 = £50.

Ratio problems: sharing and combining amounts

Sharing a quantity in a given ratio means splitting it into the number of equal parts implied by the ratio, then allocating each share the correct number of those parts – for example, sharing £40 in the ratio 3:5 means dividing by the total of 8 parts (£5 per part), then giving 3 parts (£15) and 5 parts (£25) to each side. A related but different question type asks for a missing quantity given a ratio and one known amount, which requires finding the value of one part first, then scaling up or down from there. Both question types depend on correctly identifying the total number of parts in the ratio before doing any division, which is the step most often skipped under time pressure.

Worked example: compound interest over multiple years

The routine below is an original model written for this resource, not a reproduction of any official past paper or mark scheme.

Question type: a sum is invested at a fixed compound interest
rate for several years -- find the total after n years.

Step 1 - convert the percentage rate to a decimal multiplier:
e.g. 3% per year becomes a multiplier of 1.03.

Step 2 - raise the multiplier to the power of the number of
years:
e.g. over 5 years, use 1.03^5, not 1.03 x 5.

Step 3 - multiply the original amount by this multiplier:
this gives the total after growth, in one calculation.

Step 4 - subtract the original amount if the question asks for
the interest earned rather than the total balance.

Step 2 is where most errors occur: compound interest requires raising the multiplier to a power, not multiplying the multiplier by the number of years. That product is not simple interest either: 1.03 x 5 = 5.15 would make the total more than five times the original amount. Simple interest at 3% for 5 years multiplies the original amount by 1 + 0.03 x 5 = 1.15, whereas compound interest multiplies it by 1.03^5 = 1.159 (to 3 d.p.), because each year’s interest also earns interest.

Common mistakes

Confusing direct and inverse proportion, especially when translating a word problem into an algebraic relationship. Applying a percentage change to the wrong base value in reverse-percentage questions. Using simple interest arithmetic (multiplying by the rate and the number of years) instead of the correct compound interest method (raising the multiplier to a power). Forgetting to convert units before applying a compound-unit formula, such as mixing minutes and hours within a speed calculation.

Quick revision checklist

• Practise converting confidently between fractions, decimals, percentages and ratios.
• Be able to state, from a word problem, whether a relationship is direct or inverse proportion before writing any algebra.
• Master reverse-percentage problems by identifying what percentage the given value represents of the original.
• Use the compound interest routine (decimal multiplier raised to a power) rather than simple interest arithmetic for growth/decay problems.
• Always check units are consistent before applying a compound-unit formula (speed, density, pressure, rates of pay).