HookThe bigger burger that customers thought was smaller
In the early 1980s the American burger chain A&W tried to beat McDonald's Quarter Pounder with more meat for the same money: a third-of-a-pound burger, the Third Pounder. It flopped. When the chain later dug into why, the focus groups reportedly delivered a jaw-dropping answer — customers thought they were being overcharged, because they believed a third of a pound was smaller than a quarter. The reasoning was simple and wrong: 3 is less than 4, so surely 1/3 is less than 1/4. The better-value burger lost in the market because the public could not order two fractions.
That single mistake is the whole of this section in one bite. A fraction, a decimal and a percentage are three costumes worn by the same number — 1/4, 0.25 and 25% are identical — and almost every Foundation question here is really asking you to change costume so a comparison becomes obvious, or to use one of them as an operator that acts on a quantity. You will convert between the three forms, order a jumbled mix of them, pull fractions out of ratios, and drive percentage change with a multiplier. Get fluent here and ratio, proportion and the whole of the money-maths world (VAT, sale prices, interest, tips) fall into place; stay shaky and you make the Third Pounder error on a page that costs marks instead of burgers.
ModelThree costumes, one number — converting and ordering
The safest way to compare a fraction, a decimal and a percentage is to drag them all into one form, and decimals are usually the easiest target. To turn a fraction into a decimal, divide the top by the bottom: 3/8 is 3 ÷ 8 = 0.375. To turn a percentage into a decimal, divide by 100 — because 'per cent' literally means 'per hundred' — so 37% = 0.37. Going the other way, a terminating decimal becomes a fraction by reading its place value: 0.375 is 375 thousandths, 375/1000, which cancels to 3/8. (Turning a recurring decimal back into a fraction is Higher-tier only, so on Foundation every decimal you are asked to convert will stop.)
Why does dividing the numerator by the denominator work? Because a fraction is a division waiting to happen — 3/8 means 'three wholes shared between eight', which is 0.375 of a whole each. And why can you order decimals just by scanning left to right? Place value again: the tenths column outranks the hundredths, which outranks the thousandths, so the first column where two numbers differ settles the argument for good.
The classic Foundation error is the Third Pounder error in disguise: comparing the raw digits instead of the values. 0.35 is not bigger than 0.4 just because 35 beats 4; line them up as 0.35 and 0.40 and the tenths column decides it instantly. With fractions, remember that a bigger denominator cuts the cake into more, thinner slices, so 1/4 is larger than 1/5 even though 5 is the larger number.
Order these smallest first: 0.35, 37%, 3/8, 2/5, 0.41. Convert everything to decimals so they share a costume. 37% = 0.37. Divide for the fractions: 3/8 = 3 ÷ 8 = 0.375 and 2/5 = 2 ÷ 5 = 0.4. Now the list reads 0.35, 0.37, 0.375, 0.4, 0.41 — and column-by-column ordering gives 0.35 < 0.37 < 0.375 < 0.4 < 0.41. Translated back to the original costumes, the answer is 0.35 < 37% < 3/8 < 2/5 < 0.41. On the answer line, write it as that inequality chain and show the three conversions above it — AQA mark schemes routinely award a mark for the conversions alone, before the ordering is even judged.
MechanismFractions living inside ratios
A ratio splits a whole into parts, and each part is a fraction of that whole. If a Year 11 class has a bus-to-walk ratio of 3:2, add the parts to find the whole — 3 + 2 = 5 — and then each number in the ratio sits over that total: 3/5 of the class take the bus, 2/5 walk. The denominator is the sum of all the parts, never one of the parts. That single sentence is the fix for the most common ratio slip of all, writing the bus fraction as 3/2 because the ratio said 3:2.
The machine runs both ways. Given the fractions you can rebuild the ratio, and given the total you can find the actual counts. Because 3/5 of a 30-strong class is 3/5 × 30 = 18, eighteen pupils take the bus and the remaining twelve walk — and 18:12 cancels straight back to 3:2, a free check that you set the fractions up the right way round. This is the bridge between this section and the ratio section (N11 and R5): a ratio is just a pair of fractions of the same whole, and spotting that turns a scary 'divide £480 in the ratio 5:3' question into 'find 5/8 and 3/8 of £480'.
A fruit squash is mixed with concentrate and water in the ratio 2:9. What fraction of the drink is concentrate, and how much concentrate is in a 2.2-litre jug? Total parts: 2 + 9 = 11, so concentrate is 2/11 of the drink. Then 2/11 of 2.2 litres = 2.2 ÷ 11 × 2 = 0.2 × 2 = 0.4 litres of concentrate, leaving 9/11 × 2.2 = 1.8 litres of water. Check: 0.4 + 1.8 = 2.2 litres, and 0.4 : 1.8 cancels (divide both by 0.2) to 2 : 9 — back to the ratio we started with.
ModelFractions and percentages as operators — the multiplier
The word 'of' means multiply. Three-quarters of 20 is 3/4 × 20 = 15; 15% of £80 is 0.15 × 80 = £12. A fraction or a percentage used this way is called an operator — it does a job to a quantity rather than describing a quantity itself. On a non-calculator paper the neat route to a percentage is to build it from friendly blocks: 10% of £80 is £8, so 5% is £4 and 15% is £8 + £4 = £12.
For percentage change, the professional tool is the multiplier. To increase by 20% you do not find 20% and add it on in two steps; you multiply by 1.20 in one, because the new amount is the original 100% plus 20%, i.e. 120% = 1.2 of the original. To decrease by 20% you multiply by 0.80, because you are keeping 80%. This is faster, it is harder to slip on, and — crucially — it chains: two changes in a row become one multiplication.
The multiplier also runs backwards, which is where Foundation candidates lose the most marks. If a price already includes 20% VAT and reads £54, the £54 is 120% of the pre-VAT price, so you divide by 1.2 to undo it: £54 ÷ 1.2 = £45. The tempting wrong move is to take 20% off the £54 (£54 − £10.80 = £43.20), but 20% of the bigger, post-VAT figure is not the same as 20% of the smaller original — which is exactly why the multiplier, not subtraction, is the tool the mark scheme rewards.
A £45 jumper is reduced by 20% in a January sale. Multiplier for a 20% fall is 0.8, so the sale price is 45 × 0.8 = £36. Separately, a tenant's £600 monthly rent rises 4%: multiplier 1.04 gives 600 × 1.04 = £624. Now the reverse-percentage trap: a coat costs £54 including 20% VAT — what was the price before VAT? The £54 is 120% of the original, so divide: 54 ÷ 1.2 = £45. Do not subtract 20% of £54 (that gives £43.20, which is wrong); check the right answer forwards — 45 × 1.2 = 54 — and the interpretation mark is yours.
CaseWhy up-then-down never gets you home
Chaining multipliers exposes a fact that surprises most students and delights every examiner: a percentage rise followed by an equal percentage fall does not return you to where you started. Put £200 up 10% and you reach £220; knock 10% off that and you land on £198, not £200. The reason is exactly the multiplier: ×1.1 then ×0.9 is ×0.99, a 1% net loss, because the 10% you removed was 10% of the larger £220, not of the original £200.
This is a favourite for a reason — it forces you to treat percentages as operators acting on whatever the current amount is, not as fixed quantities you can add and subtract at will. The same logic governs anything that grows and shrinks in steps: a share price, a salary after a rise and then a freeze, a population after a good year and a bad one. Whenever a question gives you two or more percentage changes in sequence, resist doing them one clumsy step at a time — multiply the multipliers, keep the arithmetic in one line, and let the single combined number do the talking.
A savings pot of £200 gains 10% in a strong year, then loses 10% the next. Year one: 200 × 1.1 = £220. Year two: 220 × 0.9 = £198. Straight to the answer with one multiplication: 200 × 1.1 × 0.9 = 200 × 0.99 = £198 — a 2% overall fall, even though the two percentages looked as if they should cancel.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
Paper 1 is the non-calculator paper, and this section is guaranteed to appear there in some form. Convert a mixed FDP list into one common costume before you order it, and show those conversions — the mark scheme frequently gives a method mark for the conversions even if the final ordering slips. For a percentage of an amount without a calculator, build it from 10%, 5% and 1% blocks rather than reaching for long multiplication: 10% is a division by ten, and everything else is addition from there.
On the calculator papers (2 and 3), the multiplier is the method examiners want to see for every percentage change. Write the multiplier down (×1.04, ×0.8) and, for reverse-percentage questions, divide by it — the single most common lost mark in this topic is undoing an increase by subtracting the percentage instead of dividing. When two or more changes stack up, multiply the multipliers in one line so the arithmetic cannot drift.
Across both papers, remember that 'of' means multiply and that a ratio is a set of fractions over the total of the parts. State your method line by line: the conversion, the multiplier, the operation. AQA marks method first, so a wrong final digit above a correct, fully-shown method still banks most of the marks, whereas a bare answer banks nothing if it is wrong.