HookToblerone got smaller and hoped you would not do the maths
In November 2016, shoppers noticed something odd about their Toblerone. Mondelez, the American company that makes it, had quietly redesigned the UK bars — the 400g bar shrank to 360g and the 170g bar to 150g — by widening the gaps between the famous triangular peaks so the bar looked almost the same length. The price on the shelf did not change. The public called it 'Toblergate', and it became the textbook example of a trick with a proper name: shrinkflation. You pay the same, but you get less.
The only way to see through shrinkflation is to do exactly what this section teaches: compare using a ratio rather than a headline figure. Dropping from 400g to 360g is 40g gone — a 10% cut in chocolate — and because the price held steady, the cost per gram jumped by about 11%. That price-per-gram is a compound unit, a rate, a ratio between two quantities, and it is the honest number the packaging hopes you will not calculate. Ratio, proportion and rates of change are the branch of the syllabus that turns 'that feels like a rip-off' into a defensible number — sharing a bill, scaling a recipe, reading a map, comparing two supermarket deals, working out interest on savings. AQA weights this section heavily across both the calculator and non-calculator papers, and it is the maths you will use more often than any other for the rest of your life.
ModelRatio — notation, simplest form and sharing out
A ratio compares quantities part to part: mixing squash 1 part cordial to 4 parts water is written 1:4. You simplify a ratio the same way you simplify a fraction — divide every part by a common factor — so 24:36 divides by 12 to give 2:3, its simplest form. A ratio is in simplest form when the parts share no common factor beyond 1.
Be careful what a ratio is telling you. In 1:4, the cordial is one part out of five total, not one quarter of the mixture — the classic Foundation error is to read the first number of a ratio as a fraction of the whole. To turn a ratio into fractions, add the parts for the total: 1:4 means 1/5 cordial and 4/5 water. This is also how you express one quantity as a fraction of another: 3 pupils absent out of 40 is 3/40 (less than 1), while 15 apples compared to 6 oranges is 15/6, or 5/2 (greater than 1) — a fraction can be bigger than 1 when the 'part' exceeds what you compare it to.
Sharing in a ratio is the section's bread-and-butter question, and it has a fixed three-step method: add the parts to get the total number of parts, divide the quantity by that total to find the value of one part, then multiply each side of the ratio by the value of one part. It works because a ratio just chops the whole into equal-sized parts and hands them out in the stated proportion — mixing paint, splitting winnings, or diluting a concentrate are all the same calculation.
Share £45 between two people in the ratio 2:3. Add the parts: 2 + 3 = 5 parts in total. Divide the amount by the number of parts: £45 ÷ 5 = £9 for one part. Multiply out each share: the first person gets 2 × £9 = £18, the second gets 3 × £9 = £27. Check the shares add back to the whole: £18 + £27 = £45 ✓, and the ratio £18:£27 does simplify to 2:3. That check — the parts summing to the original total — catches almost every arithmetic slip, and the 'value of one part' line is the method mark examiners look for.
MechanismProportion — ratios that stay equal
Proportion is the idea that two ratios are equal. If a recipe needs flour and sugar in the ratio 3:2, then whether you bake a small batch or a huge one, the ratio of flour to sugar stays 3:2 — the quantities change, the proportion does not. That constancy is what lets you scale anything up or down reliably: a scaled recipe, a bigger map, a doubled dose all keep the underlying ratio fixed.
The workhorse technique is the unitary method: find the value of one unit, then multiply up. If 4 identical pens cost £3, one pen costs £3 ÷ 4 = £0.75, so 10 pens cost 10 × £0.75 = £7.50. Finding 'the one' first is what makes every proportion problem the same shape, no matter the numbers. It also exposes the multiplicative relationship between two quantities — the heart of R6: to go from 4 people to 10 people you multiply by 10 ÷ 4 = 2.5, so every ingredient multiplies by 2.5 as well.
Ratios, fractions and straight lines are three faces of the same relationship. A ratio of 1:3 between x and y means y is always three times x — which is the linear function y = 3x, a straight line through the origin. That link is why direct proportion later plots as a straight line, and why 'equality of ratios' and 'a constant multiplier' and 'a linear graph through zero' all describe the same underlying fact. Recognising it turns ratio questions, proportion questions and graph questions into one skill.
A recipe for 4 people needs 300g of flour. How much flour for 10 people? Unitary method: for one person, 300g ÷ 4 = 75g. For 10 people, 75g × 10 = 750g. Or scale directly by the multiplier: 10 ÷ 4 = 2.5, and 300g × 2.5 = 750g — the same answer, because the ratio of flour to people never changes. Check it makes sense: 10 people is more than double 4 people, so 750g being more than double 300g is the right direction. Both routes are full-mark methods; showing the '75g per person' or the '× 2.5' line earns the working mark.
MechanismPercentages — the multiplier does all the work
A percentage means 'parts per hundred', so 37% is 37/100 = 0.37. The single most powerful idea in the topic is the multiplier: to increase by 15% you multiply by 1.15 (the original 100% plus 15%), and to decrease by 15% you multiply by 0.85 (100% minus 15%). One multiplication does the whole job, and it scales to compound problems that step-by-step methods cannot handle cleanly. To find one quantity as a percentage of another, divide and multiply by 100: 18 out of 40 is 18 ÷ 40 = 0.45 = 45%.
The examiner's favourite trap is the reverse (original value) percentage. A coat costs £54 in a '10% off' sale — what was the original price? The wrong instinct is to add 10% of £54 back on; that gives the wrong answer because the 10% was taken off the larger original, not the sale price. The right method reverses the multiplier: the sale price is 90% of the original, so original × 0.9 = 54, meaning original = 54 ÷ 0.9 = £60. Dividing by the multiplier undoes the decrease exactly, which adding a percentage back never does.
Percentages can exceed 100% — a value that triples has risen by 200% — and simple interest is the same percentage applied afresh each year to the original amount: £2000 at 3% simple interest earns 3% of £2000 = £60 every year, so after 4 years the interest is 4 × £60 = £240 and the balance is £2240. (Compound interest, where each year's interest also earns interest, comes later in the section and behaves very differently.)
Two quick percentage jobs, then the trap. Increase £80 by 15%: multiplier 1.15, so £80 × 1.15 = £92. Find 18 as a percentage of 40: 18 ÷ 40 = 0.45 = 45%. Now the reverse percentage: a jacket is £54 after 10% off — find the original price. The £54 represents 90% of the original, so the multiplier was 0.9 and you divide to undo it: £54 ÷ 0.9 = £60. Check forwards: £60 with 10% off is £60 × 0.9 = £54 ✓. Adding 10% of £54 would have given £59.40 — plausible-looking and wrong, which is exactly why the mark scheme rewards the ÷ 0.9 method.
DataCompound units — speed, density and the honest best-buy
A compound unit combines two measurements into a rate: speed is distance per unit time (miles per hour), density is mass per unit volume (grams per cubic centimetre), pressure is force per unit area, and price per gram is the number that exposed Toblerone. Each is built from a formula triangle you can rearrange: speed = distance ÷ time, so distance = speed × time and time = distance ÷ speed. The commonest error is inverting the division — writing speed = distance × time — so anchor it in units: miles ÷ hours gives miles per hour, which the multiplication would not.
Converting units keeps these rates honest. Simple conversions scale by the factor: 1 km = 1000 m, 1 hour = 3600 seconds, 1 kg = 1000 g. Compound conversions apply the factor to each part — to change 20 m/s into km/h, note that 20 metres every second is 20 × 3600 = 72000 metres every hour, which is 72 km/h (a handy shortcut is that m/s × 3.6 = km/h). Area and volume units carry hidden powers: because 1 m = 100 cm, an area of 1 m² is 100 × 100 = 10000 cm², and 1 m³ is 100 × 100 × 100 = 1000000 cm³ — squaring and cubing the length factor, a point revisited in the scale-factor block.
The everyday payoff is the best-buy comparison, which is unit pricing in disguise. To compare deals of different sizes, reduce both to the same rate — cost per gram, or per 100g, or per litre — and then, crucially, write a sentence saying which is better value. The number alone is not the answer; the comparison is.
First a compound-unit calculation: a car travels 150 miles in 2.5 hours, so its average speed is 150 ÷ 2.5 = 60 mph. A block of metal has mass 240g and volume 30cm³, so its density is 240 ÷ 30 = 8 g/cm³. Now the best-buy: coffee comes as 500g for £2.40 or 750g for £3.30 — which is better value? Work in pence per gram. Small: 240p ÷ 500 = 0.48p per gram. Large: 330p ÷ 750 = 0.44p per gram. The 750g jar is cheaper per gram, so it is the better value — and that concluding sentence is worth a mark on top of the two calculations.
CaseScale factors, maps and similar shapes
A scale factor tells you how many times bigger or smaller a diagram is than reality, and it is written as a ratio. An Ordnance Survey Explorer map at 1:25000 means every 1 cm on the paper is 25000 cm on the ground — so 4 cm across the map is 4 × 25000 = 100000 cm = 1000 m = 1 km of real hillside. Reading map distances is pure ratio: multiply map distance by the scale to get real distance, or divide real distance by the scale to get map distance. Getting the direction of the multiply-or-divide right is the whole skill, so sanity-check that the real distance comes out larger.
When a shape is enlarged, all its lengths multiply by the scale factor — a shape enlarged by scale factor 3 has every side three times longer. But area and volume do not scale the same way, and this is examined hard. Because area is two dimensions multiplied, it scales by the factor squared; because volume is three dimensions, it scales by the factor cubed. Enlarge by scale factor 3 and lengths triple, but the area becomes 3² = 9 times bigger and the volume 3³ = 27 times bigger. Comparing lengths, areas and volumes with the right power of the scale factor (R12) is where careful candidates pull ahead.
Two shapes that are the same shape but different sizes are similar, linked by a single scale factor between corresponding lengths. This is the ratio idea behind similar triangles, and it is the doorway to trigonometry — the sine, cosine and tangent ratios are simply the fixed ratios between the sides of similar right-angled triangles, so a 30° right-angled triangle always has the same side ratios no matter its size.
On a 1:25000 map, two peaks are 6 cm apart. What is the real distance? Multiply by the scale: 6 × 25000 = 150000 cm. Convert to sensible units: 150000 cm ÷ 100 = 1500 m = 1.5 km. Now a scale-factor-and-area problem: a rectangle of area 5 cm² is enlarged by scale factor 3 — find the new area. Lengths triple, but area scales by the square of the scale factor, 3² = 9, so the new area is 5 × 9 = 45 cm². If it were a solid enlarged by scale factor 3, its volume would scale by 3³ = 27. Using scale factor 3 for the area instead of 9 is the classic slip — the new area would wrongly come out as 15 cm² rather than 45 cm².
ModelDirect and inverse proportion, rates of change and compound growth
Two quantities are in direct proportion when doubling one doubles the other — they rise and fall together in a fixed ratio, y = kx, which plots as a straight line through the origin. The number k is the constant of proportionality, and the gradient of that line is the rate of change: on a graph of cost against litres of fuel, the gradient is the price per litre. This is the same y = mx idea from graphs, and it is why direct-proportion graphs always pass through (0, 0) — buy nothing, pay nothing.
Quantities are in inverse proportion when one going up sends the other down in the same ratio — double one, halve the other — because their product stays constant, xy = k. Saying 'x is inversely proportional to y' is exactly the same as saying 'x is proportional to 1/y' (R13). The classic case is workers and time: if 6 people take 4 days to build a wall, that is 6 × 4 = 24 worker-days of effort, so 8 people take 24 ÷ 8 = 3 days — more workers, less time, but the total effort is fixed. Inverse proportion plots not as a straight line but as the reciprocal curve from the graphs section, falling steeply then levelling off.
Growth and decay problems apply a percentage multiplier again and again — and here compound interest parts ways with simple interest. £1000 in a savings account at 4% compound interest is multiplied by 1.04 each year, and the interest itself earns interest, so after 3 years the balance is 1000 × 1.04 × 1.04 × 1.04 = 1000 × 1.04³. Depreciation is the same machinery with a decrease multiplier: a £12000 car losing 15% of its value each year is worth 12000 × 0.85 each year. Repeated multiplication, exactly as in geometric sequences, is what makes compound growth pull away from simple, year after year.
Compound interest: £1000 is saved at 4% compound interest for 3 years. Multiply by 1.04 for each year: 1000 × 1.04³. Working it through, 1.04² = 1.0816, and 1.0816 × 1.04 = 1.124864, so the balance is 1000 × 1.124864 = £1124.86, meaning £124.86 of interest. Simple interest at 4% would have earned only 3 × £40 = £120, so compounding adds £4.86 — small here, decades later it is enormous. Now inverse proportion: 6 workers take 4 days, so the job is 6 × 4 = 24 worker-days; with 8 workers it takes 24 ÷ 8 = 3 days. And depreciation: a £12000 car losing 15% a year is worth 12000 × 0.85² = 12000 × 0.7225 = £8670 after 2 years.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
This section is examined on both papers and rewards a visible method, so always show the structural line: the value of one part when sharing a ratio, the '× multiplier' when changing a percentage, the unit rate when comparing best-buys. Those lines are the M marks and survive an arithmetic slip that a bare answer would not. On the non-calculator paper, build percentages from 10% and 1% blocks (10% of £80 is £8, so 15% is £8 + £4 = £12); on the calculator paper, reach straight for the multiplier method and, for compound interest, the power key — 1.04³ in one step, never three separate multiplications written out and rounded early.
The reverse percentage is the question that separates grades: identify what percentage the given figure represents, then divide by that multiplier — never add the percentage back. Best-buy questions need a like-for-like rate (per gram, per litre) and an explicit sentence naming the better value, because the conclusion is a mark in its own right. For scale and enlargement, keep lengths, areas and volumes separate — scale factor for lengths, its square for areas, its cube for volumes — and always sanity-check a map conversion so the real distance ends up larger than the map distance.
Mind the units and the interpretation. Compound-unit answers need their unit attached (mph, g/cm³, £ per litre), and getting speed as distance ÷ time rather than the inverse is easiest to secure by checking the units divide correctly. Distinguish simple from compound interest by reading the word 'compound', and distinguish direct proportion (straight line through the origin, constant multiplier) from inverse proportion (constant product, reciprocal curve). Where a question gives a rate as the gradient of a graph, state what that gradient means in context — a price per litre, a speed — because naming the rate of change is exactly what the mark scheme is testing.