Percentages, increase and decrease
Maths · Key Stage 3 · The School
Percent means per hundred
A percentage is a fraction with a denominator of one hundred, so 25% is 25/100 or 0.25. To find a percentage of an amount, multiply: 25% of 80 is 0.25 × 80 = 20. Converting to a decimal first turns every percentage question into a single multiplication, which is faster and far less error-prone than the recipes.
Multipliers
To increase by 20%, multiply by 1.2 — the original 100% plus the extra 20%. To decrease by 20%, multiply by 0.8. Multipliers are worth the effort because they chain: two successive 10% rises is × 1.1 × 1.1 = × 1.21, done in one line. Reversing a change means DIVIDING by the multiplier, not applying the opposite percentage.
Why up then down does not return
A £100 item rises 10% to £110, then falls 10% — but that 10% is now of £110, so it falls £11, to £99. The percentages are taken of different amounts, so they do not cancel. This is not a trick question: it is exactly why a 50% pay cut needs a 100% rise to undo, and it appears constantly in real money.
Up 10%: £100 × 1.1 = £110. Down 10%: £110 × 0.9 = £99. Not £100. The rise was 10% of £100 = £10; the fall was 10% of £110 = £11, so you end £1 down. The multipliers show it in one line: 1.1 × 0.9 = 0.99, a 1% loss overall — and it comes out the same whichever order you apply them. This is exactly why a shop can advertise a 20% increase followed by a 20% sale.
Why a rise then an equal fall does not get you back
Increase 100 by 20% and you have 120. Now decrease 120 by 20% and you get 96, not 100. The percentages are equal and the amounts are not, because the second 20% is taken from a bigger number. This is why a shop can raise a price and then advertise the same percentage off and still be ahead, and why a fund that loses 50% needs a 100% gain to recover rather than another 50%. Multipliers make it obvious: 1.2 × 0.8 = 0.96, and 0.96 is not 1.