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Maths · GCSE · The School
The multiplier
Stop calculating the percentage and then adding it. A 15% increase is one calculation: multiply by 1.15. A 15% decrease is multiply by 0.85. This is faster, it is less error-prone, and — the real reason — it is the only sensible way to handle several changes in a row, because multipliers simply multiply together.
Repeated change compounds
A quantity changing by the same percentage each year does not change by the same AMOUNT each year, because each year acts on the new total. A town growing 4% a year is multiply by 1.04, once per year — three years is 1.04 cubed. A car losing 15% of its value a year is multiply by 0.85 each year. Do not multiply the first year change by three and add it; that answers a different question and always comes out too small.
Reversing a change
A coat costs 48 after a 20% reduction. What was it before? The commonest error is adding 20% to 48. But 48 is 80% of the original, so divide by 0.8, giving 60. Check forwards: 20% of 60 is 12, and 60 minus 12 is 48. Reverse percentage questions always divide by the multiplier that was applied.
A car is bought for 12,000 and loses 15% of its value each year. AFTER ONE YEAR: multiply by 0.85, giving 10,200. AFTER THREE YEARS: multiply by 0.85 three times — 0.85 cubed is about 0.614125 — so 12,000 times that is 7,369.50. Notice it is NOT 12,000 minus three lots of 1,800, which would be 6,600; each year takes 15% of a smaller amount than the year before, so the true value is higher. NOW REVERSE IT: a car is worth 7,369.50 after three years of losing 15% a year, and you want the original. Divide by 0.85 three times, or divide by 0.614125 once, and 12,000 comes back. One multiplier, used once, repeatedly, and backwards — which is every percentage question at this level.