Use a multiplier for any percentage change
To increase by multiply by ; to decrease by multiply by . So a rise is and a fall is .
Compound change uses powers of the multiplier
For repeated yearly change over years, . Growth (interest) uses ; depreciation uses . This is NOT the same as simple interest.
Reverse percentage: divide by the multiplier
If a price already includes a change, the stated value is NOT . Work backwards: . After a discount, original .
Drawn from real examiner reports.
Simple interest or wrong decimal for compound
Working out of the start once and multiplying by the number of years (simple interest) instead of compounding; or converting the rate wrongly, e.g. using instead of , or cubing instead of .
A commonly reported examiner note records students who used simple interest scored only 1 mark; the biggest cause of 0 marks was using 0.35 instead of 0.035, and some cubed 0.035 instead of 1.035.
% change: dividing by the wrong amount
For a percentage increase or profit, dividing the change by the NEW value instead of the ORIGINAL, so the percentage comes out wrong.
A commonly reported examiner note records the most common error was dividing the difference by the new value (806) instead of the original value (650).
Reverse % treated as a simple decrease
After a discount students apply the percentage to the sale price instead of the original — e.g. increasing the sale price by , or finding of it and subtracting, or using as the multiplier instead of dividing by .
The most common error on a recent higher-tier paper: students increased the sale price by 15% or used 1.15 rather than dividing by 0.85.
Adding successive percentage changes
Two changes do not simply add: a then is , an overall , not . Multiply the multipliers instead of adding the percentages.
No decision stated in a comparison
In a "which is better value / bigger" question, working out both figures but not stating which wins drops the conclusion mark. Always end with an explicit decision (and by how much where asked).
Decide first: forwards or backwards
If you are told the original amount, multiply by the multiplier. If you are told the amount AFTER a change and asked for the original, divide by the multiplier. Writing the multiplier down before calculating prevents the classic reverse-% slip.
Write the multiplier before calculating
Turn the percentage into a single multiplier first ( rise , fall ). Having it written down stops the forwards/backwards mix-up and speeds up compound-interest powers.
Check a reverse % answer forwards
After a reverse-percentage calculation, apply the stated change forwards to your answer — you must land back on the given value (). If not, you divided the wrong way.
Every percentage change is a single multiplier:
Example. A coat costs \8015%: $$80 \times 1.15 = \92.$$
What is the multiplier for a 15% increase? For a 15% decrease?
A laptop costs \54020%$. Work out the sale price.
