This builds on percentages and multipliers to handle real money problems: working backwards to an original price, and interest that builds up year after year. It's the maths behind sale tags, savings accounts and the falling value of a used car.
Why this matters: banks, shops and salaries all run on percentages. Reverse percentages catch the trick where a shop shows the sale price and hides the original; compound interest is why a savings pot snowballs — and why debts can too. This is the most useful money maths you'll learn.
Multipliers — the quick tool
Every percentage change has a multiplier. Adding a percentage counts up from 100%, taking one off counts down. Multiply the amount by it to go forward:
| Change | Multiplier | Reason |
|---|---|---|
| +20% | 1.2 | 100% + 20% = 120% = 1.2 |
| −20% | 0.8 | 100% − 20% = 80% = 0.8 |
| +3% | 1.03 | 100% + 3% = 103% = 1.03 |
| −10% | 0.9 | 100% − 10% = 90% = 0.9 |
Reverse percentages — finding the original
If you know the price after a change, divide by the multiplier to get back to the original — the opposite of multiplying to go forward.
A coat is £60 after a 20% reduction. What was the original?
• A 20% decrease means it was multiplied by 0.8.
• Original = 60 ÷ 0.8 = £75.
• A 20% decrease means it was multiplied by 0.8.
• Original = 60 ÷ 0.8 = £75.
Common mistake: don't just add 20% back on to £60. £60 is already after the cut, so divide by 0.8 — don't multiply by 1.2. (Check: £75 × 0.8 = £60 ✓, but £60 × 1.2 = £72, which is wrong.)
Compound interest — interest on the interest
With compound interest, each year's interest is added on, and the next year you earn interest on the new, larger amount. The same multiplier is applied each year, so we use a power:
Amount = P × (multiplier)ⁿ (P = starting amount, n = number of years)
£500 at 3% for 2 years: multiplier = 1.03, so 500 × 1.03² = 500 × 1.0609 = £530.45.
Why a power? Year 1: × 1.03. Year 2: × 1.03 again. Doing the same multiply n times is raising it to the power n. Analogy: it's a snowball rolling downhill — each turn it's bigger, so it picks up even more next turn. Depreciation (losing value) works the same way, with a multiplier below 1: a car falling 10% a year uses × 0.9 each year.
In a nutshell: to undo a % change, divide by the multiplier; for compound growth, raise the multiplier to the power of the number of years — and remember growth uses a multiplier above 1, decay below 1.