When two or more different percentage changes are applied one after another, multiply the corresponding decimal multipliers together to find the overall multiplier. A price rise of 20% followed by a fall of 20% does NOT return to the original value — the final result is 4% below the start.
Why does a rise followed by an equal fall not cancel out?
This is the key misconception to overcome. After a 20% increase, the base is larger. When you then decrease by 20%, you are taking 20% of a bigger number — so the decrease is worth more in absolute terms.
Example: Start with £100.
- After +20%: £100 × 1.2 = £120
- After −20%: £120 × 0.8 = £96
Overall change: £96 − £100 = −£4, which is a 4% decrease — not zero.
Combined multiplier = 1.2 × 0.8 = 0.96, confirming a 4% decrease.
How do you calculate successive percentage changes using multipliers?
Method — three steps:
- Convert each percentage change to a decimal multiplier: an increase of r% uses (1 + r/100); a decrease of r% uses (1 − r/100).
- Multiply all the multipliers together to find the overall multiplier.
- Multiply the starting value by the overall multiplier, or convert the multiplier to an overall percentage change.
Worked example 1:
A jacket costs £80. It is increased by 15% and then reduced in a sale by 25%. Find the final price.
Multiplier 1 (increase 15%): 1 + 15/100 = 1.15
Multiplier 2 (decrease 25%): 1 − 25/100 = 0.75
Overall multiplier: 1.15 × 0.75 = 0.8625
Final price: £80 × 0.8625 = £69
Overall change: 0.8625 corresponds to a decrease of (1 − 0.8625) × 100% = 13.75%.
Worked example 2:
A town's population grows by 8% one year, then by 5% the next year. Find the overall percentage increase.
Overall multiplier: 1.08 × 1.05 = 1.134
Overall increase: (1.134 − 1) × 100% = 13.4%
Note: 8% + 5% = 13%, but the true answer is 13.4% because in the second year the 5% applies to the already-increased population.
What is the equivalent single percentage change?
The overall multiplier gives you the equivalent single percentage change directly.
| Overall multiplier | Interpretation |
|---|---|
| Greater than 1 | Overall increase; percentage = (multiplier − 1) × 100% |
| Less than 1 | Overall decrease; percentage = (1 − multiplier) × 100% |
| Equal to 1 | No overall change |
Worked example 3:
Three successive price changes: +10%, −5%, +12%. Find the equivalent single change.
1.10 × 0.95 × 1.12 = 1.10 × 1.064 = 1.1704
Equivalent single change: (1.1704 − 1) × 100% = 17.04% increase (to 2 d.p.).
How do you find the original value before successive changes?
Work backwards: divide by each multiplier in reverse order.
Worked example 4:
After a 30% increase followed by a 10% decrease, a price is £234. Find the original price.
Overall multiplier: 1.30 × 0.90 = 1.17
Original price: £234 ÷ 1.17 = £200
Check: £200 × 1.17 = £234 ✓
Common mistakes and how to avoid them
| Mistake | What goes wrong | Correct approach |
|---|---|---|
| Adding percentage changes | 20% + 20% = 40% change | Multiply the multipliers: 1.2 × 1.2 = 1.44 → 44% |
| Thinking +r% then −r% cancel | Assuming 20% up then 20% down = 0% | 1.2 × 0.8 = 0.96 → 4% decrease |
| Applying changes to the original each time | +10% of £100 = £10; −5% of £100 = £5 | Each change applies to the current (updated) value |
| Forgetting to reverse the order when finding original | Dividing by the wrong multiplier first | Reverse the order of multipliers: last change first |
Frequently asked questions
Does the order of the percentage changes matter?
Not for the final value — multiplication is commutative. 1.2 × 0.8 = 0.8 × 1.2 = 0.96. The final value after a 20% rise then a 20% fall is the same as after a 20% fall then a 20% rise. However, the intermediate value after the first change does depend on order, and questions about finding the "value after the first change" require you to apply changes in the correct sequence.
How is this different from compound interest?
Compound interest applies the same percentage change repeatedly (e.g. 3% per year for 5 years → multiplier = 1.03⁵). Successive percentage changes applies different percentage changes (e.g. +10% then −5% then +8%). Both methods use the multiplier-multiplication approach; compound interest is the special case where all multipliers are equal.
Can I use a non-calculator method for successive percentage changes?
Yes, if the percentages are simple. For 10% then 50%, you can compute step by step: 10% of 200 = 20, so 220; then 50% of 220 = 110. Multiply 220 × 0.5 = 110. For messier numbers, the multiplier method on a calculator is far more efficient.
What happens if one of the changes is 0%?
A 0% change has a multiplier of 1.00. Multiplying by 1 changes nothing, so that period's change can be ignored. However, always include it in the overall multiplier product to avoid errors if the question is phrased as a multi-step process.
For step-by-step help with percentage problems, try Professor Pi at aitutors.me.