Percentage change measures how much a value has increased or decreased, expressed as a percentage of the original. The formula is: percentage change = (change ÷ original) × 100. If the answer is positive, it is an increase; if negative, it is a decrease. Always divide by the original value, not the new one.
What is the percentage change formula?
The formula has three steps:
- Find the change: subtract the original value from the new value (new − original).
- Divide by the original: (new − original) ÷ original.
- Multiply by 100 to convert to a percentage.
Written as a single expression:
Percentage change = ((new value − original value) ÷ original value) × 100
If the result is positive, the value has increased. If negative, it has decreased (and the magnitude is the percentage decrease).
How do you calculate a percentage increase?
Worked example 1: A jacket costs £40 in January and £46 in February. What is the percentage increase?
- Change = 46 − 40 = £6
- Divide by the original: 6 ÷ 40 = 0.15
- Multiply by 100: 0.15 × 100 = 15%
The price increased by 15%.
Check: 15% of £40 = 0.15 × £40 = £6. New price = £40 + £6 = £46. ✓
How do you calculate a percentage decrease?
Worked example 2: A phone was priced at £250 and is now selling for £200. What is the percentage decrease?
- Change = 200 − 250 = −£50 (negative because the price fell)
- Divide by the original: −50 ÷ 250 = −0.20
- Multiply by 100: −0.20 × 100 = −20%
The price decreased by 20%. We say "a 20% decrease" (dropping the negative sign when we have identified it as a decrease).
What is the most common mistake when finding percentage change?
The most frequent error is dividing by the new value instead of the original. This gives a wrong answer.
Wrong approach for Example 2: 50 ÷ 200 × 100 = 25%. This is not correct.
Why the original matters: percentage change is always measured relative to where you started, not where you ended up. The original value is always the denominator.
A quick self-check: ask yourself "What am I comparing the change to?" The answer is always the original (starting) value.
How do you apply percentage change to real-life problems?
| Context | Original value | New value | Change | % Change |
|---|---|---|---|---|
| A shop raises a £5 item to £5.50 | £5 | £5.50 | +£0.50 | (0.50÷5)×100 = 10% increase |
| A plant grows from 12 cm to 15 cm | 12 cm | 15 cm | +3 cm | (3÷12)×100 = 25% increase |
| Exam score drops from 80 to 68 | 80 marks | 68 marks | −12 marks | (12÷80)×100 = 15% decrease |
Notice: the percentage depends entirely on the original, so a change of "£0.50" can be 10% if the original was £5, or 0.5% if the original was £100.
How does percentage change differ from a percentage of an amount?
These are two separate skills:
- Percentage of an amount: "What is 15% of £80?" → Find a portion of a given value.
- Percentage change: "By what percentage did £80 change to £92?" → Find by how much a value changed, as a proportion of where it started.
The percentage change formula does the reverse of applying a percentage. If you apply 15% to £80 you get £12; if you then ask "what percentage change is £12 relative to £80?" the formula gives 15% back.
How do you find percentage change from a multiplier?
If you used the multiplier method to apply a percentage change, the percentage change can be read directly:
- Multiplier of 1.30 → 30% increase
- Multiplier of 0.75 → 25% decrease (since 1 − 0.75 = 0.25 = 25%)
Worked example 3: A salary changed by a multiplier of 0.92. What is the percentage change?
- Multiplier < 1, so it is a decrease.
- Percentage decrease = (1 − 0.92) × 100 = 0.08 × 100 = 8% decrease.
Frequently asked questions
Why do we always divide by the original value?
Because percentage change is defined as the change relative to the starting point. Using the new value would tell you what fraction the change is of where you ended up, which is a different (less useful) question. For example, a rise from 50 to 100 is a 100% increase (doubled), not a 50% increase.
What if the original value is zero?
If the original value is zero, percentage change is undefined — you cannot divide by zero. This situation rarely arises at KS3, but it is worth knowing. For example, if a business had zero profit last year and £10 000 this year, we say there is a profit this year but cannot express it as a percentage change.
How is percentage change different from percentage error?
Percentage error uses the same formula but the "change" is replaced by the error (difference between an estimated value and the true value). The denominator is always the true (accurate) value: percentage error = (|estimate − true| ÷ true) × 100.
Can percentage change be more than 100%?
Yes. If a value doubles, the percentage change is 100%. If it triples, it is 200%. There is no upper limit. A percentage decrease, however, cannot exceed 100% — a value cannot fall below zero and simultaneously decrease by more than 100% of itself.
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