Calculation
Mathematical Formula For Calculating Percentage Change
$$\text{Percentage Change} = \frac{\text{Final Value} – \text{Initial Value}}{\text{Initial Value}} \times 100$$
Example Broken Down With Steps
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\text{Given: Initial Value = 100, New Value = 120}
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$$
\text{Step 1: Subtract 100 from 120: } 120 – 100 = 20
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\text{Step 2: Divide 20 by 100: } \frac{20}{100} = 0.2
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\text{Step 3: Multiply 0.2 by 100: } 0.2 \times 100 = 20
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\text{Result: Percentage increased by 20%.}
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Further Explained
Example: A price increased from 100 to 120.
Steps:
- Find the absolute change: Subtract the original value from the new value to find how much it changed:
$$ 120 – 100 = 20 $$ - Divide the change by the original value: To find how large the change is compared to the original value:
$$ \frac{20}{100} = 0.2 $$ - Convert to a percentage: Multiply the result by 100 to express it as a percentage:
$$ 0.2 \times 100 = 20\% $$ - Conclusion: The percentage change is 20%.
Common Mistake: The Asymmetry of Percentage Changes
A percentage increase followed by the same percentage decrease does not return to the original number. The result is always slightly lower.
Why This Happens
The decrease is applied to the new, larger number — not the original. So the decrease removes a bigger absolute amount than the increase added.
The Formula
Net change after +x% then −x% = −(x² / 100)%
Example Broken Down With Steps
A product’s price increases by 20%, then decreases by 20%:
Start: 100
$$\text{Step 1: Increase by 20%: } 100 \times 1.20 = 120$$
$$\text{Step 2: Decrease by 20%: } 120 \times 0.80 = 96$$
$$\text{Result: The final value is 96, not 100. That is a 4% net loss.}$$
Using the formula: $$ -\frac{20^2}{100} = -\frac{400}{100} = -4% $$
Further Examples
Increase then equal decreaseNet result+10% then −10%−1% net loss+20% then −20%−4% net loss+30% then −30%−9% net loss+50% then −50%−25% net loss
The order does not matter either. A 20% decrease followed by a 20% increase also results in a 4% net loss.
Successive Percentage Changes
When a value changes by one percentage and then by another, you do not simply add the two percentages together. The combined effect follows this formula:
Combined change = a + b + (a × b / 100)%
Where a is the first percentage change and b is the second. Use a negative number for a decrease.
Example Broken Down With Steps
A price increases by 10% and then increases by 20%:
Step 1: Combined change = 10 + 20 + (10 × 20 / 100)
Step 2: = 10 + 20 + 2 = 32%
Result: The total increase is 32%, not 30%.
Example With a Decrease
A stock rises by 25% and then falls by 15%:
Step 1: Combined change = 25 + (−15) + (25 × (−15) / 100)
Step 2: = 25 − 15 − 3.75 = 6.25%
Result: The net result is a 6.25% gain.
Use our calculator above to verify: enter the original value, apply a 25% increase to find the new value, then calculate the percentage change after a 15% decrease.