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Increasing 100 by 20% gives 120; decreasing 120 by 20% gives 96—not 100. The two percentages use different bases, so they do not cancel. A useful percentage calculator should handle four common tasks: finding a percentage, calculating percentage change, applying a change, and reversing one.
Why a 20% increase and a 20% decrease do not cancel
A percentage is a fraction of a reference amount. After the first change, that reference amount may be different.
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- Start with 100. A 20% increase adds 20, giving 120.
- Now calculate 20% of 120: it is 24.
- Subtract 24 from 120. The result is 96, which is 4% below the starting value.
The calculation is 100 × 1.20 × 0.80 = 96. The Hong Kong Education Bureau’s Key Stage 3 mathematics curriculum makes the same distinction: going from 100 to 120 is an increase of 20%, while going from 120 to 100 is not a decrease of 20% because the decrease is measured against 120. Read the curriculum guide.
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For a rate r written as a decimal, an increase followed by an equal decrease has the multiplier (1 + r)(1 − r) = 1 − r². At 20%, r = 0.20, so the combined multiplier is 1 − 0.20² = 0.96. That leaves 96% of the starting amount, a net decrease of 4%.
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Which of the four percentage formulas do you need?
Use O for the original value, N for the new value, and r for the rate as a decimal. For example, 20% is 0.20. Choose the formula based on what you know and what you need to find.
| Task | Formula | Example |
|---|---|---|
| Find a percentage of an amount | amount × r |
20% of 100: 100 × 0.20 = 20 |
| Find the percentage change between two values | (N − O) ÷ O × 100% |
Subtract the original from the new value, then divide by the original. |
| Find the new value after a change | Increase: O × (1 + r)Decrease: O × (1 − r) |
Increase 113 by 25%: 113 × 1.25. Decrease it by 25%: 113 × 0.75. |
| Recover the original value from a changed value | After increase: O = N ÷ (1 + r)After decrease: O = N ÷ (1 − r) |
A price of $15 after 25% off: $15 ÷ 0.75 = $20. |
These are four ways to use the same relationship among the original value, the new value, and the percentage. For percentage change, the original value is the denominator. A positive result means an increase; a negative result means a decrease. Microsoft Support illustrates the percentage-change and reverse-price calculations and provides the increase/decrease multiplier patterns in its Excel percentage guide.
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How to reverse a percentage change
To undo a change, divide the changed amount by the multiplier that produced it. If a value increased by 20%, divide the new value by 1.20. If it was discounted by 20%, divide the discounted value by 0.80.
Do not simply subtract or add the same percentage of the changed amount. For example, after a 20% increase from 100, the value is 120. Subtracting 20% of 120 removes 24 and takes you to 96. To recover 100 from 120, divide by 1.20.
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How to calculate successive percentage changes
Apply each change to the amount reached at the previous step. Multiply the step-by-step multipliers rather than adding signed percentages.
- Increase by 10%, then increase by 5%: multiply by
1.10 × 1.05 = 1.155, for a net increase of 15.5%. - Increase by 10%, then decrease by 5%: multiply by
1.10 × 0.95 = 1.045, for a net increase of 4.5%.
The Hong Kong curriculum includes successive percentage changes among its applications, alongside discount, growth, depreciation, and interest.
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Percentage change is not percentage-point change
When a rate moves from 20% to 25%, the difference is 5 percentage points. Relative to the original 20% rate, it is a 25% increase: (25 − 20) ÷ 20 × 100% = 25%. State which comparison you mean; “up 5%” can be ambiguous.
Calculate percentage changes in Excel
For an increase or decrease, enter the original value and rate in the relevant cells, then use the matching formula pattern:
- Increase:
=old*(1+rate) - Decrease:
=old*(1-rate) - Signed percentage change:
=(new-old)/old
Format the percentage-change result as a percentage to display the decimal ratio as a percent. If you copy a formula down a table while keeping the rate in one fixed cell, use an absolute reference for that rate cell so it does not shift from row to row. Microsoft explains fixed references and the increase/decrease formula pattern in its Excel percentage guide.
Translate ambiguous wording into values first
Before calculating, identify the old value, the new value, and the base to which the percentage refers. “20% off” means the new amount is 80% of the original. A statement that an amount is “20% lower” also needs a clear reference amount to be unambiguous. Once the base and values are clear, choose the appropriate formula rather than relying on the wording alone.
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