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A 20% increase followed by a 20% decrease does not return you to your starting value because each percentage is calculated from a different base. Starting at 100, the increase gives 120; a 20% decrease from 120 removes 24, leaving 96—4% below the start. A useful percentage calculator therefore needs to handle four different tasks: finding a percentage, measuring a change, calculating a changed value, and reversing a change.
Why equal percentage rises and falls do not cancel
A percentage is a proportion of a reference amount, or base. After the first change, the base has changed too.
- Start with 100. A 20% increase is 100 × 0.20 = 20, so the new value is 120.
- Now calculate a 20% decrease from 120: 120 × 0.20 = 24.
- Subtract 24 from 120. The result is 96, which is 4% below 100.
The same calculation in multiplier form is 1.20 × 0.80 = 0.96. The Hong Kong Education Bureau makes the base distinction explicit in its Key Stage 3 mathematics curriculum guide: the change from 100 to 120 is a 20% increase, while the change from 120 to 100 is not a 20% decrease. These describe different reference amounts.
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Four percentage-calculator formulas
Let O be the original value, N the new value, and r the rate as a decimal—for example, 20% = 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% | From 100 to 120: (120 − 100) ÷ 100 × 100% = 20% |
| Find the new value after a rise or fall | Increase: O × (1 + r) Decrease: O × (1 − r) |
Increase 100 by 20%: 100 × 1.20 = 120 Decrease 100 by 20%: 100 × 0.80 = 80 |
| Find the original value from a changed value | After an increase: N ÷ (1 + r) After a decrease: N ÷ (1 − r) |
A price is $15 after 25% off: $15 ÷ 0.75 = $20 |
The table’s reverse-price example is also given by Microsoft Support’s percentage guidance. The formulas are variations on one idea: identify the original base, the changed value, and which one is unknown.
How to reverse a percentage change
To undo a percentage change, divide by the multiplier used to make it. If a value increased by 20%, its multiplier was 1.20; divide the result by 1.20 to recover the original. If it decreased by 20%, its multiplier was 0.80; divide the result by 0.80.
For example, a value that becomes 120 after a 20% increase started at 120 ÷ 1.20 = 100. Subtracting 20% from 120 would instead give 96, because that calculates a new decrease rather than reversing the increase.
Calculate successive percentage changes
For multiple changes, multiply the multiplier for each step in order. Do not add the signed percentages as if they all used the same base.
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A rise by rate r followed by a fall by the same rate has the combined multiplier (1 + r)(1 − r) = 1 − r². At 20%, this is 1 − 0.2² = 0.96, so the final value is 4% lower than the starting value. This rule works for a same-size rise and fall; for different rates, multiply their individual multipliers instead.
Use the formulas in Excel
Microsoft documents these percentage patterns for Excel, including Microsoft 365 and Excel 2016 through 2024. Enter the old value and rate in cells, then use a formula suited to the task:
- Increase an old value by a rate:
=old*(1+rate) - Decrease an old value by a rate:
=old*(1-rate) - Calculate the signed change from old to new:
=(new-old)/old
For the last formula, format the result cell as a percentage. The calculation itself returns a decimal ratio; percentage formatting displays it as a percent. When filling a formula down rows with a rate stored in one fixed cell, use an absolute cell reference for the rate so that reference does not shift as the formula is copied.
Check the wording before calculating
- “20% off” means the new amount is 80% of the original: multiply the original by 0.80.
- “20% lower” needs a stated comparison base. Translate the wording into an original and a new value before calculating if the base is unclear.
- Percentage change compares a difference with the original value: (new − original) ÷ original × 100%.
- Percentage points are not the same as percent change. A move from 20% to 25% is an increase of 5 percentage points, or a 25% relative increase from 20%.
The Sierra Leone Ministry of Basic and Senior Secondary Education’s JSS3 Mathematics lesson on percentage increase and decrease also presents percentage change relative to the original quantity. That reference base is the key to interpreting a percentage correctly.
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