What this tool does
Most percentage mistakes come from reversing a relationship. Knowing a 20% discount took a price from 120 to 96 does not tell you that 96 was a 20% discount — the base was 120, not 96. This calculator keeps the base value explicit so the direction of each calculation is never ambiguous.
How it works
A percentage is always relative to a base. In a discount, the base is the original price. In a profit margin, the base is cost. In a tax, the base is the pre-tax amount. Choosing the wrong base is the most common source of incorrect results, so this tool always names what each input is being measured against.
Percentage change compares an old value to a new one as (new minus old) divided by old. Because the base is the old value, a 50% drop and a 50% rise do not cancel out. Recovering the original from a percentage change means dividing by (1 plus or minus the rate), not multiplying.
When several percentage changes chain together they compound, they do not add. A 20% discount followed by 20% tax is a net change of minus 4%, not zero, because each percentage applies to a different base.
Two shortcuts are worth knowing because they appear in most exam and exam-adjacent questions. A percentage change from A to B is the same as a percentage increase from the smaller number to the larger one only when the change is an increase; for a decrease, the increase runs from the new value back to the old one, which is why recovering a pre-discount price is harder than it looks. Second, a 10% rise followed by a 10% fall returns to 99% of the original, since 1.1 multiplied by 0.9 is 0.99. Neither shortcut is a rounding issue, and neither can be undone by applying the same percentage in reverse.
Worked example
Finding the original price before a 20% discount that produced 96.
- Final price = 96
- Discount rate = 20%
- Original = 96 / (1 - 0.20) = 120
The original price was 120, so the 96 price represents a 20% discount off 120, not off 96.
Accuracy and limitations
- Percentage change is undefined when the original value is zero.
Frequently asked questions
- What is X percent of Y?
- Multiply Y by X and divide by 100. The order matters: 15% of 200 is 30, but 200% of 15 is a different calculation entirely. Always identify which number is the base.
- How do I reverse a percentage discount?
- Divide by one minus the rate. If 20% off gives you 96, the original is 96 divided by 0.8, which is 120. Multiplying instead would give 115.20, which is wrong because it treats the discounted price as the base.
- Why is a 50% loss not cancelled out by a 50% gain?
- Because the base changes each time. Losing 50% of 100 leaves 50. A 50% gain on 50 adds 25, giving 75, not 100. Losses always require a larger percentage gain to undo, which is why a loss is harder to recover from than a gain of the same size is to earn.
- Do two percentage changes add together?
- Only if they apply to the same base. A 10% rise and a 10% fall give a net change of minus 1%, not zero. This is the compounding effect and it applies anywhere sequential percentages are involved, including tax and discounts.
- What is 20% of 150?
- 150 multiplied by 0.20, which is 30. A quick mental route is to halve for 50% and quarter the result for 25%, then take the difference for 20%. The base is always the second number, so 20% of 150 is 30 while 150% of 20 is a completely different question.
- How do I calculate a percentage increase or decrease?
- Subtract the old value from the new one, divide by the old value and multiply by 100. Going from 80 to 100 is a rise of 20 divided by 80, which is 25%. Dividing by the new value instead would give 20%, which is the most common error in this calculation.