The idea behind every percentage problem
A percentage is a ratio written "out of 100": 15% is 0.15 and 150% is 1.5. Every percentage question gets easier once you decide one thing first: what is the base, the number you are measuring against?The same 20 is 20% of 100 but only 10% of 200. The four calculators above differ only in where the base sits, and most mistakes in everyday percentage work come from picking the wrong base.
The four calculations and their formulas
- A is what % of B: A Γ· B Γ 100. Use it for test scores, goal completion and expense shares. Example: 20 is 10% of 200.
- N% of a base number: base Γ N Γ· 100. Use it for discount amounts, tips, fees and taxes. Example: 15% of 45 is 6.75.
- Percent change: (new β previous) Γ· previous Γ 100. The denominator is always the previousvalue. Example: from 80 to 68 is β15%.
- Apply or reverse a percentage: to add p%, multiply by (1 + p/100); to subtract p%, multiply by (1 β p/100). To go back to the original, divide by the same factor: original = result Γ· (1 Β± p/100).
The calculator shows the working under each result, such as "68 Γ (1 β 15%)", so you can check that the tool did what you meant. Pressing Enter in any field runs that block's calculation.
Worked examples (checked by calculation)
Finding a discount rate. A jacket listed at $80 now costs $68. The change is (68 β 80) Γ· 80 Γ 100 =β15%, so it is 15% off.
From sale price back to list price. If all you know is "15% off, now $68", the list price is 68 Γ· (1 β 0.15) = $80. It is not 68 Γ 1.15 = $78.20. The discount was taken from the list price, so adding 15% to the lower figure uses the wrong base and lands short of the original.
Separating tax from a total. Suppose a sales tax of 8% applies (an assumed rate, chosen only for the example). A $50 item then carries $4 of tax for a total of $54. Going the other way, 54 Γ· 1.08 = $50. Taking 8% off the total, 54 Γ 0.92 = $49.68, gives the wrong answer for the same reason as above. Real tax rules vary by place and product, so use the rate printed on your receipt or in official guidance.
A tip. 15% of a $45 meal is $6.75, so the total is $51.75. For splitting a bill among several people, the tip calculator handles it directly.
Above 100%. A target of 3.5 and a result of 4.2 gives 4.2 Γ· 3.5 Γ 100 = 120%. Going past 100% only means the result is larger than the base; it is not an error.
Why a loss is harder to win back than it was to lose
If a value falls 10% and then rises 10%, it does not return to where it started. After the fall, the base is smaller, so a bigger percentage is needed to climb back. The rise required is 1 Γ· (1 β drop) β 1. The table below applies that formula to common drops, and it is true for prices, balances, weights or traffic alike.
| Drop | Rise needed to get back to the original |
|---|---|
| 5% | +5.3% |
| 10% | +11.1% |
| 20% | +25.0% |
| 30% | +42.9% |
| 40% | +66.7% |
| 50% | +100.0% |
| 60% | +150.0% |
| 70% | +233.3% |
| 80% | +400.0% |
| 90% | +900.0% |
A quick check: 100 falling 50% gives 50, and 50 must double (+100%) to reach 100 again. In the other direction, 100 rising 50% reaches 150, and a 50% fall from there gives 75, which is 25% below where you began.
Stacked discounts do not simply add up
"20% off, plus another 10% off" is not 30% off. The second discount applies to the already reduced price, so the final price is list Γ (1 β 0.2) Γ (1 β 0.1). On an $80 item that is 80 Γ 0.8 Γ 0.9 =$57.60, an effective discount of 28%. Some stores apply a coupon to the original price instead, so read the terms of the offer. The table gives the effective total for several pairs.
| First discount | Extra discount | Effective total discount |
|---|---|---|
| 10% | 10% | 19.0% |
| 20% | 10% | 28.0% |
| 30% | 20% | 44.0% |
| 30% | 30% | 51.0% |
| 50% | 20% | 60.0% |
| 50% | 50% | 75.0% |
Percent versus percentage points
When a rate that is already a percentage changes, there are two correct ways to describe it, and they say different things. An interest rate moving from 10% to 12.5% rose by 2.5 percentage points, which is a 25% relative increase (12.5 Γ· 10 β 1). Going from 10% to 15% is 5 points, or a 50% relative increase. For shares, rates and poll numbers, saying "points" avoids confusion, and a headline that says "5%" when it means "5 points" can give readers the wrong idea.
Common mistakes
- Using the wrong denominator. For 80 to 68, the decrease is 12 Γ· 80 = 15%, not 12 Γ· 68 = 17.6%.
- Zero as the previous value. Growth from zero has no defined percentage. The calculator shows an error; in writing, say "new" or quote the absolute change.
- Adding or subtracting a percentage to undo one. Divide instead, as in the sale-price example.
- Adding successive changes. A 15% rise followed by a 15% fall is 1.15 Γ 0.85 = 0.9775, so β2.25% overall, not zero.
- Negative starting values. If the previous value is negative, such as a loss, the sign of the percentage can seem backwards. Report the absolute change instead of a percentage.
- Rounding. Results are shown to two decimal places. For accounting-grade totals, round the way your own rules require.
Putting the result to use
- Shopping: turn the old and new prices into a real discount rate, and use the stacking table to judge coupons. The discount calculator is built for this.
- Tax and fees: to remove tax from a total, divide by (1 + rate). Thesales tax calculator covers the forward direction.
- Savings and growth: the recovery table explains why losses weigh more than gains; thecompound interest calculator shows how repeated percentages build up over time.
- Reports: give both the percent change and the absolute change, and label changes in rates as percentage points.
What this calculator does not do
It performs the arithmetic on the numbers you type and nothing else. It knows nothing about tax law, store policies or contract terms, so confirm any figure that matters financially against a receipt, an agreement or official guidance. The numbers are computed in your browser and are not sent anywhere.
Frequently Asked Questions
What does a negative percentage change mean?
It means the new value is smaller than the previous value, a decrease. For example, β15% means it dropped by 15%.
Is a percentage point (pp) different from a percent (%)?
Yes. A percentage point is the plain difference between two percentages (10% to 15% is 5 points). A percent is a relative change, so the same move is a 50% increase.
How do I get the original price from a discounted price?
Divide the sale price by (1 β discount). If 20% off gives $16, then 16 Γ· 0.8 = $20. In the fourth block above, choose "Original value, if the value is the result of a p% decrease".
Can a result be over 100%?
Yes. If A is larger than B the ratio is above 100%, and an increase of more than 100% means the value more than doubled. A decrease cannot exceed 100%, since a value of zero is already a 100% drop.
What happens if the previous value is 0?
A percentage change from zero is undefined, because you cannot divide by zero. The calculator shows an error. Describe the change in words or as an absolute amount instead.
How do I combine two successive percentage changes?
Multiply the factors; do not add the percentages. A 15% rise followed by a 15% fall is 1.15 Γ 0.85 = 0.9775, a net change of β2.25%.