Percentage change from an original base of zero is mathematically undefined.
Extreme value ranges might be subjected to floating point rounding conventions.
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About the Percentage Calculator
A percentage expresses a quantity as a fraction of one hundred, from the Latin per centum meaning by the hundred, and it is the most widely used way of communicating proportion in commerce, science, statistics and everyday life. Its usefulness comes from normalisation: expressing two quantities as percentages puts them on a common scale so they can be compared directly regardless of the underlying totals. That same convenience is what makes percentages so easy to misuse. A percentage is always a percentage of something, and when the base changes between two figures — as it does whenever a value rises and then falls — the arithmetic stops behaving the way intuition expects. A 50 percent increase followed by a 50 percent decrease does not return you to where you started. Understanding which base applies to which calculation is what separates confident percentage work from the errors that quietly propagate through spreadsheets and news reports.
Mathematical Formula & Logic
The four percentage operations, each with a different base.
1. Finding a percentage of a number:
Result = (Percentage ÷ 100) × Value
2. Expressing one number as a percentage of another:
Percentage = (Part ÷ Whole) × 100
3. Percentage change between two values:
Change % = ((New − Old) ÷ Old) × 100
The base is always the OLD value. A positive result is an increase,
a negative result a decrease.
4. Reverse percentage — finding the original before a change:
Original = Final ÷ (1 + Change ÷ 100)
So removing a 20% markup means dividing by 1.20, not subtracting 20%.
5. Successive percentage changes multiply, they do not add:
Net factor = (1 + a/100) × (1 + b/100)
6. Percentage points versus percent:
A move from 5% to 7% is a rise of 2 percentage points,
but a 40% increase in relative terms.
Step-by-Step Example
Work through the four operations on a single retail scenario:
Finding a percentage of a number
1. A jacket costs 4,500 and is discounted 30%.
2. Discount = (30 ÷ 100) × 4,500 = 1,350
3. Sale price = 4,500 − 1,350 = 3,150
Expressing one number as a percentage of another
4. The shop sold 84 of the 240 jackets in stock.
5. Percentage sold = (84 ÷ 240) × 100 = 35%
Percentage change
6. Last month's revenue was 250,000; this month's is 287,500.
7. Change = ((287,500 − 250,000) ÷ 250,000) × 100 = (37,500 ÷ 250,000) × 100 = 15% increase
Reverse percentage
8. A receipt shows 3,150 after a 30% discount. What was the original price?
9. Original = 3,150 ÷ (1 − 0.30) = 3,150 ÷ 0.70 = 4,500. Confirms step 3.
10. Note that adding 30% back to 3,150 gives 4,095, which is wrong, because the 30% was calculated on the larger original figure, not on the discounted one.
Why successive changes do not add
11. A price rises 50% then falls 50%: 100 → 150 → 75.
12. Net factor = 1.50 × 0.50 = 0.75, a 25% net loss, not a return to 100.
13. The increase was calculated on 100 while the decrease was calculated on the larger 150, so the two do not cancel.
Reference Data & Values
operation
question
method
result
Percentage of a number
What is 30% of 4,500?
(30 ÷ 100) × 4,500
1,350
Part as a percentage
84 is what percent of 240?
(84 ÷ 240) × 100
35%
Percentage increase
250,000 to 287,500
(37,500 ÷ 250,000) × 100
+15%
Percentage decrease
4,500 to 3,150
(−1,350 ÷ 4,500) × 100
−30%
Reverse percentage
3,150 after 30% off
3,150 ÷ 0.70
4,500
Successive changes
+50% then −50%
1.50 × 0.50
−25% net
Percentage points
5% rises to 7%
7 − 5
+2 points (+40%)
Frequently Asked Questions
Because the two changes are calculated on different bases. Starting at 100, a 50 percent increase adds 50 to give 150, but the subsequent 50 percent decrease is taken on 150 rather than 100, removing 75 and leaving you at 75. The increase was measured against the smaller number and the decrease against the larger one, so they cannot cancel. Percentage changes multiply rather than add: the net factor is 1.50 × 0.50 = 0.75, a 25 percent net loss. To reverse a 50 percent increase you need a decrease of 33.3 percent.
Percentage points measure the arithmetic gap between two percentages, while percent measures the relative change between them. If an interest rate moves from 5 percent to 7 percent, that is a rise of 2 percentage points, but in relative terms it is a 40 percent increase, since 2 divided by 5 is 0.4. Both statements are correct and they describe the same event, which is exactly why the distinction gets exploited: the larger-sounding figure is often chosen for effect. Reporting that uses percent where percentage points are meant is one of the most common statistical distortions in the press.
Divide rather than subtract. To strip an 18 percent tax from an inclusive total, divide by 1.18; to find the pre-discount price after 30 percent off, divide by 0.70. Subtracting the same percentage from the final figure gives the wrong answer because the percentage was originally calculated on the larger base. Taking 30 percent off 4,500 gives 3,150, but adding 30 percent back to 3,150 gives only 4,095, an error of 405. The general form is Original = Final ÷ (1 + change expressed as a decimal).
Convert each change into a multiplier and multiply them together rather than summing the percentages. A 10 percent rise followed by a 20 percent rise gives 1.10 × 1.20 = 1.32, a 32 percent net increase rather than 30 percent, with the extra 2 percent coming from the second increase applying to the already-increased amount. The same method handles mixed directions: a 25 percent rise then a 10 percent fall is 1.25 × 0.90 = 1.125, a net gain of 12.5 percent. This is compounding, and it is why annual growth rates cannot simply be added across years.
Yes, whenever the part exceeds the whole it is being compared against. If revenue grows from 50,000 to 200,000, the increase is (150,000 ÷ 50,000) × 100 = 300 percent, and the new figure is 400 percent of the old one. Note the difference between those two phrasings, since confusing an increase of 300 percent with being 300 percent of the original is a frequent error. Percentages above 100 are impossible only when describing a share of a fixed total, such as the proportion of votes cast.
Anchor on ten percent, which you get by moving the decimal point one place left, then build from there. Twenty percent is that doubled, five percent is it halved, and fifteen percent is ten plus five. For 15 percent of 240: ten percent is 24, five percent is 12, so the answer is 36. A second useful trick is that percentages are commutative, meaning x percent of y equals y percent of x, so 4 percent of 75 is the same as 75 percent of 4, which is simply 3.