Percent Calculator
Calculate percentages, percentage change, and percentage of a number.
Related calculators
About
Percent Calculator
Four calculators sit above this. The first solves the basic relationship between a percentage, a value and a result, so filling any two boxes gives you the third. The second asks the same question in the three ways people actually phrase it. The third compares two values, and the fourth applies an increase or a decrease to a starting figure. Each one prints its working line by line.
What a percentage is
A percentage is a fraction whose denominator is fixed at 100. The word comes from the Latin per centum, by the hundred, and that is the whole idea: rather than compare fractions with awkward denominators, everything is restated out of 100 so the numbers can be read against each other at a glance.
So 35% is 35 out of 100, which is the decimal 0.35 and the fraction 35/100, or 7/20 once reduced. All three say the same thing. If 25 of the 50 students in a class are male, that ratio is 25/50, which reduces to 1/2 and divides out to 0.5. Multiply by 100 and you have 50%.
A percentage carries no units. That is what makes it useful and also what makes it easy to misread: 50% of a class and 50% of a country are the same proportion and wildly different numbers of people. A percentage tells you the relationship, never the size.
The percentage formula
Every question the first calculator answers is one equation with three slots:
P × V1 = V2
P is the percentage, V1 is the value it acts on, and V2 is what comes out. Know any two and the third follows. The calculator converts your percentage into a decimal before multiplying, which is why 5% of 100 shows as 0.05 × 100 in the steps.
Solving for P is the case that catches people, because the division gives a decimal and the answer is wanted as a percent. If P × 30 = 1.5, then P = 1.5 ÷ 30 = 0.05, and 0.05 × 100 = 5%. That last multiplication is the step most often dropped, and it is the difference between answering 0.05% and 5%.
Percentage difference
Percentage difference compares two values without treating either as the starting point. The gap between them is divided by their average:
For 10 and 6, the gap is 4 and the average is 8, so the difference is 4 ÷ 8 = 0.5, or 50%. Because the average is the base, swapping the two values changes nothing. That symmetry is the point of the measure, and it is why the calculator above returns the same answer whichever box each number goes in.
Percentage change
Percentage change is the other question, and it does have a starting point. To apply a change, convert the percent to a decimal and add it to 1 for an increase or subtract it for a decrease:
500 increased by 10% = 500 × (1 + 0.1) = 550
500 decreased by 10% = 500 × (1 − 0.1) = 450
Doing it in one multiplication rather than working out 10% and then adding it is faster by hand and harder to get wrong, and it is what makes chains of changes easy to handle.
Difference and change are not the same measure
This is the most common mix-up on this page, and the third calculator above prints both answers together so the gap is visible. Going from 10 to 8 is a 20% decrease, because the drop of 2 is measured against the starting value of 10. The percentage difference between 10 and 8 is 22.22%, because the same drop of 2 is measured against their average of 9.
Change is also asymmetric. Ten down to 8 is a 20% fall, but 8 back up to 10 is a 25% rise, since the base moved. Difference has no such problem and returns 22.22% either way. Use change when one value came first. Use difference when neither did, such as comparing two measurements of the same thing.
Why a 50% fall needs a 100% rise
An investment that drops 50% has to double to get back to where it started. A 50% loss takes 100 down to 50, and going from 50 back to 100 is a 100% gain. The two percentages do not cancel because each is measured against a different base.
The same asymmetry runs through smaller numbers. A 10% fall needs an 11.1% rise to recover, a 20% fall needs 25%, and a 90% fall needs 900%. Any claim that averages a set of percentage returns is quietly assuming this does not happen.
Stacking percentages
Percentages applied one after another multiply, they do not add. Add 10% and then take 10% off and you land at 99, not 100, because the second 10% comes off the larger number: 1.1 × 0.9 = 0.99.
Three successive 10% discounts do not make 30% off. They make 0.9 × 0.9 × 0.9 = 0.729, which is 27.1% off. Retail stacking works the same way, which is why an extra 20% off sale prices is worth less than the two numbers suggest. The Discount Calculator works stacked offers through step by step.
Percentage points against percent
If a rate moves from 5% to 7%, that is a rise of 2 percentage points, and it is also a 40% increase in the rate. Both are correct and they describe the same move, so a figure quoted without saying which one it is can mean two very different things.
Finance goes a step finer with basis points, where one basis point is a hundredth of a percentage point. A rate cut of 25 basis points is a cut of 0.25 percentage points. The unit exists because a 0.25% cut reads ambiguously, and on large sums the ambiguity is expensive.
Working percentages out in your head
Ten per cent of anything is the number with the decimal point moved one place left, and 1% moves it two places. Most everyday percentages can be built from those two. A 15% tip is 10% plus half of it again: on a 48 bill, 4.80 plus 2.40 is 7.20.
The useful trick is that percentages commute. 8% of 50 is the same as 50% of 8, because both are 0.08 × 50, and the second version is arithmetic you can do without thinking. 4% of 25, 16% of 25 and 25% of 16 all fall out the same way.
Reversing a percentage means dividing, not multiplying back. If a price is 68 after 32% has come off, the original was 68 ÷ 0.68 = 100. Adding 32% to 68 gives 89.76, which is a different number and a common mistake on sale tags and invoices.
Where the symbol came from
Italian merchants writing per cento in the 15th century shortened it to "per 100", then to a p with a small circle above it, and eventually to the two circles and a stroke in use now. The abbreviation "pct" still appears in legal and financial text, and both "percent" and "per cent" are accepted spellings, the first more common in American writing and the second in British.
Percentages above 100 are valid and describe more than the whole: revenue rising from 40 to 130 is a 225% increase. Percentages below zero appear as decreases. Only a base of zero breaks the arithmetic, since nothing can be a percentage of nothing, which is why the calculators above return a message rather than a number in that case.
Reading your result
Check which question you asked before you use the answer. If one of your two numbers came first in time, percentage change is the measure you want, and the calculator reports it against that starting value. If the two numbers are simply two readings of the same thing, percentage difference is the honest one. And when you are quoting a move in something that is already a percentage, say whether you mean points or percent.
Common questions
Frequently asked questions
Convert the percent to a decimal and multiply. For 5% of 100 that is 0.05 x 100 = 5. Moving the decimal point two places left is all the conversion takes, so 5% becomes 0.05 and 140% becomes 1.4.
Change measures the gap against the starting value; difference measures it against the average of the two. Going from 10 to 8 is a 20% decrease but a 22.22% difference. Use change when one value came first, and difference when neither did.
Because each percentage is taken from a different base. 100 down 20% is 80, and 80 up 20% is 96 rather than 100, since the second 20% is a fifth of the smaller number. Getting 80 back to 100 takes a 25% rise.
A rate moving from 5% to 7% has risen 2 percentage points, which is also a 40% increase in the rate itself. Points describe the arithmetic gap between two percentages; percent describes the relative move. A basis point is a hundredth of a percentage point.
Divide rather than add back. A price of 68 after 32% off came from 68 divided by 0.68, which is 100. Adding 32% to 68 gives 89.76, and that is wrong because the 32% was a share of the original figure rather than of the reduced one.
No, they multiply. Three 10% discounts give 0.9 x 0.9 x 0.9 = 0.729, which is 27.1% off rather than 30%. Each discount applies to whatever is left after the one before it.
Find 10% by moving the decimal one place left, then build from there. A 15% tip is 10% plus half of it. It also helps that percentages commute: 8% of 50 is the same as 50% of 8, and the second is easier to do in your head.
Yes. Anything more than the whole is over 100%, so revenue rising from 40 to 130 is a 225% increase. The only value that breaks the arithmetic is a base of zero, because nothing can be a percentage of nothing.