Percentage Calculator
Seven percentage questions, each answered with the working laid out line by line
What is P% of X?
A is what percent of B?
Add P% to X, or take it off
By what percent did it go from old to new?
What was the number before the percentage was applied?
A rate moved from one percentage to another. How many points, and how many percent?
How far apart are A and B when neither one came first?
15% of 200 is 30.
The working, step by steppart = whole × percentage ÷ 100
| Step | Working | Result |
|---|---|---|
| Turn the percentage into a decimal | 15 ÷ 100 | 0.15 |
| Multiply by the whole | 0.15 × 200 | 30 |
| Check it the other way round | 200% of 15 | 30 |
| Answer | 15% of 200 | 30 |
Answers update as you type. Figures are shown to four decimal places at most, and the working keeps six so a rounding step never hides where a number came from. Negative values are allowed wherever the arithmetic still means something.
Which percentage question are you asking?
Most wrong answers come from running the right numbers through the wrong formula. Find the row that sounds like your situation, then use that tab. The last column says whether the order of your two numbers changes the answer.
| It sounds like | Use | Formula | Order of the numbers |
|---|---|---|---|
| What is 15% of 200? A tip, a commission, a share of a bill | Percent of | whole × p ÷ 100 | Either way round, 8% of 50 is 50% of 8 |
| 27 marks out of 30 as a percentage | What percent | part ÷ whole × 100 | Part on top, whole underneath |
| Add 10% to 500, or take 12.5% off 50 | Add or take off | old × (1 ± p ÷ 100) | One multiplier does it |
| A price went from 60 to 72 | Percent change | (new - old) ÷ |old| × 100 | Order matters, the old value is the base |
| What did it cost before 15% off? | Reverse | new ÷ (1 ± p ÷ 100) | Never take p% of the new figure |
| A rate moved from 4% to 6% | Points or percent | new rate - old rate | 2 points, which is a 50% rise |
| Shop A sold 200, shop B sold 240, neither came first | Difference | |a - b| ÷ average × 100 | Order does not matter |
| An estimate of 70 against an exact 80 | Percent error, worked by hand | |estimate - exact| ÷ |exact| × 100 | The exact value is the base, 12.5% |
Fractions, ratios and proportions are close relatives with their own tools. Adding or simplifying fractions lives on the fraction calculator, part to part comparisons on the ratio calculator, and growth over many periods on the compound interest calculator.
One formula, seven percentage questions
How percent of, percent change, reverse percentages, percentage points and percent difference relate, and the traps that turn a correct calculation into a wrong answer.
Every percentage question on this page is built from one relationship: a part is some percentage of a whole. Written out, percentage = 100 × part ÷ whole, part = whole × percentage ÷ 100, and whole = 100 × part ÷ percentage. The seven tabs above are that one line rearranged for the seven situations people actually meet, and the reason the calculator splits them apart is that the arithmetic is easy while choosing the right version of it is not.
The percentage formula, three ways round
Know any two of part, whole and percentage and the third follows. 80% of 30 is 24, because 30 × 80 ÷ 100 = 24. 27 out of 30 is 90%, because 27 ÷ 30 × 100 = 90. And if 10 is 5% of something, that something is 200, because 100 × 10 ÷ 5 = 200. Most of the long lists of phrasings you will see elsewhere, "Y out of what is P%", "X is what percent of Y", "what is P% of X", are these three formulas wearing different clothes.
One shortcut is worth knowing before anything else. A percent of a number is the same as that number percent of the first one, so 8% of 50 and 50% of 8 are both 4. The multiplication does not care about order. When 18% of 50 feels awkward, 50% of 18 is 9 and you are done. The Percent of tab shows this check as its third line every time.
Percent of a number, and what percent one number is of another
To find a percent of a number, turn the percentage into a decimal by dividing by 100, then multiply. 60% of 1,260 is 0.6 × 1,260 = 756. To find what percent one number is of another, divide the part by the whole and multiply by 100: 65 out of 250 is 0.26, which is 26%. The only way to go wrong here is to put the numbers upside down, so the What percent tab labels the part and the whole explicitly and warns you when the part is larger than the whole, which gives an answer above 100%.
Adding a percentage or taking one off with a multiplier
Adding 10% to 500 is 500 + 50 = 550, and taking 10% off is 450. Two steps work, but one step is cleaner. A rise of 10% is the same as multiplying by 1.10, and a fall of 12.5% is the same as multiplying by 0.875, so 50 less 12.5% is 50 × 0.875 = 43.75. The decimal multiplier is the single most useful habit in percentage work, because it turns every increase, discount, markup and tax addition into one multiplication, and it is what makes the reverse calculation further down obvious.
Percentage change: why the old value is the base
The percent change formula is (new value - old value) ÷ |old value| × 100. From 60 to 72 is a change of 12 on a base of 60, which is a 20% increase. From $1,250 to $1,445 is 195 on 1,250, a 15.6% increase. The vertical bars around the old value matter more than most pages admit. They mean you divide by the size of the starting number, ignoring its sign, so a move from 50 to -22 comes out as a 144% decrease and a move from -10 to -25 as a 150% decrease, both reading in the direction a person would describe them.
A change can be larger than 100% upward, since anything that more than doubles has risen by more than 100%, and a fall of more than 100% is possible only when the value crosses below zero. A change from zero has no percentage at all, because there is nothing to divide by. The calculator refuses that case instead of printing infinity.
The asymmetry that catches everyone
Because the base is always the starting value, the same gap is a different percentage in each direction. Hours that rise from 35 to 45.5 have gone up 30%. The same 10.5 hours taken away again, from 45.5 back to 35, is a fall of only 23.08%, because this time the divisor is 45.5. On rounder numbers: 100 to 110 is a 10% rise, and 110 back to 100 is a 9.09% fall.
Chain two changes and the effect becomes a trap. A rise of 50% followed by a fall of 50% does not bring you home. It is 1.5 × 0.5 = 0.75 of where you started, so you finish 25% down. In general a rise of p% followed by a fall of the same p% always leaves you below the start, and to undo a 50% rise you need a fall of only 33.33%. The Add or take off tab prints that undo figure on every result, and the Percent change tab shows the same gap measured backwards, so the asymmetry is on screen rather than something you have to remember. For growth over many periods, where the same effect compounds year after year, the compound interest calculator is the right tool.
Percent change, percent difference and percent error
Three formulas look alike and answer different questions. The rule that separates them is about what the two numbers mean.
| Use | When | Base | Worked case |
|---|---|---|---|
| Percent change | An old value and a new value | The old value | 200 to 240 customers is a 20% increase |
| Percent difference | Two values of the same kind, neither older or better | The average of the two | Shop A on 200, shop B on 240 is 18.18% |
| Percent error | An approximate value against an exact one | The exact value | An estimate of 70 against an actual 80 is 12.5% |
Percent difference divides the gap by the average of the two values, so it has no direction and swapping the numbers leaves it alone: 5 and 7 are 33.33% apart in either order, and 10 and 6 are 50% apart. It always lands between the two directional changes, which is why the Difference tab prints both of them beneath it. Run 200 and 240 through the Percent change tab and you get 20% one way and 16.67% the other; the Difference tab gives 18.18%, sitting in between. If someone quotes a single percentage for two numbers that have an obvious before and after, they wanted percent change. If the numbers are two shops, two labs or two quotes, they wanted percent difference.
Reverse percentages: finding the original value
A car sells for $6,800 after 15% off. What was the price? The tempting move is to take 15% of $6,800 and add it back, which gives $7,820. It is wrong by $180, because the 15% was 15% of the original price, not of the sale price. The original value is what you are looking for, so that is the base you do not yet have.
Two methods get it right. With the multiplier, a 15% discount leaves 0.85 of the original, so the original is $6,800 ÷ 0.85 = $8,000. With the 1 percent method, $6,800 is 85% of the original, so 1% is $6,800 ÷ 85 = $80, and 100% is $8,000. Both are shown as separate lines in the Reverse tab, along with the wrong answer so you can see how far off it would have been.
The same logic runs upward. A shirt costing $68.75 after 25% tax was added cost $68.75 ÷ 1.25 = $55 before tax. Attendance that rose 12% to 728 was 728 ÷ 1.12 = 650. And $44 after 10% off came from $44 ÷ 0.9 = $48.89. The pattern to remember is that the reverse of adding 25% is dividing by 1.25, not taking 25% off. The third option in the Reverse tab covers the plain version, where 36 is 45% of something: 1% is 36 ÷ 45 = 0.8, so the whole is 80.
Percentage points versus percent
When the two numbers you are comparing are themselves percentages, there are two honest answers and one common mistake. A percentage point is the plain arithmetic difference between two percentages. Unemployment moving from 5% to 7% has risen by 2 percentage points. The relative change in the rate is a percent change like any other: a rate going from 4% to 6% is up 2 points and also up 50%.
The mistake is to say the rate rose by 2%, which strictly means 2% of 4%, a move to 4.08%. Real reporting shows how far apart the two readings can be. A payroll tax cut from 6.2% to 4.2% is 2 percentage points, not a 2% cut, and it is a 32.3% fall in the rate, and so in the revenue it raises. A diagnosis rate moving from 1% to 5% is 4 points and a 400% increase. Smoking falling from 50% in 1980 to 40% in 1990 is 10 points and a 20% fall. A rule of thumb from reporters' own guidance: if both numbers are percentages, lead with the percentage points.
Percentage points are abbreviated pp, p.p. or %pt. Finance uses a smaller unit, the basis point, which is one hundredth of a percentage point, or one part in 10,000. A rate cut of 150 basis points is 1.5 percentage points. The Points or percent tab gives all three at once: points, basis points and the relative change.
The traps in one table
| Trap | What goes wrong | Right answer |
|---|---|---|
| Reverse by subtraction | 15% of $6,800 added back gives $7,820 | $6,800 ÷ 0.85 = $8,000 |
| Up then down by the same percent | Up 50% then down 50% looks like no change | You end 25% below the start |
| Points called percent | 6.2% to 4.2% described as a 2% cut | 2 points, a 32.3% cut |
| Wrong formula for the pair | Percent change used for two shops | Percent difference, 18.18% for 200 and 240 |
| Zero as the base | A change from 0 given as a percentage | No percentage exists |
Where this page stops
Percentages sit beside several close relatives that have their own tools. Adding, simplifying and converting fractions belongs to the fraction calculator. Part to part comparisons, such as 3 to 2, belong to the ratio calculator, and solving for a missing term in two equal ratios to the proportion calculator. A mean or a weighted mean of several values is on the average calculator, and splitting a bill with a tip on the tip calculator.
Frequently Asked Questions
Why does a 50 percent rise followed by a 50 percent fall leave me worse off?
Because the two changes are measured against different bases. The rise is 50% of the starting value, and the fall is 50% of the larger value you reached. Multiply the two steps together and you get 1.5 × 0.5 = 0.75, which is 25% below where you began.
The same thing happens on any pair of equal and opposite changes. 100 up to 110 is a 10% rise, but 110 back to 100 is only a 9.09% fall. To undo a 50% rise you need a fall of 33.33%, not 50%. The Add or take off tab shows that undo figure under every answer.
Should I use percent change or percent difference to compare two numbers?
Ask whether one number came first. If there is an old value and a new value, such as last year and this year, use percent change, which divides by the old value: 200 to 240 is a 20% increase.
If the two numbers are the same kind of thing and neither is the starting point, such as two shops or two quotes, use percent difference, which divides by their average: 200 and 240 are 18.18% apart in either order. If one number is an estimate and the other is exact, use percent error, which divides by the exact value: 70 against 80 is 12.5%.
How do I find the price before a discount or before tax was added?
Divide by the multiplier rather than subtracting a percentage. A price of $6,800 after 15% off is 85% of the original, so the original is $6,800 ÷ 0.85 = $8,000. A price of $68.75 with 25% tax included is 125% of the pre-tax price, so that price is $68.75 ÷ 1.25 = $55.
The common mistake is to take 15% of $6,800 and add it back, which gives $7,820 and is $180 short. The Reverse tab shows the right answer, the 1 percent method and that wrong answer side by side.
When should a change be described in percentage points instead of percent?
Whenever both numbers are already percentages, such as an interest rate, a tax rate or an unemployment rate. The gap between them is measured in percentage points: 5% to 7% is a rise of 2 percentage points.
You can also give the relative change, which is an ordinary percent change of the rate itself, so 4% to 6% is 2 points and a 50% rise. What you should not do is call 2 points a 2% rise, since 2% of 4% would only take the rate to 4.08%. A payroll tax cut from 6.2% to 4.2% is 2 points and a 32.3% fall.
What happens to percent change when the starting number is zero or negative?
A change from zero has no percentage, because the formula divides by the starting value and there is nothing to divide by. The calculator says so instead of showing infinity.
A negative start is allowed. The formula divides by the size of the starting value, ignoring its sign, so the answer points the way a person would describe it: -10 to -25 is a 150% decrease, and 50 to -22 is a 144% decrease. A fall of more than 100% can only happen when a value crosses below zero.
Why is 8 percent of 50 the same as 50 percent of 8?
Both are 8 × 50 ÷ 100, and multiplication gives the same result in either order. So x% of y always equals y% of x, and both of these come to 4.
It is a genuinely useful shortcut. 18% of 50 is awkward in your head, but 50% of 18 is simply half of 18, which is 9. The Percent of tab prints this reversed check on every answer.
How does the 1 percent method work for reverse percentages?
Write down which percentage the number you have represents, divide to find 1%, then multiply by 100. If 45% of a number is 36, then 1% is 36 ÷ 45 = 0.8 and the number is 80.
For a change, first add or subtract the change from 100%. Attendance up 12% to 728 means 728 is 112% of the original, so 1% is 6.5 and the original is 650. The most frequent slip is dividing by the change, 12, instead of by 112.
How big is a basis point compared with a percentage point?
A basis point is one hundredth of a percentage point, or one part in 10,000. To turn basis points into percentage points, divide by 100, so a cut of 150 basis points is 1.5 percentage points.
Finance uses basis points because rates often move by fractions of a point, and saying 25 basis points is clearer than saying a quarter of a percent, which could be misread as a relative change. The Points or percent tab gives the basis point figure alongside the other two.
Can a percentage be more than 100 percent?
Yes, whenever the part is larger than the whole or a value more than doubles. 150 is 150% of 100, and a rise from 1% to 5% is a 400% increase in the rate.
A decrease is different. A fall of exactly 100% takes a positive value to zero, and anything beyond that takes it below zero, so a discount of more than 100% has no sensible meaning. The Reverse tab refuses a decrease of 100% or more for that reason.