Percentage Calculator

Three questions hiding behind one word

Percent means per hundred. A percentage is a fraction with 100 fixed as the denominator, which is the only reason 15% and 0.15 and 15/100 are the same thing written three ways. Everything on this page is that one idea moved around.

What makes percentages slippery is that the word covers three different questions, and people rarely say which one they mean. The arithmetic for each is a single line:

What is X% of Y  →  Y × X ÷ 100
X is what percent of Y  →  X ÷ Y × 100
Change from A to B  →  (B − A) ÷ A × 100

The mistakes are almost never in the multiplication. They are in picking the wrong line, or in dividing by the wrong number. Ask what percent 45 is of 180 and the answer is 25%; divide the other way around and you get 400%, which is a correct answer to the question nobody asked. The two numbers are the same, the sum is the same length, and only the order tells them apart.

That is why every answer here is printed with its formula and the numbers already substituted into it. A bare 36 cannot be checked. A line reading 240 × 15 ÷ 100 = 36 can be checked by eye in a second, and if it is answering the wrong question you will see that immediately too.

Which of the five panels you want

• % of: you know the rate and the total, you want the part. Sales tax on a price, a 15% deposit, 30% of a monthly budget, the commission on a sale.
• What %: you know the part and the total, you want the rate. A test score, a completion rate, what share one line item is of an invoice.
• Change: you have a before and an after, and you want the movement between them with its direction. Last month against this month, a price rise, a headcount cut. The answer is signed, so an increase is positive and a decrease is negative.
• Discount: a price and a percent off, returning what you pay and what you keep. It is the % of calculation with the subtraction already done, because the number you actually want at the register is the one after the deduction.
• Tip: a bill, a tip rate and a number of people. It returns the tip, the total, and the share each person owes when there is more than one of you.

If you cannot tell whether you want % of or What %, look at what you are missing. Missing a plain number means the first; missing a rate means the second.

How to work out a percentage increase or decrease

1. Open the Change panel. Two boxes: A is where you started, B is where you ended up. Getting them the wrong way around is the only real hazard here.
2. Type the earlier figure into From (A). This is the number the percentage is measured against, and it is what makes the answer asymmetric.
3. Type the later figure into To (B). The answer updates as you type, so there is nothing to submit.
4. Read the sign. A positive result in green is an increase, a negative result in red is a decrease, and 0% means the two numbers match.
5. Check the second line, Difference. It is the plain subtraction, and it is often the number your reader actually wants: three more sales is clearer than a 1.4% rise.
6. Read the formula under each line to confirm which number ended up on the bottom. If A and B are swapped, the formula shows it before the answer reaches anyone else.
7. Swap the two boxes and run it again to see the asymmetry for yourself. From 50 to 80 is a 60% rise; from 80 back to 50 is a 37.5% fall. Same two numbers, same gap, different base.

How to use the discount and tip shortcuts

1. For a sale price, open Discount, enter the ticket price and the percent off. The preset chips cover the usual signs in a store window, and the box still takes anything in between.
2. Read both lines. Final price is what you pay, You save is what the sign was shouting about. They are different numbers and stores quote whichever is larger.
3. For a stacked offer, such as 20% off and then a further 10%, run it twice: put the first final price into the price box and apply the second discount to it. The combined result is 28% off, not 30%.
4. For a restaurant bill, open Tip and enter the bill before the tip. Pick a preset or type your own rate.
5. Set Split between to the number of people. Leave it at 1 and the per person line disappears, because it would only repeat the total.
6. Round the per person figure up when you hand over money. Two decimal places will not always divide cleanly into three or seven shares, and the leftover fraction has to land on somebody.
7. If the bill already includes a service charge, check before adding a tip on top. Applying a percentage to a number that already contains one is the commonest double charge in this whole area.

Percentage points, and other ways the base moves

A percentage is meaningless without the number it was taken from, and most percentage arguments are really arguments about that base.

Points versus percent

When the thing that changed is itself a percentage, there are two honest answers. A rate moving from 4% to 5% has gone up by one percentage point, and by 25% in relative terms. Headlines pick whichever sounds better. Say "points" when you mean the arithmetic difference, and reserve "percent" for the relative change.

Percentages do not add up

Two discounts of 20% and 10% make 28%, not 30%, because the second is taken from the reduced price. A 10% rise followed by a 10% fall leaves you at 99, not 100. Percentages multiply, and the order of multiplication does not matter, which is why a tax applied before a discount gives the same total as one applied after.

Reversing a percentage

To recover the original price from a total that already includes 20% tax, divide by 1.2 rather than subtracting 20%. Subtracting takes 20% of the larger number and leaves you short. The same trick recovers a pre discount price: divide the sale price by 0.8 for 20% off.

Getting the answer out of the page

• The copy button on a line puts that figure alone on the clipboard, formatted the way it appears.
• Copy with the formula takes every line of the current answer with its working, one per line. This is the one to use when somebody else has to trust the number, because it shows what was divided by what.

The same sums elsewhere

Spreadsheet, X% of Y  →  =B1*A1/100
Spreadsheet, percent change  →  =(B1-A1)/A1

• Spreadsheets store a percentage as the underlying decimal and only display the sign. In Excel, applying the percentage format to a cell that already holds 25 shows 2,500%, while typing 25 into a cell formatted beforehand gives 25%. Typing 25% or 0.25 comes out right either way.
• The percent key on a pocket calculator is not standardized. On many models 200 + 10 % gives 220, because the key silently takes 10% of the first number and adds it. On others the same sequence gives 200.1. If a physical calculator disagrees with this page, that key is usually why.
• Phone calculators: the iOS and Android built in calculators both have a percent key that divides by 100, so 15% becomes 0.15 and you then multiply.
• Local number formats: much of Europe writes 1.234,56 where the United States writes 1,234.56, and some locales put a space before the percent sign. This page reads and writes the English convention, and accepts pasted commas and spaces as separators rather than as decimal points.

What this page will not do

• It does not know your currency. There is no symbol anywhere and no exchange rate. The arithmetic is identical in dollars, euros, pounds or yen; read the output in whatever you typed in.
• It does not add tax. A discount is applied to the price you entered, nothing more. Whether tax comes before or after the discount depends on where you are, and the two orders can give different totals when rounding lands between them.
• Divide by zero has no answer. Asking what percent a number is of 0, or the change starting from 0, returns a plain message rather than a number.
• Rounding is a display choice. Percentages are shown to four decimals. Discount and tip amounts are rounded to two, with exactly half a cent going up, and each later line starts from the rounded figure above it, the way a receipt does.
• Negative bases are handled by a convention. A change from a negative starting number is divided by its absolute value, so the sign always follows the direction the number moved. The strict textbook formula divides by the signed value and flips the sign in that case. For positive starting numbers, which is nearly always, the two agree exactly.

A percentage on its own invites a wrong reading. A 200% increase in a figure that was 3 is still a small number, and "half of respondents" means something different when there were 12 of them. Quote the base alongside the rate.