How to work out any percentage
Nearly every percentage question is one of three sums. Learn the three and you can do most of them in your head.
Three common percentage questions in one place, pick a mode and type your numbers.
Percentages feel slippery until you notice something freeing: almost every percentage question you ever meet is one of just three sums. A percent means a part out of 100, and once you can spot which of the three you are looking at, the maths is quick. Here they are, with the shortcuts that let you do most of them without a calculator, plus the two trickier questions that build on them.
1. How to find a percentage of a number
This is the most common one: what is 20% of 500? Turn the percent into a decimal by dividing by 100, then multiply. So 20% becomes 0.20, and 0.20 times 500 is 100. The mental shortcut is even faster: 10% of any number just moves the decimal one place left, so 10% of 500 is 50, and 20% is double that, 100. Need 5%? Halve the 10%. Need 15%? Add the 10% and the 5%. Almost any everyday percentage can be built from those pieces.
2. What percent one number is of another
Here the question flips: 30 is what percent of 200? Divide the part by the whole, then multiply by 100. So 30 divided by 200 is 0.15, times 100 is 15%. Use this whenever you want to express a score, a share or a tip as a percentage. It is also how you turn "18 marks out of 40" into a grade: 18 divided by 40 is 0.45, so 45%.
3. Percentage increase and decrease
This one trips people up the most. A price goes from 200 to 260, what is the change? Subtract the old from the new, divide by the old, then multiply by 100. So (260 minus 200) divided by 200 is 0.30, a 30% rise. For a decrease it works the same way: a fall from 200 to 150 is (200 minus 150) divided by 200, which is 0.25, a 25% drop. The key is that the starting number is always the one you divide by.
Why does the starting number matter so much?
Because percentage change is not symmetric. Going from 200 up to 260 is a 30% rise, but coming back down from 260 to 200 is only about a 23% fall, since the base is bigger the second time. This is the single most common percentage mistake, and knowing it saves you from a lot of bad conclusions about prices and figures.
How to add or subtract a percentage (tax, tips, discounts)
Often you do not want the percentage on its own, but the new total. To add 20% tax to a 50 bill, work out the 20% (10) and add it, giving 60, or multiply straight by 1.20 in one step. To take 30% off a 80 item, multiply by 0.70 to get 56. The one-step multipliers are worth memorising: adding x% means multiplying by (1 + x/100), and taking x% off means multiplying by (1 - x/100). That single trick handles most shopping, tipping and tax maths instantly.
How to reverse a percentage (find the original price)
Here is the question that catches almost everyone: an item costs 60 after a 20% discount, what was the original price? The instinct is to add 20% back, but that is wrong, because the 20% was taken off the larger original, not the smaller sale price. Instead, divide. If 20% was taken off, the sale price is 80% of the original, so divide by 0.80: 60 divided by 0.80 is 75. The same logic reverses tax: if a total of 120 already includes 20% tax, the pre-tax figure is 120 divided by 1.20, which is 100. Whenever a percentage has already been applied and you want the number before it, you divide by the multiplier rather than adding the percentage back.
Percentage difference between two numbers
Percentage change assumes one number came first. When neither did, and you just want how far apart two values are, use percentage difference: take the gap between them, divide by their average, and multiply by 100. For 40 and 60, the gap is 20 and the average is 50, so the difference is 40%. It is the right tool for comparing two measurements where there is no "before" and "after".
The bottom line
Decide which question you are actually asking, keep the part-over-whole idea in mind, and remember to divide, not add, when you reverse a percentage. When the numbers are awkward or you just want to be sure, drop them into our percentage calculator, which handles every mode above.
Frequently asked questions
How do I work out 10% of a number quickly?
Move the decimal point one place to the left. 10% of 250 is 25. From there, 20% is double (50) and 5% is half (12.5), so you can build most percentages in your head.
How do I calculate percentage increase or decrease?
Subtract the old value from the new, divide by the old value, then multiply by 100. A rise from 200 to 260 is (260-200)/200 = 30%. A fall from 200 to 150 is (200-150)/200 = 25%. Always divide by the starting number.
How do I reverse a percentage to find the original price?
Divide, do not add. If 60 is the price after 20% off, the original is 60 divided by 0.80, which is 75. If 120 includes 20% tax, the pre-tax figure is 120 divided by 1.20, which is 100.
How do I find what percent one number is of another?
Divide the part by the whole and multiply by 100. For example, 30 out of 200 is 30 divided by 200, times 100, which is 15%. The same method turns marks out of a total into a grade.
Why is a percentage rise and fall not the same?
Because the starting number changes. Going 200 to 260 is a 30% rise, but 260 back to 200 is only about a 23% fall, since the second sum divides by the larger number.