How to use it
What is X% of Y takes a percentage and a total. 15% of 200 is 30. Reach for this one for tips, discounts and tax.
X is what percent of Y takes a part and a whole. 30 out of 120 is 25%. Test scores, shares of a budget, completion rates.
Percentage change from X to Y takes an old value and a new one. 80 to 100 is a 25% increase.
Nothing needs pressing.
Stacked discounts do not add up
A shop advertises 20% off, then another 10% off at the till. That is not 30% off. It is 28%.
The second discount comes off the already-reduced price, not the original.
Take £100 and knock off 20%, leaving £80. Take 10% off that and you lose £8, not £10, because you are discounting a smaller number now. Final price £72, so the real saving is 28%.
The gap widens as the discounts grow. 50% off followed by another 50% off leaves you paying 25% of the original, not zero. Nobody has ever got anything free that way.
The same arithmetic runs through anything that compounds. Sales tax on top of a service charge. A fee charged on a fee. Two pay rises in consecutive years.
Two rises of 10% are not a 20% rise either. They come to 21%.
Up 50% then down 50% is not where you started
Percentage change is measured against whatever the starting number happened to be, and that number keeps moving underneath you.
Start at 100. Add 50% and you are at 150. Now take 50% off and you land on 75, because that second 50% was half of 150 rather than half of 100. Down a quarter from where you began, despite two percentages that look like they should cancel.
Which is why a stock that falls 50% needs to rise 100% just to break even.
The fall is measured against the high number, the recovery against the low one. A 90% crash needs a 900% recovery.
Be suspicious anywhere you see percentage changes averaged together. Averaging them is almost always the wrong operation, and the reason is the same asymmetry: a fund that gains 20% one year and loses 20% the next has not broken even, it is down 4%, yet the average of the two changes reads as zero. Anyone quoting an average annual return calculated that way is describing something that did not happen to the money. The honest figure is the compound growth rate, which accounts for each year building on the last, and it is always lower than the simple average whenever the numbers move around. The more volatile the series, the wider that gap gets.
Percent or percentage point
These get used interchangeably in news reporting. They mean different things.
An interest rate moving from 5% to 7% has risen by 2 percentage points. It has also risen by 40%, because 2 is 40% of 5.
Both statements are true. They sound wildly different, which is exactly why you tend to see whichever one suits the argument being made.
The rule: percentage points measure the gap between two percentages, and percent measures relative change. If the underlying figure is already a percentage, say which you mean. This calculator reports relative change.
Working backwards from a total
A common question, and one the three boxes cover even without a dedicated field for it.
Say a price of £120 already includes 20% tax. The pre-tax figure is not £120 minus 20%. That gives £96, and it is wrong. Your £120 represents 120% of the original, so you divide by 1.2 and get £100.
Discounts work the same shape in reverse. If £72 is what remains after 28% off, the original was 72 divided by 0.72. Again £100.
So: divide by one plus the rate to strip an addition. Divide by one minus the rate to reverse a reduction.
Subtracting the percentage back off is the classic mistake. It always leaves you short, and the bigger the rate the further out you land.
Frequently asked questions
Why does the percent-of box show a dash when the total is zero?
Dividing by zero is undefined.
Does it handle decimals and negatives?
Yes, all three boxes accept both, and results round to a few decimal places so they stay readable.
Is 20% off then 10% off the same as 30% off?
No. It works out at 28% off, because that second discount comes off the already-reduced price rather than the original, which makes it smaller than the label suggests.
What is the difference between percent and percentage points?
Percentage points measure the gap between two percentages, while percent measures relative change. Going from 10% to 15% is 5 percentage points. It is also a 50% increase.
How do I find the original price before a discount?
Divide by one minus the rate. So £72 after 28% off means 72 divided by 0.72, giving £100. Adding the percentage back on is the classic mistake and always leaves you short.
Why does up 50% then down 50% not return to the start?
Each percentage is measured against a different number. Up 50% from 100 gives 150. Down 50% from 150 gives 75.
Results are rounded for display; for accounting or exams, double-check critical figures.