Percentage Calculator
Four kinds of percentage question, with the arithmetic shown alongside each answer.
Most percentage mistakes are not arithmetic mistakes. They come from calculating a different question than the one being asked — a share instead of a change, or a decrease measured against the wrong starting figure. This calculator separates the four common questions into four modes and prints the working underneath the answer, so you can see which one you actually asked.
How it works
Pick the question
A share of a value, a proportion, a change, or an adjustment.
Fill in the two numbers
The labels change to match the mode you chose.
Check the working
The line under the result shows the calculation that produced it.
A rise of 50% and a fall of 50% do not cancel out
Percentage change is measured against the starting value, and after a rise the starting value has changed. Take 100, add 50% to reach 150, then take 50% off: you land on 75, not 100. Getting back to 100 from 150 requires a fall of 33.3%, because the same 50 is now being measured against a larger base.
This asymmetry is behind a great deal of misleading arithmetic. An investment that loses 50% needs a 100% gain to recover. A price cut of 20% followed by a rise of 20% leaves the price 4% below where it started. And a shop advertising "50% off, then 20% off at the till" is offering 60% off in total, not 70% — the second discount applies to the already reduced price.
The reliable habit is to always name the base. "Sales rose 15%" is meaningless until you know 15% of what, measured when. The change mode here uses the first number as the base and prints it in the working line for exactly that reason.
Percentage points are not percentages
If an interest rate moves from 2% to 3%, that is a rise of one percentage point, and also a rise of 50%. Both statements are true and they describe the same event, which is why the difference between them is such a reliable source of confusion — and such a common tool in a misleading headline.
The convention is simple once stated: use percentage points for the arithmetic difference between two percentages, and percent for the relative change between them. A poll moving from 40% to 44% has gained four percentage points, or ten percent of its previous share. Reporting that as "a 10% rise" is not wrong, but it means something quite different from "a 10 point rise".
A related trap is averaging percentages. The mean of 50% and 100% is not the overall rate unless both were measured on the same sized group. If a shop converts 50% of 10 visitors one day and 100% of 90 the next, the combined rate is 95%, not 75%. Percentages can only be averaged directly when their bases are equal — otherwise you have to return to the underlying counts.