Everyday arithmetic that is quicker to check than to do in your head — percentages, fractions, ratios of speed and distance, ages, tips, and numbers written out in words.

Each shows the working, which is the useful part. Percentage change in particular is a calculation most people can do and many get backwards, and seeing the expression makes the direction obvious.

Which tool do I want?

If you…	Use
Percentage of, change between, or increase and decrease	Prozentsatzrechner
You need to simplify or convert a fraction	Bruchrechner
Two of speed, distance and time are known	Geschwindigkeit, Entfernung & Zeit
You need an exact age or a duration in years	Altersrechner
You are splitting a bill	Trinkgeldrechner
You need a figure written out for a cheque or contract	Zahl in Wörter
You are converting Roman numerals	Römische Ziffern
You are working out a grade average	Notendurchschnitt-Rechner

A 20% fall does not undo a 20% rise

Percentages are taken from whatever the current value is, so the base moves. Take 100, add 20% to get 120, then remove 20% — that 20% is now of 120, so you lose 24 and land on 96. To get back to 100 you would need 16.7%.

The same asymmetry runs through anything measured in percentages. An investment down 50% needs 100% to recover. A price cut by 30% needs a 43% increase to return to where it was.

Related, and worth keeping separate: percentage points versus percent. An interest rate moving from 2% to 3% has risen by one percentage point and by 50 percent. Both are correct, they describe the same change, and choosing whichever sounds more dramatic is a standard trick in reporting.