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Sloppy calculations using percentages

In the past I have noticed cases where calculations with percentages were carried out in a sloppy fashion. Often I accepted this in silence as an approximation, but I have always wanted to look how big the error actually is that we're making. The formulation that I found was actually quite surprising and interesting, hence the fact that I wrote it down. The result can be found in the following text. When performing sloppy calculations using percentages I can think of two operations (maybe there are more), which lead to the same formulation.

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