Stripping VAT out of a VAT-inclusive price

Stripping out VAT is the perfect example of the asymmetry between multiplying by $(1+i)$ and dividing by $(1+i)$: getting the operation wrong leads to a plausible but incorrect result.

Example — Stripping out VAT

An invoice shows $122$ EUR VAT included at $22\%$. Taxable amount: $122/1{,}22 = \boxed{100}$ EUR; VAT: $22$ EUR. Typical mistake: “I take $22\%$ off $122$”, which would give $122\cdot 0{,}78 = 95{,}16$ EUR, wrong. The asymmetric structure $(1+p)$ versus $1/(1+p)$ is the same one we saw with GDP.

Dividing by $1{,}22$ gives the correct taxable amount of $100$ EUR; multiplying by $0{,}78$ instead gives $95{,}16$ EUR, because it takes $22\%$ off $122$ and not $22\%$ of the taxable amount. It is the very same asymmetry as an increase followed by a decrease.

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Topics: Percentages
Concepts: VAT · Stripping out VAT
Skills: Calculating