Adding Percentages (%)

A percentage (%) is just a ratio, parts per 100.

You can multiply by a percentage to get a proportion of the total:

P% of T = T × P / 100

A percentage has no ‘dimension’, or units. It is a ‘pure number’. It can’t really be added or a quantity (other than to another pure number). To add a percentage really means to add a proportion of the original amount:

“A + P%” = A × (1 + P / 100)

Similarly, a percentage can’t really be subtracted:

“A – P%” = A × (1 – P / 100)

[In the following, ‘£’ could be ‘€’, ‘$’, etc.]

For example to get a gross price from a net price:

£gross = “£net + VAT%” = £net × (1 + VAT / 100)

To get a net price from a gross price, you do not subtract the rate percentage. Rather, you must reverse the above, like this:

£net = £gross / (1 + VAT / 100)

If, for example, VAT is at 17.5%, we have

£gross = £net × 1.175

£net = £gross / 1.175
£net ≈ £gross × 0.851

(So the latter works out closer to taking off 15%, not 17.5%)

Here’s how to calculate any one of £gross, £net, £tax or tax_rate from any two of the others:

to calculate given use
£gross £net £tax £gross = £net + £tax
£net tax_rate £gross = “£net + tax_rate%”
= £net × (1 + tax_rate / 100)
= £net × (100 + tax_rate) / 100
£tax tax_rate £gross = £tax × (1 + 100 / tax_rate)
£net £gross £tax £net = £gross – £tax
£gross tax_rate £net = £gross / (1 + tax_rate / 100)
= £gross / ((100 + tax_rate) / 100)
= £gross × 100 / (100 + tax_rate)
£tax tax_rate £net = £tax × 100 / tax_rate
£tax £gross £net £tax = £gross – £net
£gross tax_rate £tax = £gross / (1 + 100 / tax_rate)
£net tax_rate £tax = £net × tax_rate%
= £net × (tax_rate / 100)
tax_rate

(%)

£gross £net tax_rate = 100 × (£gross / £net – 1)
£gross £tax tax_rate = 100 / (£gross / £tax – 1)
£net £tax tax_rate = 100 × £tax / £net

If VAT (the tax_rate) = 17.5 = 35/2, i.e. if VAT% = 17.5% = 175/1000 = 7/40, then

£gross = £net × 47 / 40
£gross = £tax × 47 / 7
£net = £gross × 40 / 47
£net = £tax × 40 / 7
£tax = £gross × 7 / 47
£tax = £net × 7 / 40
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