Add 10 per cent tax to a $100 item and the price is $110.00, with $10.00 of tax. Now work backwards from $110.00 to find the tax it contains: it is still $10.00, and the pre-tax price is $100.00.
The tempting shortcut, taking 10 per cent off $110.00, gives $99.00. That is wrong, and it is wrong in a way that survives casual checking because the numbers look close.
Why subtraction fails
Adding tax multiplies by 1.1, so removing it must divide by 1.1. Subtracting 10 per cent multiplies by 0.9, and 1.1 times 0.9 is 0.99, not 1. The round trip loses one per cent every time.
This is the same asymmetry that makes a 50 per cent fall need a 100 per cent rise to recover, covered in the piece on percentage changes. Percentages are always taken from whatever base they are attached to, and the two directions use different bases.
The general rule
To add a rate, multiply by (1 + rate). To remove it, divide by (1 + rate). At 10 per cent that is multiply by 1.1 and divide by 1.1. At 20 per cent it is 1.2 in each direction, and the error from subtracting instead grows with the rate: at 20 per cent the round trip loses four per cent.
Where it shows up
Invoices are the usual place. A supplier quoting a tax inclusive figure and a buyer recording it as tax exclusive produces a small, consistent discrepancy across every line, which is exactly the kind of error that survives a long time because no single figure looks obviously wrong.
Discounts stacked with tax are the other common case. A 20 per cent discount and a 10 per cent tax give the same final figure in either order, since multiplication commutes, but the tax amount recorded differs depending on whether the discount was applied to the tax inclusive or exclusive price. If the paperwork needs a tax figure, the order matters for that line even though the total agrees.
A check that takes a second
The tax on a tax inclusive price is always a smaller share of that price than the rate suggests. At a 10 per cent rate, tax is one eleventh of the gross, which is about 9.09 per cent of it. If a calculation returns tax equal to 10 per cent of the gross figure, it has used the wrong base.
The sales tax calculator takes a setting for whether the amount entered already includes tax, which removes the choice of formula from the equation entirely.