The question you opened
Tax first, or discount first
A 45 dollar lamp gets 20 percent off, and the state adds 8 percent sales tax.
Does the total change if the tax is applied before the discount?
It stays the same
Both routes multiply 45 by 0.8 and by 1.08, so both land on 38.88. Multiplication gives the same product in either order.
Take the discount first. Twenty percent off 45 dollars leaves 36, and 8 percent tax on 36 adds 2.88, so the total is 38.88. Now reverse it. Tax on 45 adds 3.60 and brings it to 48.60, and 20 percent off 48.60 leaves 38.88. The same number twice. The reason is that both operations are multiplications. Taking 20 percent off multiplies by 0.8. Adding 8 percent tax multiplies by 1.08.
The total is 45 times 0.8 times 1.08 whichever way round the two factors are written, because multiplication is commutative. Nothing about tax or retail is involved. The intuition that fails is a real one applied to the wrong operation. Paying tax on a smaller amount genuinely does cost less tax, by 72 cents here. What gets forgotten is that the discount is then computed on a larger amount and hands back exactly the same 72 cents. Each step differs, and the endpoint does not.
Order does matter the moment one of the steps stops being a plain multiplication. A flat 5 dollar coupon is a subtraction, so taking it before tax saves the tax on those 5 dollars. A discount capped at 10 dollars behaves differently depending on what it is applied to. Rounding to the cent at each step can shift the total by a penny.
Any of these breaks the symmetry, which is why shops write the sequence into their terms. The habit worth carrying is to sort the steps into multiplications and additions. Multiplications commute freely, so their order never changes the answer. Additions and subtractions mixed among them do not, and those are the only places where arguing about sequence is worth the breath.
Technique: Multipliers can swap places