The question you opened
Half off, then 20 percent more off
A coat is marked 50 percent off, and the register takes another 20 percent off the sale price.
What is the total discount from the original price, in percent?
60 percent
Half the price survives the first cut, and four fifths of that half survives the second. You pay 0.4 of the sticker, so 60 percent has come off.
Track what stays rather than what goes. After 50 percent off, half the price remains. The extra 20 percent removes a fifth of that half, leaving four fifths of it. Half times four fifths is two fifths, so the shopper pays 40 percent of the sticker and the total discount is 60 percent. A 100 dollar coat makes it concrete. Fifty off leaves 50.
Twenty percent of 50 is 10, so the register takes 40. Sixty dollars have come off a hundred dollar coat, which is 60 percent, and the missing 10 points are the difference between what the sign implies and what the till charges. The rule for any stack is to multiply the survivors. Two cuts of a and b leave (1 minus a) times (1 minus b) of the price.
Thirty and forty leave 0.7 times 0.6, which is 0.42, so 58 percent comes off. Ten cuts of 10 percent leave 0.9 to the tenth power, about 0.35, so the price never reaches zero however many coupons are stapled together. Retailers rely on this. Stacked offers read as addition and behave as multiplication, and the gap always favours the shop.
The chaining runs upward too: two 10 percent price rises are a 21 percent rise, so the seller gains a little more than the sign says on the way up. The practical habit is to convert every percent off into the fraction that survives, multiply those fractions, then subtract from one at the end. It is one step longer to say and several steps shorter to compute, and it never produces the impossible discounts that addition invents.
Technique: Chain what is left, 0.5 times 0.8