The question you opened
Twenty percent up, twenty percent down
A jacket price rises 20% in March, then falls 20% in April.
The April price is what percent of the February price?
96%
The rise is 20% of the old price, the fall is 20% of the bigger one. 100 goes to 120, then 120 minus 24 leaves 96.
A jacket goes up 20% in March and comes back down 20% in April. The instinct says the price is where it started. It lands at 96% of the February price, and the missing 4% is worth understanding, because the same 4% shows up in salaries, discounts and investment returns. Start with a round number. Say the jacket was 100 in February. A 20% rise adds 20, so March is 120.
Now April cuts 20%, and the cut is taken from 120, not from 100. Twenty percent of 120 is 24. Take 24 off 120 and you land on 96. The reason the two moves fail to cancel is that they measure different things. The rise is a fifth of the small price. The fall is a fifth of the large one. A fifth of something bigger is bigger, so the fall wins by 4.
A shortcut worth memorising: multiply the two factors instead of adding the percentages. Up 20% means times 1.2. Down 20% means times 0.8. And 1.2 times 0.8 is 0.96, which is your 96% straight away, no matter what the starting price was. The starting price never mattered, which is why the answer is a percent rather than an amount. The same multiplication tells you something useful. Any rise followed by an equal fall lands below where you started, always, whatever the size.
Up 50% and down 50% gives 0.75. Up 10% and down 10% gives 0.99. The loss grows with the square of the move, so big swings hurt far more than small ones. Shops know this. A price marked up before a sale can be discounted by the same headline percentage and still sit above the original. Reading the two factors as a multiplication takes a couple of seconds and tells you what the pair of moves really did.
Technique: Percent moves ride on different bases