The question you opened
Ninety six dollars after 20 percent off
A pair of boots rings up at 96 dollars, and the tag said 20 percent off.
What was the price before the discount, in dollars?
$120
Twenty percent off leaves 80 percent of the tag, so 96 is 0.8 of it. Dividing 96 by 0.8 gives 120, and the saving was 24 dollars.
A 20 percent discount leaves 80 percent of the tag price, so the 96 dollars on the receipt is 0.8 of the number you are looking for. Divide rather than add: 96 over 0.8 is 120. Checking it forwards, 20 percent of 120 is 24, and 120 minus 24 is 96. The saving was 24 dollars rather than the 19.20 that adding 20 percent suggests.
Dividing by 0.8 has a friendly form. Dividing by 0.8 is the same as multiplying by 1.25, so 96 plus a quarter of 96 gives 96 plus 24, which is 120. In general, reversing a discount of d means multiplying by 1 over 1 minus d. Reversing 25 percent off means multiplying by 4 over 3. Reversing 50 percent off means doubling.
Reversing 10 percent off means adding a ninth, not a tenth. That last one is where most of the money is lost. People undo a percentage by applying the same percentage in the other direction, and the base has changed underneath them. Undoing 10 percent off with 10 percent on lands at 99 percent of the original, and the gap grows with the size of the discount.
The same reversal runs through anything quoted after a deduction. A salary of 4,000 after a 20 percent tax means gross pay of 5,000, found by dividing. A price of 120 including 20 percent value added tax means a net price of 100, again found by dividing by 1.2 rather than by subtracting 20 percent. The single sentence worth keeping is that percentages are undone by division. Any time the answer arrives through addition, the base has quietly moved.
Technique: Divide by the share that survived