The question you opened
Can you find the mistake in three steps
Step 1: 40 + 60 = 100. Step 2: cut 10%, giving 90. Step 3: add 10% back, giving 100.
Which step number is wrong?
Step 3
Step 3 adds 10% of 90, which is 9, so the total reaches 99. The second percentage is taken from the smaller number.
Three steps are written out, and one of them is wrong. Step 1 says 40 plus 60 equals 100. Step 2 says cut 10 percent, giving 90. Step 3 says add 10 percent back, giving 100. The error is in step 3. Check them in order. Forty plus sixty is one hundred, so step 1 is fine. Ten percent of one hundred is ten, and one hundred minus ten is ninety, so step 2 is fine.
Step 3 adds ten percent back, but ten percent of what. The number on the table is now 90, and ten percent of 90 is 9. Adding it gives 99, not 100. The mistake is a switched base. Both percentages are called ten percent, and they sound like the same quantity, so it feels as though one should undo the other. The first ten percent means ten percent of 100. The second means ten percent of 90.
Same word, different amount, and the difference is exactly the missing 1. To come back to 100 from 90 you would need to add 10 out of 90, which is about 11.1 percent. That asymmetry is a permanent feature of percentages rather than a quirk of these numbers. Any cut of p percent needs a bigger percentage rise to reverse it, because the rise is calculated on a smaller number. The habit that catches this is one question, asked every time a percentage appears: percent of what.
Write the base down next to it if the chain is long. In sequences of discounts, tax rates, commissions and interest, the base moves quietly at each step, and every argument about who owes what usually turns out to be an argument about which base was used. A useful shortcut for chains like this is to multiply factors instead of adding percentages. Down 10 percent is times 0.9. Up 10 percent is times 1.1. And 0.9 times 1.1 is 0.99, which is your 99 immediately.
Technique: Check what each percent is taken of