Increase 100 by 10% and it becomes 110. Reduce that by 10% and the result is 99, not 100. Rounding is not the issue. The second percentage has a new base: the first step adds one tenth of 100, or 10; the second removes one tenth of 110, or 11.
A percentage describes a proportion of a specified quantity, not a fixed amount to add or remove. Adding 10 and then subtracting 10 cancels out, but adding 10% and then subtracting 10% does not follow the same logic. Behind each percentage is a question that needs an answer: a percentage of what?
The steps can also be understood as successive scaling: multiply by 1.1, then by 0.9, giving an overall multiplier of 0.99. For any positive starting quantity, these operations therefore leave 99% of the original. This is an exact arithmetic relationship, not an empirical approximation or a claim that every real-world change follows one process.
Returning from 110 to 100 requires removing 10, which is approximately 9.09% of 110, not 10%. To reverse a change, the percentage must be calculated from the new base. When an increase and a decrease appear together, identify their respective starting quantities before deciding whether the rates cancel.
https://openstax.org/books/prealgebra-2e/pages/6-2-solve-general-applications-of-percent
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