Percentages do not add naively; 50%+50% = 125%, insofaras two fifty percent increases back-to-back will produce a 125% increase overall. 10%+10%=21%, while 1%+1%=2.01%; as you deal with smaller and smaller numbers, the addition becomes closer and closer to what might be expected.

Let me use a new symbol, to avoid confusion, and say that 1@+1@=2@, where '@' refers to a new concept of percentage. Further, as we go to smaller and smaller numbers, the '@' becomes the same as the '%', so that .01@ is very nearly .01%, but .001@ is even closer to .001%. Now a 25% increase can be represented as 22.3@; if I increase something by .1% 223 times, I get a 24.97% increase, or about 25%. (I can interpolate, i.e. if I apply another .1% increase the cumulative effect is greater than 25%, so 25% should actually be somewhere between 22.3@ and 22.4@.) If I make a 10@ increase and then a 20@ increase, this is the same as increasing by .1@ 100 times and then another 200 times, so that I get a 30@ increase. (This may be criticized as circular; it is intended to show that my conceptions are consistent.) Similarly, while a 50% decrease and a 50% increase will not cancel each other out, a 50@ increase will cancel a 50@ decrease.

The astute student of mathematics will recognized that what I'm playing with are natural logarithms; in a mathematician's terms, the logarithm of 1.25 is .223..., so that I've denoted a 25% increase, which is a multiplication by 1.25, by 22.3@, where I've multiplied the .223 by 100 to make it easier to read. A series of percentage increases is a series of multiplications, but corresponds to additions of '@'s, and, therefore, to additions of logarithms; the logarithm of the product of two numbers is equal to the sum of their logarithms.