For example, with a 100-gallon tank and a goal of changing 30 gallons each month, one might consider changing 30 gallons once, 15 gallons twice, 10 gallons three times, 5 gallons six times or 1 gallon 30 times. In the extreme case, we can imagine changing an infinitesimally small amount of water an infinitely large number of times, eventually consuming the entire 30 gallons (I actually do this in my aquarium, as I'll explain).
Aquarists often think that many small changes are not as efficient as one big change since some of the water in all subsequent changes was already replaced by earlier changes. This is a correct assertion, but it is often overstressed. After changing 10% three times, only 10% of the first 10% change was changed the second time (1% of the total). So the difference is small. We can mathematically calculate the efficiency of such changes as follows. If we use our 30% example, then one 30% change removes 30% of the impurities, assuming an equal distribution of the impurity within the water. If we do six 5% changes, then the reduction in impurities = 1-(0.95)6 = 26.5%. So it is less efficient (six 5% changes exactly equal 26.5% changed in one batch), but it is not radically less efficient. Going smaller still, the difference is even smaller. Doing 30 one percent changes removes 1-(0.99)30 = 26.0% of the impurities.
