Let me try to explain the math a little better then:
A primary goal of water changes is the lower the concentration of pollutatants in the water, e.g. nitrate.
Let's say that the current nitrate reading is 100 ppm. ppm is parts per million which is another way of saying 100 mL of nitrate per 1000 L of water. or 100 microliters of nitrate per 1 L.
Just to make the math simpler, let's say the tank is 1 L, so if the reading is 100 ppm, there is exactly 100 mcL (mcL = microliter) of nitrates in there. Now, lets do a water change of x% This reduces the amount of nitrate by that x% (just like a sale)
And, so long as you put perfectly clear water (no nitrates) you there won't be any extra nitrates in there.
So, let's look at the 70% water change (x=70%)
100 mcL to start, 70% goes out. 70% of 100mcL is 70mcL so
100-70 = 30 mcL. or 30% of the original concentration.
Now, let's look at the 3 30% changes
100 mcL to start. 30% goes out. 30% of 100mcL is 30mcL so100-30 = 70 mcL. This is the concentration after 1 water change.
Now, let's do another 30% change. 30% of 70mcl is 21mcL. so 70-21 = 49 mcL. This is the concentration after 2 water changes.
One more time 30% of 49mcL is 14.7mcL. So 49-14.7 = 34.3 mcL which is the answer above. Also it is 34.3% of the original concentration, less than one large 70% change.
This is where my comment about production comes in. If each day there is production of nitrates, this math requires a little modification; what ever the generation rate is needs to be added after the dilution step but the idea is the same.
The math in the post above is just the quick way of doing that caclulation without stopping at each point (that is what the powers do in this case -- repeated water changes). A little math shows that 3 33.1% water changes is equal to 1 70% change.
If you want to look at the equivlance of two schemes, here is the math:
1 x% water change versus z y% changes (that is perform a y% change z times)
(1-x/100) = (1-y/100)^z
The way I found the 33.1% above is by solving for y after setting x=70 and z=3
You can even generalize this more, w x% changes versus z y% changes:
(1-x/100)^w = (1-y/100)^z
Using this we can find the 3 30% changes are equal to
2 changes of 41.4%
4 changes of 23.5%
5 changes of 19.3%
6 changes of 16.3%
8 changes of 12.5%
10 changes of 10.1% and etc.