I realize this is an older thread and unlikely to be edited, but I was just pointed here in order to answer one of my questions (it was answered) and found an error. True I use BtS and the HOF mod so maybe things have changed. The error is in step 2 of the basics where it says
I have 3 cities and my commerce is not a multiple of 10 in any of them. If my gold is at 10% I should get at least 3 gold per turn according to this statement, but I get only 2. Here are the numbers. City #1 pulls in 16 commerce. Both other cities pull in 4 each, for a total of 24 commerce each turn. At 10% gold, I generate 1.6 + 0.4 + 0.4 gold which is 2.4 gold floored to 2. Or it could be that the rounding (to nearest) is happening before the addition, so let's disprove that. (By this I mean 2(1.6) + 0(0.4) + 0(0.4)). If city #1's commerce is reduced to 14 then 1.4 gold would be rounded to 1 and I'd get only 1 gold per turn. Since I get 2 we know the rounding happens after the addition. And further to prove that the type of rounding is flooring and not rounding to the nearest, we increase the commerce to 26. 2.6 still turns out 2, not 3, so 2.6 must be floored.
So, given that I have 24 commerce (and ignoring the +1 science for having a city and all other modifiers), if I set it to 100% science I get 24 science and 0 gold. If I set it to 100% gold I get 0 science and 24 gold. Unfortunately, if I set it to 90% science and 10% gold I get 21 science and 2 gold. Similarly, if I have 10% science and 90% gold I get 2 science and 21 gold.
So the correct formula for science and gold generated is, quite simply;
FLOOR (rate * commerce)
(+1 science if there is a city).
Therefore, you will often lose 1 commerce if your slider is not set to 100% of something. What I often do to achieve a 80% science rating, say, is do 1 turn of 100% gold then 4 turns of 100% science. This might not always be a good idea though.