For example, when increasing $20$ by $75\%$, we add $75\%$ of $20$, to $20$. But, why when we increase $20$ by $200\%$, we don't add $200\%$ of $20$, to $20$? (Or, do we?)
Due to this inconsistency, it almost makes me feel like this procedure was invented synthetically, but agreed upon, nevertheless.
On
Like everyone said, there is no inconsistency. However, method of calculation do differ.
You are asking about increasing $20$ by its $200\%$. A $200\%$ of any number, means you want to double up the value, because $100\%$ means the number itself.
So, $200\%$ of $y$ means $2y$, and then you are increasing $y$ by $200\%$, it becomes $y+2y$ eventually.
A $75\%$ increase in 20 will mean, multiplying $20$ by a factor of $1.75$. Why? Because it will (indirectly) mean $20 + 0.75\times 20=20(1+0.75)=1.75\times 20$
I am not sure if there is a name to this process, but that's a very good way to deal with $\%ages$.
There is no inconsistency. If you want to add $200\%$ of $20$ to $20$, then you'd do it the same way that you'd add $75\%$ of $20$ to $20$.
For example, \[ 0.75\times 20+20=20\times 1.75=35 \] \[ 2.00\times 20+20=20\times 3.00=60 \]
It's the same arithmetic with just different numbers.