jmc

The $3$-digit multiples of $20$ are $$100,120,140,160,\dots ,960,980.$$ To form the numbers in this list, we can choose any of $9$ hundreds digits and any of $5$ tens digits (but we have only one option for the units digit, which must be $0$). So, there are $9\cdot 5=45$ multiples of $20$ in our list. However, we want to exclude those which are also multiples of $55$.

The least common multiple of $20$ and $55$ is $220$, so we have to exclude the multiples of $220$ from our list. There are four such numbers: $220$, $440$, $660$, and $880$. This leaves $45-4=\boxed{41}$ three-digit multiples of $20$ which are not multiples of $55$.