There isn't a reason why you shouldn't write something like $100 * (£20 / £47) as "int dollars = 100 * 20 / 47;". Note that this expression assicates to the left instead of the right, which can be the right thing to do if doing integer arithmetic. But it would not work with a strongly typed setup as in your example.
In my experience trying to prevent accidental mistakes is a waste of time and often makes our lives miserable. Catching the rare bug by doing complicated work in the type system when it would have been easy to find in normal code anyway is not worth it.
It was just the first example from the top of my head. The expression above calculates the right thing, cast to int. In general, prescribing which units we can multiply and which not, is extremely silly if you consider how we learn it in school. You can multiply anything and everything, simply take care of the units. There isn't an obvious reason why we couldn't have 2000 dollar-pounds as a transient value in a longer computation.
The real problem is that most type systems aren't fit to track the units automatically. Solution: Don't beat yourself up, track the units in your mind / in comments / in variable names instead of the type system. And just get it right. It's not that hard - if you mix something up that's usually the type of bug that is immediately noticed and fixed.
In my experience trying to prevent accidental mistakes is a waste of time and often makes our lives miserable. Catching the rare bug by doing complicated work in the type system when it would have been easy to find in normal code anyway is not worth it.