LL(1) CFG for triple amount of b's compared to a's? I would like to construct a LL(1) context-free grammar for a formal language $A$ = "strings where the amount of character b is exactly triple compared to the amount of character a" for $\Sigma = \{a,b\}$.

I've tried to figure it out so that if we consider a string $w$ to have one a, then the acceptable presentations for $w$ considering language $A$ would be abbb, babb, bbab and bbba.

But how do I build the CFG rules for that? I have tried something like $$S \rightarrow A$$ $$A \rightarrow \epsilon | aAbbb |baAbb | bbaAb | bbbaA$$ but I think it's not very good because at least I cannot build an LL(1)-parser out of it (which would be good) because of ambiguity. And it doesn't quite work out for a string like bbbbbbaa $\in A$.

Is it possible to build a CFG for this language that can be parsed using a LL(1) parser?  If not, why not?
I believe that any string able to be represented by a CFG can also be parsed in LL or LR. That is just one of the features of CFG's. The grammar you have originally written can be parsed using LL as the upper-case $A$ will act always as your left-most variable to be expanded and simplified. 

I think your problem with constructing the grammar can be solved by also including rules in which the lower-case and upper-case $a$'s are switched, though, that is a rather clunky solution. 

$S \rightarrow A$ 

$A \rightarrow \epsilon | baAbb | bAabb | bbbaA | bbbAa | bbaAb | bbAab | aAbbb| Aabbb$