How can we use the FPTAS for problem B to solve problem A
You need a stronger guarantee on the reduction, that it be approximation preserving. Not all reductions are approximation preserving. An extreme example is given by Min Vertex Cover and Max Independent Set. The complement of a vertex cover is an independent set, and so there are simple NP-hardness reductions between these problems. However, while Min Vertex Cover has a 2-approximation algorithm, Max Independent Set cannot be approximated to within $n^{1-\epsilon}$ for any $\epsilon &gt; 0$ unless $\mathsf{P=NP}$.