Solution:

a.
Each person needs $\frac{5}{3} = \frac{20}{12}$ of a cupcake. If they cut four of the cupcakes into $\frac{5}{12}$ and $\frac{7}{12}$ and the last cupcake in half, then one person can take the four $\frac{5}{12}$ pieces, giving them
$$4 \cdot \frac{5}{12} = \frac{20}{12}$$
and the other two can each take two $\frac{7}{12}$ pieces and one $\frac{1}{2} = \frac{6}{12}$ piece, giving them as well
$$\frac{7}{12} + \frac{7}{12} + \frac{6}{12} = \frac{20}{12}$$
All pieces are larger than $\frac{1}{3}$ ounces.

b.
Each person needs $\frac{3}{5} = \frac{12}{20}$ of a cupcake. We can divide two of the cupcakes into $\frac{6}{20}$, $\frac{7}{20}$ and $\frac{7}{20}$ pieces and the last cupcake into four $\frac{1}{4} = \frac{5}{20}$ pieces. Then four of the people can get one $\frac{7}{20}$ piece and one $\frac{5}{20}$ piece and the fifth person can get the two $\frac{6}{20}$ pieces.

All pieces are larger than $\frac{1}{5}$ ounces.