When we say "half of 5/9", we mean "$$\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{5}}}{{\mathtt{9}}}}\right)$$", because "of" is multiplication in math.
Multiplying fractions is simple: you just multiply the numerators and the denominators together and the products are the new and numerators and denominators.
Then you simplify.
When we say "half of 5/9", we mean "$$\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{5}}}{{\mathtt{9}}}}\right)$$", because "of" is multiplication in math.
Multiplying fractions is simple: you just multiply the numerators and the denominators together and the products are the new and numerators and denominators.
Then you simplify.