jmc

Let the number of left-handed boys be $x$. Since there are four times as many right-handed boys, the number of right-handed boys is $4x$. Since there are twice as many left-handed girls as left-handed boys, and there are $x$ left-handed boys, there are $2x$ left-handed girls. We place all of this in a Venn diagram:



We also know that half the girls on the team are left-handed. Since there are $2x$ left-handed girls, there are also $2x$ right-handed girls.



There are a total of $$36=2x+x+4x+2x=9x$$people on the team, so $x=4$. We are trying to find the number of right-handed boys. This number is $$4x=4\cdot 4=\boxed{16}.$$