$R$ is normally distributed with mean $0\%$ and standard deviation $10\%$, with a probability of $0.8$ and mean $30\%$ and standard deviation $10\%$, with a probability of $0.2$. S is normally distributed random rate of return on another asset that has same mean and variance as R. Calculate mean and variance as R.
$E(S) = 0.8(0)+0.2(.30) =0.06$
For $V(S)$, I am confused. Is it $V(S) = 0.8(0.10^2) + 0.2(0.10^2)$
$R$ is not $R=0.8\mathcal{N}(0,10)+0.2\mathcal{N}(30,10)$. Instead, the density function of $R$ is given by the mixture $$ f(r)=0.8f_1(r)+0.2f_2(r), $$ where $f_1$ is the density of $\mathcal{N}(0,10)$ and $f_2$ is the density of $\mathcal{N}(30,10)$. From here you can calculate the mean and variance. (The mean will be equal to the one you calculated, but the variance is different.) See https://en.wikipedia.org/wiki/Mixture_distribution.