Combination of polydisperse polymer mixtures
It should not be difficult to derive either of the number-averaged molecular weight or weight-averaged molecular weight by their definitions. To avoid ambiguity, I will use $i$ as the index of each actual length of chains and $b$ as the index of batches.

First, the definitions:

$\bar{M}_\mathrm{n,final}=\frac{\Sigma_in_iw_i}{\Sigma_in_i}$ 

$\bar{M}_\mathrm{w,final}=\frac{\Sigma_in_iw_i^2}{\Sigma_in_iw_i}$ where $n$ and $w$ are number and molecular weight of each chain length.

The numerator of $\bar{M}_\mathrm{n,final}$ is, in fact, the total mass of the mixture. Hence, $\Sigma_in_iw_i=\Sigma_bw_b$. The denominator is the total number of chains. Hence, $\Sigma_in_i=\Sigma_bn_b=\Sigma_b\frac{w_b}{\bar{M}_\mathrm{n,b}}$.

It is slightly more complicated in the case of $\bar{M}_\mathrm{w,final}$ but you got it correct. Feel free to ignore my derivation. The numerator does not have a straightforward definition. It has to be derived from the $\bar{M}_\mathrm{w,b}$ instead. $\Sigma_in_iw_i^2=\Sigma_b\Sigma_{i,b}n_{i,b}w_{i,b}^2=\Sigma_b\bar{M}_\mathrm{w,b}w_b$. Since we have already defined the $\Sigma_in_iw_i$ above, we can have the final equation.

$\bar{M}_\mathrm{n,final}=\frac{\Sigma_bw_b}{\Sigma_b\frac{w_b}{\bar{M}_\mathrm{n,b}}}$ 

$\bar{M}_\mathrm{w,final}=\frac{\Sigma_b\bar{M}_\mathrm{w,b}w_b}{\Sigma_bw_b}$

$\bar{M}_\mathrm{w,final}/\bar{M}_\mathrm{n,final}=\frac{\Sigma_b\frac{w_b}{\bar{M}_\mathrm{n,b}}\Sigma_b\bar{M}_\mathrm{w,b}w_b}{(\Sigma_bw_b)^2}$