#### Answer

$\text{Powder 1 = 30 mg, and Powder 2 = 15 mg}$

#### Work Step by Step

Let us consider that $\text{Powder 1 = x mg, and Powder 2 = y mg}$
Here, we have:
$\text{Powder 1 contains 20% vitamin $B_{12}$ and 30% vitamin E and}
\\ \text{Powder 2 contains 40% vitamin $B_{12}$ and 20% vitamin E}$
We set the two equations as follows:
$0.2 x+0.4y=12~~~~~~~(1)$
$0.3x+0.2y = 12~~~~~~~(2)$
We multiply equation (1) by $\dfrac{3}{2}$ and then subtract it from equation (2) to obtain:
$-0.4 y = -6 \implies y=15 \ mg $
Now, back-substitute the value of $y$ into Equation (1) to solve for $x$
$0.2 x+(0.4)(15)=12 \implies x=30 \ mg $
Therefore, our desired results are:
$\text{Powder 1 = 30 mg, and Powder 2 = 15 mg}$