Given a 3 box model of the atmosphere,determine fraction of mass transport from lower layer P = 300mb, Pot. Equiv. Temp = 375K 



P = 600mb, Pot. Equiv. Temp = 325K



P = 900mb, Pot. Equiv. Temp = 350K  



(surface)

Given that a deep convective plume sends a portion of the bottom layer to the top:

What fraction of the mass of the lower layer must rise in order to just barely stabilize the lower layer with respect to the middle layer?  How many millibars of environmental subsidence does this imply?

I need direction on how to begin this.  Which equation will assist with this answer.

My attempt:

If I remove a chuck sized $\alpha$ from the bottom layer, it will contain $\frac{3m}{2}$ the mass of the middle layer of same size.  So, is the answer simply $\frac{2}{3}$ of the mass must move out of the lower layer?  I believe I am wrong, it seems to be more involved than this, but I cannot grasp this concept.
My results:

If an $\alpha$ sized chunk was moved up, then what remains is $(1-\alpha)$.  Because each layer is 300mb thick, they have equal parts mass.  What goes up must be compensated, and due to conservation of mass, there is no gain or loss of mass in my system.  Potential temperature is a conserved quantity, so I can say:

$$350K(1-\alpha)+325K\alpha = 325K(1-\alpha)+375K\alpha$$
$$\alpha = \frac{1}{3}$$

Further,
$\frac{1}{3}$ mass movements equates to $\frac{1}{3}$ of a $300mb$ layer, which is $100mb$ subsidence.