[Reincarnate]

Corrected version of my earlier post:

C = k * epsilon_0 * A / d

where
C = capacitance (farad)
A = area (m^2)
d = distance (m)
k = dielectic constant
epsilon_0 = 8.854 * 10^(-12) F/m

For the k1 chunk
C_1 = 4 * 8.854 * 10^(-12) F/m * (.05*.01) m^2 / (.01 m) = 1.7708 * 10^(-12) F

For the k2 chunk
C_2 = 6.5 * 8.854 * 10^(-12) F/m * (.05*.04) m^2 / (.01 m) = 1.15102 * 10^(-11) F

Add the capacitances together to get the value for the whole system:

C_system = C_1 + C_2 = 1.7708 * 10^(-12) F + 1.15102 * 10^(-11) F = 1.3281 * 10^(-11) F

Now we solve for our new k from the original equation (rearranged): k = d*C / (A * epsilon_0)

k_system = .01 m * (1.3281 * 10^(-11) F) / ((.05*.05) m^2 * 8.854 * 10^(-12) F/m) = 6

So in the end, it comes down to the following:

k_system = (1/5)*(4) + (4/5)*(6.5) = 6



Algebraic answer:

k_system = (A_1 / (A_1 + A_2)) * k_1 + (A_2 / (A_1 + A_2)) * k_2

Where A_1 is the area of the plate corresponding to the k_1 stuff and A_2 is the area of the plate corresponding to the k_2 stuff