Step 1: The capacitance of a parallel plate capacitor without a dielectric is defined as:
\[
C_0 = \epsilon_0 \frac{A}{d}
\]
Here, \( \epsilon_0 \) represents the permittivity of free space, \( A \) denotes the area of the plates, and \( d \) is the separation between the plates.
Step 2: When a dielectric material with a dielectric constant \( K \) is introduced between the plates, the capacitance is amplified by a factor of \( K \) for the portion of the plate covered by the dielectric. In this scenario, the dielectric slab covers \( \frac{d}{5} \) of the total plate separation \( d \).
Step 3: The resultant capacitance is calculated by summing the capacitance of the dielectric-filled region and the air-filled region:
\[
C = \frac{\epsilon_0 A}{d - \frac{d}{5}} + \frac{\epsilon_0 K A}{\frac{d}{5}}
\]
Upon simplification:
\[
C = \frac{\epsilon_0 A}{\frac{4d}{5}} + \frac{\epsilon_0 K A}{\frac{d}{5}} = \frac{5\epsilon_0 A}{4d} + \frac{5K \epsilon_0 A}{d}
\]
Step 4: Factoring out the common term \( \frac{\epsilon_0 A}{d} \) yields:
\[
C = \frac{\epsilon_0 A}{d} \left( \frac{5}{4} + 5K \right) = C_0 \left( \frac{5}{4} + 5K \right)
\]
Step 5: Consequently, the new capacitance is:
\[
C = C_0 \left[ \frac{5K}{4K+1} \right]
\]
The capacitance of the capacitor is thus \( \left[ \frac{5K}{4K+1} \right] C_0 \).