If the capacitance of air filled capacitor is C1=C then the capacitance of dielectric filled capacitor is C2=kC.
Here the charge on air filled capacitor is Q=CV and this charge is shared to the two capacitance. Now the charge on air filled capacitor is Q1=CV/2 and charge on dielectric filled capacitor is CV/2.
Common potential , V′=Q1+Q2C1+C2=CV/2+CV/2C+kC=CVC+kC
or V′V=11+k
or 1+k=VV′
or k=VV′−1=V−V′V′