Two metallic plates form a parallel plate capacitor. The distance...

Before the metal sheet is inserted, capacitance is given by $C = \dfrac{{ε}_{0}A}{d}$.

After the sheet is inserted, the system is equivalent to two capacitors in series, each of capacitance ${C}^{'} = \dfrac{{ε}_{0}A}{\left(\dfrac{d}{4}\right)} = 4C$.
The equivalent capacity is now $\dfrac{1}{C '} = \dfrac{1}{4C} + \dfrac{1}{4C} ⇒ C ' = 2C$.

The ratio is $\dfrac{C}{C '} = \dfrac{1}{2} = 1: 2$