When the compression process is performed suddenly or instantly then there is not enough time for heat transfer, therefore the sudden compression process is a adiabatic compression process.

Ideal gas equation for adiabatic process, $P{V}^{\gamma }=\text{Constant}$

$$\begin{gathered} \therefore P_i{V}_i^{\gamma }=P_f{V}_f^{\gamma } \\ \Rightarrow \frac{P_f}{P_i}=\frac{V_i^{\gamma }}{V_f^{\gamma }}={\left[\frac{V_i}{V_f}\right]}^{\gamma }={\left[\frac{V_i}{{\displaystyle \frac{V_i}{4}}}\right]}^{1.5} \\ \Rightarrow \frac{P_f}{P_i}=\frac{8}{1} \end{gathered}$$