$P_{2} V_{2}=P_{1} V_{1}$

$P_{2}=P_{1} \frac{V_{1}}{V_{2}}=P_{1} \frac{V_{1}}{V_{1} / 2}=2 P_{1}$

For gas in $\mathrm{B}$, when compression is adiabatic,

$P_{2}^{\prime} V_{2}^{\prime}=P_{1} V_{1}^{\gamma}$

$P_{2}^{\prime}=P_{1}\left(\frac{V_{1}}{V_{2}^{\prime}}\right)^{\gamma}=P_{1}\left(\frac{V_{1}}{V_{1} / 2}\right)^{\gamma}=2^{\gamma} P_{1}$

$\therefore \frac{P_{2}^{\prime}}{P_{2}}=\frac{2^{\gamma} P_{1}}{2 P_{1}}=2^{\gamma-1}$