$M=\frac{\rho R T}{P} \quad\left(\text { as } P=\frac{\rho R T}{M}\right)$

where $P, T$ and $\rho$ are the pressure, temperature and density of the gas respectively and $R$ is the universal gas constant.

$\therefore$ The molecular weight of $A$ is

$M_{A}=\frac{\rho_{A} R T_{A}}{P_{A}}$ and that of $B$ is $M_{B}=\frac{\rho_{B} R T_{B}}{P_{B}}$

Hence, their corresponding ratio is

$\frac{M_{A}}{M_{B}}=\left(\frac{\rho_{A}}{\rho_{B}}\right)\left(\frac{T_{A}}{T_{B}}\right)\left(\frac{P_{B}}{P_{A}}\right)$

Here, $\frac{\rho_{A}}{\rho_{B}}=1.5=\frac{3}{2}, \frac{T_{A}}{T_{B}}=1$ and $\frac{P_{A}}{P_{B}}=2$

$\therefore \quad \frac{M_{A}}{M_{B}}=\left(\frac{3}{2}\right)(1)\left(\frac{1}{2}\right)=\frac{3}{4}$