Evaluating the rate law expression

Let us divide the data into four cases $\mathcal{Case I,II,III,IV}$ respectively. From our reference to $\displaystyle{\mathcal{Case I and II}}$ on doubling the concentration of $\displaystyle{B}$ the rate doubles so we can see that it is of order $\displaystyle{1}$ w.r.t. $\displaystyle{B}$ and on halving the concentration of $\displaystyle{A}$ the rate too decreases to half. Again on doubling the concentration of both $\displaystyle{A}$ and $\displaystyle{B}$ the rate quadruples .Hence our rate law is $\boxed{\mathcal{Rate} = k[A]^{1} [B]^{1} }$