Two ideal gases A and B are contained in separate horizontal...

According to the first law of thermodynamics, $Q = \Delta U + W$.

Given that $Q_A = Q_B$ and $\Delta U_A = \Delta U_B$, the work done by both gases must be equal:
$W_A = W_B$

The work done by a gas expanding at constant pressure is $W = P \Delta V$. For a cylinder with piston radius $r$ and displacement $x$, the change in volume is $\Delta V = \pi r^2 x$.

Thus, $P_A (\pi r_A^2 x_A) = P_B (\pi r_B^2 x_B)$

Rearranging for the ratio of displacements:
$\dfrac{x_A}{x_B} = \dfrac{P_B}{P_A} \times \left(\dfrac{r_B}{r_A}\right)^2$

Substituting the given ratios $\dfrac{P_A}{P_B} = \dfrac{2}{1}$ and $\dfrac{r_A}{r_B} = \dfrac{1}{2}$:
$\dfrac{x_A}{x_B} = \left(\dfrac{1}{2}\right) \times \left(\dfrac{2}{1}\right)^2 = \dfrac{1}{2} \times 4 = \dfrac{2}{1}$

Therefore, the ratio of the displacements is $2 : 1$.

Answer: $2 : 1$