Consider the bcc unit cells of the solids 1 and 2 with the position...

If radius of $A$ is $r$ radius of $B$ will be $2r$ . Atoms in the body diagonal touch.

Therefore:
$r + 2r + 2r + r = \sqrt{3}a$

$6r = \sqrt{3}a$ or $a = \dfrac{6r}{\sqrt{3}}$

Packing Efficiency
$= \dfrac{1 \times \dfrac{4}{3}\pi{r}^{3} + 1 \times \dfrac{4}{3}\pi{( 2r )}^{3}}{{a}^{3}} = \dfrac{\dfrac{4}{3}\pi{r}^{3} \times 9}{{( 6r / \sqrt{3} )}^{3}}$
$= \dfrac{4 \times 9 \times \pi \times {r}^{3} \times 3\sqrt{3}}{3 \times 216 \times {r}^{3}} = 90.5$