Radii of A⁺ and that of X^- and Y^- have been given as 

$\begin{array}{l}{\mathrm{A}}^{+}=1.00\mathrm{pm}\\ {\mathrm{X}}^{-}=1.00\mathrm{pm}\\ {\mathrm{Y}}^{-}=2.00\mathrm{pm}\end{array}$

Thus, ratio of volume of AX and AY unit cells is

Ratio of ${\mathrm{r}}^{+}/{\mathrm{r}}^{-}$  will decide the type of unit cell. 

For $A\mathrm{X}, \frac{{\mathrm{r}}_{{\mathrm{A}}^{+}}}{{\mathrm{r}}_{{\mathrm{X}}^{-}}}=\frac{1}{1}=1.0$, hence, AX has bcc structure.

$\therefore 2\left({\mathrm{r}}_{{\mathrm{A}}^{+}}{\mathrm{r}}_{{\mathrm{X}}^{-}}\right)=\mathrm{a}\sqrt{3}$

$\text{a=}\frac{2}{\sqrt{3}}(r_{A^{+}}+r_{X^{-}})$

$=2\times \frac{2}{\sqrt{3}}=\frac{4}{\sqrt{3}}\mathrm{pm}$

Volume $\left(\mathrm{AX}\right)={\mathrm{a}}^3={\left(\frac{4}{\sqrt{3}}\right)}^3=\frac{64}{3\sqrt{3}}$

For$\mathrm{AY}\frac{{\mathrm{r}}_{{\mathrm{A}}^{+}}}{{\mathrm{r}}_{{\mathrm{Y}}^{-}}}=\frac{1}{2}=0.5$

Thus, for octahedral or square planar structure 

$\mathrm{r}=\frac{\mathrm{a}}{2}$

$\therefore \mathrm{a}=2\mathrm{r}=2({\mathrm{r}}_{{\mathrm{A}}^{+}}+{\mathrm{r}}_{{\mathrm{Y}}^{-}})=2(1+2)=6\mathrm{pm}$

Volume $\left(\mathrm{AY}\right)={\mathrm{a}}^3=216{\mathrm{pm}}^3$

$\therefore \text{Ratio=}\frac{\mathrm{Volume}\left(\mathrm{AX}\right)}{\mathrm{Volume} \left(\mathrm{AY}\right)}=\frac{64}{3\sqrt{3}\times 216}=0.057$