A cube A has side thrice as long as that of cube B.

Suppose that the length of the side of cube B is $l \mathrm{~cm}$.

Then, the length of the side of cube $\mathrm{A}$ is $3 \times l \mathrm{~cm}$.

Now, ratio $=\frac{\text { volume of cube } \mathrm{A}}{\text { volume of cube } \mathrm{B}}=\frac{(3 \times l)^{3} \mathrm{~cm}^{3}}{(l)^{3} \mathrm{~cm}^{3}}=\frac{3^{3} \times l^{3}}{l^{3}}=\frac{27}{1}$

$\therefore$ The ratio of the volume of cube $\mathrm{A}$ to the volume of cube $\mathrm{B}$ is $27: 1$.