Each side length of Cube A is $4 \mathrm{~cm}$ long. Each side length of Cube $B$ is triple the side length of Cube $A$. How many times greater is the volume of Cube $B$ than that of Cube A?

Step 1 ：Let the side length of Cube A be \(a\) and the side length of Cube B be \(b\). Given that \(a = 4\) cm and \(b = 3a = 12\) cm.

Step 2 ：Find the volume of Cube A: \(V_A = a^3 = 4^3 = 64\) cubic cm.

Step 3 ：Find the volume of Cube B: \(V_B = b^3 = 12^3 = 1728\) cubic cm.

Step 4 ：Find the ratio of the volumes: \(\frac{V_B}{V_A} = \frac{1728}{64} = 27\).

Step 5 ：\(\boxed{27}\): The volume of Cube B is 27 times greater than that of Cube A.