If we increase the side length of a cube by 3 times, the new side length will be three times the original length. Let's assume the original side length of the cube is "x". 

After increasing the side length by 3 times, the new side length becomes 3x. 

The volume of a cube is calculated by cubing its side length. 

Original volume = x^3
New volume = (3x)^3 = 27x^3

To determine how many times the volume increased, we can divide the new volume by the original volume:

(27x^3) / (x^3) = 27

Therefore, the volume of the cube is increased by 27 times when the side length is increased by 3 times.