You compress a spring by a certain distance. then you decide to compress it further so that its elastic potential energy increases by another 50%

You can determine the percent increase in the spring's compression distance.

The elastic potential energy of a spring is given by:

PE = [1/2] k * X^2, where X is the deformation of the spring (compression is this case).

Then if the initial deformation is A:

Its PE is [1/2] k * A^2

And the increased potential energy is PE' = 1.5 * PE = 1.5*[1/2] k*A^2.

If you call B the new deformation (compression distance), the new potential energy is PE' = [1/2] k * B^2

From which you can equal: 1.5*[1/2] k * A^2 = [1/2] k * B^2

You can cancell the terms that appear in both sides =>

1.5 A^2 = B^2 => (B/A)^2 = 1.5 => B/A = √1.5 = 1.225

Which means that the compression distance increased 22.5%