A particle is executing simple harmonic motion (SHM). The ratio of potential energy and kinetic energy of the particle when its displacement is half of its amplitude will be

$$U = \frac{1}{2}kx^2$$

$$K = \frac{1}{2}k(A^2 - x^2)$$

Given displacement $$x = \frac{A}{2}$$

$$U = \frac{1}{2}k\left(\frac{A}{2}\right)^2 = \frac{1}{2}k\left(\frac{A^2}{4}\right) = \frac{1}{8}kA^2$$

$$K = \frac{1}{2}k\left(A^2 - \left(\frac{A}{2}\right)^2\right) = \frac{1}{2}k\left(A^2 - \frac{A^2}{4}\right) = \frac{1}{2}k\left(\frac{3A^2}{4}\right) = \frac{3}{8}kA^2$$

$$\frac{U}{K} = \frac{\frac{1}{8}kA^2}{\frac{3}{8}kA^2} = \frac{1}{3}$$