Strain energy is (1/2) force times extension for Hooke's law.

Steps:

• Strain energy U = (1/2) F δ, where δ is extension.
• For string X: δ = x, so U_X = (1/2) F x.
• For string Y: δ = 2x, so U_Y = (1/2) F (2x) = F x.
• Ratio U_X / U_Y = [(1/2) F x] / [F x] = 1/2.

Why C is correct:

• The formula U = (1/2) F δ directly gives the ratio of extensions as 1/2 since F is the same.

Why the others are wrong:

• A: Assumes energy proportional to extension squared with same stiffness, ignoring different stiffnesses.
• B: Would imply equal energies, but Y stores twice as much due to larger extension.
• D: Results from wrongly using U = (1/2) k δ² assuming identical k, but k differs (stiffer for X).

Final answer: C