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Resistance of a wire is given by $R=\frac{\rho L}{A}$, where, $\rho$ is conductivity, $L$ is length of wire and $A$ is cross-sectional area of wire.

For a single wire, $A$ and $\rho$ are constant.

Thus, for parts $X$ and $Y$, resistance, $\frac{R_X}{R_Y}=\frac{l_X}{l_Y}$.

When wire is stretched to double of its length, then resistance becomes four times, thus, $R_W=4R_X=2R_Y$.

Hence, the ratio is $\frac{R_X}{R_Y}=\frac{1}{2}$.

So,  $\frac{l_{\mathrm{X}}}{l_{\mathrm{Y}}}=\frac{1}{2}$