A 1m long wire is broken into two unequal parts X and Y . The X...

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

For a single wire, $A$ and ρ are constant.

Thus, for parts $X$ and $Y$, resistance, $\dfrac{{R}_{X}}{{R}_{Y}} = \dfrac{{l}_{X}}{{l}_{Y}}$.

When wire is stretched to double of its length, then resistance becomes four times, thus, ${R}_{W} = 4{R}_{X} = 2{R}_{Y}$.

Hence, the ratio is $\dfrac{{R}_{X}}{{R}_{Y}} = \dfrac{1}{2}$.

So, $\dfrac{{l}_{X}}{{l}_{Y}} = \dfrac{1}{2}$