A steel wire can sustain $$100$$ kg weight without breaking. If the wire is cut into two equal parts, each part can sustain a weight of

Breaking of a wire occurs when the tensile stress in the material reaches its ultimate (breaking) stress. Breaking stress is a material property; it does not depend on the length of the wire but only on the nature of the material and its cross-sectional area.

For the original steel wire:
maximum load $$W_{\max} = 100 \text{ kgf}$$ (where $$1 \text{ kgf}=g \text{ newton}$$).
Corresponding breaking stress is
$$\sigma_{\text{break}} = \frac{W_{\max} \, g}{A},$$
where $$A$$ is the wire’s cross-sectional area.

If the wire is cut into two equal halves, the length becomes $$\tfrac{1}{2}$$ of the original, but the cross-sectional area $$A$$ remains unchanged. Under any load $$W$$ the tensile stress in either half is still $$\sigma = \dfrac{W\,g}{A}$$, the same formula as before.

Because the breaking stress of steel is unchanged and the area is unchanged, the maximum load that produces this critical stress is also unchanged. Hence each half can sustain the same load of $$100 \text{ kgf}$$ before breaking.

Therefore, each part can sustain a weight of 100 kg.

Option C which is: $$100 \text{ kg}$$