Factor of safety is a term used in engineering to determine the safety margin of a structure or component. It is defined as the ratio of the ultimate stress (also known as the breaking stress) to the working stress (also known as the allowable stress).

The correct answer is option 'D', which states that the factor of safety is the ratio of ultimate stress to working stress. This means that the factor of safety is determined by dividing the maximum stress a material can withstand (ultimate stress) by the stress that the material is allowed to experience during normal operation (working stress).

Here's an explanation of each option listed and why option 'D' is the correct answer:

a) Breaking stress to working stress:
The breaking stress refers to the maximum stress a material can withstand before it fails or breaks. The working stress, on the other hand, is the stress that the material is subjected to during normal operation. While the breaking stress is an important parameter, it does not directly relate to the safety margin of a structure. Therefore, option 'A' is not the correct answer.

b) Endurance limit to yield stress:
The endurance limit is the maximum stress a material can withstand for an infinite number of cycles without failing due to fatigue. The yield stress, on the other hand, is the stress at which a material begins to deform permanently. While both parameters are important in material design, they do not directly relate to the safety margin of a structure. Therefore, option 'B' is not the correct answer.

c) Elastic limit to ultimate stress:
The elastic limit is the maximum stress a material can withstand without undergoing permanent deformation. It represents the boundary between elastic behavior (where the material returns to its original shape after the stress is removed) and plastic behavior (where the material undergoes permanent deformation). The ultimate stress, as mentioned earlier, is the maximum stress a material can withstand before it fails or breaks. While both parameters are important in material design, they do not directly relate to the safety margin of a structure. Therefore, option 'C' is not the correct answer.

d) Ultimate stress to working stress:
As explained earlier, the factor of safety is the ratio of ultimate stress to working stress. The ultimate stress represents the maximum stress a material can withstand before it fails or breaks, while the working stress represents the stress that the material is allowed to experience during normal operation. By dividing the ultimate stress by the working stress, engineers can determine the safety margin of a structure or component. A higher factor of safety indicates a greater safety margin, meaning the structure or component is designed to withstand higher loads or stresses than it typically encounters during normal operation. Therefore, option 'D' is the correct answer.