Radioactive decay follows the principle of half-lives, which is the time required for half of the radioactive atoms in a sample to decay. After 200 years, element A, with a half-life of 100 years, will have undergone two half-lives, reducing its initial amount to one-quarter of the original. In contrast, element B, with a half-life of 50 years, will have undergone four half-lives, reducing its initial amount to one-sixteenth, resulting in a ratio of the remaining atoms of A to B of 4:1.