To understand why the correct answer is option 'C', let's first review what half-life means in the context of radioactive decay.

Half-life is the time it takes for half of the radioactive substance to decay. In other words, after one half-life, only half of the original substance remains. After two half-lives, only one-fourth of the original substance remains, and so on.

Now let's analyze each option to determine which one is correct.

a) 12 months:
If the substance has a half-life of four months, after 12 months (3 times the half-life), the substance would have gone through three half-lives. In each half-life, only half of the substance decays. Therefore, after three half-lives, only 1/8 (1/2 * 1/2 * 1/2) of the substance would remain. This is not equal to three-fourths (3/4), so option 'a' is incorrect.

b) 4 months:
After four months, the substance would have gone through one half-life. As mentioned earlier, after one half-life, only half of the substance remains. Therefore, option 'b' is also incorrect.

c) 8 months:
After eight months (two times the half-life), the substance would have gone through two half-lives. In each half-life, only half of the substance decays. Therefore, after two half-lives, only one-fourth (1/2 * 1/2) of the substance would remain. This matches the three-fourths (3/4) mentioned in the question, so option 'c' is the correct answer.

d) 3 months:
After three months, the substance has not yet completed one half-life. Therefore, option 'd' is also incorrect.

In conclusion, the correct answer is option 'c' because after eight months, the substance will have decayed to three-fourths (1/4) of its original amount, which matches the given information.