Final answer:
The specimen is 300 million years old, calculated based on the number of parent and daughter atoms present, applying the half-life decay principle of the radioactive isotope with a half-life of 100 million years.
Step-by-step explanation:
To determine the age of the mineral specimen based on the half-life of its radioactive isotope, we rely on an understanding of the decay process. In this case, the specimen contains four atoms of the parent isotope (the radioactive atoms) for every twelve atoms of the daughter isotope (the decay products). Following the concept that, with each half-life, the number of parent atoms reduces by half, we can calculate the number of half-life intervals that have passed to understand the age of the specimen.
Starting with four parent atoms, after one half-life (100 million years), there would be two parent atoms. After two half-lives (200 million years), there would be one parent atom. Given that there are currently four parent atoms and twelve daughter atoms, this indicates that three half-lives have occurred because starting with 1 parent atom (after two half-lives), the remaining one must have decayed into 2 atoms, and then those 2 decayed to the current 4, which means we had (1->2->4) multiplying by 2 each half-life: 100 million to 200 million, then 200 million to 400 million years ago. As there are three intervals (100 million years each), the mineral specimen is thus 300 million years old.