1. **Problem Statement:** We are given a radioactive substance with a half-life of 3 months. Starting with 50 grams, we want to find how much remains after 12 months. 2. **Formula:** The amount remaining after time $t$ can be calculated using the half-life decay formula: $$\text{Remaining Amount} = I \times \left(\frac{1}{2}\right)^{\frac{t}{T}}$$ where $I$ is the initial amount, $t$ is the elapsed time, and $T$ is the half-life. 3. **Given values:** - Initial amount $I = 50$ grams - Half-life $T = 3$ months - Time elapsed $t = 12$ months 4. **Calculate the number of half-lives:** $$\frac{t}{T} = \frac{12}{3} = 4$$ 5. **Calculate remaining amount:** $$\text{Remaining Amount} = 50 \times \left(\frac{1}{2}\right)^4 = 50 \times \frac{1}{16} = 3.125$$ 6. **Interpretation:** After 12 months, 3.125 grams of the substance remain. This shows exponential decay where the substance halves every 3 months, so after 4 half-lives (12 months), only $\frac{1}{16}$ of the original amount remains.