The fraction remaining of a radioactive substance is the ratio of the final amount ($N$) to the initial amount ($N_0$). It can be calculated using the formula: $$\text{Fraction remaining} = \frac{N}{N_0} = \left(\frac{1}{2}\right)^{n}$$ where: • $N$ is the final amount of the substance remaining. • $N_0$ is the initial amount of the substance. • $n$ is the number of half-lives that have passed, calculated as $n = \frac{t}{T_{1/2}}$. • $t$ is the total time elapsed. • $T_{1/2}$ is the half-life of the substance. Example: A radioactive sample has a half-life of $5$ years. What fraction of the original sample will remain after $15$ years? Step 1: Determine the number of half-lives ($n$) that have passed. $$n = \frac{t}{T_{1/2}}$$ $$n = \frac{15 \text{ years}}{5 \text{ years/half-life}} = 3 \text{ half-lives}$$ Step 2: Use the formula for the fraction remaining. $$\text{Fraction remaining} = \left(\frac{1}{2}\right)^{n}$$ $$\text{Fraction remaining} = \left(\frac{1}{2}\right)^{3}$$ $$\text{Fraction remaining} = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}$$ $$\text{Fraction remaining} = \frac{1}{8}$$ After $15$ years, $\boxed{\frac{1}{8}}$ of the original sample will remain.