Physics

The key is tracking how many nuclei remain after each half-life, then subtracting from the initial amount. Element A decays twice as fast (20-min half-life vs. 40-min), so after 80 minutes, A completes 4 half-lives while B completes only 2. Use $N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$ to find survivors, then the number decayed is $N_0 - N(t)$ for each element—the ratio of decayed nuclei follows directly.

The key is tracking how many nuclei remain after each half-life, then subtracting from the initial amount. Element A decays twice as fast (20-min half-life vs. 40-min), so after 80 minutes, A completes 4 half-lives while B completes only 2. Use $N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$ to find survivors, then the number decayed is $N_0 - N(t)$ for each element—the ratio of decayed nuclei follows directly.