Half-lives of two radioactive elements \(A\) and \(B\) are \(20\) minutes and \(40\) minutes, respectively. Initially, the samples have equal number of nuclei. After \(80\) minutes, the ratio of decayed number of \(A\) and \(B\) nuclei will be

\(\therefore \) Nuclei remaining \(=\frac{\mathrm{N}_{\mathrm{o}}}{2^{4}} .\) Therefore nuclei decayed \(=\mathrm{N}_{0}-\frac{\mathrm{N}_{0}}{2^{4}}\)

For \(\mathrm{B}_{\mathrm{t} / 2}=40 \mathrm{min} ., \mathrm{t}=80 \mathrm{min},\) number of half lifes \(\mathrm{n}=2\)

\(\therefore \) Nuclei remaining \(=\frac{\mathrm{N}_{\mathrm{o}}}{2^{2}} .\) Therefore nuclei decayed \(=\mathrm{N}_{0}-\frac{\mathrm{N}_{\mathrm{o}}}{2^{2}}\)

\(\therefore\) Required ratio\( = \frac{{{\text{No}} - \frac{{{\text{No}}}}{{{2^4}}}}}{{{\text{No}} - \frac{{{\text{No}}}}{{{2^2}}}}} = \) \(\frac{{1 - \frac{1}{{16}}}}{{1 - \frac{1}{4}}} = \frac{{15}}{{16}} \times \frac{4}{3} = \frac{5}{4}\)