# Half-lives of two radioactive nuclei A and B are 10 minutes and 20 minutes, respectively. If, initially a sample has an equal number of nuclei, then after 60 minutes, the ratio of decayed numbers of nuclei A and B will be: Option 1) 3 : 8 Option 2) 1 : 8 Option 3) 8 : 1 Option 4) 9 : 8

Number of nuclei in terms of half life -

$N=\frac{N_{0}}{2^{t/t_{1/2}}}$

- wherein

Very useful to determine number of nuclei in terms of half life

 A B $t_{1/2}$ 10 m 20 m t 60 m 60 m =6 $t_{1/2}$ =3 $t_{1/2}$

after 60 minutes.

undecay $N_{A}=\frac{N_{0}}{2^{t/t_{1/2}}}=\frac{N_{0}}{2^{6}}=N_{B}=\frac{N_{0}}{2^{3}}$

undecay  $N^{1}_{A}=N_{0}-\frac{N_{0}}{2^{6}}=\frac{63N_{0}}{64}$

$N^{1}_{B}=N_{0}-\frac{N_{0}}{8}=\frac{7N_{0}}{8}$

So,  $\frac{N^{1}_{A}}{N^{1}_{B}}=\frac{63}{64}\times \frac{8}{7}=\frac{9}{8}$

$\Rightarrow N_{A}^{1}:N_{B}^{1}=9:8$

Option 1)

3 : 8

Option 2)

1 : 8

Option 3)

8 : 1

Option 4)

9 : 8

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