The mean lives of a radioactive sample are $30$ years and $60$ years for $\alpha$-emission and $\beta$-emission respectively. If the sample decays by both $\alpha$-emission and $\beta$-emission simultaneously,the time after which only one-fourth of the sample remains is:

$A$ radioactive substance of half-life $138.6 \text{ days}$ is placed in a box. After $n$ days, only $20\%$ of the substance is present. Then the value of $n$ is $[\ln(5) = 1.61]$.

$A$ radioactive substance decays by $10\%$ in $5$ days. What percentage of the original substance will remain after $20$ days (in $\%$)?

$A$ radioactive sample decays by two modes: $\alpha$-decay and $\beta$-decay. $66.6\%$ of the time it decays by $\alpha$-decay and $33.3\%$ of the time it decays by $\beta$-decay. If the effective half-life of the sample is $60 \text{ years}$,what will be the half-life of the sample if it decays only by $\alpha$-decay? (in years)

In a nuclear reactor,the activity of a radioactive substance is $2000 / s$. If the mean life of the products is $50 \text{ minutes}$,then in the steady power generation,the number of radionuclides is:

$A$ radioactive sample $S_{1}$ having the activity $A_{1}$ has twice the number of nuclei as another sample $S_{2}$ of activity $A_{2}$. If $A_{2} = 2 A_{1}$,then the ratio of half-life of $S_{1}$ to the half-life of $S_{2}$ is