In a sample of radioactive material,what fraction of the initial number of active nuclei will remain undisintegrated after half of a half-life of the sample?

Carbon-$14$ decays with a half-life of about $5,800$ years. In a sample of bone,the ratio of Carbon-$14$ to Carbon-$12$ is found to be $\frac{1}{4}$ of what it is in free air. This bone may belong to a period about $x$ centuries ago,where $x$ is nearest to:

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

Using a nuclear counter, the count rate of emitted particles from a radioactive source is measured. At $t = 0$, it was $1600$ counts per second, and at $t = 8 \, s$, it was $100$ counts per second. The count rate observed, in counts per second, at $t = 6 \, s$ is close to:

The initial activity of a certain radioactive isotope was measured as $16000 \ counts \ min^{-1}$. Given that the only activity measured was due to this isotope and that its activity after $12 \ h$ was $2000 \ counts \ min^{-1}$,its half-life,in hours,is nearest to

Half-life of a radioactive sample is $24 \,h$. If a newly prepared radioactive sample shows $4$ times the allowed and safe value of radioactivity, the minimum time after which one can work safely with the source is (in $\,h$)