Decay constants of two radioactive samples $A$ and $B$ are $15x$ and $3x$ respectively. They have an equal number of initial nuclei. The ratio of the number of nuclei left in $A$ and $B$ after a time $t = \frac{1}{6x}$ is

$A$ and $B$ are two radioactive elements. Their half-lives are $1 \, year$ and $2 \, years$ respectively. Initially,$10 \, g$ of $A$ and $1 \, g$ of $B$ are taken. After how many years will their remaining quantities be equal?

Half-life of a substance is $10$ years. In what time,it becomes $\frac{1}{4}$ of the initial amount?

$A$ radioactive nucleus decays by two different processes. The half-life of the first process is $5$ minutes and that of the second process is $30\,s$. The effective half-life of the nucleus is calculated to be $\frac{\alpha}{11}\,s$. The value of $\alpha$ is $..............$

The half-life of ${ }_{84}^{209} Po$ is $103 \text{ years}$. The time it takes for a $100 \text{ g}$ sample of ${ }_{84}^{209} Po$ to decay to $3.125 \text{ g}$ is

The fossil bone has a ${}^{14}C:{}^{12}C$ ratio,which is $\frac{1}{16}$ of that in a living animal bone. If the half-life of ${}^{14}C$ is $5730 \, years$,then the age of the fossil bone is .......... $years$.