$A$ radioactive material has a half-life of $10$ days. What fraction of the material would remain after $30$ days?

$A$ sample of radioactive material has mass $m$,decay constant $\lambda$,and molar mass $M$. If the Avogadro constant is $N_A$,what is the initial activity of the sample?

If the half-life of a radioactive nucleus is $3$ days,nearly what fraction of the initial number of nuclei will decay on the third day? (Given,$\sqrt[3]{0.25} \approx 0.63$)

The half-life of a radioactive sample is $5 \,s$. If the initial mass of the sample is $60 \,g$, then the time required to reduce the sample to $7.5 \,g$ is (in $\,s$)

Two radioactive materials $X_1$ and $X_2$ have decay constants $5 \lambda$ and $\lambda$ respectively. Initially,they have the same number of nuclei. After time $t$,the ratio of the number of nuclei of $X_1$ to that of $X_2$ is $\frac{1}{e}$. Then $t$ is equal to:

$A$ radioactive sample consists of two distinct species having an equal number of $N_0$ atoms initially. The mean-life of one species is $\tau$ and of the other is $5\tau$. The decay products in both cases are stable. The total number of radioactive nuclei at $t = 5\tau$ is