$A$ radioactive sample of $U^{238}$ decays to $Pb$ through a process for which the half-life is $4.5 \times 10^9$ years. Find the ratio of the number of nuclei of $Pb$ to $U^{238}$ after a time of $1.5 \times 10^9$ years (given $2^{1/3} = 1.26$).

$A$ radioactive element decays to form a stable nuclide. The graph representing the number of radioactive nuclei $(N)$ versus time $(t)$ is:

$A$ 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)

At time $t = 0$,the mass of a radioactive element sample is $10 \, g$. After a time interval equal to two mean lives,what mass of the sample will approximately remain in $g$?

The half-life of radium is $1600$ years. Its mean life will be ....... years.

What mass in $\mu g$ of $Th^{227}$ is required to produce an activity of $1 \ mCi$ if its half-life is $1.9 \ \text{years}$?