A radioactive element having half-life 30 min . is undergoing beta decay. The fraction of radioactive…

The fraction of radioactive material remaining after time $t$ is determined by the decay formula:

$\frac{N}{N_0} = \left(\frac{1}{2}\right)^{t/T_{1/2}}$

Given a half-life $T_{1/2} = 30\ \text{min}$ and time $t = 90\ \text{min}$, the elapsed time spans $3$ half-lives.

Substituting into the decay law yields:

$\frac{N}{N_0} = \left(\frac{1}{2}\right)^3 = \frac{1}{8}$

The fraction remaining undecayed is therefore $\frac{1}{8}$.