The half-life of a radioactive nuclide is $100$ hours. The fraction of original activity that will remain after $150$ hours would be :

Given,

${\mathrm{T}}_{\frac{1}{2}}=100 \mathrm{hours}$

$\mathrm{t}=150 \mathrm{hours}$

As we know that the number of nuclide remain after $n$ half life is given by,

 $N=\frac{N_0}{2^n}$

$\because 100$ hours $=1$ half life

$150$ hours$=\frac{1}{100}\times 150$

$=\frac{3}{2}$ half life

So, Fraction remain after $\frac{3}{2}$ half life

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