The half-life of a radioactive nucleus is $5$ years. The fraction of the original sample that would decay in $15$ years is :

The formula to calculate the number of half lives covered by the radioactive material is given by

$n=\frac{t}{T_{{\displaystyle \frac{1}{2}}}} ...\left(1\right)$

Substitute the values of the known parameters into equation (1) to calculate the number of half lives covered.

$\begin{array}{rcl}n& =& \frac{15 \mathrm{years}}{5 \mathrm{years}}\\ & =& 3\end{array}$

The relation between the amount of radioactive substance after certain time can be written as

$N=N_0{2}^{-n} ...\left(2\right)$

Substitute the value of $n$ into equation (2) to obtain the fraction of the remaining substance.

$\begin{array}{rcl}N& =& N_0{2}^{-3}\\ & \Rightarrow & \frac{N}{N_0}=\frac{1}{8}\end{array}$

Hence, the amount of the decayed substance can be written as

$\begin{array}{rcl}1-\frac{N}{N_0}& =& 1-\frac{1}{8}\\ & =& \frac{7}{8}\end{array}$