The half-life of a radioactive nucleus is 5 years. The fraction of...

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

$n = \dfrac{t}{{T}_{\dfrac{1}{2}}} . . . \left(1\right)$

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

$\begin{cases}n & = & \dfrac{15years}{5years} \\ & = & 3\end{cases}$

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.

$\begin{cases}N & = & {N}_{0}{2}^{- 3} \\ & ⇒ & \dfrac{N}{{N}_{0}} = \dfrac{1}{8}\end{cases}$

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

$\begin{cases}1 - \dfrac{N}{{N}_{0}} & = & 1 - \dfrac{1}{8} \\ & = & \dfrac{7}{8}\end{cases}$