Method 1
using the relation
\(\frac{N}{N_{0}}=\left(\frac{1}{2}\right)^{n}\)
where \(n=\) number of half lives
\(n=\frac{t}{t_{1 / 2}}\)
and \(t=\) time \(t >0=36\) days
\(t_{1 / 2}=\) half life \(=9\) days
\(N=\) no of atom at time \(t\)
\(N_{0}=\) original or initial no of atom and \(\frac{N}{N_{0}}=\) fraction that remained
whilst \(1-\frac{N}{N_{0}}=\) fraction decayed
Now
$$
\begin{aligned}
&\frac{N}{N_{0}}=\left(\frac{1}{2}\right)^{\frac{1}{r / 2}}=\left(\frac{1}{2}\right)^{\frac{36}{9}} \\
&\frac{N}{N_{0}}=\left(\frac{1}{2}\right)^{4}=2^{-4} \\
&\frac{N}{N_{0}}=\frac{1}{2^{4}}=\frac{1}{16}
\end{aligned}
$$raction that decayed
\(=1-\frac{N}{N_{0}}\)
\(=1-\frac{1}{16}=\frac{15}{16}\)
method - 2
\(\left(\begin{array}{l} N _{0} \stackrel{9 \text { days }}{\longrightarrow} \frac{N_{0}}{2} \stackrel{9 \text { days }}{\longrightarrow} \frac{N_{0}}{4} \\ \stackrel{9 \text { dyss }}{\longrightarrow} \frac{N_{0}}{8} \stackrel{9 \text { dyys }}{\longrightarrow} \frac{N_{0}}{16},\end{array}\right)\)raction remaining \(=\frac{1}{16}\)
Hence,raction decayed \(=1-\frac{1}{16}=\frac{15}{16}\)
Note: method 2 is preferable when time \(t>0\) is not so large. If however \(t>0\) is very large, then method 2 will not be conservative, so we employ method.