$PV = RT \Rightarrow V =\left(\frac{ R }{ P }\right) T$

$V \propto T$ (at constant pressure).

Hence, $\frac{ V _{1}}{ V _{2}}=\frac{ T _{1}}{ T _{2}}$

$\Rightarrow \frac{ V _{2}}{ V _{1}}=\frac{ T _{2}}{ T _{1}}$

where, $V _{2}$ is the final volume.

$\frac{ V _{2}}{ V _{1}}-1=\frac{ T _{2}}{ T _{1}}-1$

$\Rightarrow \frac{ V _{2}- V _{1}}{ V _{1}}=\frac{ T _{2}- T _{1}}{ T _{1}} \quad\left[\because T _{2}- T _{1}=1 K \right]$

$\Rightarrow \frac{ V _{2}- V _{1}}{ V _{1}}=\frac{1}{ T _{1}}=\frac{1}{ T }$