The rate of a first order reaction is 0.08$mol{L}^{-1}$ at 20 min. and 0.06$mol{L}^{-1}$ at 40 min. after start of reaction the half-life of reaction is 

As, $\text{ rate }=\mathrm{k}\left[\mathrm{A}\right];0.08=\mathrm{k}[\mathrm{A}{]}_{10};0.06=\mathrm{k}[\mathrm{A}{]}_{20}$

$\therefore \frac{{\left[A\right]}_{10}}{{\left[A\right]}_{20}}=\frac{0.08}{0.06}=\frac{4}{3}$

For first order reaction: $t=\frac{2.303}{k}\log \frac{{\left[A\right]}_{10}}{{\left[A\right]}_{20}}$ 

When t=(40-20) min; $\therefore 20=\frac{2.303}{k}\log \frac{4}{3}$

$\therefore k=\frac{2.303}{20}\log \frac{4}{3}=0.0144{\min }^{-1}$

$\therefore t_{1/2}=\frac{0.693}{k}\log \frac{0.693}{0.0144}=48.13\min$