A and B decompose via first order kinetics with half-lives 54 ....

Given $A = {t}_{1/2} = 54min$

$B = {t}_{1/2} = 18min$

$t = 0$ '$x$' $M t = 0$ '$x$' $M$

$⇒$ To calculate : \left[{A}_{t}\right] = 16 \times \left[{B}_{t}\right] time $= ?</p><p>$⇒$For$I$order kinetic : \left[{A}_{t}\right] = \dfrac{{A}_{0}}{{\left(2\right)}^{n}}$

$n \to$ no. of Half lives

$⇒$ Now from the relation $( 1 )$

$⇒ \dfrac{x}{{\left(2\right)}^{{n}_{1}}} = \dfrac{x}{{\left(2\right)}^{{n}_{2}}} \times 16 ⇒ {\left(2\right)}^{{n}_{2}} = {\left(2\right)}^{{n}_{1}} \times {\left(2\right)}^{4}$

$⇒ {n}_{2} = {n}_{1} + 4 ⇒ \dfrac{t}{{\left({t}_{1/2}\right)}_{2}} = \dfrac{t}{{\left({t}_{1/2}\right)}_{1}} + 4$

$⇒ t\left(\dfrac{1}{18} - \dfrac{1}{54}\right) = 4 ⇒ t = \dfrac{4 \times 18 \times 54}{36}$

$⇒ t = 108min$