$21-2$ In "cold$-$shot$-$cooling" or "quench" operation of a three$-$stage fixed$-$bed catalytic reactor $10$r an exothermic reaction $($see Figure $21\mathrm{.}11),$ temperature is controlled by splitting the feed to the various stages. For the reaction $A$ products, suppose the feed of pure $A$ is divided so that $45\%$ enters the first stage and $30\%$ enters the second stage, that is$,$ joining the outlet from the first stage to form the feed to the second. No reaction occurs between stages, and $f_A$ is based on original feed.

$($a$)$ If the fraction of A converted in the stream leaving the first stage $(f_{Al})$ is $0\mathrm{.}40,$ what is the fraction of A converted $(f_{AO2})$ in the feed entering the second stage?

$($b$)$ If $f_{A2}=0\mathrm{.}60,$ what is $f_{A03}?$