$11\mathrm{.}15$ Enzyme purification using two$-$phase aqueous partitioning

Leucine dehydrogenase is recovered from $150$ litres of Bacillus cereus homogenate using an aqueous two$-$phase polyethylene glycol$-$salt system. The homogenate initially contains $3\mathrm{.}2$ units of enzyme $m{l}^{-1}.$ A polyethylene glycol$-$salt mixture is added and two phases form. The enzyme partition coefficient is $3\mathrm{.}5.$

$($a$)$ What volume ratio of upper and lower phases must be chosen to achieve $80\%$ recovery of enzyme in a single extraction step?

$($b$)$ If the volume of the lower phase is $100$ litres, what is the concentration factor for $80\%$ recovery?