Aqueous two$-$phase extraction is used to purify a recombinant Beta$-$galactosidase fusion

peptide produced in Bacteria. For optimum separation, $500$ kg of a mixture of $17\mathrm{.}7\%$ w$/$w

polyethylene glycol $($PEG$)$ and $19\mathrm{.}7\%$ w$/$w potassium phosphate salt in water is needed.

Left over from previous pilot$-$plant trials is $50$ kg mixture of $20\%$ w$/$w PEG in water, and

$200$ kg of a mixture of $20\%$ w$/$w PEG of a and $25\%$ w$/$w salt in water. Also on hand is $200$

kg of an aqueous stock solution of $50\%$ w$/$w PEG, $200$ kg of an aqueous stock solution of

$40\%$ w$/$w salt, and an unlimited supply of extra water. If all of both leftover mixtures must

be used, how much of each stock solution and additional water is required?