An aqueous feed solution of $2000k\frac{g}{h}$ of toluene$-$benzene solution contains $30\mathrm{.}0wt\%$ toluene and is to be extracted in a countercurrent multistage process with pure acetone with a flow rate of $2000k\frac{g}{h}$ to reduce the toluene concentration to $2\mathrm{.}0wt\%$ in the final raffinate. $($Use equilibrium data given below and please use a graph paper for the solution.$)$

$\backslash$table$[[$Weight $\%$ in acetone layer,Weight $\%$ in benzene layer$],[$Acetone$,$Toluene,Benzene,Acetone,Toluene,Benzene$],[98\mathrm{.}45,0,1\mathrm{.}55,2\mathrm{.}12,0,97\mathrm{.}88],[95\mathrm{.}45,2\mathrm{.}85,1\mathrm{.}70,2\mathrm{.}80,1\mathrm{.}87,95\mathrm{.}33],[85\mathrm{.}80,11\mathrm{.}70,2\mathrm{.}50,5\mathrm{.}40,8\mathrm{.}90,85\mathrm{.}70],[75\mathrm{.}70,20\mathrm{.}50,3\mathrm{.}80,9\mathrm{.}20,17\mathrm{.}30,73\mathrm{.}50],[67\mathrm{.}80,26\mathrm{.}20,6\mathrm{.}00,14\mathrm{.}50,24\mathrm{.}60,60\mathrm{.}90],[55\mathrm{.}00,32\mathrm{.}80,12\mathrm{.}20,22\mathrm{.}00,30\mathrm{.}80,47\mathrm{.}20],[31\mathrm{.}00,34\mathrm{.}60,34\mathrm{.}4,31\mathrm{.}00,34\mathrm{.}60,34\mathrm{.}40]]$

a$)$ Calculate the composition of the mixing point, $M.$

b$)$ Calculate the flow rates and compositions of $L_N$ and $V_1.$

c$)$ Calculate the number of theoretical stages required.

d$)$ Please discuss how the number of theoretical stages can be decreased.