A mixture of $55$ mole percent benzene with $45$ mole percent toluene is to be distilled in a column to give a top product of $96$ mole percent benzene and a bottom product of $4$ mole percent benzene. The feed enters the column with a molar percent vaporization equal to the distillate$-$to$-$feed ratio.

The equilibrium data at atmospheric pressure are:

$\backslash$table$[[,,,,,,,,,],[x,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}4,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}7,0\mathrm{.}8,0\mathrm{.}9],[y,0\mathrm{.}20,0\mathrm{.}38,0\mathrm{.}51,0\mathrm{.}63,0\mathrm{.}71,0\mathrm{.}78,0\mathrm{.}85,0\mathrm{.}91,0\mathrm{.}96]]$

$($a$)(10\%)$ Using the McCabe and Thiele method, please find the minimum reflux ratio. Please write the equation of top operating line and the equation of q$-$line. Please also draw the top operating line and the q$-$line.

$($b$)(10\%)$ Using the McCabe and Thiele method, please also find the number of plates required using a reflux ratio of $1\mathrm{.}3$ times the minimum reflux ratio. Please write the equation of top operating line. Please also draw the top operating line, the bottom operating line and the q$-$line.