A mixture of ethanol and water containing $16\%$ mole of ethanol is continuously distilled in a plate$-$fractionating column to give a product containing $77\%$ mole ethanot and a waste of $2\%$ mole ethanol. It is proposed to withdraw $25\%$ of ethanol in the entering stream as a side stream containing $50\%$ mole of ethanol. The equilibrium mole fraction data for ethanol is given below: Determine the number of theoretical plates required and the plate from which side the stream should be withdrawn if the feed is liquid at its boiling point and a reflux ratio of $2$ is used.