A $100$ mol$/$h liquid feed containing $57\%-$pentane and $43\%-$hexane by mole will be distilled in a distillation column. Calculate the top and bottom product flow rates and their compositions for the following cases:

a$)$ If there is $93\%$ n$-$pentane in the top product and $92\%-$hexane in the bottom product

b$)$ If $91\%$ of the n$-$pentane in the feed is recovered in the top product and $89\%$ of the $-$hexane in the feed is recovered in the bottom product

c$)$ If $90\%$ of the n$-$pentane in the feed is recovered and $-$pentane in the top product is $97\%$

d$)$ If the bottom product flow rate is $60$ mol$/$h and $97\%$ of the $-$hexane in the feed is recovered in the bottom product