Linda receives utility from buying clothes and grocery items (At the JBS Grocery shop). Her budget is

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Question:

Linda receives utility from buying clothes and grocery items (At the JBS Grocery shop). Her budget is $304. The JBS shop is offering a $16 coupon if the amount spent on groceries is equal or greater than $100. Let Y be the quantity of units of clothes consumed with a price of $1 per unit and X the quantity of grocery items consumed with a price of $2 per unit/item. Linda's utility function is: U (X,Y) = 3/4LnY + 1/4 LnX

1) Find the optimal allocation of groceries (X) and clothes (Y) for Linda if her budget is $304.

2) Find the optimal allocation of groceries (X) and clothes (Y) if Linda's budget is $320. (NOTE***Here i do understand how the answer is x= 40 units of groceries and y=240 units of clothing BUT i do not understand how it will look graphically because at 40units of groceries means Linda spent 40units*2$/unit= 80$ but then how did she get the 16$ coupon if she spent 80$ and not a 100$. It doesn't make sense to me?

3) Based on your answers of part a and b, Could you tell what would Linda's optimal allocation be if she receives the offer from ABC shop? (Note*** Here I do not understand why the answer is x=40 and y=240 is it because linda is in theory richer?)

4) Could Linda's utility be higher if she chooses to buy 50 grocery items ? Could you explain your answer drawing an indifference curve? Would you consider this result a corner solution? (Note*** I think her utility stays the same but then again i do not know why. Would really appreciate some explanation. Please also let me know if this is a corner solution!)