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Question 73

My son adores chocolates. He likes biscuits too, but he hates apples. I told him that he could buy as many chocolates as he wanted, but for every chocolate he bought, he had to buy twice as many biscuits. He also had to buy more apples than the total number of chocolates and biscuits together. Each chocolate costs ₹1, an apple costs twice as much as a chocolate, and four biscuits cost the same as one apple. If the numbers of chocolates, biscuits and apples bought are all positive integers, which of the following could be the total amount I spent on him that evening?

Let the number of chocolates be 'x'. Then the number of biscuits is '2x'. Since the number of apples must be greater than the total number of chocolates and biscuits, $$\text{apples}>x+2x=3x.$$

Also, each chocolate costs ₹1, each apple costs ₹2, and four biscuits cost the same as one apple. Since one apple costs ₹2, $$4\text{ biscuits}=₹2$$ so each biscuit costs ₹0.50.

Therefore, if the number of apples is a, the total amount spent is $$x+2x(0.5)+2a =2x+2a.$$

Now, we evaluate each option and check its validity. 

Option A: Since a>3x, the smallest possible value of a is (3x+1). Hence, $$\text{Total cost}>2x+2(3x+1)=8x+2.$$

Taking (x=4). Then the number of biscuits is 8, and the number of apples must be greater than 12. Taking 13 apples gives $$ 4+8(0.5)+13(2) =4+4+26 =\boxed{₹34}. $$ Hence, 34 is a valid amount spent.  

Option B: For ₹33, the total cost (2x+2a) is necessarily even, so ₹33 is impossible.

Option C: For (x=1), the minimum possible expenditure is 8(1)+2=₹10, so ₹8 is also not possible.

Hence, the correct option is option A.

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