Question No 65.By giving 2 bananas free with every 5 apples bought, a shopkeeper makes a profit of 25%. However, by giving 5 bananas free with every 2 apples bought, he suffers a loss of 25%. What is the maximum number of bananas he can give away with 8 apples and still make a profit?

The problem can be solved by first understanding the cost price and selling price of the apples and bananas.

1. When the shopkeeper gives 2 bananas free with every 5 apples bought, he makes a profit of 25%. This means that the selling price of 5 apples is equal to the cost price of 5 apples and 2 bananas, and this selling price is 125% of the cost price of 5 apples. Let's denote the cost price of an apple as A and the cost price of a banana as B. We can write this as:

   5A = 1.25 * (5A + 2B)

   Solving this equation gives us B = 0.5A.

2. When the shopkeeper gives 5 bananas free with every 2 apples bought, he suffers a loss of 25%. This means that the selling price of 2 apples is equal to the cost price of 2 apples and 5 bananas, and this selling price is 75% of the cost price of 2 apples. We can write this as:

   2A = 0.75 * (2A + 5B)

   Substituting B = 0.5A from the first equation into this equation gives us A = 2B.

From these two equations, we can see that the cost price of an apple is twice the cost price of a banana.

3. Now, we want to find out the maximum number of bananas the shopkeeper can give away with 8 apples and still make a profit. This means that the selling price of 8 apples should be more than the cost price of 8 apples and the given number of bananas. We can write this as:

   8A > 8A + nB

   where n is the number of bananas. Substituting A = 2B into this equation gives us:

   8 * 2B > 8 * 2B + nB

   Simplifying this equation gives us n < 16.

Therefore, the maximum number of bananas the shopkeeper can give away with 8 apples and still make a profit is 15.