# How many apples are there in a string bag if there are more than 11 in a string bag? Find ALL possible solutions.  A shopkeeper gets apples from two suppliers: one delivers apples in boxes, the other in bags. There are always the same number of apples in each of the boxes. This is also true when talking about bags though a box and a bag may contain a diffamount number of apples.Everyday when the shopkeeper gets the apples from the suppliers, he repacks them into string bags so that there are always the same number of apples in each of the string bags. One day, the shopkeeper received 9 boxes and 3 bags. When he repacked the apples into 11 string bags, he had LESS than 97 apples left. Another day, he got 6 boxes and 2 bags. After he repacked them into 8 string bags, he had AT LEAST 56 apples left. How many apples are there in a string bag if there are more than 11 in a string bag? Find ALL possible solutions.   Note that there are LESS THAN 97 apples left, and AT LEAST 56 apples. I don't know to prove that a set of solutions are the only possible solutions, and assistance would be appreciated!

Let us say box contain x number of apple, bag contain y number of apple and string bag contain z number of apple.

In first day shop keeper receives (9x+3y) apples.

When he repacks he use 11 string bags. At that time he had less than 97 apples left without packing in string bags.

So;

[Apples received > apples in string bags +97]

Because we have less than 97 apples left.

9x+3y > 11z+97------(1)

On second day he receives (6x+2y) apples.

When he repack he use 8 string bags. At that time he had lat least 56 apples left without packing in string bags.

So;

[Apples received >= apples in string bags +56]

Because we have at least 56 apples left.

6x+2y >= 8z+56

We can divide this by 2.

3x+2y >= 4z+28------(2)

(2)*3

9x+3y >= 12z+84------(3)

Now we have two expressions for (9x+3y). (1) is absolutely greater than (9x+3y) but (3) is greater than or equal to (9x+3y).

So we can be sure that (1)>=(3)

11z+97>=12z+84

13 >= z

So in each string bag we can have apples less than or equal to 13. It is given that a string bag contains more than 11 apples.

So z = 12 or z = 13

So a string bag contains either 12 apples or 13 apples.

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