The three children of a farm couple are sent to the market to sell 90 apples. Karen, the oldest, carries 50 apples; Bill, the middle one, carries 30; and John, the youngest, carries only 10. The parents have stipulated five rules: 

(a) The selling price is either $1 for 7 apples or $3 for 1 apple, or a combination of the two prices. 

(b) Each child may exercise one or both options of the selling price. 

(c) Each of the three children must return with exactly the same amount of money. 

(d) Each child’s income must be in whole dollars (no cents allowed). 

(e) The amount received by each child must be the largest possible under the stipulated conditions. Given that the three kids are able to sell all they have, use ILP to show how they can satisfy the parents’ conditions.