Problema Solution
Angelia has several bottles. Of them, 5 are full, 5 are half full, and 5 are empty. She wants to divide these bottles among 3 boxes that each box holds 5 bottles and she wants each box to weigh the same. How would she divide the bottles? How many different ways can you find?
Answer provided by our tutors
let x represent the weight of a bottle and let 'y' represent the weight of its contents when half-full, therefore '2y' represents the weight of the contents when full
the weights of the empty, half-full and full bottles are then x, x+y and x+2y, respectively
the average weight of each bottle is:
(5x + 5(x+y) + 5(x+2y))/15
which simplifies to:
(3x+3y)/3 = x+y
the total weight of each box should be 5x+5y, which is '5 bottles' plus '5 half-contents', or '2 1/2 contents'
so no box has more than two full bottles and no box has any empty bottles, one box must therefore have 3 half-filled bottles and at least one box must have two full bottles
box A: x+2y, x+2y, x+y, x, x
box B: x+y, x+y, x+y, x+2y, x
box C: x+2y, x+2y, x+y, x, x
two of the boxes should have two full bottles and one half bottle, the remaining box should have three half bottles and one full bottle, all remaining slots are filled with empty bottles