A ranch owner has 18 bales of hay to distribute in 3 cow pastures and 3 horse pastures. He wants each cow pasture to have the same number of bales and each horse pasture to have the same number of bales. He wants at least 1 bale in each pasture. How many different ways can he distribute the hay among the pastures. make a table to show your reasoning.
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He can place three bales in three cow pastures and three in three horse pastures. He can place two bales in three cow pastures and and four in each of the three horse pastures. He can place one in each of the cow pastures and 5 in each of the horse pastures. In other words, as you go down from three to one for the cow pastures, the horse pastures remain consistent with the specifications. Building up, four also works because it leaves six or two for each of the horse pastures. Likewise, five works because it leaves one for each of the horse pastures--but nothing higher than five for each cow pasture can work. We can now follow the same logic for the horse pastures, moving from five to one. It all works out the same because the condition are the same. As such, your table would feature I-5 on the left side of a square, 1-5 on the bottom side of the square, and cow pastures on top with horse pastures on the right. This is an exact description of what the table would look like.
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Oct 1st, 2015
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