A rancher buys 100 live animals for $100.00. Chickens cost .50 each goats $3.50 each and cows $10.00 each. How many of each did the rancher buy? Is there more than one possiblity?
Changing the animals:
A farmer has $100.00 and he needs to buy 100 animals. Sheep cost $10, goats
>cost $3.50, and chickens cost .50 cents. He has to but at least 1 of each
>animal. How many of each does he buy?
Let s, g and c be the respective numbers of each animal.
Then, s + g + c = 100.
Also, 10s + 3.5g + .5c = 100 or 20s + 7g + c = 200.
Subtracting the first expression from the second yields 19s + 6g = 100.
Dividing through by the lowest coefficient yields g + 3s + s/6 = 16 + 4/6.
(s - 4)/6 must be an integer k making s = 6k + 4.
Substituting back into 19s + 6g = 100 yields g = 4 - 19k.
Clearly, k can only be zero making s = 4, g = 4 and c = 92.
Checking: 10(4) + 3.5(4) + .5(92) = $100.00.
