# a farmer bought chickens, goats and sheep. the cost of a chicken is .50

goat is \$3.50 and sheep is \$10
He bought 100 animals and spent \$100. how many of each animal did he buy

Since you have two equations and three unknowns, there are many combinations of c (the number of chickens), g (the number of goats) and s (the number of sheep) that meet the requirements
c + g + s = 100 (total number)
0.5 c + 3.5 g + 10 s = 100 (total cost)

However, the only acceptable solutions are the ones for which c, g and s are all integers. I had to resort to trial and error to make that happen. g = 0, 1 , and 3 did not work. However, if you choose g = 4, then
c + 4 + s = 100
0.5 g + 14 + 10 s = 100
10 c + 40 + 10 s = 1000
9.5 c + 26 = 900
9.5 c = 874
c = 92 ; s = 100 -4 -92 = 4 ; g = 4

There may be other solutions as well.

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.

Question ID
321

Created
August 19, 2005 1:11am UTC

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