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Answer :  5 cows =$50
                 1 pig =$3
                 94 ducks =$47

Explanation:

let number of cows be x
let the number of pigs be y
then the number of chickens = 100-x-y

10x + 3y + .5(100-x-y) < 100
times 2
20x + 6y + 100 – x – y < 200
19x + 5y < 100

consider 19x + 5y = 100

But , what is the question?

Do you want the different combinations you can buy? Do we want as close as possible to $100?

Remember x and y have to be positive integers.
so x has to be between 1 and 5

form a table with columns
x, y, and 100-x-y, and cost

1 16 83 99.50
1 15 82 86.00
…
5 1 94 100

x=5, y=1 is the only combination that will use all the money

He should buy 5 cows, 1 pig and 94 chickens

5+1+94 = 100
$50 + $3 + $47 = $100