Math CentralQuandaries & Queries


Question from Jessica, a student:

You have exactly $\$100.00$ to spend. You must get 100 animals.
The chicks cost $\$0.10$ each.
The pigs cost $\$2.00$ each.
The sheep cost $\$5.00$ each.
You must get some of each animal.
How many of each animal can you get?

Hi Jessica,

Solving this problem involves some guess and check but a little algebra can reduce the amount of guessing and checking.

Suppose the number of chicks purchased is $C,$ the number of pigs purchased is $p$ and the number of sheep purchased is $s.$ Since the total number of animals purchased is 100 you know that

\[c + p + s = 100.\]

You also know that the total cost was $\$100$ and hence

\[0.1 c + 2 p + 5 s = 100.\]

Multiplying the second equation by $10$ and subtracting the first equation from the result yields

\[19 p + 49 s = 900\]


\[19 p = 900 - 49s.\]

$s$ and $p$ are positive integers and $s$ can't be larger then 18. Do you see why? Now comes the guess and check. Try possible values of $s$ until you find an $s$ so that $900 - 49s$ is a multiple of 19. Does this value of $s$ give you a solution to the problem?


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