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April 12th, 2014, 11:01 PM   #1
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Word Problem

Ok, this isn't homework. One of my coworkers asked the question. I was able to come up with the answer by trial and error, but could not figure out how to show it in mathematical terms.

Chickens are 2 for a dollar. Pigs are 3 dollars each. Cows are 10 dollars each.

You have to buy exactly 100 animals and spend exactly 100 dollars.
How many of each animal do you buy?

Answer: 5 Cows, 1 Pig, and 94 Chickens

How do I express that in mathematical terms?

Last edited by Darkwing; April 12th, 2014 at 11:08 PM.
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April 13th, 2014, 12:36 AM   #2
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c --> Chickens
p --> Pigs
w --> cows

c/2 + p/3 + w/10 = 100
c+p+w = 100

Last edited by sivela; April 13th, 2014 at 12:44 AM.
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April 13th, 2014, 01:36 AM   #3
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Erm, not quite, the first equation is:
c/2 + 3p + 10w = 100

Otherwise you'd have to sell some pigs or cows
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April 13th, 2014, 03:39 PM   #4
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So it doesn't appear that you can completely solve the equation without plugging in numbers to see what gives you a whole number for ALL animals. That's basically how I solved it the first time, because you have two parameters you have to satisfy, 100 total animals and 100 dollars.
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April 13th, 2014, 08:40 PM   #5
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No, you have two equations for three variables, so there's some trial and error to it. On the other hand, we know that $c$, $p$ and $w$ are all integers. And we can easily see that $0 \le w \le 10$, $0 \le p \le 33$ and so $57 \le c \le 200$. Furthermore, $c$ is even. Some of these are pretty simply to discount too ($w \gt 5$ does't allow us to reach 100 animals, for example).

So we have a good number of constraints that limit our choices. Even better, we can calculate both $c$ and $p$ in terms of $w$, giving a very limited number of choices to verify.
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