April 12th, 2014, 11:01 PM  #1 
Newbie Joined: Apr 2014 From: Texas Posts: 2 Thanks: 0  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. 
April 13th, 2014, 12:36 AM  #2 
Senior Member Joined: Jan 2009 Posts: 344 Thanks: 3 
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. 
April 13th, 2014, 01:36 AM  #3 
Senior Member Joined: Apr 2014 From: UK Posts: 918 Thanks: 331 
Erm, not quite, the first equation is: c/2 + 3p + 10w = 100 Otherwise you'd have to sell some pigs or cows 
April 13th, 2014, 03:39 PM  #4 
Newbie Joined: Apr 2014 From: Texas Posts: 2 Thanks: 0 
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.

April 13th, 2014, 08:40 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra 
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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