Farmer Buying Livestock

Date: 9/12/96 at 18:26:47
From: Anonymous
Subject: Farmer Buying Livestock

This problem has me stumped. 

A farmer spends $100 total to get 100 animals. The cost of cows is 
$10, pigs $3, and chickens $0.50. The farmer buys at least one of each 
animal.

I have gotten this far:

100 = 10X + 3Y +.5Z
100 = X + Y + Z

It's that third variable that is confusing. Also, do you have any
general tips for solving problems with more than two variables?

Thanks.

Date: 9/14/96 at 17:14:29
From: Doctor Ana
Subject: Re: Farmer Buying Livestock

Usually, we need as many equations as we have variables, but there is 
a solution to this problem. 

First of all, let's solve the second equation for z. We get 
z = 100 - x - y. This makes sense because if we know how many cows 
and pigs we are going to buy, we can figure out how many chickens we 
are going to buy by subtracting the number of cows and pigs from 100. 

The next part gets a little trickier. Because we only have 2 equations 
and 3 unknowns, there are going to be lots and lots of answers to the
2 equations that are MATHEMATICALLY correct. But there is only going 
to be 1 solution that will make sense, and you'll see why. 

Let's substitute 100-x-y  for z in the first equation. This means that 
we are going to replace z with something that has the same value.

This gives us

100 = 10x + 3y + .5(100 - x - y)

We'll come back to z (chickens) once we know what x and y are. Let's 
simplify the equation.

100 = 10x + 3y + 50 - .5x - .5y
(you can finish simplifying this on your own)

and then let's solve the equation for y. I'm going to show you the 
equation solved for y, but I want you to simplify the equation and 
solve for y on your own, too.

y = 20 - 3.8x

Now, what we are going to do is to start guessing values for x and see 
what sort of values we get for y. Because these are animals, we will 
need to have integers for our answers. (The farmer can't buy half a 
cow!) Why don't you finish the table that I started below.

x   y
-   -
0   20
1   16.2
2   12.4
3
4
5
6
7
8
9
 
Now, how many of these answers make sense? Is there just one? Could 
you keep going with more and more cows and eventually find another 
number that worked?

When you think you know how many cows and pigs you will buy, find out 
how many chickens you will buy. When you're done, be sure that your 
answers make sense and be sure that you can explain how you got your 
answer. If you have any more questions please write back.

-Doctor Ana,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/