SOLUTION: You must buy 100 chickens for exactly $100. You must buy at least one chicken from each store. The first store charges 5 cents/chicken, the second charges $1/chicken and the third

You must buy 100 chickens for exactly $100. You must buy at least one chicken
from each store. The first store charges 5 cents/chicken, the second charges
$1/chicken and the third charges $5/chicken. How many chickens should you buy
from each store?  I know the answer is 80 for .05, 19 for $1, and 1 for$5.  Is
there any mathematical equation you can apply to solve it though, because I
just used guess and check. 

You have the number of chickens for $1 and 5$ reversed.
The answer is 80 for .05, 1 for $1, and 19 for $5.

Yes it can be done by algebra, since we know that the 
solutions must be in positive integers.  

Let x = the number of chickens at  $.05 each
Let y = the number of chickens at $1.00 each
Let z = the number of chickens at $5.00 each

The system is

.05x + 1.00y + 5.00z = 100
           x + y + z = 100

Since both left sides equal to 100, they are equal to 
each other. Setting those equal:

.05x + 1.00y + 5.00z = x + y + z

                  4z = .95x

Multiplying both sides by 100

                400z = 95x

Dividing both sides by 5

                 80z = 19x

Divide through by 80x

                 z/x = 19/80

So the fraction z/x must be the same as or it must reduce to 19/80.

So let z = 19n and x = 80n where n is a positive integer, since only
fractions of the form (19n)/(80n) will be the same as or reduce to 19/80.
Substitute those in:

    x + y + z = 100
80n + y + 19n = 100
      99n + y = 100
            y = 100 - 99n

Since y must be positive, n can only be 1. So

            y = 100 - 99(1)
            y = 1

   z = 19n  and x = 80n 
   z = 19*1 and x = 80*n         
   z = 19   and x = 80   

Edwin