Question 1109307: I have to spend $100 to buy 100 animal dog cost $10 cat cost $3 mice cost $.50 I have to use at least one of each. How much animal of each I will need to equal $100
Found 2 solutions by josmiceli, greenestamps:
Answer by josmiceli(19441)   (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
(1)  the total number of dogs, cats, and mice is 100
(2)  the total cost of the dogs ($10 each), cats ($3 each), and mice ($.50 each) is $100
How to solve a problem like this:
.. eliminate one variable to give you one equation in two variables.
.. solve that equation for either variable in terms of the other.
.. use the fact that the variables must have whole number values to find the solution(s)
The way this problem is stated, there should only be one answer. In similar problems, there might be several solutions.
Multiply equation (2) by 2 and subtract equation (1) from the result, eliminating variable c:
(2) 
(1)
(3) 
A little logical analysis at this point can find the answer. So let's do that; but then I will show you the formal algebraic way to finish the problem -- so you can apply the method the next time you see a problem like this.
In equation 3, "5b" is a multiple of 5, because b is a whole number; and 100 is also a multiple of 5. That means "19a" must be a multiple of 5, with a being a whole number. Since 19 is a prime number, that means a must be a multiple of 5. And with the two equations we started with, it is clear that a must be 5.
So a=5; equation (3) then tells us that b is 1; and then equation (1) tells us c is 94.
Answer: 5 dogs, 1 cat, 94 mice.
And here is the formal algebraic method for finishing the problem, starting with equation (3).
 [do this so that (100-20a) is evenly divisible by 5]
In this form of the equation, b and a are whole numbers, so (20-4a) is a whole number; that means (a/5) must be a whole number.
So, as with the logical reasoning we used above, we see that a must be a multiple of 5. And, as before, the conditions of the problem only allow one choice for the value of a: a=5.
And once again (of course), the solution is 5 dogs, 1 cat, and 94 mice.
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