Mixture of mixtures. I am formulating this problem for work, so it is important that I get it right.  As of right now I am only considering the case where the number of chemicals is equal to the number of pre-made mixtures.
Pretend that Mr. Smith has three chemical blends on hand:  Blend A, Blend B, and Blend C.  Let c1, c2, c3 be chemical 1, chemical 2, chemical 3, repectively.  Blend A is made up of 10 percent c1, 34 percent c2, the rest plain water.  Blend B is made up of 9 percent c1, 18 percent c2, 9 percent c3, the rest plain water.  Blend C is made up of 28 percent c1, the rest plain water.  What proportions of Blend A, Blend B, and Blend C should Mr. Smith mix together to create a new blend, Blend D, that will have 8 percent c1, 12 percent c2, 4 percent c3, the rest plain water.
My idea was to set up a system of linear equations
\begin{align*}
10A+9B+28C&=8\\
34A+18B&=12\\
9B&=4
\end{align*}
Then solve for A,B,C.  When I did this, I got approximately $A=.12$, $B=.44$ and $C=.10$.  I interpreted this to mean that Mr. Smith should mix 12 parts A, 44 parts B, and 10 parts C and 34 parts just water to achieve the desired blend.
Here are my questions:
1) Is this method correct?  If not I need to get it right, as soon as possible, and I would appreciate your help.
2) What should I do if the number of blends does not equal the number of chemicals.  Then my system of equations would not be solvable?
3) When would a solution like the one above not be feasible?  Obviously if we get a negative number, but what if the sum of A,B,C is greater than 1?  Is this bad?
4) Approximations to this problem are good enough.  That is, I can tweak Mr. Smith's requirements somewhat if it will get me a feasible solution.  Are there any methods that can make use of this fact?  Thank you.
 A: *

*Your set up is mostly correct, except that you asked to mix the blend A,B,C, and your equations are really for mixing A,B,C, but after you solve them, you said you are going to mix chemicals $c_1, c_2,c_3$. I suppose you really want to mix the blend $A,B,C$, otherwise there are nothing to solve. So I will assume the variables to be $A,B,C$.

*Your equation actually implicitly assumed that $A+B+C=1$, because you talked about proportions.

*Here then comes the complicated explanation. If there are more blends than chemicals, you can get one or even more solutions, which are fine, just choose one proportion to mix them. If in your case, equal or less blends than chemicals, it seems it does not work, unless the proportions happen to fit that of the blends. Or you might consider adding water into it to make it fit.

*For approximation, it of course comes from the cases where the solution does not exist. You may consider least square approximation. But the solution might not be good enough.  
An extreme case example for you to see what happens when the sum is bigger than $1$. Suppose we have blends $A,B$, chemicals $c_1,c_2$. Blend $A$ contains $10\%$ $c_1$ and $20\%$ $c_2$; blend $B$ contains $5\%$ $c_1$ and $5\%$ $c_2$. Suppose we want to create blend $D$ that has $30\%$ $c_1$ and $40\%$ $c_2$. You can see why this is extreme case. Since with so little percentage in $A,B$, you just cannot create a high percentage chemical blend $D$. Setting up equations and solve gives you $A=1, B=4$. But since we are not adding the condition $A+B=1$, this only gives you that $1 g$ of $A$ and $4 g$ of $B$ gives you $0.3 g$ of $c_1$ in $5 g$ of $D$, where the percentage does not really agree. 
So I guess you have to do something about the blends to make it happen. 
Basically, if all blends have a percentage of one chemical lower than the desired one, you just cannot get the desired percentage.
