Can this whiskey be a result of combining whiskey A, B, and C in any ratio? There is a whiskey made up of 64% corn, 32% rye, and 4% barley that was made by blending other whiskies together. I am trying to figure out if there is a chance the ratio of this whiskey could be the result of blending two, maybe three whiskies of different ratios.
The possible whiskies:
Whiskey A is 60% corn, 36% rye and 4% barley.
Whiskey B is 81% corn, 15% rye, and 4% barley. 
Whiskey C is 75% corn, 21% rye, and 4% barley
I have a feeling there is a possibility because these whiskies all have 4% barley, but I can't figure out if the other percentages match up in any 1:2:3 ratio in the blend. Any help will be greatly appreciated. Thank you.
 A: Yes, and in infinitely many different ways!
Row reduce $\begin{bmatrix}
60&81&75&64\\
36&15&21&32\\
4&4&4&4\\
\end{bmatrix}$ to get $\begin{bmatrix}
1&0&\frac{2}{7}&\frac{17}{21}\\
0&1&\frac{5}{7}&\frac{4}{21}\\
0&0&0&0\\
\end{bmatrix}$.
Let's say:
$A$=ratio of Whiskey A in mixture
$B$=ratio of Whiskey B in mixture
$C$=ratio of Whiskey C in mixture
From the row reduction we get the relations: $$(*)A=\frac{17}{21}-\frac{2}{7}C, B=\frac{4}{21}-\frac{5}{7}C$$
Since $A,B,C$ are all between $0$ and $1$, these relations put a restriction on $C$: namely $C\leq\frac{4}{15}$. So choose your favorite value for $C$ in the appropriate range, plug it into the equations in $(*)$. That will give you the amounts of whiskeys A and B you need to get the desired mixture.
A: Hint Consider the three base whiskies as three unit vectors over the three-dimensional vector space of corn, rye, and barley. Mixing whiskeys then corresponds to simple operations on vectors, and you should be able to express this mixing operation in terms of operations you have already studied.  Then see if the vector $(64, 32, 4)$ can be expressed in terms of these operations on $A,B,$ and $C$.
A: You could any of


*

*Whiskey D made up of $\dfrac{17}{21}$ of Whiskey A and $\dfrac{4}{21}$ of Whiskey B 

*Whiskey E made up of $\dfrac{11}{15}$ of Whiskey A and $\dfrac{4}{15}$ of Whiskey C

*Any blend of whiskeys D and E
and you would have your 64%, 32% and 4% fractions.
A: For simplification, without any lost of information, I shall consider that the mixture you will make will be in the ratio 1:x:y.  
Now, let us forget about percentage and consider what were percentages as quantities and let us mix 1 liter of A plus "x" liters of B plus "y" liters of C. So, the total volume of the blend is (1+x+y) liters of D. This volume then contains   
(60 + 81 x + 75 y) of corn
(36 + 15 x + 21 y) of rye
( 4 +  4 x +  4 y) of barley  
Since these was the content of (1+x+y) liters of D, then one liter of D contains    
(60 + 81 x + 75 y) / (1 + x + y) percents of corn
(36 + 15 x + 21 y) / (1 + x + y) percents of of rye
and 4 percents of barley.  
So, for your example, if you take x=2 and y=3 (your 1:2:3 ratio), the final mixture contains
(60 + 2 * 82 + 3 * 75) / (1 + 2 + 3) = 447 / 6 = 74.5 % of corn
(36 + 2 * 15 + 3 * 21) / (1 + 2 + 3) = 129 / 6 = 21.5 % of rye
to which 4.0 % of barley have to be added.   
If you want a mixture containing 64 percents of corn and 32 percents of rye, you just need to solve two equations for two unknowns, namely
(60 + 81 x + 75 y) / (1 + x + y) = 64
(36 + 15 x + 21 y) / (1 + x + y) = 32  
the solution of which ... does not exist. From these two equations, the only thing you can extract is   
y = (4 - 17 x) / 11
Then, any blend made mixing 1 liter of A, x liters of B and (4 - 17 x) /11 liters of C will give you the desired composition. For sure,the value of x cannot be larger than 4 / 17. So you can generate "many" mixtures satisfying your requirements. 
You could have made the problem more general not considering the percentage of barley to be the same in each A, B and C. The approach would stay the same.  
By, the way, since the drink is ready, Cheers !
