Mixture to determine optimal blend of three variables with three different attributes (see description)

Question

Lets say you have 3 substances (A, B, and C)

And now say each substance is made up of three mixes of varying percentages (X, Y, Z)

We will say substance A is 10% X, 20% Y, and 70% Z

Substance B is 80% X, 10% Y, and 10% Z

Substance C is 10% X, 75% Y, and 15% Z

Now lets say I want to blend the three substances, creating a new substance D, that is say 30% X, 35% Y, and 35% Z.

How would I go about finding a blend that utilizes all three substances (if possible of course) and matches the desired percentages of X, Y, and Z?

Answer 1

Since Alligation is used as a tag, I'm assuming that this is the desired method to use to solve this problem, as opposed to some other algebraic approach.

To prepare a blend that is 30% X as required, you would have the following:

A 10% X.......5......0.....5

B 80% X.......2......2.....4

C 10% X ......0......5.....5

So, this blend would require mixing A B and C in a ratio of 1.25 : 1 : 1.25

 

To prepare a blend that is 35% Y as required, you would have the following:

A 20% Y......4......0......4

B 10% Y......0......4......4

C 75% Y......1.5...2.5...4

So, this blend would require A B and C in a ratio of 1 :1 : 1 

 

This ratio is different from the first, so it appears that these three cannot be combined to give the desired result. At least, that's the result I get from using alligation the way I know how to use it.  I'd be curious to see if there is another way to do this and get a viable solution.  Are you sure this problem has a legitimate answer?