Seeing as the correct solution has been posted in the comments to the question, I don't think giving hints to an answer that is already spelled out makes a lot of sense. Therefore, I'll fully explain the answer below.
The trick of this puzzle is
that it's not an algebra question, even if it looks like one. It's a geometry puzzle!
Suppose you put each blend in a Cartesian plane, where x = % of orange and y = % of apple. Then, if you have only two blends, it's not hard to convince yourself that the possible results of mixtures of both lie on the line segment between these two points:
Similarly, if we had three blends, the possible mixtures would form a triangle with the three points as vertices, and for $n$ blends we would have a polygon:

Hopefully, this should give enough intuition to understand the following possible strategy to find the correct mixing:
Draw all points corresponding to all given blends on the plane. Find a triangle of given blends that contains the target (this should be easy to do visually). By our earlier observations, this means those 3 bottles can be used to form the perfect mix.
Then draw a line connecting one of the vertices of the triangle and the target point, and measure the following segments with a ruler:

Mixing $m$ units of bottle C, $n$ units of bottle A, and $\frac{s(m+n)}{r}$ units of bottle B gives a perfect mix. The reasoning for this is that we can form the perfect blend by mixing B and a mixture of A and C (somewhere on the line segment between A and C). By drawing the segment, we can find exactly what that mixture is. All that remains is finding the proportions, which we do by measuring the segments.
An alternative solution, which I didn't know about and was suggested by @McFry in the comments, is more symmetric but follows pretty much the same idea:
Again, find a triangle that contains the wanted point. Draw lines from all vertices to the central point and measure the segments:
(The company-supplied mixes are always in the larger right triangle.)
Using $\frac{y}{x+y}$ units of bottle A, $\frac{s}{r+s}$ units of bottle B and $\frac{n}{m+n}$ units of bottle C, we have a perfect mix. These values are known as barycentric coordinates of a point.
And to answer the question about the number of bottles, from the strategy it should be clear that:
We only need to mix at most 3 bottles. If a mix is covered by a triangle, we already know how to obtain it. The informal proof is that the reachable mixes form a polygon with the given bottles as vertices, and every polygon can be triangulated, so we know that for every reachable mix there must be a triangle that covers it.