Let's make Mango Orange Apple Juice!

Question

Annie recently took a temp job at the famous juice shop, All About Juice. As is the case with most juice shops, creative mixtures are the norm, but the store's most famous product is still its traditional mango orange apple juice. The key component of the formula is the proportion of mango, orange and apple juice in the final blend, and as a trade secret she never told me what it was, exactly.

She did tell me a funny story, though: as a way of hiding the secret, instead of providing pure apple/mango/orange juice to the servers for mixing, every day they receive several bottles with different proportions of the 3 juices in each (perfectly mixed and inseparable) and have to work out how much of each bottle to use to get the perfect blend! These are quite weird mixes too, for example one of them had 43% orange, 26% mango and 31% apple. The company guarantees that the correct proportion is mixable from the given bottles, but figuring out a way of obtaining it is always a fun exercise in the morning (if you like math, that is).

Lately, Annie had been trying to figure out a fast way to do the mixing. What is the highest amount of bottles she will need to mix to make the juice in the worst case? Can you show her a quick way to find the correct mix on paper, without computers or calculators?

Answer 1

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.