Estimate the cost of an individual item in a mixed bag, knowing only the total value of the bag (multiple samples with different mixes allowed)

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Lets say I have a swimming pool filled with differently coloured balls. There are a fixed number of different colours. Each differently coloured ball has set cost (not necessarily unique) that I don't know.

I can take as many random samples of the pool as required (e.g. I can fill 1000 bags with 100 randomly sampled balls from the pool). I know the total cost of each bag and how many balls of each colour there are in each bag.

How can I determine the cost for each ball colour?

(Apologies in advance if I didn't put the correct tag, math newb here - corrections are most welcome)