A mixture X is made by removing a certain fraction of orange juice from a jar filled with pure orange juice and replacing it with the same amount of water. The same fraction of mixture X is then removed and replaced with water to form a new mixture Y. If the ratio of orange juice to water in mixture Y is 9 : 16, what is the ratio of water in mixture X to that in mixture Y?

Let the initial pure juice be 1 and the replaced fraction be x. After the first replacement, mixture X contains (1 - x) juice and x water. After the second replacement, the remaining juice is (1 - x)^2. Since the final ratio of juice to water is 9 : 16, (1 - x)^2 / (1 - (1 - x)^2) = 9/16. Solving gives (1 - x)^2 = 9/25, so 1 - x = 3/5, meaning X had 3/5 juice and 2/5 water. Mixture Y has 9/25 juice and 16/25 water, making the ratio of water in X to water in Y 2/5 : 16/25, which simplifies to 5 : 8.

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