By what percent did the median annual cost of health insurance in City

The answer should be E) , since we don't know the annual cost of health insurance in City Y in either of the two years 1990 and 2000.

Official Solution:

By what percent did the median annual cost of health insurance in City \(X\) increase from 1990 to 2000?

We need to determine the percent increase of a certain median annual cost between 1990 and 2000.

Statement 1 tells us that in 1990, the median annual cost in City \(Y\) (call it \(Y_{90}\)) was \(\frac{2}{3}\) of the median annual cost in City \(X\) (call it \(X_{90}\)). Write an equation to express this: \(X_{90} = \frac{2}{3}Y_{90}\). This tells us the relationship between the cost in the two cities in 1990, but it tells us nothing about City \(X\) in 2000, so there is no way to calculate the percent increase in the cost of health insurance in City \(X\) between the two years. Statement 1 is NOT sufficient to answer the question. Eliminate answer choices A and D. The correct answer choice is B, C, or E.

Statement 2 gives us an equation relating the median annual costs in the two cities in 2000: \(X_{00} = \frac{6}{7}Y_{00}\). Again, though this equation gives us the relationship between the two cities in 2000, it says nothing about City \(X\) in 1990, and thus it is impossible to calculate the percent increase. Statement 2 is NOT sufficient to answer the question. Eliminate answer choice B. The correct answer choice is either C or E.

Taking the statements together: Statement 1 gives us \(X_{90} = \frac{2}{3}Y_{90}\), and statement 2 gives us \(X_{00} = \frac{6}{7}Y_{00}\). However, no information is given relating \(X_{90}\) and \(X_{00}\), or \(Y_{90}\) and \(Y_{00}\), and so we are unable to establish a relationship between the cost in City \(X\) in 1990 and the cost in City \(X\) in 2000. Thus, the statements together are NOT sufficient.

Answer: E.

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