2014 AIME I Problems/Problem 2

Problem 2

An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is 0.58. Find $N$ .

Solution

First, we find the probability both are blue, then the probability both are green, and add the two probabilities which equaling $0.58$ . The probability both are blue is $\frac{4}{10}\cdot\frac{16}{16+N}$ , and the probability both are green is $\frac{6}{10}\cdot\frac{N}{16+N}$ , so $\frac{4}{10}\cdot\frac{16}{16+N}+\frac{6}{10}\cdot\frac{N}{16+N}=\frac{29}{50}.$ Solving this equation, we get $N=144$ .

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2014 AIME I Problems/Problem 2

Problem 2

Solution