AIME I 2014 P2

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Problem Statement

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 $N$.

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Solution

$$\text{There are 10 balls in the first urn and 16+N balls in the second urn.}$$ $$\implies \text{There are } (10)(16+N) \text{ total possible ways to pick two balls.}$$ $$\text{Let } W_1 \text{ be the number of ways to pick two green balls}$$ $$\therefore W_1 = (4)(16) = 64$$ $$\text{Let } W_2 \text{ be the number of ways to pick two blue balls}$$ $$\therefore W_2 = (6)(N) = 6N$$ $$\dfrac{W_1 + W_2}{(10)(16+N)} = 0.58$$ $$64 + 6N = 92.8 + 5.8N$$ $$N = 144$$