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Algebra Difficulty 4.5 AIME Find the answer

Let a,b,c,na, b, c, n be positive real numbers such that a+ba=3,b+cb=4\frac{a+b}{a}=3, \frac{b+c}{b}=4, and c+ac=n\frac{c+a}{c}=n. Find nn.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We have 1=ba⋅cb⋅ac=(3−1)(4−1)(n−1)1=\frac{b}{a} \cdot \frac{c}{b} \cdot \frac{a}{c}=(3-1)(4-1)(n-1) Solving for nn yields n=76n=\frac{7}{6}.

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