HMMT November

Algebra Difficulty 4.5 AIME Find the answer

Let $a, b, c, n$ be positive real numbers such that $\frac{a+b}{a}=3, \frac{b+c}{b}=4$ , and $\frac{c+a}{c}=n$ . Find $n$ .

Your answer

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

Solution

We have $1=\frac{b}{a} \cdot \frac{c}{b} \cdot \frac{a}{c}=(3-1)(4-1)(n-1)$ Solving for $n$ yields $n=\frac{7}{6}$ .

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