Welcome 2016! Part 6

Algebra Level 4

\[\dfrac{ab}{a+b}=3 \ , \ \dfrac{bc}{b+c}=4 \ , \ \dfrac{ca}{c+a}=5 \]

Let \(a,b\) and \(c\) be real numbers satisfying all 3 equations above. Given that \( \dfrac{abc}{ab+ac+bc} \) is equal to \( \dfrac mn\), where \(m\) and \(n\) are coprime positive integers, find the value of \(m+n\).

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