Question : If $\frac{2+a}{a}+\frac{2+b}{b}+\frac{2+c}{c}=4$, then the value of $\frac{ab+bc+ca}{abc}$ is:
Option 1: $2$
Option 2: $1$
Option 3: $0$
Option 4: $\frac{1}{2}$
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Correct Answer: $\frac{1}{2}$
Solution : Given that $\frac{2+a}{a}+\frac{2+b}{b}+\frac{2+c}{c}=4$, we can rewrite this equation as: $⇒\frac{2}{a}+1+\frac{2}{b}+1+\frac{2}{c}+1=4$ Simplifying, we get: $⇒\frac{2}{a}+\frac{2}{b}+\frac{2}{c}=1$ Multiplying through by $abc$, we get: $⇒2bc+2ac+2ab=abc$ Rearranging terms, we get: $⇒abc=2(ab+bc+ca)$ $\therefore\frac{ab+bc+ca}{abc}=\frac{1}{2}$ Hence, the correct answer is $\frac{1}{2}$.
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