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Find positive integers a,b and c such that

\[ \dfrac{3}{11} = \frac 1{a+ \frac1{b+ \frac1c}}. \]

 Jul 26, 2021
 #1
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Find positive integers a,b and c such that
\(\dfrac{3}{11} = \frac {1} {a+ \frac1{b+ \frac1c}} \).

 

\(\begin{array}{|r|rcl|rcl|} \hline \dfrac{3}{11} = \frac {1} {a+ \frac1{b+ \frac1c}} &\dfrac{11}{3} &=& a+ \frac1{b+ \frac1c} \\\\ &{\color{red}3}+\dfrac{2}{3} &=& {\color{red}a}+ \frac1{b+ \frac1c}~\Rightarrow \mathbf{a=3} \\\\ &\dfrac{2}{3} &=& \frac1{b+ \frac1c} &\dfrac{3}{2} &=& b+ \frac1c \\\\ &&&&{\color{red}1}+\dfrac{1}{2} &=& {\color{red}b}+ \frac1c ~\Rightarrow \mathbf{b=1} \\\\ &&&&\dfrac{1}{2} &=& \dfrac1c \\\\ &&&& \mathbf{2} &=& \mathbf{c} \\ \hline \end{array}\)

 

\(\dfrac{3}{11} = \frac {1} {3+ \frac1{1+ \frac12}}.\)

 

laugh

 Jul 27, 2021
edited by heureka  Jul 27, 2021

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