Fractions with 3 diffferent variables

Question

Calculate the values of $a$, $b$ and $c$ if: $$\frac{5}{13} = \frac{1}{a+\frac{1}{b+\frac{2}{c}}}$$ Can anyone give me a hint and not the answer? Thanks.

Answer 1

Hint: $$\dfrac{5}{13}=\dfrac{1}{a+\dfrac{c}{bc+2}}=\dfrac{bc+2}{a(bc+2)+c}\\ \implies a=\dfrac{13(bc+2)-5c}{5(bc+2)}=\dfrac{13}{5}-\dfrac{c}{bc+2}$$

Answer 2

Just swing the terms at LHS and RHS as follows $$ \begin{align} \frac{5}{13} &= \frac{1}{a+\frac{1}{b+\frac{2}{c}}}\\ a+\frac{1}{b+\frac{2}{c}}&=\frac{13}{5}\\ \frac{1}{b+\frac{2}{c}}&=\frac{13}{5}-a\\ \frac{1}{b+\frac{2}{c}}&=\frac{13-5a}{5}\\ b+\frac{2}{c}&=\frac{5}{13-5a}\\ \frac{bc+2}{c}&=\frac{5}{13-5a}\\ (bc+2)(13-5a)&=5c. \end{align} $$ I think it is enough. The rest I leave it to you.

$$\\$$

---

$$\Large\color{blue}{\text{# }\mathbb{Q.E.D.}\text{ #}}$$