If $$a + \frac{1}{b} = 1$$   and $$b + \frac{1}{c} = 1$$   then $$c + \frac{1}{a}$$  is equal to?

Solution (By Examveda Team)

$$\eqalign{ & a + \frac{1}{b} = 1{\text{, }}b + \frac{1}{c} = 1{\text{, }}c + \frac{1}{a} = ? \cr & {\text{Put values, }} \cr & a = \frac{1}{2},b = 2,{\text{ }}c = - 1 \cr & c + \frac{1}{a} \cr & = - 1 + \frac{1}{{\left( {\frac{1}{2}} \right)}} \cr & = - 1 + 2 \cr & = 1 \cr & \cr & {\bf{Alternate:}} \cr & \Rightarrow a + \frac{1}{b} = 1 \cr & \Rightarrow a = 1 - \frac{1}{b} = \boxed{\frac{{b - 1}}{b}} \cr & \frac{1}{a} = \frac{b}{{b - 1}} \cr & \Rightarrow b + \frac{1}{c} = 1 \cr & \Rightarrow \frac{1}{c} = 1 - b,\boxed{c = \frac{1}{{1 - b}}} \cr & \therefore c + \frac{1}{a} \cr & = \frac{1}{{1 - b}} + \frac{b}{{b - 1}} \cr & = \frac{1}{{1 - b}} - \frac{b}{{1 - b}} \cr & = \frac{{1 - b}}{{1 - b}} \cr & = 1 \cr} $$