If a + 1/b = 1 and b + 1/c = 1 , then c + 1/a is equal to = ?

Answer: Option C

$$\begin{array}{rl}& a+\frac{1}{b}=1\\ & \Rightarrow ab+1=b\\ & \Rightarrow ab-b=-1\\ & \Rightarrow b\left(a-1\right)=-1\\ & \Rightarrow b=\frac{1}{\left(1-a\right)}\\ & \\ & b+\frac{1}{c}=1\\ & \Rightarrow bc+1=c\\ & \Rightarrow bc-c=-1\\ & \Rightarrow c\left(b-1\right)=-1\\ & \Rightarrow c=\frac{1}{\left(1-b\right)}\\ & \\ & \therefore c+\frac{1}{a}=\frac{1}{\left(1-b\right)}+\frac{1}{a}\\ & c+\frac{1}{a}=\frac{1}{1-\left(\frac{1}{1-a}\right)}+\frac{1}{a}\\ & c+\frac{1}{a}=\frac{1}{\frac{\left(1-a\right)-1}{\left(1-a\right)}}+\frac{1}{a}\\ & c+\frac{1}{a}=\frac{\left(1-a\right)}{-a}+\frac{1}{a}\\ & c+\frac{1}{a}=\frac{\left(a-1\right)}{a}+\frac{1}{a}\\ & c+\frac{1}{a}=\frac{a-1+1}{a}\\ & c+\frac{1}{a}=\frac{a}{a}\\ & c+\frac{1}{a}=1\end{array}$$