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

### Concept & Formula This problem involves a cyclic system of equations. The goal is to express both variables in the final expression ($c$ and $a$) in terms of a single intermediate variable (in this case, $b$) using the given equations, and then substitute them to find a constant value. $$ \text{Substitution Goal: Express } c = f(b) \text{ and } \frac{1}{a} = g(b) $$ ### Step-by-Step Solution * **Given:** * Equation 1: $a + \frac{1}{b} = 1$ * Equation 2: $b + \frac{1}{c} = 1$ * Target expression: $c + \frac{1}{a}$ * **Calculation:** 1. From Equation 1, solve for $\frac{1}{a}$ in terms of $b$. First, isolate $a$: $a = 1 - \frac{1}{b}$. Find a common denominator: $a = \frac{b - 1}{b}$. Take the reciprocal to find $\frac{1}{a}$: $\frac{1}{a} = \frac{b}{b - 1}$. 2. From Equation 2, solve for $c$ in terms of $b$. Isolate $\frac{1}{c}$: $\frac{1}{c} = 1 - b$. Take the reciprocal to find $c$: $c = \frac{1}{1 - b}$. To match the denominator of $\frac{1}{a}$, factor out a $-1$: $c = \frac{-1}{b - 1}$. 3. Substitute these expressions back into the target equation: $c + \frac{1}{a}$. $c + \frac{1}{a} = \left(\frac{-1}{b - 1}\right) + \left(\frac{b}{b - 1}\right)$. 4. Since they share a common denominator, add the numerators. $\frac{-1 + b}{b - 1} = \frac{b - 1}{b - 1}$. 5. Simplify the fraction. Any non-zero number divided by itself is $1$. Result is $1$. ### Exam Strategy & Shortcut Use the Value Assumption strategy. Pick a number for $b$ that makes calculation easy, such as $b = 2$. If $b = 2$, then from Equation 1: $a + \frac{1}{2} = 1 \implies a = \frac{1}{2}$. If $b = 2$, then from Equation 2: $2 + \frac{1}{c} = 1 \implies \frac{1}{c} = -1 \implies c = -1$. Now, calculate the target $c + \frac{1}{a}$: Target = $(-1) + \frac{1}{\frac{1}{2}} = -1 + 2 = 1$. This confirms the answer instantly without complex algebraic manipulation. ### Common Pitfall When rearranging terms, students often mishandle reciprocals, incorrectly assuming that if $a = 1 - \frac{1}{b}$, then $\frac{1}{a} = 1 - b$. You must combine the terms into a single fraction before taking the reciprocal. ### Final Answer Therefore, the correct answer is 1.