Linked reciprocal relations: If a + 1/b = 1 and b + 1/c = 1 (with a, b, c non-zero), then find the exact value of c + 1/a.

Introduction / Context:
This algebra problem involves chained relations among a, b, and c that include reciprocals. The goal is to compute c + 1/a without solving for all variables explicitly. Such problems reward comfort with transforming equations and handling compound fractions.

Given Data / Assumptions:

• a + 1/b = 1 and b + 1/c = 1.
• a, b, c ≠ 0 so that all reciprocals are defined.
• We seek c + 1/a.

Concept / Approach:
First isolate a and b in terms of b and c. Then substitute to express 1/a in terms of c alone. A compact cancellation will appear, making the final evaluation immediate.

Step-by-Step Solution:

Verification / Alternative check:
Pick any convenient c ≠ 0, 1 (e.g., c = 2). Then b = (2 − 1)/2 = 1/2, a = 1 − 1/b = 1 − 2 = −1, hence 1/a = −1, and c + 1/a = 2 − 1 = 1.

Why Other Options Are Wrong:
The larger integers (3, 4, 24) and 0 ignore the exact cancellation found after substitution; only 1 satisfies the identity for all permitted c.

Common Pitfalls:
Dropping the negative sign when computing b − 1, or inverting a compound fraction incorrectly while finding 1/a.

Final Answer:
1