Difficulty: Medium
Correct Answer: 1
Explanation:
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:
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:
From a + 1/b = 1 ⇒ a = 1 − 1/b = (b − 1)/b ⇒ 1/a = b/(b − 1).From b + 1/c = 1 ⇒ b = 1 − 1/c = (c − 1)/c.Compute b − 1 = (c − 1)/c − 1 = (c − 1 − c)/c = −1/c.Thus 1/a = b/(b − 1) = ((c − 1)/c) / (−1/c) = −(c − 1).Finally, c + 1/a = c − (c − 1) = 1.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
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