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If a, b, c are non-zero and satisfy a + (1/b) = 1 and b + (1/c) = 1, determine the exact value of the product abc.

Difficulty: Medium

Correct Answer: -1

Explanation:


Introduction / Context:
This problem involves linked linear-fractional relations between a, b, and c. The goal is to deduce the product abc without solving for each variable individually, by expressing the variables in terms of a single parameter.



Given Data / Assumptions:

  • a + 1/b = 1.
  • b + 1/c = 1.
  • a, b, c ≠ 0.


Concept / Approach:
Isolate a and b in terms of b and c respectively, then substitute progressively to express a in terms of c alone. Multiply a, b, and c to compute abc. Watch the algebraic signs carefully.



Step-by-Step Solution:

From a + 1/b = 1 ⇒ a = 1 − 1/b = (b − 1)/b.From b + 1/c = 1 ⇒ b = 1 − 1/c = (c − 1)/c.Express a in terms of c: a = (b − 1)/b = (( (c − 1)/c ) − 1) / ((c − 1)/c).Simplify numerator: (c − 1 − c)/c = (−1)/c; divide by (c − 1)/c gives a = −1/(c − 1).Compute product: abc = [−1/(c − 1)] * [(c − 1)/c] * c = −1.


Verification / Alternative check:
Pick any convenient c ≠ 0,1 (e.g., c = 2) and compute b and a from the relations; the product abc numerically evaluates to −1 every time, confirming the derivation.



Why Other Options Are Wrong:
−3, 1, 3, and 0 do not satisfy the derived identity; abc is constant and equals −1 regardless of the choice of c (excluding singular cases).



Common Pitfalls:
Dropping negative signs during simplification or forgetting to divide by the compound fraction when solving for a in terms of c.



Final Answer:
-1

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