Multiple choice

The efficiencies of A, B and C are in harmonic progression with distinct terms. If C works at double of his efficiency, then the number of days taken by A and C to complete a work, working together, will be one-third the number of days taken by B alone to complete the work. In how many days can A alone complete the work?

  1. 2

  2. 4

  3. 3

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the efficiencies be 1/x, 1/y, 1/z in harmonic progression. The problem provides insufficient constraints to uniquely determine the individual efficiencies or the time taken by A alone, as the harmonic progression condition and the work-rate equation do not yield a unique solution for the variables.

AI explanation

Let the efficiencies of A, B, and C be a, b, and c. Since they are in harmonic progression, their times taken to complete the work alone, which are 1/a, 1/b, and 1/c, are in arithmetic progression. This means 2/b = 1/a + 1/c, or b = 2ac/(a + c). The condition that C working at double efficiency with A takes one-third the time of B alone means that a + 2c = 3b. Substituting b into the second equation gives a + 2c = 6ac/(a + c). This simplifies to a^2 + ac + 2ac + 2c^2 = 6ac, or a^2 - 3ac + 2c^2 = 0. Factoring yields (a - c)(a - 2c) = 0. Because the terms of the harmonic progression are distinct, A and C must have different efficiencies, so a cannot equal c; therefore, a = 2c. The condition a = 2c means A takes half the time of C, but without knowing the exact value of C's efficiency, A's time cannot be determined. The result is Cannot be determined.

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