B takes 12 more hours than A to complete a task. If they work together, they…

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B takes 12 more hours than A to complete a task. If they work together, they take 16 fewer hours than B would take to complete the task. How long will it take A and B together to complete a task twice as difficult as the first one?

  1. A.

    16 hrs

  2. B.

    12 hrs

  3. C.

    14 hrs

  4. D.

    18 hrs

Attempted by 10 students.

Show answer & explanation

Correct answer: A

Let x be the number of hours A takes to finish the original task alone.

Then B takes x + 12 hours. Together they take 16 fewer hours than B alone, so together time = (x + 12) - 16 = x - 4 hours.

Rates (work per hour): A = 1/x, B = 1/(x+12), together = 1/(x-4). Set up the equation:

  1. 1/x + 1/(x+12) = 1/(x-4).

  2. Combine the left side: (2x + 12)/[x(x+12)] = 1/(x-4). Cross-multiply:

  3. (2x + 12)(x - 4) = x(x + 12).

  4. Expand and simplify: 2x^2 + 4x - 48 = x^2 + 12x => x^2 - 8x - 48 = 0.

  5. Solve quadratic: (x - 12)(x + 4) = 0, so x = 12 or x = -4. Reject -4 (negative time).

Thus A takes 12 hours and B takes 24 hours for the original task. Together they take x - 4 = 8 hours to finish the original task.

For a task twice as difficult, they need twice the time: 2 × 8 = 16 hours.

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