To solve this problem, we need to calculate how much work is done per day when A works continuously and B and C work alternately. Here’s the step-by-step solution:

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Step 1: Work done by A, B, and C individually

• A can complete the work in 12 days, so A's efficiency = 1/12 work/day.
• B can complete the work in 15 days, so B's efficiency = 1/15 work/day.
• C can complete the work in 20 days, so C's efficiency = 1/20 work/day.

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Step 2: Combined work for a 2-day cycle

• In the first day, A and B work together:

  • Work done = (1/12) + (1/15) = (5 + 4)/60 = 9/60 = 3/20 work.

• On the second day, A and C work together:

  • Work done = (1/12) + (1/20) = (5 + 3)/60 = 8/60 = 2/15 work.

• Total work done in 2 days = (3/20 + 2/15):

  • LCM of 20 and 15 = 60.
  • Convert fractions:
    • 3/20 = 9/60, and 2/15 = 8/60.
  • Total work in 2 days = 9/60 + 8/60 = 17/60 work.

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Step 3: Total days required to complete the work

• Total work = 1 unit (the entire task).
• Work done in 2 days = 17/60 units.
• Number of full 2-day cycles = (1 ÷ (17/60)) = 60/17 ≈ 3.53 cycles.

This means that after 3 full 2-day cycles, 3 × 17/60 = 51/60 = 0.85 units of work will be completed.

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Step 4: Work left and final calculations

After 3 cycles, the remaining work = 1 - 51/60 = 9/60 = 3/20 units.

• On the next day, A and B will work:
  • Work done in one day by A and B = 3/20 units (exactly matches the remaining work).

Thus, the total time taken = 6 days (for 3 cycles) + 1 day = 7 days.

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Final Answer:

The work will be completed in 7 days.

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Would you like me to explain any step in more detail or solve a variation of this problem?

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Related Questions:

1. How is efficiency calculated for multiple workers completing a task?
2. What happens if B and C swap roles every third day instead of alternating daily?
3. How do we calculate work for cases with uneven schedules for workers?
4. What if one of the workers takes a break for a certain number of days? How would it affect the solution?
5. How can you represent this calculation using an algebraic equation for cycles?

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Tip:

When dealing with alternate working schedules, always calculate the work done in a full cycle (e.g., 2 days here) to simplify calculations.