Working together, A, B, and C can complete a task in 12 days. If A alone can complete the task in 55 days and B alone can complete the task in 66 days, then in how many days can C alone complete the same task?

Introduction / Context:

This time and work problem provides the combined completion time for three workers and the individual completion times for two of them. The goal is to determine the time required by the third worker alone. This tests comfort with setting up and solving linear equations involving work rates, a common theme in quantitative aptitude sections.

Given Data / Assumptions:

• A, B, and C together finish the task in 12 days.
• A alone can finish the task in 55 days.
• B alone can finish the task in 66 days.
• We assume constant work rates and take total work as one unit.

Concept / Approach:

We treat the completion times as reciprocals of daily work rates. The combined rate of A, B, and C is 1/12 of the total job per day. A's rate is 1/55 per day and B's rate is 1/66 per day. We set up an equation that the combined rate equals the sum of individual rates. Solving this equation gives C's daily work rate, and its reciprocal is the time C alone would need to finish the work.

Step-by-Step Solution:

Verification / Alternative Check:

We can verify by recomputing the combined rate using the derived value for C. Using a common denominator of 660, 1/55 = 12/660, 1/66 = 10/660, and 1/20 = 33/660. The sum is 12 + 10 + 33 = 55 over 660, which simplifies to 1/12. This matches the given completion time of 12 days for A, B, and C together, confirming that C taking 20 days alone is correct.

Why Other Options Are Wrong:

If C took 22, 40, or 44 days, the rate of C would be different and the sum of all three rates would no longer be exactly 1/12, so the combined completion time would not remain 12 days. Among the options, only 20 days leads to a combined rate equal to 1/12 per day and therefore satisfies all given conditions.

Common Pitfalls:

Some test takers mistakenly average the times of A and B or try to take an average with the joint time instead of working with reciprocal rates. Another frequent error is incorrect simplification of fractions when calculating 1/55 + 1/66, which leads to an incorrect equation for C. Careful fraction arithmetic and correct use of the rates equation avoids these mistakes.

Final Answer:

Worker C alone can complete the work in 20 days.