A can paint a house in 55 days and B can paint the same house in 66 days. When they work together with a third painter C, all three of them complete the painting in just 12 days. If C worked alone from start to finish, in how many days would C be able to paint the entire house?

Introduction / Context:

This question applies the time and work concept to a painting job involving three workers. Two individual completion times are given, and the combined completion time when all three work together is also provided. The task is to determine the individual efficiency of the third worker. Such problems test a candidate's ability to handle work rate equations and isolate an unknown rate from a combined rate.

Given Data / Assumptions:

• A can paint the house alone in 55 days.
• B can paint the same house alone in 66 days.
• A, B, and C together finish the house in 12 days.
• We assume the size of the work is one unit and all workers maintain constant rates.

Concept / Approach:

We convert each worker's completion time into a daily work rate. The combined rate of A, B, and C working together is 1 divided by the total days taken together. This combined rate is the sum of their individual rates. Since we know A's and B's rates, we can subtract these from the combined rate to find C's rate. Finally, the reciprocal of C's rate gives the time taken by C alone to finish the job.

Step-by-Step Solution:

Verification / Alternative Check:

Check by recomputing the combined rate using C's rate. The sum 1/55 + 1/66 + 1/20 should equal 1/12. Using a common denominator of 660: 1/55 = 12/660, 1/66 = 10/660, and 1/20 = 33/660. Their sum is 12 + 10 + 33 = 55 over 660, which simplifies to 1/12. This matches the combined rate required to finish the house in 12 days, confirming that the derived rate for C is consistent.

Why Other Options Are Wrong:

If C took 24, 33, or 44 days, then C's rate would be smaller than 1/20, and the sum of the three rates would be less than 1/12, meaning the work would take more than 12 days. Hence these options contradict the given condition. The only time that correctly reproduces the combined 12 day completion time is 20 days.

Common Pitfalls:

Common mistakes include adding the given days directly instead of adding rates, or forgetting to subtract A and B's rates from the combined rate before taking the reciprocal. Some students also make errors in fractional arithmetic, especially when finding common denominators. Carefully setting up the rate equation and simplifying step by step avoids these issues.

Final Answer:

If C works alone from start to finish, the house will be painted in 20 days.