A can paint a house in 45 days and B can paint the same house in 15 days. Along with a third painter C, they can finish the painting in 5 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 problem is a classic example of a time and work question involving three workers with different efficiencies. We are given the individual completion times of two workers and the combined completion time when all three work together. The objective is to determine the time required by the third worker alone. This tests the ability to set up and solve an equation involving sums of work rates.

Given Data / Assumptions:

• A alone can paint the house in 45 days.
• B alone can paint the house in 15 days.
• A, B, and C together finish the painting in 5 days.
• We assume steady work rates and take the entire painting job as one unit of work.

Concept / Approach:

We interpret the given days as reciprocals of daily work rates. The combined rate of A, B, and C is 1/5 of the work per day. The individual rates of A and B are 1/45 and 1/15 respectively. The sum of these plus C's rate must equal the combined rate. Once C's rate is found, taking its reciprocal gives the time C alone would require to finish the job from start to finish.

Step-by-Step Solution:

Verification / Alternative Check:

Check by recomputing the combined rate with C's rate included. A's rate is 1/45, B's rate is 1/15 which is 3/45, and C's rate is 1/9 which is 5/45. Together they work at (1 + 3 + 5) / 45 = 9/45 = 1/5 of the work per day, so they finish in 5 days as given. This confirms that the computed rate for C is consistent with the problem statement.

Why Other Options Are Wrong:

If C took 12 or 15 days, then C's rate would be smaller and the combined rate would be less than 1/5, resulting in more than 5 days to finish. An 8 day completion time would give a rate higher than 1/9, causing the joint completion time to be less than 5 days, again contradicting the given data. Only 9 days produces the required joint rate of 1/5 per day.

Common Pitfalls:

A common mistake is to take the arithmetic mean of the given times or to average them instead of dealing with reciprocal rates. Some may also incorrectly add 45, 15, and 5 or subtract days directly. The correct method is always to convert to rates, set up the equation, and then take reciprocals at the end.

Final Answer:

If C worked alone, C would finish painting the house in 9 days.