A can paint a house in 42 days and B can paint the same house in 21 days. Along with C, they finish painting the house in 7 days. In how many days can C alone paint the house?

Introduction / Context:
This is a classic three person time and work question. We are given the individual times for A and B, as well as the joint time for A, B and C together. The objective is to deduce the time required by C alone to paint the house, which is done by using the relationship between their individual and combined work rates.

Given Data / Assumptions:

• A alone can paint the house in 42 days.
• B alone can paint the house in 21 days.
• A, B and C together paint the house in 7 days.
• Each painter works at a constant rate with no interruptions.

Concept / Approach:
As usual, we express each worker's efficiency as a rate equal to 1 divided by the time taken to complete the job. We know the combined rate of A, B and C together and subtract the known individual rates of A and B to isolate C's rate. The reciprocal of C's rate then gives the time required for C alone to complete the work.

Step-by-Step Solution:
Step 1: Let the total painting work be 1 unit.Step 2: Rate of A = 1 / 42 of the house per day.Step 3: Rate of B = 1 / 21 of the house per day.Step 4: Rate of A, B and C together = 1 / 7 of the house per day.Step 5: Combined rate of A and B = 1 / 42 + 1 / 21.Step 6: Convert to common denominator 42: 1 / 42 = 1 / 42 and 1 / 21 = 2 / 42, so combined rate of A and B = 3 / 42 = 1 / 14.Step 7: Rate of C alone = combined rate of A, B and C minus rate of A and B.Step 8: Rate of C = 1 / 7 - 1 / 14.Step 9: Convert 1 / 7 to denominator 14: 1 / 7 = 2 / 14, so rate of C = 2 / 14 - 1 / 14 = 1 / 14 of the house per day.Step 10: Time taken by C alone to complete the painting = 1 divided by (1 / 14) = 14 days.

Verification / Alternative check:
We can verify that the combined work in 7 days equals the whole job. In one day, A does 1 / 42, B does 1 / 21 and C does 1 / 14 of the work. Their total daily rate is 1 / 42 + 1 / 21 + 1 / 14. Using denominator 42, this is 1 / 42 + 2 / 42 + 3 / 42 = 6 / 42 = 1 / 7. Therefore, in 7 days they do 7 * (1 / 7) = 1 full job, confirming that the calculation is consistent and C alone needs 14 days.

Why Other Options Are Wrong:

• 9 days: This would imply a faster rate and would change the combined rate of A, B and C, so the total time would not be 7 days.
• 12 days: This gives rate 1 / 12 which is different from 1 / 14 and does not fit the given combined completion time.
• 15 days: This rate 1 / 15 makes the combined rate smaller, leading to a total time greater than 7 days for the three workers.

Common Pitfalls:
A frequent mistake is to average the individual times instead of working with rates. Another error is failing to convert fractions to a common denominator when adding or subtracting. Always remember that for multiple workers, it is the rates that add and subtract, and careful fraction handling is essential.

Final Answer:
C alone can paint the house in 14 days.