A alone can paint a house in 25 days and B alone can paint the same house in 10 days. Working together with a third painter C, they finish the job in 6.25 days. In how many days can C alone complete the job if working at the same constant rate?

Introduction / Context:
This problem involves three workers, A, B, and C, who together complete a painting job in a given time. We know the individual times of A and B, and the combined time of all three, and we must find C's individual time. This is a classic application of the concept of work rate addition and subtraction, where we isolate the unknown worker's rate by subtracting the known rates from the combined rate.

Given Data / Assumptions:

• A alone can paint the house in 25 days.
• B alone can paint the house in 10 days.
• A, B, and C together complete the house in 6.25 days (which is 25/4 days).
• All painters work at constant rates.
• We need the time taken by C alone to finish the house.

Concept / Approach:
We treat the entire house as 1 unit of work. Each painter has a rate equal to 1 divided by their respective time. The combined rate of A, B, and C is 1 divided by 6.25 days. We first compute the individual rates of A and B, sum them, and subtract that sum from the combined rate to find C's rate. Finally, we take the reciprocal of C's rate to obtain the time C needs alone.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: A's rate = 1/25 of the work per day. Step 3: B's rate = 1/10 of the work per day. Step 4: Time of A + B + C together = 6.25 days = 25/4 days, so combined rate = 1 / (25/4) = 4/25 per day. Step 5: Sum of A and B's rates = 1/25 + 1/10 = (2/50 + 5/50) = 7/50 per day. Step 6: C's rate = combined rate − (A + B) rate = 4/25 − 7/50. Step 7: Convert 4/25 to 8/50, so C's rate = 8/50 − 7/50 = 1/50 of the work per day. Step 8: Time taken by C alone = 1 / (1/50) = 50 days.

Verification / Alternative check:
We can verify by recombining the rates. A's rate is 1/25, B's rate is 1/10, and C's rate we found as 1/50. Their combined rate should be 1/25 + 1/10 + 1/50 = (2/50 + 5/50 + 1/50) = 8/50 = 4/25. This is exactly the rate corresponding to 25/4 days, or 6.25 days, which matches the given data. Hence our calculation for C is consistent and correct.

Why Other Options Are Wrong:
40 and 30 days would correspond to higher rates for C, which would make the combined time less than 6.25 days, contradicting the question. 60 and 25 days either make C too slow or duplicate A's speed, which is not implied. Only 50 days gives a rate that, when combined with A and B, reproduces the correct total time of 6.25 days.

Common Pitfalls:
A common mistake is to average the times (25, 10, and 6.25) directly instead of working with rates. Another error is to mishandle fractional time like 6.25 days when converting it to a fraction. Writing 6.25 as 25/4 is more precise and simplifies the calculation. Carefully subtracting known rates from the total combined rate is essential to isolate the unknown worker's rate correctly.

Final Answer:
C alone can complete the painting job in 50 days.