A can do a piece of work in 20 days and B can do the same work in 15 days. With the help of a third person C, they finish the entire work in 5 days. In how many days can C alone complete the whole work?

Introduction / Context:
This time and work question involves three workers, A, B, and C. We know the individual times of A and B, and the combined time when all three work together. From this information we must deduce the efficiency of C and then convert that efficiency into the number of days C alone would need to complete the job.

Given Data / Assumptions:
- A alone completes the work in 20 days.
- B alone completes the work in 15 days.
- A, B, and C together complete the work in 5 days.
- Total work is assumed to be 1 unit.
- Work rates add linearly when workers cooperate.

Concept / Approach:
We first calculate the individual daily work rates of A and B. Next we find the combined daily rate of all three from the fact that they finish the work in 5 days. The daily rate of C is then obtained by subtracting the known rates of A and B from the combined rate. Finally, we compute the reciprocal of C’s rate to find C’s individual time to complete the entire job alone.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Rate of A = 1 / 20 work per day. Step 3: Rate of B = 1 / 15 work per day. Step 4: A, B, and C together finish in 5 days, so combined rate of A + B + C = 1 / 5 work per day. Step 5: Compute rate of A + B: 1 / 20 + 1 / 15. Step 6: Take LCM of 20 and 15 which is 60, so 1 / 20 = 3 / 60 and 1 / 15 = 4 / 60. Step 7: Rate of A + B = 3 / 60 + 4 / 60 = 7 / 60 work per day. Step 8: Rate of C = combined rate - rate of A + B = 1 / 5 - 7 / 60. Step 9: Convert 1 / 5 to denominator 60: 1 / 5 = 12 / 60. Step 10: Rate of C = 12 / 60 - 7 / 60 = 5 / 60 = 1 / 12 work per day. Step 11: Time taken by C alone = 1 / (1 / 12) = 12 days.

Verification / Alternative check:
Check by recombining rates: A does 1 / 20, B does 1 / 15, C does 1 / 12 per day. Adding gives 1 / 20 + 1 / 15 + 1 / 12. The LCM of 20, 15, and 12 is 60. So the combined rate is 3 / 60 + 4 / 60 + 5 / 60 = 12 / 60 = 1 / 5, which matches the given time of 5 days. Hence C’s computed rate and time are consistent.

Why Other Options Are Wrong:
- 5 and 6 days: These would imply C is extremely fast and make the combined rate larger than 1 / 5 per day.
- 10 days: Close but leads to a slight mismatch in the combined rate when all three are considered.
- 15 days: This is slower than indicated by the difference between the combined rate and the sum of A and B’s rates.

Common Pitfalls:
Many learners mistakenly average the given times instead of working with rates. Another common error is to forget to subtract both A and B’s rates from the joint rate to isolate C’s rate. Always convert each time into a rate, use linear addition of rates, and then revert rates back into times by taking reciprocals.

Final Answer:
C alone can complete the whole work in 12 days.