A can do a piece of work in 10 days, B can do it in 12 days and C can do it in 15 days. If all three work together from the start, in how many days will they complete the work?

Introduction / Context:
This problem involves three workers, A, B, and C, each with different individual times to complete the same task. We are asked how long all three will take if they work together from the beginning. This is a straightforward application of the concept of adding work rates to obtain a combined rate.

Given Data / Assumptions:
- A alone can complete the work in 10 days.
- B alone can complete the work in 12 days.
- C alone can complete the work in 15 days.
- All three work together until the job is completed.
- Total work is considered as 1 unit.

Concept / Approach:
We convert each person’s time into a daily work rate by taking the reciprocal of the time. Then we add the three rates to find the combined rate when A, B, and C work together. Finally, we take the reciprocal of this combined rate to determine the total time required to finish the work jointly.

Step-by-Step Solution:
Step 1: Assume total work = 1 unit. Step 2: Rate of A = 1 / 10 work per day. Step 3: Rate of B = 1 / 12 work per day. Step 4: Rate of C = 1 / 15 work per day. Step 5: Combined rate = 1 / 10 + 1 / 12 + 1 / 15. Step 6: LCM of 10, 12, and 15 is 60. Step 7: Convert each rate: 1 / 10 = 6 / 60, 1 / 12 = 5 / 60, 1 / 15 = 4 / 60. Step 8: Combined rate = 6 / 60 + 5 / 60 + 4 / 60 = 15 / 60 = 1 / 4 work per day. Step 9: Time taken together = 1 / (1 / 4) = 4 days.

Verification / Alternative check:
We can verify by computing how much work is completed in 4 days. At a rate of 1 / 4 work per day, in 4 days they complete 4 * 1 / 4 = 1 unit of work, exactly matching the total work. Also, the combined time of 4 days is logically less than each individual time of 10, 12, and 15 days, which makes sense because more workers should complete the job faster.

Why Other Options Are Wrong:
- 6, 5 1/4, and 4 4/11 days: These values correspond to slower combined rates and would not match the correct sum of daily work contributions from A, B, and C.
- 7 days: This is even higher and would mean a combined work rate smaller than any single worker, which is impossible when all three are working together.

Common Pitfalls:
Common mistakes include averaging the given times instead of summing work rates, or mishandling fraction addition by not using the correct LCM. Another error is to forget that combined time must always be shorter than the smallest individual time when everyone works simultaneously on the same job.

Final Answer:
A, B, and C working together will complete the work in 4 days.