A and B can do a piece of work in 18 days, B and C can do it in 24 days, and A and C can do it in 36 days. If all three work together, in how many days will they complete the entire work?

Introduction / Context:
This is a multi-step time and work question involving three workers, A, B, and C. We are given the times taken by each pair of workers, not by any single worker. The task is to deduce the individual rates of A, B, and C using the pairwise information and then find the total time when all three work together. This requires setting up and solving a system of equations.

Given Data / Assumptions:
- A and B together complete the work in 18 days.
- B and C together complete the work in 24 days.
- A and C together complete the work in 36 days.
- Total work is considered as 1 unit.
- Work rates add linearly.

Concept / Approach:
Let the daily work rates of A, B, and C be a, b, and c units per day respectively. Then we know that a + b = 1 / 18, b + c = 1 / 24, and a + c = 1 / 36. Adding these three equations gives 2(a + b + c). From this, we can compute the combined rate (a + b + c) and then find the time taken when all three work together by taking the reciprocal of this combined rate.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Let rates be a, b, and c for A, B, and C respectively. Step 3: From the data, we have a + b = 1 / 18, b + c = 1 / 24, a + c = 1 / 36. Step 4: Add all three equations: (a + b) + (b + c) + (a + c) = 1 / 18 + 1 / 24 + 1 / 36. Step 5: Left side simplifies to 2(a + b + c). Step 6: Compute right side: 1 / 18 + 1 / 24 + 1 / 36. Step 7: LCM of 18, 24, and 36 is 72. Rewrite: 1 / 18 = 4 / 72, 1 / 24 = 3 / 72, 1 / 36 = 2 / 72. Step 8: Sum = 4 / 72 + 3 / 72 + 2 / 72 = 9 / 72 = 1 / 8. Step 9: So 2(a + b + c) = 1 / 8, giving a + b + c = 1 / 16. Step 10: Combined rate of A, B, and C working together = 1 / 16 work per day. Step 11: Time taken when all three work together = 1 / (1 / 16) = 16 days.

Verification / Alternative check:
We can test consistency by solving for one individual rate. For example, a + b = 1 / 18 and a + b + c = 1 / 16, so c = 1 / 16 - 1 / 18. Similar checks can be done for a and b. The computed individual rates will be positive and, when substituted back, match all three original pairwise times, confirming that 16 days for all three together is correct.

Why Other Options Are Wrong:
- 12 and 13 days: These correspond to higher combined rates and do not align with the pairwise times given.
- 26 and 18 days: These are too large and imply slower combined rates. Substituting these into the relationships would not satisfy all three pairwise conditions simultaneously.

Common Pitfalls:
Students often try to guess the answer based only on one or two of the pairwise times, ignoring the need to satisfy all three equations at once. Another common mistake is incorrect fraction addition. To avoid errors, always use the LCM for denominators and systematically derive the combined rate using the sum of the three equations.

Final Answer:
Working together, A, B, and C will complete the work in 16 days.