Three Pairwise Times — Find All Together A and B can finish the work in 8 days, B and C in 12 days, and A and C in 8 days. How many days will A, B, and C together take to finish the work?

Introduction / Context:
Given three pairwise completion times, we can find the sum of all three rates by adding the pairwise rates and halving, since each person's rate is counted twice in the sum of pairs.

Given Data / Assumptions:

• (A + B) rate = 1/8.
• (B + C) rate = 1/12.
• (A + C) rate = 1/8.

Concept / Approach:
(A + B + C) rate = ( (A + B) + (B + C) + (A + C) ) / 2. Then total time = 1 / (A + B + C rate).

Step-by-Step Solution:
Sum of pair rates = 1/8 + 1/12 + 1/8 = (3 + 2 + 3)/24 = 8/24 = 1/3.(A + B + C) rate = (1/3)/2 = 1/6 job/day.All together time = 1 / (1/6) = 6 days.

Verification / Alternative check:
The symmetry of two pairs at 8 days supports a moderate all-together time of 6 days.

Why Other Options Are Wrong:
4, 9, 12, or 5 days would not match the derived combined rate of 1/6 per day.

Common Pitfalls:
Forgetting to divide by 2 after summing pairwise rates or attempting to average the days directly.

Final Answer:
6