Pipes and Cisterns – Pairwise fill times given, find time for all three together: Three pipes A, B and C are connected to a tank. A and B together fill the tank in 6 hours, B and C together in 10 hours, and A and C together in 12 hours. How long will A, B and C take to fill the tank when all three operate together from the start?

Introduction / Context:
When pairwise times are known, we can add the corresponding rates to deduce the total rate for all three together. This uses linearity of rates and a simple identity.

Given Data / Assumptions:

• (A + B) time = 6 h ⇒ rate = 1/6 per h.
• (B + C) time = 10 h ⇒ rate = 1/10 per h.
• (A + C) time = 12 h ⇒ rate = 1/12 per h.

Concept / Approach:
Add the three pair rates: (A + B) + (B + C) + (A + C) = 2(A + B + C). Hence 2(A + B + C) = 1/6 + 1/10 + 1/12. Divide by 2 to get the combined rate.

Step-by-Step Solution:
Sum pair rates: LCM 60 ⇒ 1/6 + 1/10 + 1/12 = 10/60 + 6/60 + 5/60 = 21/60 = 7/20.Therefore, A + B + C rate = (7/20) / 2 = 7/40 per h.Time together = 1 / (7/40) = 40/7 h ≈ 5 h 43 min.

Verification / Alternative check:
Check reasonableness: 40/7 ≈ 5.71 h lies lower than any pair time (6, 10, 12), which is expected since three pipes beat any two.

Why Other Options Are Wrong:
9 h and 6 h contradict the derived rate; 7 h 36 min is not equal to 40/7 h.

Common Pitfalls:
Taking an average of times (invalid) instead of working with rates, or forgetting to divide the sum by 2.

Final Answer:
40/7 h