Two inlets A and B can fill a cistern in 12 minutes and 14 minutes, respectively, while an outlet C can empty it in 8 minutes. If all three taps are opened simultaneously on an empty cistern, what fraction of the cistern remains unfilled at the end of 7 minutes?

Introduction / Context:
Compute the net rate with all three taps, multiply by 7 minutes to get the filled fraction, then subtract from 1 to get the unfilled fraction.

Given Data / Assumptions:

• A = 1/12 tank/min.
• B = 1/14 tank/min.
• C = −1/8 tank/min.

Concept / Approach:
Net rate r = 1/12 + 1/14 − 1/8. Filled in 7 minutes = 7r. Unfilled = 1 − 7r.

Step-by-Step Solution:

Verification / Alternative check:
Converting to decimals yields the same result; 19/24 ≈ 0.7917 unfilled, consistent.

Why Other Options Are Wrong:
5/24 is the filled fraction, not the unfilled; 7/24 and 17/24 do not match the exact arithmetic.

Common Pitfalls:
Confusing “remaining unfilled” with “filled.” Always subtract from 1.

Final Answer:
19/24