Two taps can fill a cistern in 10 minutes and 15 minutes, respectively. With the waste pipe (outlet) also open, the cistern is filled in 18 minutes. How long does the waste pipe alone take to empty a full cistern?
Correct Answer: 9 minutes
Introduction / Context:This is similar to standard “pipes and cisterns” problems. We compute the combined inlet rate, subtract the waste rate to get the observed net rate, and then isolate the waste rate to find its emptying time.
Given Data / Assumptions:
- Inlet rates: 1/10 and 1/15 tank/min.
- Net filling time with waste open = 18 min ⇒ net rate = 1/18 tank/min.
Concept / Approach:If w is the waste rate (tank/min), then (1/10 + 1/15) − w = 1/18. Solve for w and invert to get time to empty a full cistern.
Step-by-Step Solution:
1/10 + 1/15 = (3 + 2)/30 = 1/6.1/6 − w = 1/18 ⇒ w = 1/6 − 1/18 = 2/18 = 1/9 tank/min.Emptying time = 1 / (1/9) = 9 minutes.Verification / Alternative check:Net = 1/6 − 1/9 = (3 − 2)/18 = 1/18, as given.
Why Other Options Are Wrong:7/13/23 minutes produce wrong net rates and fail the 18-minute condition.
Common Pitfalls:Arithmetic with fractions; prefer a common denominator to avoid mistakes.
Final Answer:9 minutes