One pipe can fill a tank three times as fast as another pipe. If together the two pipes can fill the tank in 36 minutes, then in how many minutes will the slower pipe alone fill the tank?

Introduction / Context:
This is a classic pipes and cistern time and work question. Two pipes working together fill a tank in a known time, and one pipe is three times as fast as the other. We must express their rates in terms of a single variable, use the combined rate, and then find the time for the slower pipe.\n

Given Data / Assumptions:
- Let the slower pipe be pipe S and the faster pipe be pipe F
- Pipe F is three times as fast as pipe S
- Together S and F fill the tank in 36 minutes
- Tank is assumed empty at the start and there is no leakage

Concept / Approach:
If a pipe fills a tank in T minutes, its rate is 1/T tank per minute. We set up rates in terms of a single variable for the slower pipe, add the rates for the combined work, and equate this to the given overall filling rate. Then we solve for the time of the slower pipe.

Step-by-Step Solution:
Step 1: Let time taken by the slower pipe alone be x minutes, so its rate is 1/x tank per minute. Step 2: Faster pipe is three times as fast, so its rate is 3/x tank per minute and its time alone would be x/3 minutes. Step 3: Combined rate = 1/x + 3/x = 4/x tank per minute. Step 4: Together they fill the tank in 36 minutes, so combined rate is 1/36 tank per minute. Step 5: Set 4/x = 1/36, so x = 4 * 36 = 144 minutes.

Verification / Alternative check:
If the slower pipe takes 144 minutes, the faster pipe takes 144/3 = 48 minutes. Rates are 1/144 and 1/48. Their sum equals 1/144 + 1/48 = (1 + 3) / 144 = 4/144 = 1/36, confirming that together they indeed fill the tank in 36 minutes.

Why Other Options Are Wrong:
81 minutes: This would make the faster pipe time 27 minutes, giving a combined time different from 36 minutes.
108 minutes: Gives a combined rate higher than required, leading to a filling time less than 36 minutes.
192 minutes: Too slow, leading to a combined time greater than 36 minutes.

Common Pitfalls:
A common mistake is to assume that if one pipe is three times as fast, then its time is simply 36 / 3 without setting up the equation, which leads to wrong results. Another error is to mix up time and rate, or to add times directly instead of adding rates. Always work with rates when dealing with combined work problems.

Final Answer:
The slower pipe alone will fill the tank in 144 minutes.