Two pipes fill a tank, and one pipe can fill the tank three times as fast as the other. If together the two pipes can fill the tank in 36 minutes, in how many minutes will the slower pipe alone fill the tank completely?

Introduction / Context:
This problem is a straightforward application of work and time concepts to pipes and cisterns. It checks whether the student can translate a speed comparison statement like "three times as fast" into an algebraic relation, and then combine the individual rates to match the given joint time. Questions of this type are very common in entrance exams and aptitude tests.

Given Data / Assumptions:

There are two pipes filling the same tank.

The faster pipe fills the tank three times as fast as the slower pipe.

Working together, both pipes fill the tank in 36 minutes.

All flow rates are assumed to be constant, and there are no leaks.

Concept / Approach:
Work and time problems are handled by assigning a variable for the time taken by one agent, then expressing the other agent's time or rate in terms of that variable. The combined rate is the sum of individual rates. Since time is given for both pipes working together, we equate the combined rate to one tank divided by the combined time, and solve for the unknown time of the slower pipe. Using minutes as the unit throughout keeps the calculation consistent.

Step-by-Step Solution:
Let the time taken by the slower pipe alone to fill the tank be t minutes.Then the rate of the slower pipe = 1/t tank per minute.The faster pipe fills three times as fast, so its rate is 3 * (1/t) = 3/t tank per minute.Therefore, the combined rate of both pipes working together is 1/t + 3/t = 4/t tank per minute.Given that together they fill the tank in 36 minutes, the combined rate is also 1/36 tank per minute.So we set 4/t = 1/36.Cross multiplying gives t = 4 * 36 = 144.Hence, the slower pipe alone will fill the tank in 144 minutes.
Verification / Alternative check:
Check the rates: slower pipe rate = 1/144 tank per minute.Faster pipe rate = 3/144 = 1/48 tank per minute.Combined rate = 1/144 + 1/48 = (1 + 3) / 144 = 4/144 = 1/36 tank per minute.Time for one tank at this rate is 36 minutes, which matches the problem statement. Therefore, the solution is consistent.
Why Other Options Are Wrong:
108 minutes, 96 minutes or 72 minutes would make the slower pipe too fast, which would increase the combined rate and reduce the joint filling time below the given 36 minutes. On the other hand, 180 minutes would make the slower pipe too slow, decreasing the combined rate and increasing the joint filling time beyond 36 minutes. Only 144 minutes produces the exact combined time required.

Common Pitfalls:
Some students misinterpret “three times as fast” as adding three minutes or halving instead of forming a clear multiplicative relation. Others may equate times instead of rates, writing 1/t + 3t or similar incorrect expressions. Another common mistake is inverting the equations and writing t/4 = 36, which would lead to t = 144/ rather than 144, or mixing up units. Keeping the notion that rate = 1/time and being consistent with that throughout prevents these errors.

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