One filling pipe A is three times as fast as a second filling pipe B and therefore takes 32 minutes less than B to fill a cistern. If both pipes A and B are opened together when the cistern is empty, in how many minutes will the cistern be completely full?

Introduction / Context:
This problem uses a relationship between the speeds of two filling pipes. Pipe A is three times as fast as pipe B, and we know the difference between their individual filling times. By expressing both times in terms of one variable, we can find each individual time and then their combined time to fill the cistern.

Given Data / Assumptions:
- Pipe A fills the cistern three times as fast as pipe B
- Time taken by pipe A is 32 minutes less than time taken by pipe B
- The cistern is initially empty
- Both pipes are opened together, and we want the time until the cistern is full

Concept / Approach:
Let the time taken by the slower pipe B be x minutes. Since A is three times as fast, A will take x/3 minutes. The difference between these times is given as 32 minutes. Once we find x and x/3, we convert them into rates and add the rates to find the combined filling time.

Step-by-Step Solution:
Step 1: Let time taken by pipe B alone be x minutes. Step 2: Time taken by pipe A alone = x/3 minutes because it is three times as fast. Step 3: Given time difference: x - x/3 = 32. Step 4: Simplify: (2/3)x = 32, so x = 32 * 3 / 2 = 48 minutes. Step 5: Therefore, time for A alone = 48 / 3 = 16 minutes. Step 6: Rate of A = 1/16 tank per minute; rate of B = 1/48 tank per minute. Step 7: Combined rate = 1/16 + 1/48 = (3 + 1) / 48 = 4/48 = 1/12 tank per minute. Step 8: Time taken together = 1 divided by 1/12 = 12 minutes.

Verification / Alternative check:
Check the difference in individual times: B takes 48 minutes, A takes 16 minutes, and 48 - 16 = 32 minutes which matches the problem statement. Also, in 12 minutes A alone would fill 12/16 = 3/4 of the tank and B alone would fill 12/48 = 1/4 of the tank, so together they indeed fill one full tank in 12 minutes.

Why Other Options Are Wrong:
6, 8 or 10 minutes: These times imply combined rates that are inconsistent with the derived individual rates and the given 32 minute difference. They would result in either more or less than one full tank being filled.

Common Pitfalls:
One common mistake is to assume that the faster pipe time is three times longer instead of three times shorter, which reverses the relation. Another error is to mis-handle the fraction when solving x - x/3 = 32. Always write out the equation clearly and simplify step by step.

Final Answer:
Together, pipes A and B will fill the cistern in 12 minutes.