The efficiencies of two filling pipes are in the ratio 4 : 5. A third pipe is an emptying pipe whose efficiency is two thirds of the average efficiency of the first two pipes, and it can empty a completely full tank in 36 minutes. In how many minutes can the two filling pipes together fill the empty tank?

Introduction / Context:
This problem relates efficiencies of three pipes: two inlets and one outlet. The outlet pipe rate is linked to the average rate of the first two pipes, and its emptying time is given. From this information we can derive the absolute rates of the filling pipes and then find the time they need to fill the tank together.

Given Data / Assumptions:
- Efficiency ratio of two filling pipes A and B = 4 : 5
- Average efficiency of A and B = (4 + 5) / 2 = 4.5 parts
- Emptying pipe C has efficiency = two thirds of this average = 2/3 * 4.5 = 3 parts
- Pipe C alone empties a full tank in 36 minutes
- All efficiencies are measured in the same units of tank per minute

Concept / Approach:
We treat the given ratio numbers as proportional to actual rates. Since C empties the tank in 36 minutes, we can equate its proportional efficiency to the real rate 1/36 and find the scale factor. Using this factor we find actual rates for A and B. Then we add the rates of A and B to get their combined rate and find the time to fill one tank.

Step-by-Step Solution:
Step 1: Let the common efficiency unit be k. Then rates are: A = 4k, B = 5k, C = 3k. Step 2: Pipe C empties a full tank in 36 minutes, so 3k = 1/36 tank per minute. Step 3: Therefore k = 1 / (36 * 3) = 1/108. Step 4: Actual rate of A = 4k = 4/108 = 1/27 tank per minute. Step 5: Actual rate of B = 5k = 5/108 tank per minute. Step 6: Combined rate of A and B = 1/27 + 5/108 = 4/108 + 5/108 = 9/108 = 1/12 tank per minute. Step 7: Time required by A and B together = 1 divided by 1/12 = 12 minutes.

Verification / Alternative check:
We can check quickly: if A and B together fill the tank in 12 minutes, they fill 1/12 of the tank per minute. Using k = 1/108, the sum of their proportional rates 4k + 5k equals 9k which is 9/108 = 1/12, exactly matching the calculated combined rate.

Why Other Options Are Wrong:
16, 14 and 20 minutes: These times correspond to smaller combined rates than 1/12 tank per minute. When back substituted, they are inconsistent with the emptying information that C empties the tank in 36 minutes based on efficiency two thirds of the average of A and B.

Common Pitfalls:
Many students forget that the given efficiencies are in ratio form and must be linked to an actual rate using the emptying condition. Another mistake is computing the average of 4 and 5 incorrectly, or using one third instead of two thirds. Careful reading and consistent use of proportional reasoning are essential.

Final Answer:
Both filling pipes together can fill the tank in 12 minutes.