Taps X and Y can fill a tank in 30 minutes and 40 minutes respectively, while tap Z can empty a full tank in 60 minutes. If all three taps are opened together and kept running continuously, in how many minutes will the tank be completely filled?

Introduction / Context:
This question involves two inlet taps and one outlet tap operating at the same time. We need to find the net filling rate and then the time required to fill the tank completely. It is a typical pipes and cistern situation with both filling and emptying effects acting simultaneously.

Given Data / Assumptions:
- Tap X fills the tank in 30 minutes
- Tap Y fills the tank in 40 minutes
- Tap Z empties a full tank in 60 minutes
- All three taps are opened together and kept open
- The tank is initially empty and has no other leakage

Concept / Approach:
If a tap fills a tank in T minutes, its rate is 1/T of the tank per minute. If a tap empties the tank in T minutes, its rate is -1/T of the tank per minute. When taps operate together, their rates add algebraically. After we compute the net rate, time is simply the reciprocal of this rate.

Step-by-Step Solution:
Step 1: Rate of X = 1/30 tank per minute. Step 2: Rate of Y = 1/40 tank per minute. Step 3: Rate of Z (emptying) = -1/60 tank per minute. Step 4: Net rate when all three are open = 1/30 + 1/40 - 1/60. Step 5: Take LCM of 30, 40 and 60 which is 120. Then rates are 4/120, 3/120 and -2/120 respectively. Step 6: Net rate = (4 + 3 - 2) / 120 = 5/120 = 1/24 tank per minute. Step 7: Time required to fill the tank = 1 divided by 1/24 = 24 minutes.

Verification / Alternative check:
We can also reason proportionally. In 24 minutes, X would fill 24/30 = 4/5 of the tank, Y would fill 24/40 = 3/5 of the tank, and Z would empty 24/60 = 2/5 of the tank. Net filled fraction = 4/5 + 3/5 - 2/5 = 5/5 = 1 whole tank, which confirms the time of 24 minutes.

Why Other Options Are Wrong:
48 minutes and 72 minutes: These assume much slower net filling rates; if used, the net fraction filled would exceed 1 for the given inflows and outflow.
None of these: This is incorrect because 24 minutes is an exact solution and matches the given options list.

Common Pitfalls:
The main mistake is to add or subtract the times instead of adding the rates. Some may also forget that the emptying tap has a negative contribution. Miscalculating the LCM or the fraction operations can also lead to wrong net rates. Writing each rate explicitly and verifying the arithmetic helps avoid these errors.

Final Answer:
When all three taps run together, the tank will be completely filled in 24 minutes.