Two pipes can fill a tank in 12 hours and 18 hours, respectively, while a third pipe can empty it in 8 hours. How long (in hours) will it take to fill the empty tank if all three pipes are opened simultaneously?

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Accepted Answer

Step 1: Calculate the filling rates of the two pipes and the emptying rate of the third pipe.
- First pipe fills the tank in 12 hours, so its rate = 1/12 tank per hour.
- Second pipe fills the tank in 18 hours, so its rate = 1/18 tank per hour.
- Third pipe empties the tank in 8 hours, so its rate = 1/8 tank per hour (emptying means negative contribution).

Step 2: Calculate the net rate when all three pipes are open simultaneously.
Net rate = (1/12) + (1/18) - (1/8)

Step 3: Find a common denominator for 12, 18, and 8.
- LCM of 12, 18, and 8 is 72.

Step 4: Convert each rate to have denominator 72.
- 1/12 = 6/72
- 1/18 = 4/72
- 1/8 = 9/72

Step 5: Calculate net rate:
Net rate = (6/72) + (4/72) - (9/72) = (6 + 4 - 9)/72 = 1/72 tank per hour.

Step 6: Calculate time to fill the tank at net rate:
Time = 1 / (1/72) = 72 hours.

Step 7: Re-check the calculation as the answer options include 48, 24, 36, and 72.

Re-calculation:
Net rate = (1/12) + (1/18) - (1/8)
= (3/36) + (2/36) - (4.5/36) [converted to denominator 36]
= (3 + 2 - 4.5)/36 = 0.5/36 = 1/72

So the time is indeed 72 hours.

Therefore, the correct answer is 72 hours.

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