A tank can be filled by two separate pipes in 12 hours and 16 hours respectively. A third pipe can drain the tank completely in 30 hours. If all three pipes are operating simultaneously, how many hours will it take to fill the tank completely?

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Let's calculate the rate at which each pipe fills or empties the tank.

The first pipe fills 1/12 of the tank in one hour.

The second pipe fills 1/16 of the tank in one hour.

The third pipe empties 1/30 of the tank in one hour.

When all three pipes operate simultaneously, the net filling rate per hour is:
Rate = 1/12 + 1/16 - 1/30

To find a common denominator, we need the LCM of 12, 16, and 30, which is 240.

Converting all fractions to the denominator of 240:
1/12 = 20/240
1/16 = 15/240
1/30 = 8/240

Now we can calculate the net rate:
Rate = 20/240 + 15/240 - 8/240 = 27/240

This means that 27/240 of the tank gets filled in one hour when all three pipes operate together.

To fill the entire tank, the time required would be:
Time = 1 ÷ (27/240) = 240/27 = 80/9 = 8 8/9 hours

Therefore, it will take 8 hours and 8/9 of an hour (approximately 8 hours and 53 minutes) to fill the tank completely.

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