A pipe can fill a tank in 3 hours. There are two outlet pipes from the tank which can empty it in 7 and 10 hours respectively. If all three pipes are opened simultaneously, determine how long it will take to fill the tank.

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To solve this problem, we need to calculate the net rate at which the tank fills when all three pipes are operating simultaneously.

First, let's determine the rates for each pipe:
- Inlet pipe can fill the tank in 3 hours → Rate = 1/3 tank per hour
- First outlet pipe can empty the tank in 7 hours → Rate = -1/7 tank per hour
- Second outlet pipe can empty the tank in 10 hours → Rate = -1/10 tank per hour

The net rate at which the tank fills is the sum of all three rates:
Net rate = 1/3 - 1/7 - 1/10

To add these fractions, we need a common denominator. For 7 and 10, the LCM is 70:
1/7 = 10/70 and 1/10 = 7/70

So, 1/7 + 1/10 = 10/70 + 7/70 = 17/70

Now, our equation becomes:
Net rate = 1/3 - 17/70

To subtract these fractions, we need a common denominator. The LCM of 3 and 70 is 210:
1/3 = 70/210 and 17/70 = 51/210

Net rate = 70/210 - 51/210 = 19/210 tank per hour

The time taken to fill the tank is the reciprocal of the net rate:
Time = 1 ÷ (19/210) = 210/19 ≈ 11 hours

Therefore, it will take approximately 11 hours to fill the tank when all three pipes are opened simultaneously.

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