A tank is fitted with 8 pipes, some of them fill the tank and the rest are waste pipes, meant to empty the tank. Each of the pipes that fill the tank can fill it in 8 hours, while each of those that empty the tank can empty it in 6 hours. If all the pipes are left open when the tank is full, it will take exactly 6 hours for the tank to empty. How many of these pipes fill the tank?

Let x be the number of pipes that fill the tank, so the number of waste pipes is 8 minus x. Using the net work rate formula, each filling pipe does 1/8 of the work per hour and each waste pipe does -1/6 of the work per hour. Because all open pipes empty the full tank in exactly 6 hours, their combined rate is -1/6 per hour, giving the equation x(1/8) + (8 - x)(-1/6) = -1/6. Multiplying the entire equation by 24 yields 3x - 32 + 4x = -4, which simplifies to 7x = 28. Solving for x gives the number of filling pipes as 4.

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