A tank has four pipes. Through the first three pipes, the tank can be filled in 12 minutes, through the second, the third and the fourth pipe, it can be filled in 15 minutes and through the first and the fourth pipe in 20 minute. How much time will it take all four pipe to fill up the tank?(in min)

Let each pipe's rate be a, b, c, d (tanks per minute). From the conditions: a + b + c = 1/12 (fills in 12 min), b + c + d = 1/15, a + d = 1/20. Subtract third from first: (a+b+c) - (a+d) = 1/12 - 1/20, giving b + c - d = 1/30. Add this to second equation: (b+c+d) + (b+c-d) = 1/15 + 1/30, so 2(b+c) = 3/30 = 1/10, meaning b + c = 1/20. Now all four together: a + b + c + d = (a + d) + (b + c) = 1/20 + 1/20 = 1/10. So all four pipes fill the tank in 10 minutes.