An inlet pipe A can fill a tank in x minutes, while an outlet pipe B can empty the fully filled tank in (x − 4) minutes. If both pipes are opened together, they can empty a fully filled tank in 63 minutes. Find the time taken by pipe A alone to completely fill the tank.

Let the capacity of the tank be 1 unit. Rate of filling by pipe A = 1/x (tank per minute) Rate of emptying by pipe B = 1/(x − 4) (tank per minute) When both pipes are opened and the tank is full, it is getting emptied. Net rate of emptying = (rate of B) − (rate of A) So, 1/(x − 4) − 1/x = 1/63 Now solve this equation. 1/(x − 4) − 1/x = [x − (x − 4)] / [x(x − 4)] = 4 / [x(x − 4)] So, 4 / [x(x − 4)] = 1/63 Cross-multiply: x(x − 4) = 4 × 63 x(x − 4) = 252 x² − 4x − 252 = 0 Solve the quadratic: Discriminant D = 4² + 4×252 = 16 + 1008 = 1024 = 32² x = [4 ± 32] / 2 So, x = (4 + 32) / 2 = 36 / 2 = 18 or x = (4 − 32) / 2 = −28 / 2 = −14 (not possible, time cannot be negative) Therefore, the time taken by pipe A alone to fill the tank is = 18 minutes