MAT Exam 2023 Paper

Concept:Let the total time taken to fill the tank be $x$ hours. Use the unitary method: the fraction of the tank filled in 1 hour by a pipe is the reciprocal of its filling time.Explanation:Pipe $T$ fills the tank in 30 hours, so its rate is $\frac{1}{30}$ per hour.Pipe $H$ fills the tank in 20 hours, so its rate is $\frac{1}{20}$ per hour.For the first $\frac{x}{2}$ hours, only pipe $H$ is open.Work done in this period:$$\frac{x}{2} \times \frac{1}{20} = \frac{x}{40}$$For the remaining $\frac{x}{2}$ hours, both pipes are open.Their combined rate is:$$\frac{1}{30} + \frac{1}{20} = \frac{2+3}{60} = \frac{1}{12}$$Work done in this period:$$\frac{x}{2} \times \frac{1}{12} = \frac{x}{24}$$Total work must equal 1 full tank:$$\frac{x}{40} + \frac{x}{24} = 1$$Taking LCM = 120:$$\frac{3x + 5x}{120} = 1$$$$\frac{8x}{120} = 1$$$$x = \frac{120}{8} = 15$$Thus, the tank will be completely full in 15 hours.Answer:Option A. 15 hours.