1. Problem statement: A pump fills a tank in 2 hours, but with a leak, it takes 2.5 hours to fill the tank. Find the time in hours for the leak alone to empty the full tank. 2. Define rates: - Pump rate = $\frac{1}{2}$ tank/hour (fills tank in 2 hours) - Combined filling rate with leak = $\frac{1}{2.5} = \frac{1}{\frac{5}{2}} = \frac{2}{5}$ tank/hour 3. Let the leak's emptying rate be $\frac{1}{x}$ tank/hour, where $x$ is the number of hours to empty a full tank. 4. The net filling rate (pump rate minus leak rate) equals combined filling rate: $$\frac{1}{2} - \frac{1}{x} = \frac{2}{5}$$ 5. Solve for $\frac{1}{x}$: $$\frac{1}{2} - \frac{2}{5} = \frac{1}{x}$$ Find common denominator: $$\frac{5}{10} - \frac{4}{10} = \frac{1}{x}$$ $$\frac{1}{10} = \frac{1}{x}$$ So, $$x = 10$$ 6. The leak can empty the tank in 10 hours. Final answer: $10$ hours