Two pipes fill a tank separately in \(x\) hours and \((x+4)\) hours, respectively. Working together, they fill the tank in \(\frac{24}{5}\) hours. How many hours will the faster pipe take to fill the tank alone?

Correct answer: 8 hours

Adding the filling rates gives \(\frac{1}{x}+\frac{1}{x+4}=\frac{5}{24}\). On simplifying, \(24(2x+4)=5x(x+4)\), so \(5x^2-28x-96=0\). Factoring gives \((x-8)(5x+12)=0\), yielding \(x=8\) or \(x=-\frac{12}{5}\). Since time cannot be negative, \(x=8\) hours. The other pipe takes 12 hours, so the 8-hour pipe is the faster one. Exam tip: In filling-rate problems, add the reciprocals of the individual times and reject any negative time value.