\( 1<\begin{array}{l}\text { A winery has a vat with two pipes leading to it. The inlet pipe can fill the vat in } 6 \\ \text { hours, while the outlet pipe can empty it in } 10 \text { hours. How long will it take to fill } \\ \text { the vat if both pipes are left open? }\end{array} \)

Let's break this down! The inlet pipe fills the vat at a rate of \(\frac{1}{6}\) of the vat per hour, while the outlet pipe empties it at \(\frac{1}{10}\) of the vat per hour. When both pipes are open, their combined effect is calculated by subtracting the outlet's rate from the inlet's rate: \[ \text{Net rate} = \frac{1}{6} - \frac{1}{10} \] To subtract these fractions, first find a common denominator (which is 30): \[ \frac{1}{6} = \frac{5}{30}, \quad \frac{1}{10} = \frac{3}{30} \] Now, subtract: \[ \text{Net rate} = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15} \] This means that when both pipes are open, they fill \(\frac{1}{15}\) of the vat in one hour. Therefore, to fill the entire vat, it would take: \[ \text{Time} = 15 \text{ hours.} \] So, if both pipes are left open, it will take 15 hours to fill the vat!