After raining for \( \frac{3}{4} \) of an hour, a rain gauge is \( \frac{2}{5} \) filled. If it continues to rain at that rate for 15 more minutes, what fraction of the rain gauge will be filled?

To find how much more the rain gauge will fill, we first determine the rate at which the rain fills the gauge. The gauge is \( \frac{2}{5} \) filled after \( \frac{3}{4} \) of an hour. 1. The rate of filling per hour can be calculated as follows: \[ \text{Rate} = \frac{\text{Amount filled}}{\text{Time}} = \frac{\frac{2}{5}}{\frac{3}{4}} = \frac{2}{5} \cdot \frac{4}{3} = \frac{8}{15} \] This means the rain gauge fills at a rate of \( \frac{8}{15} \) of its capacity per hour. 2. Next, we convert 15 minutes into hours: \[ 15 \text{ minutes} = \frac{15}{60} = \frac{1}{4} \text{ hours} \] 3. Now we calculate how much more the gauge will fill in \( \frac{1}{4} \) hour: \[ \text{Amount filled in } \frac{1}{4} \text{ hour} = \text{Rate} \times \text{Time} = \frac{8}{15} \times \frac{1}{4} = \frac{8}{60} = \frac{2}{15} \] 4. Finally, we sum the fractions filled: \[ \text{Total filled} = \frac{2}{5} + \frac{2}{15} \] To add these, we need a common denominator: \[ \frac{2}{5} = \frac{6}{15} \] So, \[ \text{Total filled} = \frac{6}{15} + \frac{2}{15} = \frac{8}{15} \] Therefore, after it continues to rain for 15 more minutes, the fraction of the rain gauge that will be filled is \( \frac{8}{15} \).