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?
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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 \( \frac{8}{15} \).