To find out how much farther Gabrielle lives from work compared to her brother, we need to compare the distances each of them lives from their respective workplaces.
Gabrielle's distance from the hospital is: \[ 3 \frac{7}{10} \text{ miles} \] We can convert this into an improper fraction: \[ 3 \frac{7}{10} = \frac{3 \times 10 + 7}{10} = \frac{30 + 7}{10} = \frac{37}{10} \]
Her brother's distance from the law firm is: \[ 2 \frac{2}{5} \text{ miles} \] Similarly, we convert this to an improper fraction: \[ 2 \frac{2}{5} = \frac{2 \times 5 + 2}{5} = \frac{10 + 2}{5} = \frac{12}{5} \]
Next, we need a common denominator to subtract these distances. The least common multiple of 10 and 5 is 10. We can express \(\frac{12}{5}\) with a denominator of 10: \[ \frac{12}{5} = \frac{12 \times 2}{5 \times 2} = \frac{24}{10} \]
Now we can find the difference between the two distances: \[ 3 \frac{7}{10} - 2 \frac{2}{5} = \frac{37}{10} - \frac{24}{10} = \frac{37 - 24}{10} = \frac{13}{10} \]
Finally, we convert \(\frac{13}{10}\) back to a mixed number: \[ \frac{13}{10} = 1 \frac{3}{10} \]
Thus, Gabrielle lives \(\boxed{1 \frac{3}{10}}\) miles farther from work than her brother.
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