To find out how much farther Rhianna lives from work compared to her brother, we need to subtract the distance her brother lives from the distance Rhianna lives.
Rhianna's distance: \(9 \frac{1}{2}\) miles Brother's distance: \(6 \frac{2}{3}\) miles
First, we will convert these mixed numbers to improper fractions.
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For \(9 \frac{1}{2}\): \[ 9 \frac{1}{2} = \frac{9 \times 2 + 1}{2} = \frac{18 + 1}{2} = \frac{19}{2} \]
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For \(6 \frac{2}{3}\): \[ 6 \frac{2}{3} = \frac{6 \times 3 + 2}{3} = \frac{18 + 2}{3} = \frac{20}{3} \]
Now we need to subtract \(\frac{20}{3}\) from \(\frac{19}{2}\). To do that, we need a common denominator. The least common multiple of 2 and 3 is 6.
Next, we convert both fractions to have a denominator of 6:
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Convert \(\frac{19}{2}\) to have a denominator of 6: \[ \frac{19}{2} = \frac{19 \times 3}{2 \times 3} = \frac{57}{6} \]
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Convert \(\frac{20}{3}\) to have a denominator of 6: \[ \frac{20}{3} = \frac{20 \times 2}{3 \times 2} = \frac{40}{6} \]
Now we can subtract: \[ \frac{57}{6} - \frac{40}{6} = \frac{57 - 40}{6} = \frac{17}{6} \]
Finally, we convert \(\frac{17}{6}\) to a mixed number: \[ \frac{17}{6} = 2 \frac{5}{6} \]
Thus, Rhianna lives \(2 \frac{5}{6}\) miles farther from work than her brother.
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