To determine how far Ernesto plans to run this week, we first need to calculate \( 4 \frac{3}{4} \) times the distance he ran last week, which is \( 2 \frac{1}{5} \) miles.
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Convert the mixed numbers to improper fractions:
For \( 2 \frac{1}{5} \): \[ 2 \frac{1}{5} = 2 \times 5 + 1 = 10 + 1 = 11 \quad \text{(improper fraction: } \frac{11}{5}) \]
For \( 4 \frac{3}{4} \): \[ 4 \frac{3}{4} = 4 \times 4 + 3 = 16 + 3 = 19 \quad \text{(improper fraction: } \frac{19}{4}) \]
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Now calculate \( \frac{19}{4} \times \frac{11}{5} \): \[ \frac{19}{4} \times \frac{11}{5} = \frac{19 \times 11}{4 \times 5} = \frac{209}{20} \]
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We can convert \( \frac{209}{20} \) to a mixed number:
- Divide \( 209 \) by \( 20 \): \[ 209 \div 20 = 10 \quad \text{(with a remainder of } 9\text{)} \]
- So, \( 209 = 20 \times 10 + 9 \), which yields the mixed number: \[ 10 \frac{9}{20} \]
Thus, the distance Ernesto plans to run this week is \( \boxed{10 \frac{9}{20}} \).