To find out how much Alana's dog weighs, we start by calculating the dog's weight as a multiple of the cat's weight.
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First, we convert Alana's cat's weight into an improper fraction. The cat weighs \(8 \frac{1}{2}\) pounds: \[ 8 \frac{1}{2} = \frac{17}{2} \text{ pounds} \]
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Next, we need to calculate the weight of Alana's dog, which is \(8 \frac{5}{6}\) times the weight of the cat: \[ \text{Dog's weight} = \frac{8}{1} \times \frac{17}{2} \times \frac{5}{6} \]
To handle the multiplication, we can first express \(8\) as a fraction: \[ 8 = \frac{8}{1} \]
Now multiply: \[ \frac{8}{1} \times \frac{17}{2} = \frac{8 \times 17}{1 \times 2} = \frac{136}{2} = 68 \]
Now multiply that by \(\frac{5}{6}\): \[ 68 \times \frac{5}{6} = \frac{68 \times 5}{6} = \frac{340}{6} \]
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Now, simplify \(\frac{340}{6}\): \[ \frac{340 \div 2}{6 \div 2} = \frac{170}{3} \]
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To express \(\frac{170}{3}\) as a mixed number, we perform the division: \[ 170 \div 3 = 56 \quad \text{remainder } 2 \] Thus, we can write: \[ \frac{170}{3} = 56 \frac{2}{3} \]
Therefore, Alana's dog weighs \(\boxed{56 \frac{2}{3}}\) pounds.
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