To determine how many whole boards of length \( \frac{4}{5} \) yard can be cut from a piece of wood that is \( 10 \frac{1}{4} \) yards long, we'll first convert \( 10 \frac{1}{4} \) yards into an improper fraction.
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Convert \( 10 \frac{1}{4} \) to an improper fraction: \[ 10 \frac{1}{4} = \frac{10 \times 4 + 1}{4} = \frac{40 + 1}{4} = \frac{41}{4} \]
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Now, we need to divide \( \frac{41}{4} \) by \( \frac{4}{5} \): \[ \frac{41}{4} \div \frac{4}{5} = \frac{41}{4} \times \frac{5}{4} = \frac{41 \times 5}{4 \times 4} = \frac{205}{16} \]
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Now, perform the division to find how many whole boards this represents: \[ 205 \div 16 \approx 12.8125 \]
This means Andre can cut a total of \( 12 \) whole boards from the piece of wood.
Therefore, the final answer is: \[ \boxed{12} \]
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