To find out how many birdhouses Sammie can make, we need to divide the total length of the board by the amount of wood used for each birdhouse.
First, let's convert the mixed numbers to improper fractions.
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Convert \(10 \frac{2}{5}\) to an improper fraction: \[ 10 \frac{2}{5} = \frac{10 \times 5 + 2}{5} = \frac{50 + 2}{5} = \frac{52}{5} \]
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Convert \(1 \frac{3}{10}\) to an improper fraction: \[ 1 \frac{3}{10} = \frac{1 \times 10 + 3}{10} = \frac{10 + 3}{10} = \frac{13}{10} \]
Now we can divide the total length of the board by the length needed for each birdhouse: \[ \text{Number of birdhouses} = \frac{52/5}{13/10} \]
To divide by a fraction, we multiply by its reciprocal: \[ \frac{52}{5} \div \frac{13}{10} = \frac{52}{5} \times \frac{10}{13} \]
Now we multiply the fractions: \[ = \frac{52 \times 10}{5 \times 13} = \frac{520}{65} \]
Now we simplify \(\frac{520}{65}\): \[ 520 \div 65 = 8 \]
Thus, Sammie can make 8 birdhouses.