To calculate the area of the vegetable garden, you'll need to multiply the length by the width.
First, convert the mixed numbers to improper fractions:
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For the width: \[ 5 \frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{20 + 1}{4} = \frac{21}{4} \text{ ft} \]
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For the length: \[ 10 \frac{1}{2} = \frac{10 \times 2 + 1}{2} = \frac{20 + 1}{2} = \frac{21}{2} \text{ ft} \]
Now, we can calculate the area: \[ \text{Area} = \text{length} \times \text{width} = \left(\frac{21}{2}\right) \times \left(\frac{21}{4}\right) \]
To multiply the fractions, multiply the numerators and the denominators: \[ \text{Area} = \frac{21 \times 21}{2 \times 4} = \frac{441}{8} \text{ ft}^2 \]
To convert \(\frac{441}{8}\) to a mixed number, divide 441 by 8: \[ 441 \div 8 = 55 \quad \text{(remainder 1)} \] So, \(\frac{441}{8} = 55 \frac{1}{8}\) ft².
The total area of the vegetable garden is: \[ \boxed{55 \frac{1}{8}} \text{ ft}^2 \]