To express the relationship between the area of the original square and the dilated square, we first need to calculate the areas of both squares.
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Area of the original square: The side length is 4 units, so we calculate the area: \[ \text{Area}_{\text{original}} = \text{side}^2 = 4^2 = 16 \text{ square units} \]
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Area of the dilated square: The side length of the dilated square is 6 units, so we calculate this area: \[ \text{Area}_{\text{dilated}} = 6^2 = 36 \text{ square units} \]
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Creating the fraction representing the relationship: We want the fraction to have the area of the dilated square as the numerator and the area of the original square as the denominator: \[ \text{Fraction} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{36}{16} \]
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Simplifying the fraction: To simplify \(\frac{36}{16}\), we can divide both the numerator and denominator by their greatest common divisor (GCD), which is 4: \[ \frac{36 \div 4}{16 \div 4} = \frac{9}{4} \]
So the fraction that expresses the relationship is \(\frac{9}{4}\).
Since it looks like the answer choices do not include \(\frac{9}{4}\), let’s evaluate them.
A. \( \frac{16}{36} = \frac{4}{9} \)
B. \( 2 \frac{1}{4} = \frac{9}{4} \)
C. \( \frac{4}{9} \)
D. \( \frac{2}{3} \)
The proper relationship in simplest terms is indeed \( \frac{9}{4} \), and that corresponds with option B when considering the fraction representation.
Therefore, the final answer is: \[ \boxed{2 \frac{1}{4}} \]