To find the fraction that expresses 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:
- Side length = 4 units
- Area = \( \text{side}^2 = 4^2 = 16 \) square units
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Area of the dilated square:
- Side length = 6 units
- Area = \( \text{side}^2 = 6^2 = 36 \) square units
Next, we express the relationship by forming a fraction with the area of the dilated square (36) as the numerator and the area of the original square (16) as the denominator:
\[ \frac{36}{16} \]
Now, we simplify this fraction:
\[ \frac{36 \div 4}{16 \div 4} = \frac{9}{4} \]
Thus, the fraction that expresses the relationship between the area of the dilated square and the area of the original square is:
\[ \frac{9}{4} \]
However, since the problem asks for a fraction with the numerator expressing the area of the dilated square and the denominator expressing the area of the original square, it is typically expressed as follows:
\[ \frac{\text{Area of Dilated Square}}{\text{Area of Original Square}} = \frac{36}{16} \]
This fraction can also be evaluated numerically to fit with the provided answer choices. Converting \( \frac{36}{16} \) gives:
\[ \frac{36 \div 4}{16 \div 4} = \frac{9}{4} \]
Looking at the choices provided, none exactly match \( \frac{9}{4} \). Therefore, if the intended context was to present this fraction in any specific form or relationship, please ensure the correct setup aligns with your input options or clarify if another fraction representation is needed.
If we take \( \frac{Area_{dilated}}{Area_{original}} \) directly as provided:
\[ \frac{36}{16} \rightarrow \frac{9}{4} \]
However, none listed corresponded to direct simplification. Therefore, the specific fraction expressed as relationships must follow the established theory but also could simply track \( \frac{Area_{original}}{Area_{dilated}} \) if asked inversely.
Thus, \( \frac{36}{16} \) can be seen as \( \frac{9}{4} \) but answer accuracy to your friendly fraction might observe towards \( \frac{4}{9} \) inversely, noting that the aligning does provide structurally close association rather than responding sharply as highlighted.