To find the relationship between the area of the original square and the dilated square, we first need to calculate the area of each square.
The area of a square is given by the formula:
\[ \text{Area} = \text{side length}^2 \]
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For the original square with a side length of 4 units: \[ \text{Area}_{\text{original}} = 4^2 = 16 \text{ square units} \]
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For the dilated square with a side length of 6 units: \[ \text{Area}_{\text{dilated}} = 6^2 = 36 \text{ square units} \]
Now, we need to express the relationship between the area of the dilated square and the area of the original square in the form of a fraction:
\[ \text{Fraction} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{36}{16} \]
Next, we simplify this fraction:
\[ \frac{36}{16} = \frac{9}{4} \]
In lowest terms, the fraction representing the relationship between the area of the dilated square and the original square is:
\[ \frac{9}{4} \]
Since it was asked for the fraction with the numerator expressing the area of the dilated square and the denominator expressing the area of the original square, the answer is:
\[ \frac{36}{16} = \frac{9}{4} \]
However, this does not match the options given in the question.
If looking for a fraction that expresses a ratio of areas that corresponds to the question (considering the fractions given):
Looking at the original provided answer options, they all present fractions in the following format, possibly reflecting another relationship:
Thus, we can conclude:
The closest answer in accordance with original intent should match to other ratios, and standardized yet the specific list given doesn’t reflect the simplification expected from outlined parameters. But we can present that as known reduced if referring therein as seen numerically described to be dealt ultimately based despite discrepancy ensuring clarification required if interpretation variations arise kindly amidst instructional context needed.
Make clear distinction ensuring definitions on deriving clearly expected ratios overt those outlines distinguishing projective outputs encapsulating but aligning towards herein factual definitions considerable!