To find the fraction that expresses the relationship between the area of the original square and the area of the dilated square, we first calculate the areas of both squares.
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Area of the original square:
- The sides of the original square are 4 units.
- The area \( A_1 \) is given by: \[ A_1 = \text{side}^2 = 4^2 = 16 \text{ square units} \]
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Area of the dilated square:
- The sides of the dilated square are 6 units.
- The area \( A_2 \) is given by: \[ A_2 = \text{side}^2 = 6^2 = 36 \text{ square units} \]
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Forming the fraction:
- The fraction that expresses the relationship between the area of the dilated square (the numerator) and the area of the original square (the denominator) is: \[ \frac{A_2}{A_1} = \frac{36}{16} \]
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Simplifying the fraction:
- We can simplify \( \frac{36}{16} \) by dividing the numerator and denominator by their greatest common divisor, which is 4: \[ \frac{36 \div 4}{16 \div 4} = \frac{9}{4} \]
Therefore, the fraction expressing the relationship between the area of the original square and the dilated square is:
\[ \frac{9}{4} \]
This can be expressed as "Start Fraction 9 over 4 End Fraction", which is not in the provided responses. However, if we are expressing the relationship of the dilated area compared to the original area, the fraction is:
\[ \frac{36}{16} \to \frac{9}{4} \]
If you only want the relationships between the areas (1:4, for instance), you would represent it as \( 36:16 \) simplified to \( 9:4 \) if you were to reverse it to express the other way, but here we need \( \frac{A_dilated}{A_original} = \frac{36}{16} = \frac{9}{4} \) which is the area of the dilated square over the original square, leading me to suspect that answer \( 9 \) starts with a potential confusion in interpretation.
The options are incorrect based on original question context; if we are to summarize, \(\frac{36}{16} or \frac{9}{4}\) fits the question of being expressed in lowest terms, hence chose \( \frac{9}{4}\) if provided or similar clear representation.
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