To find the fraction expressing the relationship between the area of the dilated square and the original square, we first need to calculate the areas of both squares.
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Area of the original square: The side length of the original square is 4 units. The area \( A_1 \) of a square is given by the formula: \[ A_1 = \text{side}^2 = 4^2 = 16 \text{ square units} \]
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Area of the dilated square: The side length of the dilated square is 6 units. The area \( A_2 \) of the dilated square is: \[ A_2 = \text{side}^2 = 6^2 = 36 \text{ square units} \]
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Expressing the relationship as a fraction: The fraction expressing the relationship with the area of the dilated square in the numerator and the area of the original square in the denominator is: \[ \frac{A_2}{A_1} = \frac{36}{16} \]
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Simplifying the fraction: We can simplify the fraction \( \frac{36}{16} \) by finding the greatest common divisor (GCD) of 36 and 16, which is 4: \[ \frac{36 \div 4}{16 \div 4} = \frac{9}{4} \]
Thus, the fraction in lowest terms that expresses the relationship between the area of the dilated square and the original square is: \[ \boxed{\frac{9}{4}} \]