To find the area of the original rectangle with sides measuring \( x \) inches and \( y \) inches, we use the formula for the area of a rectangle, which is given by:
\[ \text{Area} = x \times y \]
Substituting the values \( x = 7 \) and \( y = 3 \), we have:
\[ \text{Area} = 7 \times 3 = 21 \text{ square inches} \]
Next, when DeeDee dilates the rectangle using a scale factor of \( \frac{1}{4} \), the new dimensions of the rectangle become:
\[ \text{New length} = x \times \frac{1}{4} = 7 \times \frac{1}{4} = \frac{7}{4} \text{ inches} \] \[ \text{New width} = y \times \frac{1}{4} = 3 \times \frac{1}{4} = \frac{3}{4} \text{ inches} \]
Now, we can find the area of the dilated rectangle:
\[ \text{Area of dilated rectangle} = \frac{7}{4} \times \frac{3}{4} = \frac{21}{16} \text{ square inches} \]
To compare the areas, we calculate how many times greater the area of the original rectangle is compared to the area of the dilated rectangle. We divide the area of the original rectangle by the area of the dilated rectangle:
\[ \text{Comparison} = \frac{21}{\frac{21}{16}} = 21 \times \frac{16}{21} = 16 \]
Thus, the area of the original rectangle is 16 times greater than the area of the dilated rectangle. In summary, the original rectangle's area is 21 square inches, while the area of the dilated rectangle is \( \frac{21}{16} \) square inches, demonstrating that the original area is significantly larger due to the impact of the dilation scale factor of \( \frac{1}{4} \).
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