The area of a rectangle is calculated using the formula \( \text{Area} = \text{length} \times \text{width} \). For DeeDee's original rectangle with sides measuring \( x \) inches and \( y \) inches, the area can be expressed as:
\[ \text{Area}_{\text{original}} = x \times y. \]
When she dilates the rectangle using a scale factor of \( \frac{1}{4} \), the new dimensions become:
\[ \text{New length} = \frac{x}{4} \quad \text{and} \quad \text{New width} = \frac{y}{4}. \]
The area of the dilated rectangle is then calculated as follows:
\[ \text{Area}_{\text{dilated}} = \left(\frac{x}{4}\right) \times \left(\frac{y}{4}\right) = \frac{xy}{16}. \]
To compare the areas of the original and dilated rectangles, we can calculate the ratio of the original area to the dilated area:
\[ \text{Ratio} = \frac{\text{Area}{\text{original}}}{\text{Area}{\text{dilated}}} = \frac{xy}{\frac{xy}{16}} = 16. \]
This means the area of the original rectangle is \( 16 \) times greater than the area of the dilated rectangle. Essentially, when a shape is dilated, both its dimensions and area are affected by the square of the scale factor. Since the scale factor is \( \frac{1}{4} \), the area is reduced by a factor of \( \left(\frac{1}{4}\right)^2 = \frac{1}{16} \). Thus, the area of the original rectangle is \( 16 \) times that of the dilated rectangle, confirming the result of our calculations.