To find the area of the original rectangle, we use the formula for the area of a rectangle, which is:
\[ \text{Area} = \text{length} \times \text{width} \]
In this case, the length is \(x\) inches and the width is \(y\) inches. Therefore, the area of the original rectangle is:
\[ \text{Area}_{\text{original}} = x \times y = xy \text{ square inches} \]
Next, we need to calculate the area of the dilated rectangle after applying a scale factor of \(\frac{1}{4}\). When a shape is dilated by a scale factor \(k\), the new dimensions are obtained by multiplying each dimension by \(k\).
For DeeDee's rectangle:
- The new length after dilation is \(\frac{1}{4}x\)
- The new width after dilation is \(\frac{1}{4}y\)
Now, we can calculate the area of the dilated rectangle:
\[ \text{Area}_{\text{dilated}} = \left(\frac{1}{4}x\right) \times \left(\frac{1}{4}y\right) \]
This simplifies to:
\[ \text{Area}_{\text{dilated}} = \frac{1}{4}x \cdot \frac{1}{4}y = \frac{1}{16}xy \text{ square inches} \]
Now we know the areas of both rectangles:
- Area of the original rectangle: \(xy\)
- Area of the dilated rectangle: \(\frac{1}{16}xy\)
To find how many times greater the area of the original rectangle is compared to the area of the dilated rectangle, we can set up the ratio:
\[ \text{Ratio} = \frac{\text{Area}{\text{original}}}{\text{Area}{\text{dilated}}} = \frac{xy}{\frac{1}{16}xy} \]
When we simplify this ratio, we have:
\[ \text{Ratio} = \frac{xy \cdot 16}{xy} = 16 \]
Thus, the area of the original rectangle is 16 times greater than the area of the dilated rectangle.
In conclusion:
- The area of the original rectangle is \(xy\) square inches.
- The area of the dilated rectangle is \(\frac{1}{16}xy\) square inches.
- The area of the original rectangle is 16 times greater than the area of the dilated rectangle.
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