To find the area of the original rectangle, we can use the formula for the area of a rectangle, which is given by:
\[ \text{Area} = \text{length} \times \text{width} \]
In this case, the sides of the rectangle are \(x\) inches and \(y\) inches. Therefore, the area \(A\) of the original rectangle is:
\[ A = x \cdot y \]
Next, we need to determine the area of the rectangle after it has been dilated by a scale factor of \(\frac{1}{4}\). When a shape is dilated, each dimension of the shape is multiplied by the scale factor. Therefore, the new dimensions of the rectangle after dilation will be:
\[ \text{New length} = x \cdot \frac{1}{4} = \frac{x}{4} \] \[ \text{New width} = y \cdot \frac{1}{4} = \frac{y}{4} \]
Now, we can find the area \(A'\) of the dilated rectangle:
\[ A' = \text{New length} \times \text{New width} = \left(\frac{x}{4}\right) \cdot \left(\frac{y}{4}\right) \] \[ A' = \frac{xy}{16} \]
Next, we need to find out how many times greater the area of the original rectangle is compared to the area of the dilated rectangle. To do this, we can take the ratio of the area of the original rectangle to the area of the dilated rectangle:
\[ \text{Ratio} = \frac{A}{A'} = \frac{xy}{\frac{xy}{16}} = xy \cdot \frac{16}{xy} = 16 \]
Thus, the area of the original rectangle is \(16\) times greater than the area of the dilated rectangle.
To summarize:
- The area of the original rectangle is \(xy\) square inches.
- The area of the dilated rectangle is \(\frac{xy}{16}\) square inches.
- The area of the original rectangle is \(16\) times greater than the area of the dilated rectangle.
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