Question

DeeDee draws a rectangle with sides measuring x inches and y inches. She dilated the rectangles using scale factor of 1/4. What is the area of the original rectangle in terms of x and y? How many times greater is the area of the original rectangle compared to the area of the dilated rectangle? Explain your answer and show all your work.

Answer 1

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.