To solve the problem, we start by finding the area of the original square.
The side length of the original square is given as 5 cm. The area \( A \) of a square is calculated using the formula:
\[ A = \text{side length}^2 \]
Thus, the area of the original square is:
\[ A_{\text{original}} = 5^2 = 25 \text{ cm}^2 \]
Next, we apply the dilation. The square is dilated using a scale factor of 4. This means that each side length of the square will be multiplied by 4. Therefore, the new side length \( s_{\text{dilated}} \) of the dilated square is:
\[ s_{\text{dilated}} = 5 \times 4 = 20 \text{ cm} \]
We then calculate the area of the dilated square using the same area formula:
\[ A_{\text{dilated}} = (s_{\text{dilated}})^2 = 20^2 = 400 \text{ cm}^2 \]
Now, we want to find the ratio of the area of the dilated square to the area of the original square. This ratio \( R \) is given by:
\[ R = \frac{A_{\text{dilated}}}{A_{\text{original}}} = \frac{400}{25} \]
Calculating this ratio gives:
\[ R = \frac{400}{25} = 16 \]
Thus, the ratio of the area of the dilated square to the area of the original square is:
\[ \boxed{16} \]
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