To find the ratio of the base area to the curved surface area of the cylinder, follow these steps:

1. Base Area:
The base of the cylinder is a circle with radius \( r = 5 \) cm.
\[
\text{Base Area} = \pi r^2 = \pi \times 5^2 = 25\pi \text{ cm}^2
\]

2. Curved Surface Area:
The curved surface area of the cylinder is given by:
\[
\text{Curved Surface Area} = 2 \pi r h = 2 \pi \times 5 \times 10 = 100\pi \text{ cm}^2
\]

3. Ratio of Base Area to Curved Surface Area:
\[
\text{Ratio} = \frac{\text{Base Area}}{\text{Curved Surface Area}} = \frac{25\pi}{100\pi} = \frac{25}{100} = \frac{1}{4}
\]

So, the ratio of the base area to the curved surface area is:

C. 1:4