Solution:

The gravitational force between two masses \( m_1 \) and \( m_2 \) is:

\[
F = G \frac{m_1 \cdot m_2}{r^2}
\]

To maximize \( F \), we set \( m_1 = x \) and \( m_2 = 100 - x \), then maximize the product \( m_1 \cdot m_2 = x(100 - x) \).

This product is maximized when \( x = 50 \). So, the masses should be split equally.

The ratio of masses is:

\[
\frac{m_1}{m_2} = \frac{50}{50} = 1
\]