Given:
Mass of object 1, \( m_1 = 4 \, \text{g} \)
Mass of object 2, \( m_2 = 25 \, \text{g} \)

Let \( v_1 \) be the velocity of object 1 and \( v_2 \) be the velocity of object 2.

Kinetic energy (\( KE \)) of an object is given by the formula:

\[ KE = \frac{1}{2} \cdot m \cdot v^2 \]

Since the kinetic energies of both objects are the same, we have:

\[ \frac{1}{2} \cdot m_1 \cdot v_1^2 = \frac{1}{2} \cdot m_2 \cdot v_2^2 \]

Given that the kinetic energies are equal, we can cancel out the factor of \( \frac{1}{2} \) from both sides:

\[ m_1 \cdot v_1^2 = m_2 \cdot v_2^2 \]

We need to find the ratio of linear momentum, which is given by:

\[ \text{Ratio of linear momentum} = \frac{p_1}{p_2} = \frac{m_1 \cdot v_1}{m_2 \cdot v_2} \]

From the equation \( m_1 \cdot v_1^2 = m_2 \cdot v_2^2 \), we can solve for \( \frac{v_1}{v_2} \):

\[ \frac{v_1}{v_2} = \sqrt{\frac{m_2}{m_1}} \]

Substituting this into the ratio of linear momentum equation:

\[ \text{Ratio of linear momentum} = \frac{m_1 \cdot \sqrt{\frac{m_2}{m_1}}}{m_2} \]

\[ \text{Ratio of linear momentum} = \sqrt{\frac{m_1}{m_2}} \]

Substituting the given values of \( m_1 \) and \( m_2 \):

\[ \text{Ratio of linear momentum} = \sqrt{\frac{4 \, \text{g}}{25 \, \text{g}}} = \sqrt{\frac{4}{25}} = \frac{2}{5} \]

Therefore, the ratio of linear momentum is \( \frac{2}{5} \), which corresponds to option (b).