1. The collision frequency (f) is given by the formula :

$$ f = \sqrt{2} \pi d^2 v n_v $$

Where:

- $d$ is the diameter of the molecule,

- $v$ is the average velocity of the molecules, and

- $n_v$ is the number density (number of molecules per unit volume).

2. From this equation, we can see that the collision frequency (f) is directly proportional to the number density ($n_v$), because all the other variables ($d$ and $v$) are constant :

$$ f \propto n_v $$

3. Therefore, the ratio of two different collision frequencies ($f_1$ and $f_2$) is equal to the ratio of their corresponding number densities ($n_{v1}$ and $n_{v2}$):

$$ \frac{f_1}{f_2} = \frac{n_{v1}}{n_{v2}} $$

4. Given that the number density increased from $3 \times 10^{19}$ to $12 \times 10^{19}$, we can substitute these values into the equation to find the ratio of the collision frequencies:

$$ \frac{f_1}{f_2} = \frac{3 \times 10^{19}}{12 \times 10^{19}} $$

5. Simplifying this equation gives:

$$ \frac{f_1}{f_2} = 0.25 $$

So, the ratio of the collision frequency of air molecules before and after the increase in number is 0.25.