Partial Pressure Ratio of Nitrogen to Oxygen :

The partial pressure of a gas is calculated using its mole fraction. The mole fraction is the ratio of the number of moles of the gas to the total number of moles of all gases.

For nitrogen ($ \text{N}_2 $), the mole fraction $\frac{n_{\text{N}_2}}{n_{\text{N}_2} + n_{\text{O}_2} + n_{\text{Ar}}}$ can be simplified as $\frac{70/28}{70/28 + 27/32 + 3/40}$.

For oxygen ($ \text{O}_2 $), the mole fraction is $\frac{27/32}{70/28 + 27/32 + 3/40}$.

Therefore, the ratio of the partial pressure of nitrogen to oxygen is :

$$ \frac{P_{\text{N}_2}}{P_{\text{O}_2}} = \frac{\text{mole fraction of } \text{N}_2}{\text{mole fraction of } \text{O}_2} = \frac{70/28}{27/32} = 2.96 $$

Partial Pressure Ratio of Oxygen to Argon :

For argon ($ \text{Ar} $), the mole fraction is $\frac{3/40}{70/28 + 27/32 + 3/40}$.

The ratio of the partial pressure of oxygen to argon is :

$$ \frac{P_{\text{O}_2}}{P_{\text{Ar}}} = \frac{\text{mole fraction of } \text{O}_2}{\text{mole fraction of } \text{Ar}} = \frac{27/32}{3/40} = 11.25 $$