The ratio of $\sigma$ bond in ${\mathrm{P}}_4{\mathrm{O}}_{10}$ and ${\mathrm{P}}_4{\mathrm{O}}_6$ is respectively.

Structure of ${\mathrm{P}}_4{\mathrm{O}}_{10}$ is shown below which shows that it has 16 $\sigma$ bonds.Structure of ${\mathrm{P}}_4{\mathrm{O}}_6$ is shown below which shows that it has 12 $\sigma$ bonds.

Ratio of sigma bond of P₄O₁₀ and P₄O₆ is:

$\begin{array}{ccc}\frac{\mathrm{Sigma} \mathrm{bonds} \mathrm{of} {\mathrm{P}}_4{\mathrm{O}}_{10}}{\mathrm{Sigma} \mathrm{bonds} \mathrm{of} {\mathrm{P}}_4{\mathrm{O}}_6}& =& \frac{16 \mathrm{\sigma } \mathrm{bonds}}{12 \mathrm{\sigma } \mathrm{bonds}}\\ & =& \frac{4}{3}\end{array}$