A physical quantity X is defined by the relation X = (a b^2)/(c +...

According to the principle of homogeneity of dimensions, quantities added together must have the same dimensions. From the denominator $(c + d^2)$:
$[d^2] = [c] = [\mathrm{L}]$
$[d] = [\mathrm{L}^{1/2}]$

The dimensional formula of the entire equation is:
$[X] = \dfrac{[a][b^2]}{[c]}$

We are given that $X$ is pressure and $a$ is force. Substituting their dimensional formulas:
$[\mathrm{M} \mathrm{L}^{-1} \mathrm{T}^{-2}] = \dfrac{[\mathrm{M} \mathrm{L} \mathrm{T}^{-2}][b^2]}{[\mathrm{L}]}$

Simplifying the right side:
$[\mathrm{M} \mathrm{L}^{-1} \mathrm{T}^{-2}] = [\mathrm{M} \mathrm{T}^{-2}][b^2]$

Solving for $[b^2]$:
$[b^2] = \dfrac{[\mathrm{M} \mathrm{L}^{-1} \mathrm{T}^{-2}]}{[\mathrm{M} \mathrm{T}^{-2}]} = [\mathrm{L}^{-1}]$
$[b] = [\mathrm{L}^{-1/2}]$

The dimensional formula of the ratio $\dfrac{b}{d}$ is:
\left[\dfrac{b}{d}\right] = \dfrac{[\mathrm{L}^{-1/2}]}{[\mathrm{L}^{1/2}]} = [\mathrm{L}^{-1}]$<br><br>Answer: \left[\mathrm{L}^{-1}\right]$