M_3 A_2 is a sparingly soluble salt of molar mass y g mol ^ -1 and...

The molar solubility $S$ of the salt is given by $S = \dfrac{x}{y}$ mol L$^{-1}$.

The dissociation of the sparingly soluble salt $M_3A_2$ is:

$M_3A_2 \rightleftharpoons 3M^{2+} + 2A^{3-}$

The molar concentration of the anion $A^{3-}$ is:

$[A^{3-}] = 2S$

The solubility product $K_{sp}$ of the salt is:

$K_{sp} = [M^{2+}]^3 [A^{3-}]^2 = (3S)^3 (2S)^2 = 27S^3 \times 4S^2 = 108S^5$

The ratio of the molar concentration of the anion to the solubility product is:

$\dfrac{[A^{3-}]}{K_{sp}} = \dfrac{2S}{108S^5} = \dfrac{1}{54S^4}$

Substituting $S = \dfrac{x}{y}$ into the expression, we get:

$\text{Ratio} = \dfrac{1}{54 \left(\dfrac{x}{y}\right)^4} = \dfrac{1}{54} \cdot \dfrac{y^4}{x^4}$

Answer: $\dfrac{1}{54} \cdot \dfrac{y^4}{x^4}$