If $\frac{4}{5}:\frac{12}{10}::\frac{x}{6}:\frac{9}{2}$, find the value of $x$.

Given the proportion relation:
$$\frac{\left(\frac{4}{5}\right)}{\left(\frac{12}{10}\right)}=\frac{\left(\frac{x}{6}\right)}{\left(\frac{9}{2}\right)}$$

Simplify the left-hand side (LHS):

$$\text{LHS}=\frac{4}{5}\times \frac{10}{12}=\frac{40}{60}=\frac{2}{3}$$

Simplify the right-hand side (RHS):

$$\text{RHS}=\frac{x}{6}\times \frac{2}{9}=\frac{2x}{54}=\frac{x}{27}$$

Now, set LHS equal to RHS:

$$\frac{2}{3}=\frac{x}{27}$$

$$3x=2\times 27$$

$$3x=54⟹x=18$$