Which of the following fractions is smaller than the fraction $\frac{64}{18}$?

First, we have to convert the given fraction into its simplest form.

$\frac{64}{18}=\frac{64÷2}{18÷2}=\frac{32}{9}$

If the denominators are the same, then the fraction with the smaller numerator is the lesser fraction.

$$\begin{gathered} \frac{12}{9}<\frac{32}{9} \\ \frac{32}{9}=\frac{32}{9} \\ \frac{54}{9}>\frac{32}{9} \\ \frac{76}{9}>\frac{32}{9} \end{gathered}$$

Therefore, only $\frac{12}{9}$ is smaller than $\frac{64}{18}$.