The fractions $\frac{3}{10}$ and $\frac{5}{23}$ are each less than $\frac{1}{2}$ (which is choice $(E)$) so cannot be the greatest among the choices. The fractions $\frac{4}{7}$ and $\frac{2}{3}$ are each greater than $\frac{1}{2}$, so $\frac{1}{2}$ cannot be the greatest among the choices. This means that the answer must be either $\frac{4}{7}$ or $\frac{2}{3}$. Using a common denominator of $3 \times 7=21$, we re-write these fractions as $\frac{4}{7}=\frac{12}{21}$ and $\frac{2}{3}=\frac{14}{21}$ which shows that $\frac{2}{3}$ has the greatest value among the five choices.