CEMC Cayley, 2022

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. (Note that $\frac{1}{2} = \frac{5}{10} = \frac{11.5}{23}$ which we can use to compare the given fractions
to $\frac{1}{2}$.)

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 gretest
value among the five choices.

(Alternatively, we could have converted the fractions into decimals and
used their decimal approximations to compare their sizes.)