Which of the following fractions has the greatest value?
(A) $\frac{3}{10}$
(B) $\frac{4}{7}$
(C) $\frac{5}{23}$
(D) $\frac{2}{3}$
(E) $\frac{1}{2}$
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.)

ANSWER: (D)