Which of the following fractions has the greatest value?
(A) $\frac{44444}{55555}$
(B) $\frac{5555}{6666}$
(C) $\frac{666}{777}$
(D) $\frac{77}{88}$
(E) $\frac{8}{9}$
We write each of the choices in lowest terms:

$$
\frac{44444}{55555}=\frac{4(11111)}{5(11111)}=\frac{4}{5} \quad \frac{5555}{6666}=\frac{5(1111)}{6(1111)}=\frac{5}{6} \quad \frac{666}{777}=\frac{6(111)}{7(111)}=\frac{6}{7} \quad \frac{77}{88}=\frac{7(11)}{8(11)}=\frac{7}{8} \quad \frac{8}{9}
$$

(The last choice was already in lowest terms.)

Next, we note that $\frac{4}{5}=1-\frac{1}{5}$ and $\frac{5}{6}=1-\frac{1}{6}$ and $\frac{6}{7}=1-\frac{1}{7}$ and $\frac{7}{8}=1-\frac{1}{8}$ and $\frac{8}{9}=1-\frac{1}{9}$.

The fraction with the greatest value will be the one that is equal to 1 minus the smallest fraction.

Since $\frac{1}{9}<\frac{1}{8}<\frac{1}{7}<\frac{1}{6}<\frac{1}{5}$, then the fraction with the greatest value is $\frac{8}{9}$.

ANSWER: (E)