Comparing fractions

$\frac{44444}{55555} = \frac{4}{5}$

$\frac{5555}{6666} = \frac{5}{6}$

$\frac{666}{777} = \frac{6}{7}$

$\frac{77}{88} = \frac{7}{8}$

$\frac{8}{9} = \frac{8}{9}$

This gives us that $\frac{8}{9}$ is greater than all. Thus $\frac{8}{9}$ is the answer.

$\begin{cases} A = \dfrac {44444}{55555} & = \dfrac 45 & = \dfrac {2016}{2520} \\ B = \dfrac {5555}{6666} & = \dfrac 56 & = \dfrac {2100}{2520} \\ C = \dfrac {666}{777} & = \dfrac 67 & = \dfrac {2160}{2520} \\ D = \dfrac {77}{88} & = \dfrac 78 & = \dfrac {2205}{2520} \\ E = \dfrac 89 & & = \dfrac {2240}{2520} \end{cases}$

The largest fraction is $\boxed{E}$ .

Let's generalize this. For:

$\begin{aligned} r&= \frac n{n+1} \\ &= \frac {n+1-1}{n+1} \\ &= 1- \frac 1{n+1}\end{aligned}$

This means that for $n>0$ , $r$ increases with $n$ .