AMC 12 2009b Problem 10

A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?

\textbf{(A)}\ \frac12 \textbf{(B)}\ \frac58 \textbf{(C)}\ \frac34 \textbf{(D)}\ \frac56 \textbf{(E)}\ \frac {9}{10}