2009 AMC 10B Problem 19

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?

\text{(A) } \frac 12 \qquad \text{(B) } \frac 58 \qquad \text{(C) } \frac 34 \qquad \text{(D) } \frac 56 \qquad \text{(E) } \frac 9{10}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.