To find out how many days late the book was, we need to divide the total fines paid by the daily penalty for overdue books.
The total fines paid is $1.08, and the penalty per day is $0.09.
Calculating the number of days:
\[ \text{Number of days late} = \frac{\text{Total fines}}{\text{Penalty per day}} = \frac{1.08}{0.09} \]
Calculating the division:
\[ \frac{1.08}{0.09} = 12 \]
Thus, the book was 12 days late.
The correct answer is 12 days.