2021 AMC 12A Problem 5

When a student multiplied the number 66 by the repeating decimal, \underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}}, where a and b are digits, he did not notice the notation and just multiplied 66 times \underline{1}.\underline{a} \ \underline{b}. Later he found that his answer is 0.5 less than the correct answer. What is the 2-digit number \underline{a} \ \underline{b}?

\textbf{(A) }15 \qquad \textbf{(B) }30 \qquad \textbf{(C) }45 \qquad \textbf{(D) }60 \qquad \textbf{(E) }75

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