AMC 10 2021a Problem 8

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 $\underline{a} \ \underline{b}?$ $\textbf{(A) }15 \textbf{(B) }30 \textbf{(C) }45 \textbf{(D) }60 \textbf{(E) }75$