Video solution:

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Written solution:

We can express $\underline{1}.\overline{\underline{a} \ \underline{b}}$ as an infinite geometric sum: $$\underline{1}.\overline{\underline{a} \ \underline{b}} = 1 + .\underline{a} \ \underline{b} + .00 \ \underline{a} \ \underline{b} + \cdots$$ We can therefore use the formula for the sum of a geometric sum: $$S = \dfrac{\text{first term}}{1 - \text{ratio}} = \dfrac{.\underline{a} \ \underline{b}}{1 - \dfrac{1}{100}}$$ $$= \dfrac{100}{99} \left(. \underline{a}\ \underline{b}\right) = \dfrac{\underline{a}\ \underline{b}}{99}.$$ We also know that $$1. \underline{a}\ \underline{b} = 1 + \dfrac{\underline{a}\ \underline{b}}{100}$$

Then $$\begin{align*} 66\left(1 + \dfrac{\underline{a}\ \underline{b}}{100}\right) + .5 &= 66\left(1 + \dfrac{\underline{a}\ \underline{b}}{99}\right) \\ \dfrac{66}{100}\underline{a}\ \underline{b} + .5 &= \dfrac{66}{99}\underline{a}\ \underline{b} \\ \dfrac{1}{150}\underline{a}\ \underline{b} &= .5 \\ \underline{a}\ \underline{b} &= 75. \end{align*}$$

Thus, E is the correct answer.