Periods of repeating decimals!

\[\large{A = 0.\overline{19} + 0.\overline{199} \qquad , \qquad B = 0.\overline{19} \times 0.\overline{199}}\]

Recall that \(0.\overline{19}\) stands for the repeating decimal \(0.19191919...\) for example, and that the period of a repeating decimal is the number of digits in the repeating part. In this case, the period of \(0.\overline{19}\) is 2. Find the sum of the periods of \(A\) and \(B\).

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