Periods of repeating decimals!

${A = 0.\overline{19} + 0.\overline{199}, \quad B = 0.\overline{19} \times 0.\overline{199}}$

Recall that $0.\overline{19},$ for example, stands for the repeating decimal $0.19191919...$ 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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