A repeating decimal number 5.2222... is given.

For a number $5.222...$, express it as $5 + 0.\overline{2}$. The repeating decimal $0.\overline{2}$ is simply $\frac{2}{9}$. Thus, calculate $5 + \frac{2}{9} = \frac{5 \times 9 + 2}{9} = \frac{47}{9}$.

Many students mistake the fraction for $\frac{5222}{1000}$ instead of identifying it as a pure repeating decimal, or they forget to include the integer part in the numerator calculation.

This concept is a precursor to understanding geometric series where the sum of an infinite geometric progression is $S = \frac{a}{1-r}$, which is how the rule for recurring decimals is derived.