Answer
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Step 1:: Understand the problem
We are given a repeating decimal, $0.02 \overline{2}$, and we need to express it as a fraction.
Step 2:: Find the pattern in the repeating decimal
The given repeating decimal $0.02 \overline{2}$ has the repeating digit '2' after the decimal point.
Step 3:: Calculate the difference between the decimal and its repeating part
100x = 2. \overline{2}
Then,
Step 4:: Subtract the two equations to eliminate the repeating part
99x = 2. \overline{2} - 0.02 \overline{2}
Step 5:: Calculate the difference on the right side of the equation
2. \overline{2} - 0.02 \overline{2} = 2.222 - 0.02222 = 2.2000
Step 6:: Divide both sides by 99
\frac{2.2000}{99} = \frac{2200}{9900}
Step 7:: Simplify the fraction
\frac{2200}{9900} = \frac{11}{49.5}
Final Answer
0.02 \overline{2} = \boxed{\frac{11}{49.5}}
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