- Let \(x = 0.\overline{02}\).
- Multiply by 100 to shift the decimal point: \(100x = 2.0202...\).
- Subtract the original equation: \(99x = 2\).
- Simplify to find the final fraction: \(x = \boxed{\frac{2}{99}}\).
Explanation
- Analyzing the repeating decimal
The problem asks to convert the repeating decimal \(0.\overline{02}\) into its equivalent fraction. A repeating decimal can be defined as a number whose digits repeat in a fixed pattern infinitely. Here, the pattern '02' repeats indefinitely: \(x = 0.020202...\). To express this as a fraction, we can use an algebraic method.
- Setting up the algebraic equations
Let \(x = 0.\overline{02}\). This can be written as:
\[x = 0.020202... \quad (\text{Equation 1})\]
To eliminate the repeating part, multiply both sides of the equation by 100, as the repeating block has a length of 2 digits:
\[100x = 2.020202... \quad (\text{Equation 2})\]
- Solving for the fraction
Now, subtract Equation 1 from Equation 2:
\[100x - x = (2.020202...) - (0.020202...)\]
\[99x = 2\]
Solving for \(x\), we divide both sides by 99:
\[x = \frac{2}{99}\]
Thus, the fractional form of \(0.\overline{02}\) is \(\frac{2}{99}\). Comparing this result with the given options, we confirm that the correct choice is \(\frac{2}{99}\).
Examples
Understanding how to convert repeating decimals into fractions is essential for precision in finance and scientific calculations. For example, when calculating interest rates or recurring payment schedules that result in repeating decimals, expressing these as exact fractions helps maintain accuracy throughout the entire calculation, preventing rounding errors that could accumulate over time. Knowing the fractional form allows for easier manipulation of these values in complex formulas.