To convert the repeating decimal \(0.02\overline{02}\) to a fraction, we can follow these steps:
Let \(x = 0.02\overline{02}\).
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First, we recognize that we can express this repeating decimal as: \[ x = 0.02020202 \ldots \]
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We multiply \(x\) by 100 (because the repeating part has 2 digits): \[ 100x = 2.02020202 \ldots \]
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Next, we can subtract the original \(x\) from this equation: \[ 100x - x = 2.02020202 \ldots - 0.02020202 \ldots \] Simplifying this gives: \[ 99x = 2 \]
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Now we solve for \(x\): \[ x = \frac{2}{99} \]
The fraction equivalent to \(0.02\overline{02}\) is \(\frac{2}{99}\).
However, checking the options you provided, none of them is \(\frac{2}{99}\).
To conform to the available answers:
- First option \(15 = \frac{1}{5}\) results in \(0.2\) which is incorrect.
- Second option \(150 = \frac{1}{50}\) results in \(0.02\) which is incorrect.
- Third option \(2101 = \frac{2}{101}\) evaluates to approximately \(0.019801980198...\), which is close but not repeating the same digits.
- Fourth option \(299\) at first glance seems to be void of calculations.
Thus, while \(\frac{2}{99}\) represents \(0.02\overline{02}\), please double-check if further clarifications or conversions exist that match the available answer choices.
My final answer for \(0.02\overline{02}\) is \(\frac{2}{99}\).
1 answer