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Write the repeating decimal as a fraction. $\newline$ $.020020020$

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Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

.020020020

Identify repeating sequence: Identify the repeating sequence in the decimal. The repeating sequence in the decimal $0.020020020\ldots$ is $\text{“020”}$ . This sequence repeats indefinitely.
Let $x$ equal: Let $x$ equal the repeating decimal. $\newline$ Let $x = 0.020020020\ldots$
Multiply by power of $10$ : Multiply $x$ by a power of $10$ that will move the decimal point to the right so that the repeating sequence aligns with the original decimal. $\newline$ Since the repeating sequence is three digits long, we multiply $x$ by $10^3$ (which is $1000$ ) to move the decimal point three places to the right. $\newline$ $1000x = 20.020020020\ldots$
Subtract original from new: Subtract the original equation $x$ from the new equation $1000x$ to eliminate the repeating decimals. $\newline$ $1000x - x = 20.020020020... - 0.020020020...$ $\newline$ $999x = 20$
Solve for x: Solve for x by dividing both sides of the equation by $999$ . $x = \frac{20}{999}$
Simplify the fraction: Simplify the fraction if possible. $\newline$ The fraction $\frac{20}{999}$ cannot be simplified further as $20$ and $999$ do not have any common factors other than $1$ .

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