Write the repeating decimal as a fraction. .09090909 ______

Identify Repeating Pattern: Identify the repeating pattern in the decimal. The repeating pattern in the decimal 0.09090909... is 09.Convert to Fraction: Let x equal the repeating decimal, so x = 0.09090909... To convert this into a fraction, we will create an equation that isolates the repeating part.Multiply by 100: Multiply x by a power of 10 that matches the length of the repeating pattern. Since the repeating pattern is two digits long (09), we multiply x by 100. So, 100x = 9.09090909...Subtract Original: Subtract the original x from the 100x to get rid of the decimal part. 100x - x = 9.09090909... - 0.09090909...Isolate Repeating Decimal: Perform the subtraction to isolate the repeating decimal. 99x = 9Divide by 99: Divide both sides of the equation by 99 to solve for x. x = 9 / 99Simplify Fraction: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 9. x = (9/9) / (99/9)Find Final Fraction: Complete the simplification to find the fraction. x = 1 / 11 So, the repeating decimal 0.09090909... as a fraction is 1/11.

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Write the repeating decimal as a fraction. $\newline$ $.09090909$

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Algebra 2

Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

.09090909

Identify Repeating Pattern: Identify the repeating pattern in the decimal. $\newline$ The repeating pattern in the decimal $0.09090909\ldots$ is $09$ .
Convert to Fraction: Let $x$ equal the repeating decimal, so $x = 0.09090909...$ To convert this into a fraction, we will create an equation that isolates the repeating part.
Multiply by $100$ : Multiply $x$ by a power of $10$ that matches the length of the repeating pattern. Since the repeating pattern is two digits long ( $09$ ), we multiply $x$ by $100$ . So, $100x = 9.09090909\ldots$
Subtract Original: Subtract the original $x$ from the $100x$ to get rid of the decimal part. $100x - x = 9.09090909... - 0.09090909...$
Isolate Repeating Decimal: Perform the subtraction to isolate the repeating decimal. $99x = 9$
Divide by $99$ : Divide both sides of the equation by $99$ to solve for $x$ . $x = \frac{9}{99}$
Simplify Fraction: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $9$ . $\newline$ $x = \left(\frac{9}{9}\right) / \left(\frac{99}{9}\right)$
Find Final Fraction: Complete the simplification to find the fraction. $x = \frac{1}{11}$ So, the repeating decimal $0.09090909\ldots$ as a fraction is $\frac{1}{11}$ .