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Write the repeating decimal as a fraction.\newline.09090909.09090909

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Full solution

Q. Write the repeating decimal as a fraction.\newline.09090909.09090909
  1. Identify Repeating Pattern: Identify the repeating pattern in the decimal.\newlineThe repeating pattern in the decimal 0.09090909…0.09090909\ldots is 0909.
  2. Convert to Fraction: Let xx equal the repeating decimal, so x=0.09090909...x = 0.09090909... To convert this into a fraction, we will create an equation that isolates the repeating part.
  3. Multiply by 100100: Multiply xx by a power of 1010 that matches the length of the repeating pattern. Since the repeating pattern is two digits long (0909), we multiply xx by 100100. So, 100x=9.09090909…100x = 9.09090909\ldots
  4. Subtract Original: Subtract the original xx from the 100x100x to get rid of the decimal part.100x−x=9.09090909...−0.09090909...100x - x = 9.09090909... - 0.09090909...
  5. Isolate Repeating Decimal: Perform the subtraction to isolate the repeating decimal. 99x=999x = 9
  6. Divide by 9999: Divide both sides of the equation by 9999 to solve for xx.x=999x = \frac{9}{99}
  7. Simplify Fraction: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 99. \newlinex=(99)/(999)x = \left(\frac{9}{9}\right) / \left(\frac{99}{9}\right)
  8. Find Final Fraction: Complete the simplification to find the fraction. x=111x = \frac{1}{11} So, the repeating decimal 0.09090909…0.09090909\ldots as a fraction is 111\frac{1}{11}.

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