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Write the repeating decimal as a fraction.\newline.969969969.969969969

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Full solution

Q. Write the repeating decimal as a fraction.\newline.969969969.969969969
  1. Denote repeating decimal as xx: Let's denote the repeating decimal 0.969696...0.969696... as xx.
    x=0.969696...x = 0.969696...
    To convert this repeating decimal into a fraction, we will multiply xx by a power of 1010 that matches the length of the repeating pattern. Since the repeating pattern is two digits long (9696), we will multiply xx by 100100.
    100x=96.969696...100x = 96.969696...
  2. Multiply by power of 1010: Now, we subtract the original xx from 100x100x to get rid of the repeating decimal part.\newline100x−x=96.969696...−0.969696...100x - x = 96.969696... - 0.969696...\newline99x=9699x = 96
  3. Subtract original xx from 100x100x: Next, we divide both sides of the equation by 9999 to solve for xx.x=9699x = \frac{96}{99}
  4. Divide by 9999: We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 33. \newlinex=(963)/(993)x = (\frac{96}{3}) / (\frac{99}{3})\newlinex=3233x = \frac{32}{33}

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