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Write the repeating decimal as a fraction.\newline.18181818.18181818

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Q. Write the repeating decimal as a fraction.\newline.18181818.18181818
  1. Denote Decimal as xx: Let's denote the repeating decimal 0.18181818…0.18181818\ldots by xx.
    x=0.18181818…x = 0.18181818\ldots
    To convert this repeating decimal into a fraction, we can use algebra. We'll multiply xx by a power of 1010 that matches the length of the repeating pattern to shift the decimal point to the right so that the repeating digits align.
    Since the repeating pattern is two digits long (1818), we'll multiply xx by 100100.
    100x=18.18181818…100x = 18.18181818\ldots
  2. Multiply by 100100: Now, we subtract the original xx from 100x100x to get rid of the repeating decimal part.\newline100x−x=18.18181818...−0.18181818...100x - x = 18.18181818... - 0.18181818...\newlineThis simplifies to:\newline99x=1899x = 18
  3. Subtract and Simplify: To find the value of xx, we divide both sides of the equation by 9999.x=1899x = \frac{18}{99}
  4. Divide by 9999: We can simplify the fraction by finding the greatest common divisor (GCD) of 1818 and 9999. The GCD of 1818 and 9999 is 99. \newlinex=18/999/9x = \frac{18/9}{99/9}\newlinex=211x = \frac{2}{11}

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