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Write the repeating decimal as a fraction.\newline.199199199.199199199

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Q. Write the repeating decimal as a fraction.\newline.199199199.199199199
  1. Denote Repeating Decimal: Let's denote the repeating decimal 0.199199199…0.199199199\ldots by xx.x=0.199199199…x = 0.199199199\ldots
  2. Multiply by Power of 1010: To convert this repeating decimal into a fraction, we multiply xx by a power of 1010 that matches the repeating pattern. In this case, the repeating pattern is 199199, which is three digits long, so we multiply xx by 10310^3 (which is 10001000).\newline1000x=199.199199199…1000x = 199.199199199\ldots
  3. Subtract Original Decimal: Now we subtract the original xx from 1000x1000x to get rid of the repeating decimal part.1000x−x=199.199199199…−0.199199199…1000x - x = 199.199199199\ldots - 0.199199199\ldots
  4. Perform Subtraction: Perform the subtraction on the left side of the equation. 1000x−x=999x1000x - x = 999x
  5. Simple Equation Without Decimals: Perform the subtraction on the right side of the equation.\newline199.199199199…−0.199199199…=199199.199199199\ldots - 0.199199199\ldots = 199
  6. Solve for x: Now we have a simple equation without decimals.\newline999x=199999x = 199
  7. Simplify the Fraction: To solve for xx, we divide both sides of the equation by 999999.x=199999x = \frac{199}{999}
  8. Simplify the Fraction: To solve for xx, we divide both sides of the equation by 999999.x=199999x = \frac{199}{999}We can simplify the fraction by finding the greatest common divisor (GCD) of 199199 and 999999. The GCD of 199199 and 999999 is 11, so the fraction is already in its simplest form.

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