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Write the repeating decimal as a fraction.\newline.429429429.429429429

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Full solution

Q. Write the repeating decimal as a fraction.\newline.429429429.429429429
  1. Denote Repeating Decimal as xx: Let's denote the repeating decimal 0.429429429…0.429429429\ldots as xx.\newlinex=0.429429429…x = 0.429429429\ldots\newlineTo convert this repeating decimal into a fraction, we will first express xx in a way that isolates the repeating part.
  2. Multiply by 10001000: Multiply xx by 10001000, since the repeating part has three digits (429429), to shift the decimal point three places to the right.\newline1000x=429.429429429…1000x = 429.429429429\ldots\newlineNow we have the same repeating decimal part on both sides of the decimal point.
  3. Subtract to Eliminate Repeating Part: Subtract the original number xx from 1000x1000x to get rid of the repeating part.\newline1000x−x=429.429429429...−0.429429429...1000x - x = 429.429429429... - 0.429429429...\newlineThis subtraction will give us 999x999x on the left side and 429429 on the right side.
  4. Isolate 999x999x: Perform the subtraction to isolate 999x999x.\newline999x=429999x = 429\newlineNow we have a simple equation to solve for xx.
  5. Divide by 999999: Divide both sides of the equation by 999999 to solve for xx.x=429999x = \frac{429}{999}This fraction represents the repeating decimal.
  6. Find GCD and Simplify: Simplify the fraction by finding the greatest common divisor (GCD) of 429429 and 999999. The GCD of 429429 and 999999 is 33. Divide both the numerator and the denominator by 33 to simplify the fraction. x=429/3999/3x = \frac{429 / 3}{999 / 3}
  7. Perform Division: Perform the division to simplify the fraction. \newlinex=143333x = \frac{143}{333}\newlineThis is the fraction in its simplest form that represents the repeating decimal 0.429429429…0.429429429\ldots

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