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Write the repeating decimal as a fraction.\newline.893893893.893893893

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Full solution

Q. Write the repeating decimal as a fraction.\newline.893893893.893893893
  1. Identify repeating part: Let xx equal the repeating decimal 0.893893893…0.893893893\ldotsx=0.893893893…x = 0.893893893\ldotsIdentify the repeating part of the decimal.The repeating part is 893893.
  2. Shift decimal point: Multiply xx by 10001000 to shift the decimal point three places to the right, since the repeating part has three digits.\newline1000x=893.893893…1000x = 893.893893\ldots\newlineNow we have the same repeating decimal part on both sides of the equation.
  3. Eliminate repeating part: Subtract the original equation x=0.893893893...x = 0.893893893... from the new equation 1000x=893.893893...1000x = 893.893893... to eliminate the repeating part.\newline1000x−x=893.893893...−0.893893893...1000x - x = 893.893893... - 0.893893893...\newlineThis will give us an equation without the repeating decimal.
  4. Find value of 999x999x: Perform the subtraction to find the value of 999x999x.999x=893999x = 893 Now we have an equation without the repeating decimal.
  5. Solve for x: Divide both sides of the equation by 999999 to solve for xx.\newlinex=893999x = \frac{893}{999}\newlineThis gives us the fraction form of the repeating decimal.
  6. Simplify fraction: Simplify the fraction if possible.\newlineBoth the numerator and the denominator have a common factor of 11, so the fraction is already in its simplest form.\newlinex=893999x = \frac{893}{999}

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