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Write the repeating decimal as a fraction.\newline.440440440.440440440

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Full solution

Q. Write the repeating decimal as a fraction.\newline.440440440.440440440
  1. Identify Repeating Pattern: Identify the repeating pattern in the decimal.\newlineThe repeating pattern in the decimal is ＂440＂＂440＂.
  2. Assign Variable: Let xx be the repeating decimal, so x=0.440440440…x = 0.440440440\ldots\newlineWrite this equation down to use it later.
  3. Shift Decimal Point: Multiply xx by 10001000 to shift the decimal point three places to the right, since the repeating pattern has three digits.\newline1000x=440.440440440…1000x = 440.440440440\ldots
  4. Subtract Decimals: Subtract the original xx from 1000x1000x to get rid of the repeating decimals.\newline1000x−x=440.440440440...−0.440440440...1000x - x = 440.440440440... - 0.440440440...\newlineThis results in 999x=440999x = 440.
  5. Solve for x: Solve for x by dividing both sides of the equation by 999999.\newlinex=440999x = \frac{440}{999}
  6. Simplify Fraction: Simplify the fraction by finding the greatest common divisor (GCD) of 440440 and 999999. The GCD of 440440 and 999999 is 11, so the fraction is already in its simplest form.

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