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Write the repeating decimal as a fraction.\newline.228228228.228228228

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Full solution

Q. Write the repeating decimal as a fraction.\newline.228228228.228228228
  1. Assign xx value: Let xx equal the repeating decimal 0.228228228…0.228228228\ldots \newlinex=0.228228228…x = 0.228228228\ldots
  2. Shift decimal point: Multiply xx by 10001000 to shift the decimal point three places to the right, since the repeating block has three digits (228228).\newline1000x=228.228228228…1000x = 228.228228228\ldots
  3. Subtract to eliminate decimals: Subtract the original xx from 1000x1000x to eliminate the repeating decimals.\newline1000x−x=228.228228228...−0.228228228...1000x - x = 228.228228228... - 0.228228228...\newline999x=228999x = 228
  4. Divide by 999999: Divide both sides of the equation by 999999 to solve for xx.x=228999x = \frac{228}{999}
  5. Simplify fraction: Simplify the fraction by finding the greatest common divisor (GCD) of 228228 and 999999. The GCD of 228228 and 999999 is 33. \newlinex=228/3999/3x = \frac{228 / 3}{999 / 3}\newlinex=76333x = \frac{76}{333}

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