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Write the repeating decimal as a fraction.\newline.262262262.262262262

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Full solution

Q. Write the repeating decimal as a fraction.\newline.262262262.262262262
  1. Represent Decimal with Variable: Let's represent the repeating decimal 0.262262262…0.262262262\ldots with a variable.\newlineLet x=0.262262262…x = 0.262262262\ldots
  2. Multiply by 10001000: To isolate the repeating part, we multiply xx by 10001000 because there are three digits in the repeating sequence.\newline1000x=262.262262262…1000x = 262.262262262\ldots
  3. Subtract and Simplify: Now we subtract the original xx from 1000x1000x to get rid of the decimal part.\newline1000x−x=262.262262262...−0.262262262...1000x - x = 262.262262262... - 0.262262262...\newlineThis simplifies to:\newline999x=262999x = 262
  4. Divide by 999999: To find the value of xx, we divide both sides of the equation by 999999.x=262999x = \frac{262}{999}
  5. Check for Simplification: We can check if the fraction can be simplified by looking for common factors between the numerator and the denominator. In this case, there are no common factors other than 11, so the fraction is already in its simplest form.

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