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Write the repeating decimal as a fraction.\newline.433433433.433433433

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Full solution

Q. Write the repeating decimal as a fraction.\newline.433433433.433433433
  1. Identify Repeating Pattern: Let's first identify the repeating pattern in the decimal. The repeating decimal given is 0.433433433…0.433433433\ldots, where ＂433433＂ is the repeating sequence.
  2. Convert to Fraction: To convert the repeating decimal to a fraction, let's denote the repeating decimal as xx: x=.433433433…x = .433433433\ldots
  3. Multiply by 10001000: Multiply xx by 10001000 (since there are three digits in the repeating sequence) to shift the decimal point three places to the right: 1000x=433.433433433…1000x = 433.433433433\ldots
  4. Subtract Original Number: Now, subtract the original number xx from the result of the multiplication to eliminate the repeating part: 1000x−x=433.433433433…−.433433433…1000x - x = 433.433433433\ldots - .433433433\ldots
  5. Perform Subtraction: Perform the subtraction: 999x=433999x = 433
  6. Solve for x: Now, solve for x by dividing both sides of the equation by 999999: x=433999x = \frac{433}{999}
  7. Simplify the Fraction: Simplify the fraction by looking for common factors. Both the numerator and the denominator are divisible by 433433: x=433999=1999433x = \frac{433}{999} = \frac{1}{\frac{999}{433}}
  8. Further Simplify: Further simplify the fraction by dividing 999999 by 433433: x=1/(999/433)=1/(3×3×3×37/433)x = 1 / (999/433) = 1 / (3 \times 3 \times 3 \times 37 / 433)
  9. Final Simplified Fraction: Since 433433 is a prime number and does not share any factors with 999999 other than 11, the fraction is already in its simplest form: x=433999x = \frac{433}{999}

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