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Convert the decimal below to a fraction in simplest form. 0.519 Answer:

Convert the decimal below to a fraction in simplest form.\newline0.519 0.519 \newlineAnswer:

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Full solution

Q. Convert the decimal below to a fraction in simplest form.\newline0.519 0.519 \newlineAnswer:
  1. Question Prompt: Question prompt: Convert the decimal 0.5190.519 to a fraction in simplest form.
  2. Recognize Decimal Type: Recognize that 0.5190.519 is a repeating decimal, where the digit 99 repeats indefinitely. To convert it to a fraction, we can use the method for converting repeating decimals to fractions.
  3. Assign Variable xx: Let x=0.519519519…x = 0.519519519\ldots (the decimal repeats indefinitely).
  4. Multiply by 10001000: Multiply xx by 10001000 to shift the decimal point three places to the right, since there are three digits in the repeating part: 1000x=519.519519…1000x = 519.519519\ldots
  5. Subtract Original xx: Subtract the original xx from this new equation to get rid of the repeating part: 1000x−x=519.519519…−0.519519519…1000x - x = 519.519519\ldots - 0.519519519\ldots
  6. Perform Subtraction: Perform the subtraction: 999x=519999x = 519
  7. Divide by 999999: Divide both sides by 999999 to solve for xx: x=519999x = \frac{519}{999}
  8. Simplify Fraction: Simplify the fraction by finding the greatest common divisor (GCD) of 519519 and 999999. The GCD of 519519 and 999999 is 33.
  9. Divide by GCD: Divide both the numerator and the denominator by the GCD to simplify the fraction: (5193)/(9993)=173333(\frac{519}{3}) / (\frac{999}{3}) = \frac{173}{333}
  10. Check for Further Simplification: Check if the fraction can be simplified further. The GCD of 173173 and 333333 is 11, so the fraction is already in its simplest form.

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