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

Full solution

Q. Convert the decimal below to a fraction in simplest form.

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0.519

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Answer:

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