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$Convert the following repeating decimal to a fraction in simplest form. .9 bar(4) Answer:$

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $.9 \overline{4}$ $\newline$ Answer:

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Full solution

Q. Convert the following repeating decimal to a fraction in simplest form.

\newline

.9 \overline{4}

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Answer:

Write Variable Equation: Let $x$ be the repeating decimal $0.9$ with a repeating $4$ ( $0.9444\ldots$ ). $\newline$ We will use algebra to turn this repeating decimal into a fraction. $\newline$ First, write down the repeating decimal with an unknown variable: $\newline$ $x = 0.9444\ldots$
Shift Decimal Point: Multiply $x$ by $10$ to shift the decimal point one place to the right. This will allow us to set up an equation that will help eliminate the repeating part. $\newline$ $10x = 9.444\ldots$
Eliminate Repeating Decimals: Multiply $x$ by $1000$ to shift the decimal point three more places to the right, which will line up the repeating digits with those in the equation from Step $2$ . $\newline$ $1000x = 944.444\ldots$
Solve for x: Subtract the equation from Step $2$ from the equation in Step $3$ to eliminate the repeating decimals. $\newline$ $1000x - 10x = 944.444... - 9.444...$ $\newline$ $990x = 935$
Simplify Fraction: Solve for $x$ by dividing both sides of the equation by $990$ . $x = \frac{935}{990}$
Check for Further Simplification: Simplify the fraction by finding the greatest common divisor (GCD) of $935$ and $990$ and dividing both the numerator and the denominator by the GCD. $\newline$ The GCD of $935$ and $990$ is $5$ . $\newline$ $x = (\frac{935}{5}) / (\frac{990}{5})$ $\newline$ $x = \frac{187}{198}$
Check for Further Simplification: Simplify the fraction by finding the greatest common divisor (GCD) of $935$ and $990$ and dividing both the numerator and the denominator by the GCD. $\newline$ The GCD of $935$ and $990$ is $5$ . $\newline$ $x = (935 / 5) / (990 / 5)$ $\newline$ $x = 187 / 198$ Check if the fraction can be simplified further. Since there are no common factors between $187$ and $198$ other than $1$ , the fraction is already in its simplest form.