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$Convert the following repeating decimal to a fraction in simplest form. .8 bar(6) Answer:$

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $.8 \overline{6}$ $\newline$ Answer:

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Roots of rational numbers

Full solution

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

\newline

.8 \overline{6}

\newline

Answer:

Assign $x$ as decimal: Let $x$ equal the repeating decimal $0.8$ with $6$ repeating. $x = 0.8\overline{6}$
Shift decimal right: Multiply $x$ by $10$ to shift the decimal point one place to the right. $\newline$ $10x = 8.6666\ldots$
Align repeating digits: Multiply $x$ by $100$ to shift the decimal point two places to the right, which aligns the repeating digits with those in $10x$ . $\newline$ $100x = 86.6666\ldots$
Eliminate repeating decimals: Subtract the equation for $10x$ from the equation for $100x$ to eliminate the repeating decimals. $\newline$ $100x - 10x = 86.6666... - 8.6666...$ $\newline$ $90x = 78$
Divide by $90$ : Divide both sides of the equation by $90$ to solve for $x$ . $x = \frac{78}{90}$
Simplify fraction: Simplify the fraction by finding the greatest common divisor (GCD) of $78$ and $90$ , which is $6$ . $\newline$ $x = \frac{78 \div 6}{90 \div 6}$ $\newline$ $x = \frac{13}{15}$

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