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$Convert the following repeating decimal to a fraction in simplest form. .0 bar(8) Answer:$

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $.0 \overline{8}$ $\newline$ Answer:

Full solution

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

\newline

.0 \overline{8}

\newline

Answer:

Define $x$ as decimal: Let $x$ be the repeating decimal $0.\overline{8}$ . $\newline$ $x = 0.\overline{8}$
Convert to fraction: To convert the repeating decimal to a fraction, we can use the fact that $0.\overline{8}$ is equal to $0.08$ repeating indefinitely. $\newline$ Multiply $x$ by $10$ to shift the decimal point one place to the right. $\newline$ $10x = 0.\overline{8}$
Multiply by $10$ : Now, subtract the original $x$ from $10x$ to get rid of the repeating decimal. $10x - x = 0.8\overline{8} - 0.0\overline{8}$ $9x = 0.8$
Subtract original $x$ : Divide both sides of the equation by $9$ to solve for $x$ . $x = \frac{0.8}{9}$
Divide by $9$ : Convert the decimal $0.8$ to a fraction. Since $0.8$ is the same as $\frac{8}{10}$ or $\frac{4}{5}$ after simplification, we can write: $\newline$ $x = \frac{\frac{4}{5}}{9}$
Convert decimal to fraction: Multiply the fraction by $1$ in the form of $\frac{5}{5}$ to get a common denominator. $\newline$ $x = \left(\frac{4}{5}\right) * \left(\frac{5}{5}\right) / 9$ $\newline$ $x = \left(\frac{4\times5}{5\times9}\right)$ $\newline$ $x = \frac{20}{45}$
Multiply by common denominator: Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is $5$ . $x = \frac{20}{5} / \frac{45}{5}$ $x = \frac{4}{9}$