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$Convert the following repeating decimal to a fraction in simplest form. .0 bar(6) Answer:$

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $.0 \overline{6}$ $\newline$ Answer:

Full solution

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

\newline

.0 \overline{6}

\newline

Answer:

Define repeating decimal: Let $x$ be the repeating decimal $0.0$ with a repeating $6$ , which we can write as $x = 0.0\overline{6}$
Multiply by $10$ : To convert this repeating decimal to a fraction, we can multiply $x$ by a power of $10$ that will move the decimal point to the right so that the repeating digits line up vertically. Since we have one repeating digit, we multiply by $10$ to get $10x$ . $\newline$ So, $10x = 0.666\ldots$
Subtract original number: Now, we subtract the original number $x$ from $10x$ to get rid of the repeating part: $\newline$ $10x - x = 0.666... - 0.0666...$ $\newline$ This simplifies to $9x = 0.6$
Divide by $9$ : To find the value of $x$ , we divide both sides of the equation by $9$ : $x = \frac{0.6}{9}$
Convert to fraction: Now we simplify the fraction $0.6/9$ . Since $0.6$ is the same as $6/10$ or $3/5$ , we can write: $\newline$ $x = \frac{3}{5} / 9$
Simplify fraction: We can simplify this further by multiplying the numerator by the reciprocal of the denominator: $x = \left(\frac{3}{5}\right) * \left(\frac{1}{9}\right)$
Multiply fractions: Multiplying the fractions, we get: $x = \frac{3}{45}$
Simplify final fraction: Finally, we simplify the fraction $\frac{3}{45}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $3$ : $\newline$ $x = \frac{3}{45} \div \frac{3}{3}$ $\newline$ $x = \frac{1}{15}$