Write the repeating decimal as a fraction. .667667667 ______

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The digits ＂667＂ repeat indefinitely. Pattern identified: 0.667667667... = 0.667 + 0.000667 + 0.000000667 + ...Express Terms as Fractions: Express each term in the pattern as a fraction. 0.667667667... = 0.667 + 0.000667 + 0.000000667 + ... = 667/1000 + 667/1000000 + 667/1000000000 + ...Recognize Geometric Series: Recognize that the series 667/1000 + 667/1000000 + 667/1000000000 + ... forms a geometric series. Find the common ratio (r) of the geometric series by dividing two consecutive terms. (667/1000000) / (667/1000) = 667/1000000 * 1000/667 = 1/1000 Common Ratio (r): 1/1000Find Common Ratio: Write the repeating decimal as a fraction using the formula for the sum of an infinite geometric series, which is a_1 / (1 - r), where a_1 is the first term. Substitute a_1 = 667/1000 and r = 1/1000 into the formula. (667/1000) / (1 - 1/1000) = (667/1000) / (999/1000) = 667/1000 * 1000/999Write as Fraction: Simplify the fraction. 667/1000 * 1000/999 = 667/999 So, 0.667667667... = 667/999

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Write the repeating decimal as a fraction. $\newline$ $0.667667667$

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Algebra 2

Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

0.667667667

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The digits ＂ $667$ ＂ repeat indefinitely. $\newline$ Pattern identified: $0.667667667\ldots = 0.667 + 0.000667 + 0.000000667 + \ldots$
Express Terms as Fractions: Express each term in the pattern as a fraction. $\newline$ $0.667667667... = 0.667 + 0.000667 + 0.000000667 + ...$ $\newline$ $= \frac{667}{1000} + \frac{667}{1000000} + \frac{667}{1000000000} + ...$
Recognize Geometric Series: Recognize that the series $\frac{667}{1000} + \frac{667}{1000000} + \frac{667}{1000000000} + \ldots$ forms a geometric series. $\newline$ Find the common ratio $(r)$ of the geometric series by dividing two consecutive terms. $\newline$ $(\frac{667}{1000000}) / (\frac{667}{1000}) = \frac{667}{1000000} \times \frac{1000}{667} = \frac{1}{1000}$ $\newline$ Common Ratio $(r): \frac{1}{1000}$
Find Common Ratio: Write the repeating decimal as a fraction using the formula for the sum of an infinite geometric series, which is $a_1 / (1 - r)$ , where $a_1$ is the first term. $\newline$ Substitute $a_1 = 667/1000$ and $r = 1/1000$ into the formula. $\newline$ $(667/1000) / (1 - 1/1000) = (667/1000) / (999/1000) = 667/999$
Write as Fraction: Simplify the fraction. $\newline$ $\frac{667}{1000} \times \frac{1000}{999} = \frac{667}{999}$ $\newline$ So, $0.667667667\ldots = \frac{667}{999}$