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Write the repeating decimal as a fraction. $\newline$ $.889889889$

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Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

.889889889

Identify repeating pattern: Let's identify the repeating pattern in the decimal. The repeating pattern is $889$ . $\newline$ Pattern followed by splitting the decimal: $\newline$ $0.889889889\ldots = 0.889 + 0.000889 + 0.000000889 + \ldots$
Express in fraction form: Now, let's express each term in fraction form. $\newline$ $0.889889889\ldots$ $\newline$ = $0.889 + 0.000889 + 0.000000889 + \ldots$ $\newline$ = $\frac{889}{1000} + \frac{889}{1000000} + \frac{889}{1000000000} + \ldots$
Find common ratio: The series $\frac{889}{1000} + \frac{889}{1000000} + \frac{889}{1000000000} + \ldots$ forms a geometric series. $\newline$ Find the common ratio $(r)$ in the geometric series. $\newline$ Two consecutive terms are $\frac{889}{1000}$ and $\frac{889}{1000000}$ . $\newline$ $\left(\frac{889}{1000000}\right) / \left(\frac{889}{1000}\right)$ $\newline$ $= \frac{889}{1000000} \times \frac{1000}{889}$ $\newline$ $= \frac{1}{1000}$ $\newline$ Common Ratio $(r)$ : $\frac{1}{1000}$
Write as fraction: 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 and $r$ is the common ratio. $\newline$ Substitute $a_1 = 889/1000$ and $r = 1/1000$ into the formula. $\newline$ = $(889/1000) / (1 - 1/1000)$ $\newline$ = $(889/1000) / (999/1000)$ $\newline$ = $889/1000 \times 1000/999$ $\newline$ = $889/999$

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