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Write the repeating decimal as a fraction. $\newline$ $.380380380$

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Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

.380380380

Identify Repeating Pattern: Identify the repeating pattern in the decimal. $\newline$ The repeating pattern in the decimal is $\text{“}380\text{”}$ .
Express as Sum: Express the repeating decimal as a sum of its repeating parts. $\newline$ $0.380380380\ldots = 0.380 + 0.000380 + 0.000000380 + \ldots$
Convert to Fractions: Convert each part of the sum into a fraction. $\newline$ $0.380 = \frac{380}{1000}$ $\newline$ $0.000380 = \frac{380}{1000000}$ $\newline$ $0.000000380 = \frac{380}{1000000000}$ $\newline$ ...
Recognize Geometric Series: Recognize that the sum forms a geometric series. The first term $a_1$ is $\frac{380}{1000}$ and the common ratio $r$ is $\frac{1}{1000}$ .
Use Series Formula: Use the formula for the sum of an infinite geometric series to write the repeating decimal as a fraction. $\newline$ The formula is $S = \frac{a_1}{(1 - r)}$ , where $S$ is the sum of the series, $a_1$ is the first term, and $r$ is the common ratio.
Substitute Values: Substitute the values of $a_1$ and $r$ into the formula. $\newline$ $S = \frac{380}{1000} / \left(1 - \frac{1}{1000}\right)$
Simplify Expression: Simplify the expression. $\newline$ $S = \frac{380}{1000} / \frac{999}{1000}$ $\newline$ $S = \frac{380}{1000} \times \frac{1000}{999}$ $\newline$ $S = \frac{380}{999}$
Check for Simplification: Check the fraction for any possible simplification. $380$ and $999$ do not have any common factors other than $1$ , so the fraction is already in its simplest form.

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