Write the repeating decimal as a fraction. .993993993 ______

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal .993993993... The repeating pattern is 993.Express as Sum: Let's express the repeating decimal as a sum of its repeating parts. .993993993... = 0.993 + 0.000993 + 0.000000993 + ...Express as Fraction: Now, let's express each term as a fraction. 0.993 = 993/1000 0.000993 = 993/1000000 0.000000993 = 993/1000000000 ... This is a geometric series with the first term a_1 = 993/1000 and the common ratio r = 1/1000.Use Geometric Series Formula: To find the sum of an infinite geometric series, we use the formula S = a_1 / (1 - r), where S is the sum, a_1 is the first term, and r is the common ratio. Substitute a_1 = 993/1000 and r = 1/1000 into the formula. S = (993/1000) / (1 - 1/1000)Perform Calculation: Now, let's perform the calculation. S = (993/1000) / (999/1000) S = 993/1000 * 1000/999 S = 993/999Simplify Fraction: We can simplify the fraction 993/999 by dividing both the numerator and the denominator by their greatest common divisor, which is 3. S = (993 ÷ 3) / (999 ÷ 3) S = 331/333

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Write the repeating decimal as a fraction. $\newline$ $.993993993$

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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

.993993993

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal $0.993993993\ldots$ The repeating pattern is $993$ .
Express as Sum: Let's express the repeating decimal as a sum of its repeating parts. $\newline$ $.993993993... = 0.993 + 0.000993 + 0.000000993 + ...$
Express as Fraction: Now, let's express each term as a fraction. $\newline$ $0.993 = \frac{993}{1000}$ $\newline$ $0.000993 = \frac{993}{1000000}$ $\newline$ $0.000000993 = \frac{993}{1000000000}$ $\newline$ ... $\newline$ This is a geometric series with the first term $a_1 = \frac{993}{1000}$ and the common ratio $r = \frac{1}{1000}$ .
Use Geometric Series Formula: To find the sum of an infinite geometric series, we use the formula $S = \frac{a_1}{1 - r}$ , where $S$ is the sum, $a_1$ is the first term, and $r$ is the common ratio. $\newline$ Substitute $a_1 = \frac{993}{1000}$ and $r = \frac{1}{1000}$ into the formula. $\newline$ $S = \frac{\frac{993}{1000}}{1 - \frac{1}{1000}}$
Perform Calculation: Now, let's perform the calculation. $\newline$ $S = \frac{993}{1000} / \frac{999}{1000}$ $\newline$ $S = \frac{993}{1000} \times \frac{1000}{999}$ $\newline$ $S = \frac{993}{999}$
Simplify Fraction: We can simplify the fraction $\frac{993}{999}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $3$ . $\newline$ $S = (\frac{993 \div 3}{999 \div 3})$ $\newline$ $S = \frac{331}{333}$