Write the repeating decimal as a fraction. .47474747 ______

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The digits ＂47＂ repeat indefinitely. Pattern identified: 0.47474747... = 0.47 + 0.0047 + 0.000047 + ...Express Terms as Fractions: Express each term in the pattern as a fraction. 0.47474747... = 47/100 + 47/10000 + 47/1000000 + ...Recognize Geometric Series: Recognize that the series 47/100 + 47/10000 + 47/1000000 + ... is a geometric series. To find the common ratio (r), we divide a term by the term before it. (47/10000) / (47/100) = 47/10000 * 100/47 = 1/100 Common Ratio (r): 1/100Write as Fraction Using Formula: 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 = 47/100 and r = 1/100 into the formula. (47/100) / (1 - 1/100) = (47/100) / (99/100) = 47/100 * 100/99Simplify the Fraction: Simplify the fraction. 47/100 * 100/99 = 47/99 So, 0.47474747... = 47/99

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Write the repeating decimal as a fraction. $\newline$ $.47474747$

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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

.47474747

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The digits ＂ $47$ ＂ repeat indefinitely. $\newline$ Pattern identified: $0.47474747\ldots = 0.47 + 0.0047 + 0.000047 + \ldots$
Express Terms as Fractions: Express each term in the pattern as a fraction. $\newline$ $0.47474747\ldots = \frac{47}{100} + \frac{47}{10000} + \frac{47}{1000000} + \ldots$
Recognize Geometric Series: Recognize that the series $\frac{47}{100} + \frac{47}{10000} + \frac{47}{1000000} + \ldots$ is a geometric series. $\newline$ To find the common ratio $(r)$ , we divide a term by the term before it. $\newline$ $(\frac{47}{10000}) / (\frac{47}{100}) = \frac{47}{10000} \times \frac{100}{47} = \frac{1}{100}$ $\newline$ Common Ratio $(r): \frac{1}{100}$
Write as Fraction Using Formula: 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 $a_1 = 47/100$ and $r = 1/100$ into the formula. $\newline$ $(47/100) / (1 - 1/100) = (47/100) / (99/100) = 47/100 \times 100/99$
Simplify the Fraction: Simplify the fraction. $\frac{47}{100} \times \frac{100}{99} = \frac{47}{99}$ So, $0.47474747\ldots = \frac{47}{99}$