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Write the repeating decimal as a fraction. $\newline$ $.63636363$

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Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

.63636363

Rephrase the Problem: Let's first rephrase the ＂How can we express the repeating decimal $0.63636363\ldots$ as a fraction?＂
Identify Repeating Pattern: Identify the repeating pattern in the decimal. The digits ＂ $63$ ＂ repeat indefinitely, so we can write the decimal as $0.63636363\ldots = 0.63 + 0.0063 + 0.000063 + \ldots$
Express as Fractions: Express each term in the pattern as a fraction. The decimal can be written as $0.63636363\ldots = \frac{63}{100} + \frac{63}{10000} + \frac{63}{1000000} + \ldots$
Recognize Geometric Series: Recognize that the series $\frac{63}{100} + \frac{63}{10000} + \frac{63}{1000000} + \ldots$ is a geometric series with the first term $a_1 = \frac{63}{100}$ and a common ratio $r = \frac{1}{100}$ .
Calculate Common Ratio: Calculate the common ratio $r$ by dividing a term in the series by the previous term. For example, $\frac{63}{10000} / \frac{63}{100} = \frac{63}{10000} \times \frac{100}{63} = \frac{1}{100}$ . So, the common ratio $r$ is indeed $\frac{1}{100}$ .
Use Infinite Series Formula: Use 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. Substitute $a_1 = \frac{63}{100}$ and $r = \frac{1}{100}$ into the formula.
Perform Calculation: Perform the calculation: $(63/100) / (1 - 1/100) = (63/100) / (99/100) = 63/100 \times 100/99 = 63/99$ .
Simplify Fraction: Simplify the fraction $\frac{63}{99}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $9$ . $63 \div 9 = 7$ and $99 \div 9 = 11$ .
Final Fraction: The simplified fraction is $\frac{7}{11}$ . Therefore, the repeating decimal $0.63636363\ldots$ can be expressed as the fraction $\frac{7}{11}$ .

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