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Write the repeating decimal as a fraction. $\newline$ $.48484848$

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Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

.48484848

Denote Repeating Decimal: Let's denote the repeating decimal $0.48484848\ldots$ by $x$ . $\newline$ $x = 0.48484848\ldots$
Convert to Fraction: To convert this repeating decimal into a fraction, we first express it as an infinite sum of its repeating parts. $x = 0.48 + 0.0048 + 0.000048 + \ldots$
Express as Geometric Series: Notice that each term is $100$ times smaller than the previous term. This is a geometric series with the first term $a = 0.48$ and the common ratio $r = \frac{1}{100}$ .
Apply Sum Formula: The sum of an infinite geometric series can be found using the formula $S = \frac{a}{(1 - r)}$ , where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio.
Express First Term as Fraction: Before we apply the formula, we need to express the first term as a fraction. The first term $a = 0.48$ can be written as $\frac{48}{100}$ , which simplifies to $\frac{12}{25}$ .
Apply Formula for Sum: Now we can apply the formula for the sum of an infinite geometric series: $\newline$ $x = \frac{a}{1 - r}$ $\newline$ $x = \frac{\frac{12}{25}}{1 - \frac{1}{100}}$
Simplify Denominator: Simplify the denominator: $1 - \left(\frac{1}{100}\right) = \frac{100}{100} - \frac{1}{100} = \frac{99}{100}$
Substitute Values: Now we can substitute the values into the formula: $x = \frac{12}{25} / \frac{99}{100}$
Divide by Fraction: To divide by a fraction, we multiply by its reciprocal: $x = \frac{12}{25} \times \frac{100}{99}$
Multiply Numerators and Denominators: Multiply the numerators and denominators: $x = (12 \times 100) / (25 \times 99)$ $x = 1200 / 2475$
Simplify Fraction: Simplify the fraction by finding the greatest common divisor (GCD) of $1200$ and $2475$ , which is $25$ . $\newline$ $x = \frac{1200}{25} / \frac{2475}{25}$ $\newline$ $x = \frac{48}{99}$

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