Write the repeating decimal as a fraction. .74747474 ______

Denote Repeating Decimal: Let's denote the repeating decimal 0.74747474... as x. x = 0.74747474...Convert to Fraction: To convert this repeating decimal into a fraction, we first express it as an infinite sum of its repeating parts. 0.74747474... = 0.74 + 0.0074 + 0.000074 + ...Express as Infinite Sum: Notice that each term after the first is 1/100 times the previous term, which means this is a geometric series with the first term a = 0.74 and the common ratio r = 1/100.Identify Geometric Series: The sum S of an infinite geometric series with first term a and common ratio r (where |r| < 1) is given by S = a / (1 - r).Apply Sum Formula: We apply the formula to find the sum of the series, which represents our repeating decimal. S = 0.74 / (1 - 1/100)Perform Calculation: Now we perform the calculation. S = 0.74 / (99/100) S = 0.74 * (100/99)Write Decimal as Fraction: To avoid dealing with decimals, we can write 0.74 as 74/100. S = (74/100) * (100/99)Simplify Fraction: We simplify the fraction by canceling out the common factor of 100. S = 74/99Final Answer: The fraction 74/99 cannot be simplified further, so this is our final answer.

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Write the repeating decimal as a fraction. $\newline$ $.74747474$

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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

.74747474

Denote Repeating Decimal: Let's denote the repeating decimal $0.74747474\ldots$ as $x$ . $x = 0.74747474\ldots$
Convert to Fraction: To convert this repeating decimal into a fraction, we first express it as an infinite sum of its repeating parts. $\newline$ $0.74747474\ldots = 0.74 + 0.0074 + 0.000074 + \ldots$
Express as Infinite Sum: Notice that each term after the first is $\frac{1}{100}$ times the previous term, which means this is a geometric series with the first term $a = 0.74$ and the common ratio $r = \frac{1}{100}.$
Identify Geometric Series: The sum $S$ of an infinite geometric series with first term $a$ and common ratio $r$ (where |r| < 1) is given by $S = \frac{a}{1 - r}$ .
Apply Sum Formula: We apply the formula to find the sum of the series, which represents our repeating decimal. $S = \frac{0.74}{1 - \frac{1}{100}}$
Perform Calculation: Now we perform the calculation. $\newline$ $S = \frac{0.74}{99/100}$ $\newline$ $S = 0.74 \times \frac{100}{99}$
Write Decimal as Fraction: To avoid dealing with decimals, we can write $0.74$ as $\frac{74}{100}$ . $S = \left(\frac{74}{100}\right) \times \left(\frac{100}{99}\right)$
Simplify Fraction: We simplify the fraction by canceling out the common factor of $100$ . $S = \frac{74}{99}$
Final Answer: The fraction $\frac{74}{99}$ cannot be simplified further, so this is our final answer.