Write the repeating decimal as a fraction. .81818181 ______

Rephrase Problem: Let's first rephrase the problem into a single ＂How can we express the repeating decimal 0.81818181... as a fraction?＂Identify Repeating Pattern: Identify the repeating pattern in the decimal. The digits ＂81＂ repeat indefinitely, so we can write the decimal as 0.81818181... = 0.81 + 0.0081 + 0.000081 + ...Express Terms as Fractions: Express each term in the pattern as a fraction. The first term is 0.81, which is 81/100. The second term is 0.0081, which is 81/10000, and so on. This gives us the series: 81/100 + 81/10000 + 81/1000000 + ...Recognize Geometric Series: Recognize that the series forms a geometric series, where each term is 1/100 times the previous term. The first term (a_1) is 81/100, and the common ratio (r) is 1/100.Use Sum 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 = 81/100 and r = 1/100 into the formula.Calculate Sum: Calculate the sum of the series: (81/100) / (1 - 1/100) = (81/100) / (99/100) = 81/100 * 100/99 = 81/99.Simplify Fraction: Simplify the fraction 81/99. Both the numerator and the denominator are divisible by 9. Dividing both by 9 gives us 9/11.

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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.

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.81818181

Rephrase Problem: Let's first rephrase the problem into a single ＂How can we express the repeating decimal $0.81818181\ldots$ as a fraction?＂
Identify Repeating Pattern: Identify the repeating pattern in the decimal. The digits ＂ $81$ ＂ repeat indefinitely, so we can write the decimal as $0.81818181\ldots = 0.81 + 0.0081 + 0.000081 + \ldots$
Express Terms as Fractions: Express each term in the pattern as a fraction. The first term is $0.81$ , which is $\frac{81}{100}$ . The second term is $0.0081$ , which is $\frac{81}{10000}$ , and so on. This gives us the series: $\frac{81}{100} + \frac{81}{10000} + \frac{81}{1000000} + \ldots$
Recognize Geometric Series: Recognize that the series forms a geometric series, where each term is $\frac{1}{100}$ times the previous term. The first term $(a_1)$ is $\frac{81}{100}$ , and the common ratio $(r)$ is $\frac{1}{100}$ .
Use Sum 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 $a_1 = \frac{81}{100}$ and $r = \frac{1}{100}$ into the formula.
Calculate Sum: Calculate the sum of the series: $(\frac{81}{100}) / (1 - \frac{1}{100}) = (\frac{81}{100}) / (\frac{99}{100}) = \frac{81}{100} \times \frac{100}{99} = \frac{81}{99}$ .
Simplify Fraction: Simplify the fraction $\frac{81}{99}$ . Both the numerator and the denominator are divisible by $9$ . Dividing both by $9$ gives us $\frac{9}{11}$ .