Express each repeating decimal as a fraction in lowest terms. 0.47‾=47100+4710,000+471,000,000+⋯0.\overline{47}=\frac{47}{100}+\frac{47}{10,000}+\frac{47}{1,000,000}+\cdots

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 47

Chapter 9, Problem 47

Express each repeating decimal as a fraction in lowest terms. $0. \overline{47} = \frac{47}{100} + \frac{47}{10, 000} + \frac{47}{1, 000, 000} + \dots$

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Let $ x = 0.474747\ldots $ where the digits "47" repeat indefinitely.

Since the repeating block has 2 digits, multiply $ x $ by $ 10^{2} = 100 $ to shift the decimal point two places to the right: $ 100x = 47.474747\ldots $.

Set up the equation by subtracting the original $ x $ from this new expression to eliminate the repeating part: $ 100x - x = 47.474747\ldots - 0.474747\ldots $.

Simplify the subtraction to get $ 99x = 47 $, because the repeating decimals cancel out.

Solve for $ x $ by dividing both sides by 99: $ x = \frac{47}{99} $. Then, check if the fraction can be simplified to lowest terms.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Repeating Decimals

A repeating decimal is a decimal number in which a digit or group of digits repeats infinitely. For example, 0.474747... has '47' repeating endlessly. Understanding the pattern of repetition is essential to convert such decimals into fractions.

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. In the context of repeating decimals, the infinite sum of the repeating parts forms a geometric series, which can be summed using a formula to find an exact fraction.

Fraction Simplification

After converting a repeating decimal into a fraction, it is important to simplify the fraction to its lowest terms by dividing numerator and denominator by their greatest common divisor. This ensures the fraction is expressed in the simplest and most standard form.

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Express each repeating decimal as a fraction in lowest terms. 0.47‾=47100+4710,000+471,000,000+⋯0.\(\overline{47}\)=\(\frac{47}{100}\)+\(\frac{47}{10,000}\)+\(\frac{47}{1,000,000}\)+\(\cdots\)

Verified video answer for a similar problem:

Key Concepts

Repeating Decimals

Geometric Series

Fraction Simplification

Express each repeating decimal as a fraction in lowest terms. $0. \overline{47} = \frac{47}{100} + \frac{47}{10, 000} + \frac{47}{1, 000, 000} + \dots$