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

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 47

9장, 문제 47

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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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