Express each repeating decimal as a fraction in lowest terms. 0.5‾=510+5100+51000+510,000+⋯0.\overline{5}=\frac{5}{10}+\frac{5}{100}+\frac{5}{1000}+\frac{5}{10,000}+\cdots

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 45

9장, 문제 45

Express each repeating decimal as a fraction in lowest terms. $0. \overline{5} = \frac{5}{10} + \frac{5}{100} + \frac{5}{1000} + \frac{5}{10, 000} + \dots$

검증된 단계별 안내

Let the repeating decimal be represented by the variable $x$, so $x = 0.5555\ldots$ where the digit 5 repeats indefinitely.

Multiply both sides of the equation by 10 to shift the decimal point one place to the right: $10x = 5.5555\ldots$

Subtract the original equation from this new equation to eliminate the repeating part: $10x - x = 5.5555\ldots - 0.5555\ldots$

Simplify the subtraction: $9x = 5$

Solve for $x$ by dividing both sides by 9: $x = \frac{5}{9}$. This fraction is already in lowest terms.

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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, 0.555... has the digit 5 repeating endlessly. Understanding this pattern is essential to convert such decimals into fractions.

Geometric Series

A geometric series is the sum of terms where each term is a constant multiple (common ratio) of the previous one. In repeating decimals, the infinite sum of the repeating digits can be expressed as a geometric series, which helps in finding the fractional equivalent.

Fraction Simplification

After expressing the repeating decimal as a fraction, simplifying it to lowest terms involves dividing numerator and denominator by their greatest common divisor (GCD). This ensures the fraction is in its simplest and most understandable form.

Express each repeating decimal as a fraction in lowest terms. 0.5‾=510+5100+51000+510,000+⋯0.\(\overline{5}\)=\(\frac{5}{10}\)+\(\frac{5}{100}\)+\(\frac{5}{1000}\)+\(\frac{5}{10,000}\)+\(\cdots\)