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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 45
Chapter 9, Problem 45

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

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1
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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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, 0.555... has the digit 5 repeating endlessly. Understanding this pattern is essential to convert such decimals into fractions.
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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.
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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.
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Related Practice
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Textbook Question

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