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Chapter 8: Problem 45

Express each repeating decimal as a fraction in lowest terms. $$0.5=\frac{5}{10}+\frac{5}{100}+\frac{5}{1000}+\frac{5}{10,000}+\dots$$

Short Answer

Expert verified
The repeating decimal \(0.5\) can be represented as the fractional value in its simplest form: \( \frac{5}{9} \)

Step by step solution

01

Identify the repeating part

The repeating part of the decimal \(0.5\) is \(5\).
02

Set up the geometric series equation

The decimal can be expressed as an infinite geometric series. Each term in the series is obtained by multiplying the previous term by \(\frac{1}{10}\), so the common ratio is \(\frac{1}{10}\). This gives us the series: \(0.5 = \frac{5}{10} + \frac{5}{100} + \frac{5}{1000} + \frac{5}{10,000} + \dots\)
03

Sum up the series

A geometric series with first term \(a\) and common ratio \(r\) can be summed up as follows: \[S = \frac{a}{1-r}\] Applying this to our series where \(a = \frac{5}{10}\) and \(r = \frac{1}{10}\), we get: \(S = \frac{5/10}{1-(1/10)} = \frac{5/10}{9/10} = \frac{5}{9}\)
04

Simplify Fraction

The fraction \( \frac{5}{9} \) is already in its simplest form with no common factors other than 1. Thus, the decimal \(0.5\) can be represented as the fraction \( \frac{5}{9} \)

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