Problem 45 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the repeating part

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

Step 2: 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: \(0.5 = \frac{5}{10} + \frac{5}{100} + \frac{5}{1000} + \frac{5}{10,000} + \dots\)

Step 3: 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: \(S = \frac{5/10}{1-(1/10)} = \frac{5/10}{9/10} = \frac{5}{9}\)

Step 4: 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 \( \frac{5}{9} \)