Problem 45 - College Algebra

Question

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

Step-by-Step Solution

Answer

The repeating decimal \(0.\overline{5}\) can be expressed as the fraction \(5/9\).

Step 1: Represent the repeating decimal as an infinite geometric series

Represent \(0.\overline{5}\) as the geometric series \(5/10+5/100+5/1000+5/10,000+\ldots\). Here, the common ratio, represented by \(r\), is \(1/10\). This is because each term in the series is obtained by multiplying the preceding term by \(1/10\). The first term, represented by \(a\), is \(5/10\) (or \(1/2\)). The fraction \(5/10\) is reduced to \(1/2\) because it is a requirement to express the fraction in lowest terms at all stages.

Step 2: Apply the sum of an infinite geometric series formula

The sum, \(S\), of an infinite geometric series where \(|r|<1\) is given by the formula \(S=a/(1-r)\). Here, \(a\) is the first term and \(r\) is the common ratio. Therefore, the sum of the series representing \(0.\overline{5}\) is \(S=1/2/(1-1/10)\).

Step 3: Simplify the fraction

Now, simplify the fraction to obtain the lowest terms. \(S=1/2/(1-1/10)=1/2/(10/10-1/10)=1/2/(9/10)=(1/2)*(10/9)=5/9\). Therefore, \(0.\overline{5}\) can be expressed as the fraction \(5/9\).

Key Concepts

Infinite Geometric Series

Understanding how repeating decimals such as \(0.\overline{5}\) can be expressed as fractions often starts with comprehending the concept of an infinite geometric series. An infinite geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant called the common ratio.

Consider the series \(5/10, 5/100, 5/1000, \ldots\), it never ends and each term gets smaller since it's being multiplied by a fraction. When we have a repeating decimal, this concept becomes especially useful because the pattern within the decimal repeats indefinitely, much like the unending terms of an infinite series.

Each additional term represents going one deeper decimal place into the repeating pattern, and therefore, each contributes less and less to the total sum. Understanding this will help you see why these series can indeed sum up to a finite number, which in the case of repeating decimals, can be accurately represented by a simple fraction.

Sum of an Infinite Series

To find the sum of an infinite geometric series, certain conditions must be met: the absolute value of the common ratio, \(|r|\), must be less than 1. This ensures that the terms decrease in magnitude and approach zero, allowing a sum to be calculated.

The generic formula for the sum, \(S\), is \(S = \frac{a}{1-r}\), where \(a\) is the first term, and \(r\) is the common ratio. This formula relies on the terms getting progressively smaller, eventually becoming negligible, so that we can consider their sum after an infinite number of terms.

In the exercise, applying this formula to the repeating decimal \(0.\overline{5}\) leads us to the sum of a series that captures the essence of the repeating decimal in a tidy fractional form.

Common Ratio

A pivotal element in solving geometric series problems is identifying the common ratio \(r\). This ratio indicates the factor by which successive terms in the series are multiplied to obtain the next term.

For the repeating decimal \(0.\overline{5}\) expressed as \(5/10 + 5/100 + 5/1000 + \ldots\), the common ratio is \(1/10\). This can be seen by dividing any term in the series by its previous term. Understanding the common ratio allows us to use the infinite series sum formula since the sum exists only when the absolute value of the common ratio is less than one, guaranteeing that the series converges to a finite number. By grasping the concept of the common ratio, students can more confidently tackle problems involving infinity and beyond.

Simplify Fraction

Once the sum of the infinite geometric series is calculated, it's crucial to express the result in the simplest form as the fraction in lowest terms. This process of simplification ensures that the fraction is presented in its most basic and understandable form without changing its value.

To simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD). In the context of our example, after applying the sum formula, we arrived at the fraction \(\frac{1/2}{9/10}\) which simplifies to \(\frac{5}{9}\) by multiplying the numerator and denominator by the reciprocal of \(9/10\). As a result, the repeating decimal \(0.\overline{5}\) is elegantly distilled down to its essence as the fraction \(5/9\), its simplest form.