Problem 61 - Prealgebra

Question

Explain how 0.5 and\(0 . \overline{5}\)are different. Which is greater?

Step-by-Step Solution

Answer

0.5 is less than 0.\overline{5} because \(0.\overline{5}\) equals \( \frac{5}{9} \), which is greater than \( \frac{1}{2} \).

Step 1: Understanding 0.5

The number 0.5 is a decimal representation, which is equivalent to the fraction \( \frac{1}{2} \). This equates to 50%, which represents one half of a whole.

Step 2: Understanding 0.\overline{5}

The notation \(0.\overline{5}\) signifies a repeating decimal, which means the digit 5 repeats indefinitely. This can also be represented as the fraction \( \frac{5}{9} \) because converting a repeating decimal to a fraction involves using the formula where if \( x = 0.\overline{n} \), then \( x = \frac{n}{9} \).

Step 3: Comparing the Two Numbers

To compare 0.5 and \(0.\overline{5}\), convert them to fractions. We have 0.5 as \( \frac{1}{2} \) and \(0.\overline{5}\) as \( \frac{5}{9} \). Find a common denominator to compare: \( \frac{1}{2} = \frac{9}{18} \) and \( \frac{5}{9} = \frac{10}{18} \). Since \( \frac{10}{18} > \frac{9}{18} \), \(0.\overline{5}\) is greater than 0.5.

Key Concepts

Understanding Fractions

Fractions represent parts of a whole or a collection. They consist of two parts: a numerator, which is the number above the line, and a denominator, which is the number below the line. The fraction indicates how many parts of a divided whole are being considered. For example, in the fraction \( \frac{1}{2} \), the numerator is 1 and the denominator is 2, meaning you have one out of two equal parts.

• The numerator shows how many parts you have.
• The denominator shows into how many parts the whole is divided.

Fractions can represent proper fractions (like \( \frac{1}{2} \)), improper fractions (like \( \frac{9}{4} \)), and mixed numbers (such as 2\( \frac{1}{3} \)). When comparing fractions, it is often helpful to find a common denominator, which makes it easier to see which fraction is larger or smaller.

Repeating Decimals Explained

Repeating decimals, also known as recurring decimals, consist of one or more repeating digits after the decimal point. These are numbers like \( 0.\overline{5} \), where the digit '5' repeats endlessly. Understanding repeating decimals helps in converting them into fractions. For instance, to convert \( 0.\overline{5} \) into a fraction:

• First, let \( x = 0.\overline{5} \).
• Multiply by 10 to shift the decimal point: \( 10x = 5.\overline{5} \).
• Subtract the original \( x \) from this equation: \( 10x - x = 5.\overline{5} - 0.\overline{5} \).
• This simplifies to \( 9x = 5 \), giving \( x = \frac{5}{9} \).

This process can be used for any repeating decimal. Knowing how to convert help you better understand mathematical results and determine decimal equivalencies.

Comparing Numbers in Decimal Form

When comparing numbers like decimals, it is crucial to view them in a format that makes comparison easiest, often fractions. For comparing \( 0.5 \) and \( 0.\overline{5} \):

• Express \( 0.5 \) as a fraction: \( \frac{1}{2} \).
• Express \( 0.\overline{5} \) as a fraction: \( \frac{5}{9} \).

To accurately compare these fractions, find a common denominator. Here, \( 0.5 = \frac{9}{18} \) and \( 0.\overline{5} = \frac{10}{18} \). This shows \( 0.\overline{5} \) is greater because \( \frac{10}{18} \) is more than \( \frac{9}{18} \).
Comparing decimals by converting them into fractions provides a clear view on which number represents a larger or smaller value, aiding in mathematical precision and understanding in various applications.