Problem 30 - Prealgebra

Question

Write each decimal as a fraction or mixed number in simplest form. $$0 . \overline{32}$$

Step-by-Step Solution

Answer

The decimal \(0.\overline{32}\) is equal to the fraction \(\frac{32}{99}\).

Step 1: Understand the Repeating Decimal

The given decimal is \(0.\overline{32}\), which means it is a repeating decimal where '32' is the repeating part. This can be represented as \(0.323232...\). Our goal is to express this repeating decimal as a fraction.

Step 2: Set Up the Equation for the Repeating Decimal

Let \(x = 0.323232...\). Multiply both sides by 100 to shift the decimal point two places to the right, resulting in \(100x = 32.323232...\). Now you have two equations: \(x = 0.323232...\) and \(100x = 32.323232...\).

Step 3: Subtract to Eliminate the Repeating Part

Subtract the first equation from the second: \[100x - x = 32.323232... - 0.323232...\] which simplifies to \[99x = 32\].

Step 4: Solve for x

Divide both sides of the equation by 99 to solve for \(x\): \[x = \frac{32}{99}\]. This fraction cannot be simplified further, as 32 and 99 have no common factors other than 1.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics. It involves reducing a fraction to its simplest form, meaning that the numerator and the denominator have no common factors other than 1. Simply put, you want the fraction to be as concise as possible without changing its value. Here's how you can simplify a fraction:

• Identify any common factors the numerator and denominator share.
• Divide both the numerator and the denominator by their greatest common factor (GCF).
• If the greatest common factor is 1, the fraction is already in its simplest form.

In the context of the exercise, after obtaining the fraction representing the repeating decimal, we checked if it could be reduced further. However, the numbers 32 and 99 only share the factor 1, indicating that \( \frac{32}{99} \) is already in its simplest form. This process ensures that your final answer is as neat and straightforward as possible.

Repeating Decimals

Repeating decimals are types of decimals where one or more digits repeat infinitely. They can be a bit tricky at first, but converting them to fractions can make them easier to work with. These decimals are represented with a bar over the repeating digits, such as \( 0.\overline{32} \). In mathematic terms, this notation indicates that the series of numbers under the bar will continue to be written in a looping pattern.
To convert a repeating decimal to a fraction, follow these steps:

• Let \( x \) be the repeating decimal.
• Adjust the decimal point (if necessary) by multiplying by a power of 10 to form a parallel equation.
• Subtract the original equation from this new equation to eliminate the repeating part.
• Solve for \( x \) to convert the repeating decimal into a fraction.

By following these steps, a seemingly complicated repeating decimal can be represented as a simple fraction, making it more manageable in calculations.

Fraction Representation

Expressing numbers as fractions is a versatile and reliable representation in many areas of mathematics. A fraction consists of two parts: a numerator and a denominator. It essentially represents a division of the numerator by the denominator. For repeating decimals, once you resolve the repeating pattern into a fraction, you're often expected to express that fraction in its simplest form for clarity.
Here’s why fraction representation is useful:

• It provides exact values, unlike decimal approximations which can sometimes be inexact.
• Facilitates comparisons between numbers making them easier to work with.
• Enables operations like addition, subtraction, multiplication, and division to be performed more straightforwardly in many cases.

Once we converted the repeating decimal \( 0.\overline{32} \) to the fraction \( \frac{32}{99} \), we made sure that it was in the lowest terms, validating its clarity and precision. This clear depiction helps in understanding the true value of the number without approximation errors.